Abstract. The purpose of a consensus protocol is to keep a distributed network of nodes “in sync,” even in the presence of an unpredictable communication network and adversarial behavior by some of the participating nodes. In the permissionless setting relevant to modern blockchain protocols, these nodes may be operated by a large number of unknown players, with each player free to use multiple identifiers and to start or stop running the protocol at any time. Establishing that a permissionless consensus protocol is “secure” thus requires both a distributed computing argument (that the protocol guarantees consistency and liveness unless the fraction of adversarial participation is sufficiently large) and an economic argument (that carrying out an attack would be prohibitively expensive for a potential attacker). There is a mature toolbox for assembling arguments of the former type; the goal of this paper is to lay the foundations for arguments of the latter type. For example, the Ethereum protocol is oft-claimed to be “more economically secure” after “the merge,” meaning in its current proof-of-stake incarnation relative to the (proof-of-work) original. What, formally, does this assertion mean? Is it true? Could there be alternative protocols that are “still more economically secure” than Ethereum? How do the answers depend on the assumptions imposed on, for example, the reliability of message delivery or the active participation of non-malicious players? An ideal permissionless consensus protocol would, in addition to satisfying standard consistency and liveness guarantees, render consistency violations prohibitively expensive for the attacker without collateral damage to honest participants—for example, by programatically confiscating an attacker’s resources without reducing the value of honest participants’ resources, as is the intention for slashing in a proof-of-stake protocol. We make this idea precise with our notion of the EAAC (expensive to attack in the absence of collapse) property and prove the following results: (1) In the synchronous and dynamically available setting (in which the communication network is reliable but nonmalicious players may be periodically inactive), with an adversary that controls at least one-half of the overall resources, no protocol can be EAAC. In particular, this result rules out EAAC for all typical longest-chain protocols (be they proof-of-work or proof-of-stake). (2) In the partially synchronous and quasi-permissionless setting (in which resource-controlling non-malicious players are always active but the communication network may suffer periods of unreliability), with an adversary that controls at least one-third of the overall resources, no protocol can be EAAC. In particular, slashing in a proof-of-stake protocol cannot achieve its intended purpose if message delays cannot be bounded a priori. (3) In the synchronous and quasi-permissionless setting, there is a proof-of-stake protocol with slashing that, provided the adversary controls less than two-thirds of the overall stake, satisfies the EAAC property. Thus, while only “classical security” is possible in the dynamically available or partially synchronous settings, proof-of-stake protocols with slashing can obtain additional “economic security” in the quasi-permissionless and synchronous settings. All three results are optimal with respect to the size of the adversary. With respect to Ethereum, our work formalizes the potential security benefits of proof-of-stake sybil-resistance coupled with slashing and the common belief that the merge has increased Ethereum’s economic security. Our work also provides mathematical justifications for several key design decisions behind the post-merge Ethereum protocol, ranging from long cooldown periods for unstaking to economic penalties for inactivity.

Blockchain is an emerging technology that gained a lot of attention in the last years. Many different consensus protocols have been proposed to improve both the scalability and the resilience of existing blockchain. However, all these solutions have been defined for rather static settings. We propose a modular approach for analysing and comparing different consensus protocols used in blockchain under churn.

Safety and liveness are the two classical security properties of consensus protocols. Recent works have strengthened safety with accountability: should any safety violation occur, a sizable fraction of adversary nodes can be proven to be protocol violators. This paper studies to what extent analogous accountability guarantees are achievable for liveness. To reveal the full complexity of this question, we introduce an interpolation between the classical synchronous and partially-synchronous models that we call the x-partially-synchronous network model in which, intuitively, at most an x fraction of the time steps in any sufficiently long interval are asynchronous (and, as with a partially-synchronous network, all time steps are synchronous following the passage of an unknown ''global stablization time''). We prove a precise characterization of the parameter regime in which accountable liveness is achievable: if and only if x < 1/2 and ƒ < n/2, where n denotes the number of nodes and ƒ the number of nodes controlled by an adversary. We further refine the problem statement and our analysis by parameterizing by the number of violating nodes identified following a liveness violation, and provide evidence that the guarantees achieved by our protocol are near-optimal (as a function of x and ƒ). Our results provide rigorous foundations for liveness-accountability heuristics such as the ''inactivity leaks'' employed in Ethereum.

We study the risks of validator reuse across multiple services in a restaking protocol. We characterize the robust security of a restaking network as a function of the buffer between the costs and profits from attacks. For example, our results imply that if attack costs always exceed attack profits by 10%, then a sudden loss of .1% of the overall stake (e.g., due to a software error) cannot result in the ultimate loss of more than 1.1% of the overall stake. We also provide local analogs of these overcollateralization conditions and robust security guarantees that apply specifically for a target service or coalition of services. All of our bounds on worst-case stake loss are the best possible. Finally, we bound the maximum-possible length of a cascade of attacks. Our results suggest measures of robustness that could be exposed to the participants in a restaking protocol. We also suggest polynomial-time computable sufficient conditions that can proxy for these measures.

In an auction, a seller may masquerade as one or more bidders in order to manipulate the clearing price. We characterize single-item auction formats that are shill-proof in the sense that a profit-maximizing seller has no incentive to submit shill bids. We distinguish between strong shill-proofness, in which a seller with full knowledge of bidders' valuations can never profit from shilling, and weak shill-proofness, which requires only that the expected equilibrium profit from shilling is nonpositive. The Dutch auction (with a suitable reserve) is the unique (revenue-)optimal and strongly shill-proof auction. Moreover, the Dutch auction (with no reserve) is the unique prior-independent auction that is both efficient and weakly shill-proof. While there are multiple ex-post incentive compatible, weakly shill-proof, and optimal auctions; any optimal auction can satisfy only two properties in the set {static, ex-post incentive compatible, weakly shill-proof}.

In repeated interaction problems with adaptive agents, our objective often requires anticipating and optimizing over the space of possible agent responses. We show that many problems of this form can be cast as instances of online (nonlinear) control which satisfy \textit{local controllability}, with convex losses over a bounded state space which encodes agent behavior, and we introduce a unified algorithmic framework for tractable regret minimization in such cases. When the instance dynamics are known but otherwise arbitrary, we obtain oracle-efficient $O(\sqrt{T})$ regret by reduction to online convex optimization, which can be made computationally efficient if dynamics are locally \textit{action-linear}. In the presence of adversarial disturbances to the state, we give tight bounds in terms of either the cumulative or per-round disturbance magnitude (for \textit{strongly} or \textit{weakly} locally controllable dynamics, respectively). Additionally, we give sublinear regret results for the cases of unknown locally action-linear dynamics as well as for the bandit feedback setting. Finally, we demonstrate applications of our framework to well-studied problems including performative prediction, recommendations for adaptive agents, adaptive pricing of real-valued goods, and repeated gameplay against no-regret learners, directly yielding extensions beyond prior results in each case.

The purpose of a consensus protocol is to keep a distributed network of nodes "in sync," even in the presence of an unpredictable communication network and adversarial behavior by some of the participating nodes. In the permissionless setting relevant to modern blockchain protocols, these nodes may be operated by a large number of unknown players, with each player free to use multiple identifiers and to start or stop running the protocol at any time. Establishing that a permissionless consensus protocol is "secure" thus requires both a distributed computing argument (that the protocol guarantees consistency and liveness unless the fraction of adversarial participation is sufficiently large) and an economic argument (that carrying out an attack would be prohibitively expensive for a potential attacker). There is a mature toolbox for assembling arguments of the former type; the goal of this paper is to lay the foundations for arguments of the latter type. For example, the Ethereum protocol is oft-claimed to be "more economically secure" after "the merge," meaning in its current proof-of-stake incarnation relative to the (proof-of-work) original. What, formally, does this assertion mean? Is it true? Could there be alternative protocols that are "still more economically secure" than Ethereum? How do the answers depend on the assumptions imposed on, for example, the reliability of message delivery or the active participation of non-malicious players?

An ideal permissionless consensus protocol would, in addition to satisfying standard consistency and liveness guarantees, render consistency violations prohibitively expensive for the attacker without collateral damage to honest participants---for example, by programatically confiscating an attacker's resources without reducing the value of honest participants' resources, as is the intention for slashing in a proof-of-stake protocol. We make this idea precise with our notion of the EAAC (expensive to attack in the absence of collapse) property, and prove the following results:

Users bid in a transaction fee mechanism (TFM) to get their transactions included and confirmed by a blockchain protocol. Roughgarden (EC'21) initiated the formal treatment of TFMs and proposed three requirements: user incentive compatibility (UIC), miner incentive compatibility (MIC), and a form of collusion-resilience called OCA-proofness. Ethereum's EIP-1559 mechanism satisfies all three properties simultaneously when there is no contention between transactions, but loses the UIC property when there are too many eligible transactions to fit in a single block. Chung and Shi (SODA'23) considered an alternative notion of collusion-resilience, called c-side-contract-proofness (c-SCP), and showed that, when there is contention between transactions, no TFM can satisfy UIC, MIC, and c-SCP for any c ≥ 1. OCA-proofness asserts that the users and a miner should not be able to "steal from the protocol." On the other hand, the c-SCP condition requires that a coalition of a miner and a subset of users should not be able to profit through strategic deviations (whether at the expense of the protocol or of the users outside the coalition).

Turing-complete blockchain protocols approximate the idealized abstraction of a "computer in the sky" that is open access, runs in plain view, and, in effect, has no owner or operator. This technology can, among other things, enable stronger notions of ownership of digital possessions than we have ever had before. Building the computer in the sky is hard (and scientifically fascinating). In this talk I'll highlight some of my recent research on this challenge, emphasizing the diversity of mathematical tools required, the immediate practical impact that mathematical work on this topic has had, and some open problems of interest to theoretical computer scientists.

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by \(\tfrac{1}{\sigma }\) times that of the uniform distribution; nature then samples an input from this distribution. Here, σ is a parameter that interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm’s current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: (1) Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of σ-smooth distributions and the hypothesis class has VC dimension d . We bound the regret by \(\tilde{O}(\sqrt {T d\ln (1/\sigma)} + d\ln (T/\sigma))\) and provide a near-matching lower bound. Our result shows that under smoothed analysis, learnability against adaptive adversaries is characterized by the finiteness of the VC dimension. This is as opposed to the worst-case analysis, where online learnability is characterized by Littlestone dimension (which is infinite even in the extremely restricted case of one-dimensional threshold functions). Our results fully answer an open question of Rakhlin et al. [ 64 ]. (2) Online discrepancy minimization: We consider the setting of the online Komlós problem, where the input is generated from an adaptive sequence of σ-smooth and isotropic distributions on the ℓ 2 unit ball. We bound the ℓ ∞ norm of the discrepancy vector by \(\tilde{O}(\ln ^2(\frac{nT}{\sigma }))\) . This is as opposed to the worst-case analysis, where the tight discrepancy bound is \(\Theta (\sqrt {T/n})\) . We show such \(\mathrm{polylog}(nT/\sigma)\) discrepancy guarantees are not achievable for non-isotropic σ-smooth distributions. (3) Dispersion in online optimization: We consider online optimization with piecewise Lipschitz functions where functions with ℓ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is \(({\sigma }/{\sqrt {T\ell }}, \tilde{O}(\sqrt {T\ell }))\) -dispersed. That is, every ball of radius \({\sigma }/{\sqrt {T\ell }}\) is split by \(\tilde{O}(\sqrt {T\ell })\) of the partitions made by these functions. This result matches the dispersion parameters of Balcan et al. [ 13 ] for oblivious smooth adversaries, up to logarithmic factors. On the other hand, worst-case sequences are trivially (0, T )-dispersed. 1

When trying to solve a computational problem, we are often faced with a choice between algorithms that are guaranteed to return the right answer but differ in their runtime distributions (e.g., SAT solvers, sorting algorithms). This paper aims to lay theoretical foundations for such choices by formalizing preferences over runtime distributions. It might seem that we should simply prefer the algorithm that minimizes expected runtime. However, such preferences would be driven by exactly how slow our algorithm is on bad inputs, whereas in practice we are typically willing to cut off occasional, sufficiently long runs before they finish. We propose a principled alternative, taking a utility-theoretic approach to characterize the scoring functions that describe preferences over algorithms. These functions depend on the way our value for solving our problem decreases with time and on the distribution from which captimes are drawn. We describe examples of realistic utility functions and show how to leverage a maximum-entropy approach for modeling underspecified captime distributions. Finally, we show how to efficiently estimate an algorithm’s expected utility from runtime samples.

We present the first nontrivial procedure for configuring heuristic algorithms to maximize the utility provided to their end users while also offering theoretical guarantees about performance. Existing procedures seek configurations that minimize expected runtime. However, very recent theoretical work argues that expected runtime minimization fails to capture algorithm designers' preferences. Here we show that the utilitarian objective also confers significant algorithmic benefits. Intuitively, this is because mean runtime is dominated by extremely long runs even when they are incredibly rare; indeed, even when an algorithm never gives rise to such long runs, configuration procedures that provably minimize mean runtime must perform a huge number of experiments to demonstrate this fact. In contrast, utility is bounded and monotonically decreasing in runtime, allowing for meaningful empirical bounds on a configuration's performance. This paper builds on this idea to describe effective and theoretically sound configuration procedures. We prove upper bounds on the runtime of these procedures that are similar to theoretical lower bounds, while also demonstrating their performance empirically.

For each of $T$ time steps, $m$ experts report probability distributions over $n$ outcomes; we wish to learn to aggregate these forecasts in a way that attains a no-regret guarantee. We focus on the fundamental and practical aggregation method known as *logarithmic pooling* -- a weighted average of log odds -- which is in a certain sense the optimal choice of pooling method if one is interested in minimizing log loss (as we take to be our loss function). We consider the problem of learning the best set of parameters (i.e. expert weights) in an online adversarial setting. We assume (by necessity) that the adversarial choices of outcomes and forecasts are consistent, in the sense that experts report calibrated forecasts. Imposing this constraint creates a (to our knowledge) novel semi-adversarial setting in which the adversary retains a large amount of flexibility. In this setting, we present an algorithm based on online mirror descent that learns expert weights in a way that attains $O(\sqrt{T} \log T)$ expected regret as compared with the best weights in hindsight.

We consider mechanisms for truthfully eliciting probabilistic predictions from a group of experts. The standard approach --- using a proper scoring rule to separately reward each expert --- is not robust to collusion: experts may collude to misreport their beliefs in a way that guarantees them a larger total reward no matter the eventual outcome. It is a long-standing open question whether there is a truthful elicitation mechanism that makes any such collusion (also called "arbitrage") impossible. We resolve this question positively, exhibiting a class of strictly proper arbitrage-free contract functions. These contract functions have two parts: one ensures that the total reward of a coalition of experts depends only on the average of their reports; the other ensures that changing this average report hurts the experts under at least one outcome.

The problem of aggregating expert forecasts is ubiquitous in fields as wide-ranging as machine learning, economics, climate science, and national security. Despite this, our theoretical understanding of this question is fairly shallow. This paper initiates the study of forecast aggregation in a context where experts' knowledge is chosen adversarially from a broad class of information structures. While in full generality it is impossible to achieve a nontrivial performance guarantee, we show that doing so is possible under a condition on the experts' information structure that we call projective substitutes. The projective substitutes condition is a notion of informational substitutes: that there are diminishing marginal returns to learning the experts' signals. We show that under the projective substitutes condition, taking the average of the experts' forecasts improves substantially upon the strategy of trusting a random expert. We then consider a more permissive setting, in which the aggregator has access to the prior. We show that by averaging the experts' forecasts and then extremizing the average by moving it away from the prior by a constant factor, the aggregator's performance guarantee is substantially better than is possible without knowledge of the prior. Our results give a theoretical grounding to past empirical research on extremization and help give guidance on the appropriate amount to extremize.

Blockchain protocols come with a variety of security guarantees. For example, BFT-inspired protocols such as Algorand tend to be secure in the partially synchronous setting, while longest chain protocols like Bitcoin will normally require stronger synchronicity to be secure. Another fundamental distinction, directly relevant to scalability solutions such as sharding, is whether or not a single untrusted user is able to point to certificates, which provide incontrovertible proof of block confirmation. Algorand produces such certificates, while Bitcoin does not. Are these properties accidental? Or are they inherent consequences of the paradigm of protocol design? Our aim in this paper is to understand what, fundamentally, governs the nature of security for permissionless blockchain protocols. Using the framework developed in [12], we prove general results showing that these questions relate directly to properties of the user selection process, i.e. the method (such as proof-of-work or proof-of-stake) which is used to select users with the task of updating state. Our results suffice to establish, for example, that the production of certificates is impossible for proof-of-work protocols, but is automatic for standard forms of proof-of-stake protocols. As a byproduct of our work, we also define a number of security notions and identify the equivalences and inequivalences among them.

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by a multiplicative factor from the uniform distribution; nature then samples an input from this distribution. This interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of an input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm's current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: Online learning, Online discrepancy and Dispersion in online optimization. We show that in these setting, we can get bounds that match bounds we can get for non-adaptive adversaries.

This paper forges a strong connection between two seemingly unrelated forecasting problems: incentive-compatible forecast elicitation and forecast aggregation. Proper scoring rules are the well-known solution to the former problem. To each such rule s we associate a corresponding method of aggregation, mapping expert forecasts and expert weights to a "consensus forecast," which we call quasi-arithmetic (QA) pooling with respect to s. We justify this correspondence in several ways: QA pooling with respect to the two most well-studied scoring rules (quadratic and logarithmic) corresponds to the two most well-studied forecast aggregation methods (linear and logarithmic). Given a scoring rule s used for payment, a forecaster agent who sub-contracts several experts, paying them in proportion to their weights, is best off aggregating the experts' reports using QA pooling with respect to s, meaning this strategy maximizes its worst-case profit (over the possible outcomes). The score of an aggregator who uses QA pooling is concave in the experts' weights. As a consequence, online gradient descent can be used to learn appropriate expert weights from repeated experiments with low regret. The class of all QA pooling methods is characterized by a natural set of axioms (generalizing classical work by Kolmogorov on quasi-arithmetic means).

Demand for blockchains such as Bitcoin and Ethereum is far larger than supply, necessitating a mechanism that selects a subset of transactions to include "on-chain" from the pool of all pending transactions. EIP-1559 is a proposal to make several tightly coupled changes to the Ethereum blockchain's transaction fee mechanism, including the introduction of variable-size blocks and a burned base fee that rises and falls with demand. These changes are slated for deployment in Ethereum's "London fork," scheduled for late summer 2021, at which point it will be the biggest economic change made to a major blockchain to date. The first goal of this paper is to formalize the problem of designing a transaction fee mechanism, taking into account the many idiosyncrasies of the blockchain setting (ranging from off-chain collusion between miners and users to the ease of money-burning). The second goal is to situate the specific mechanism proposed in EIP-1559 in this framework and rigorously interrogate its game-theoretic properties. The third goal is to suggest competing designs that offer alternative sets of trade-offs. The final goal is to highlight research opportunities for the EC community that could help shape the future of blockchain transaction fee mechanisms.

Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of online adversarial and differentially private learning algorithms. The primary quantity that characterizes learnability in these settings is the Littlestone dimension of the class of hypotheses [Ben-David et al., 2009, Alon et al., 2019]. This characterization is often interpreted as an impossibility result because classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper, we apply the framework of smoothed analysis [Spielman and Teng, 2004], in which adversarially chosen inputs are perturbed slightly by nature. We show that fundamentally stronger regret and error guarantees are possible with smoothed adversaries than with worst-case adversaries. In particular, we obtain regret and privacy error bounds that depend only on the VC dimension and the bracketing number of a hypothesis class, and on the magnitudes of the perturbations.

We initiate the study of computing (near-)optimal contracts in succinctly representable principal-agent settings. Here optimality means maximizing the principal's expected payoff over all incentive-compatible contracts—known in economics as “second-best” solutions. We also study a natural relaxation to approximately incentive-compatible contracts. We focus on principal-agent settings with succinctly described (and exponentially large) outcome spaces. We show that the computational complexity of computing a near-optimal contract depends fundamentally on the number of agent actions. For settings with a constant number of actions, we present a fully polynomial-time approximation scheme (FPTAS) for the separation oracle of the dual of the problem of minimizing the principal's payment to the agent, and use this subroutine to efficiently compute a δ-incentive-compatible (δ-IC) contract whose expected payoff matches or surpasses that of the optimal IC contract. With an arbitrary number of actions, we prove that the problem is hard to approximate within any constant c. This inapproximability result holds even for δ-IC contracts where δ is a sufficiently rapidly-decaying function of c. On the positive side, we show that simple linear δ-IC contracts with constant δ are sufficient to achieve a constant-factor approximation of the “first-best” (full-welfare-extracting) solution, and that such a contract can be computed in polynomial time.

In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $\frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $\frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular function is also coordinate-wise concave along all coordinates, we provide a different $\frac{1}{2}$-approximation algorithm that runs in quasi-linear time. Both these results improve upon prior work (Bian et al. 2017; Soma and Yoshida, 2017). Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.

We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a $c$-closed graph, where for every pair of vertices $u,v$ with at least $c$ common neighbors, $u$ and $v$ are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to $c$ on $c$-closed graphs. Our results carry over to weakly $c$-closed graphs, which only require a vertex deletion ordering that avoids pairs of nonadjacent vertices with $c$ common neighbors. Numerical experiments show that well-studied social networks with thousands of vertices tend to be weakly $c$-closed for modest values of $c$.

We initiate the study of the communication complexity of fair division with indivisible goods. We focus on some of the most well studied fairness notions (envy-freeness, proportionality, and approximations thereof) and valuation classes (submodular, subadditive, and unrestricted). We show that for more than two players (and any combination of other parameters), determining whether a fair allocation exists requires exponential communication (in the number of goods). For two players, tractability depends heavily on the specific combination of parameters, and most of the paper is focused on the two-player setting. Taken together, our results completely resolve whether the communication complexity of computing a fair allocation (or determining that none exists) is polynomial or exponential, for every combination of fairness notion, valuation class, and number of players, for both deterministic and randomized protocols.

Proof-of-work blockchains reward each miner for one completed block by an amount that is, in expectation, proportional to the number of hashes the miner contributed to the mining of the block. Is this proportional allocation rule optimal? And in what sense? And what other rules are possible? In particular, what are the desirable properties that any "good" allocation rule should satisfy? To answer these questions, we embark on an axiomatic theory of incentives in proof-of-work blockchains at the time scale of a single block. We consider desirable properties of allocation rules including: symmetry; budget balance (weak or strong); sybil-proofness; and various grades of collusion-proofness. We show that Bitcoin's proportional allocation rule is the unique allocation rule satisfying a certain system of properties, but this does not hold for slightly weaker sets of properties, or when the miners are not risk-neutral. We also point out that a rich class of allocation rules can be approximately implemented in a proof-of-work blockchain.

We prove that the evolution of weight vectors in online gradient descent can encode arbitrary polynomial-space computations, even in very simple learning settings. Our results imply that, under weak complexity-theoretic assumptions, it is impossible to reason efficiently about the fine-grained behavior of online gradient descent.

Over the last twenty years, computer science has relied on concepts borrowed from game theory and economics to reason about applications ranging from internet routing to real-time auctions for online advertising. More recently, ideas have increasingly flowed in the opposite direction, with concepts and techniques from computer science beginning to influence economic theory and practice. In this lecture, I will illustrate this point with a detailed case study of the 2016-2017 Federal Communications Commission incentive auction for repurposing wireless spectrum. Computer science techniques, ranging from algorithms for NP-hard problems to nondeterministic communication complexity, have played a critical role both in the design of the reverse auction (with the government procuring existing licenses from television broadcasters) and in the analysis of the forward auction (when the procured licenses sell to the highest bidder).

We initiate the study of the communication complexity of fair division with indivisible goods. We focus on some of the most well-studied fairness notions (envy-freeness, proportionality, and approximations thereof) and valuation classes (submodular, subadditive and unrestricted). Within these parameters, our results completely resolve whether the communication complexity of computing a fair allocation (or determining that none exist) is polynomial or exponential (in the number of goods), for every combination of fairness notion, valuation class, and number of players, for both deterministic and randomized protocols.

We consider a basic problem at the interface of two fundamental fields: {\em submodular optimization} and {\em online learning}. In the {\em online unconstrained submodular maximization (online USM) problem}, there is a universe $[n]=\{1,2,\ldots,n\}$ and a sequence of $T$ nonnegative (not necessarily monotone) submodular functions arrive over time. The goal is to design a computationally efficient online algorithm, which chooses a subset of $[n]$ at each time step as a function only of the past, such that the accumulated value of the chosen subsets is as close as possible to the maximum total value of a fixed subset in hindsight. Our main result is a polynomial-time no-$\frac12$-regret algorithm for this problem, meaning that for every sequence of nonnegative submodular functions, the algorithm’s expected total value is at least $\frac12$ times that of the best subset in hindsight, up to an error term sublinear in $T$. The factor of $\tfrac 12$ cannot be improved upon by any polynomial-time online algorithm when the submodular functions are presented as value oracles. Previous work on the offline problem implies that picking a subset uniformly at random in each time step achieves zero $\frac14$-regret. A byproduct of our techniques is an explicit subroutine for the two-experts problem that has an unusually strong regret guarantee: the total value of its choices is comparable to twice the total value of either expert on rounds it did not pick that expert. This subroutine may be of independent interest.

In this paper we study the fundamental problems of maximizing a continuous non monotone submodular function over a hypercube, with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first 1/2 approximation algorithm for continuous submodular function maximization; this approximation factor of is the best possible for algorithms that use only polynomially many queries. For the special case of DR-submodular maximization, we provide a faster 1/2-approximation algorithm that runs in (almost) linear time. Both of these results improve upon prior work [Bian et al., 2017, Soma and Yoshida, 2017, Buchbinder et al., 2012]. Our first algorithm is a single-pass algorithm that uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm is a faster single-pass algorithm that exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.

The goal of fair division is to distribute resources among competing players in a “fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.

The goal of this article is to identify fundamental limitations on how efficiently algorithms implemented on platforms such as MapReduce and Hadoop can compute the central problems in motivating application domains, such as graph connectivity problems. We introduce an abstract model of massively parallel computation, where essentially the only restrictions are that the “fan-in” of each machine is limited to s bits, where s is smaller than the input size n , and that computation proceeds in synchronized rounds, with no communication between different machines within a round. Lower bounds on the round complexity of a problem in this model apply to every computing platform that shares the most basic design principles of MapReduce-type systems. We prove that computations in our model that use few rounds can be represented as low-degree polynomials over the reals. This connection allows us to translate a lower bound on the (approximate) polynomial degree of a Boolean function to a lower bound on the round complexity of every (randomized) massively parallel computation of that function. These lower bounds apply even in the “unbounded width” version of our model, where the number of machines can be arbitrarily large. As one example of our general results, computing any nontrivial monotone graph property—such as connectivity—requires a super-constant number of rounds when every machine receives only a subpolynomial (in n ) number of input bits s . Finally, we prove that, in two senses, our lower bounds are the best one could hope for. For the unbounded-width model, we prove a matching upper bound. Restricting to a polynomial number of machines, we show that asymptotically better lower bounds would separate P from NC 1 .

We study the problem of setting a price for a potential buyer with a valuation drawn from an unknown distribution $D$. The seller has “data” about $D$ in the form of $m \ge 1$ independent and identically distributed samples, and the algorithmic challenge is to use these samples to obtain expected revenue as close as possible to what could be achieved with advance knowledge of $D$. Our first set of results quantifies the number of samples $m$ that are necessary and sufficient to obtain a $(1-\epsilon)$-approximation. For example, for an unknown distribution that satisfies the monotone hazard rate (MHR) condition, we prove that $\tilde{\Theta}(\epsilon^{-3/2})$ samples are necessary and sufficient. Remarkably, this uses fewer samples than is necessary to accurately estimate the expected revenue obtained for such a distribution by even a single reserve price. We also prove essentially tight sample complexity bounds for regular distributions, bounded-support distributions, and a wide class of irregular distributions. Our lower bound approach, which applies to all randomized pricing strategies, borrows tools from differential privacy and information theory, and we believe it could find further applications in auction theory. Our second set of results considers the single-sample case. While no deterministic pricing strategy is better than $\tfrac{1}{2}$-approximate for regular distributions, for MHR distributions we show how to do better: there is a simple deterministic pricing strategy that guarantees expected revenue at least 0.589 times the maximum possible. We also prove that no deterministic pricing strategy achieves an approximation guarantee better than $\frac{e}{4} \approx .68$.

Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders' valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations.

Two genres of heuristics that are frequently reported to perform much better on "real-world" instances than in the worst case are greedy algorithms and local search algorithms. In this paper, we systematically study these two types of algorithms for the problem of maximizing a monotone submodular set function subject to downward-closed feasibility constraints. We consider perturbation-stable instances, in the sense of Bilu and Linial [11], and precisely identify the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution. Byproducts of our work include the first definition of perturbation-stability for non-additive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems.

We consider the problem of binary prediction with expert advice in settings where experts have agency and seek to maximize their credibility. This paper makes three main contributions. First, it defines a model to reason formally about settings with selfish experts, and demonstrates that ``incentive compatible'' (IC) algorithms are closely related to the design of proper scoring rules. Second, we design IC algorithms with good performance guarantees for the absolute loss function. Third, we give a formal separation between the power of online prediction with selfish experts and online prediction with honest experts by proving lower bounds for both IC and non-IC algorithms. In particular, with selfish experts and the absolute loss function, there is no (randomized) algorithm for online prediction---IC or otherwise---with asymptotically vanishing regret.

We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of items, and the aim of a mechanism is to improve the social welfare by arranging purchases and sales of the items. A mechanism is given prior distributions on the agents' valuations of the items, but not the actual valuations; thus the aim is to maximise the expected social welfare over these distributions. As in previous work, we are interested in the worst-case ratio between the social welfare achieved by a truthful mechanism, and the best social welfare possible.

Deferred-acceptance (DA) auctions} are mechanisms that are based on backward-greedy algorithms and possess a number of remarkable incentive properties, including implementation as an obviously-strategyproof ascending auction. All existing work on DA auctions considers only binary single-parameter problems, where each bidder either ``wins'' or ``loses.'' This paper generalizes the DA auction framework to non-binary settings, and applies this generalized framework to obtain approximately welfare-maximizing DA auctions for a number of basic mechanism design problems: multiunit auctions, problems with polymatroid constraints or multiple knapsack constraints, and the problem of scheduling jobs to minimize their total weighted completion time. Our results require the design of novel backward-greedy algorithms with good approximation guarantees.

Computational complexity has already had plenty to say about the computation of economic equilibria. However, understanding when equilibria are guaranteed to exist is a central theme in economic theory, seemingly unrelated to computation. In this talk we survey our main results presented at EC'15, which show that the existence of equilibria in markets is inextricably connected to the computational complexity of related optimization problems, such as revenue or welfare maximization. We demonstrate how this relationship implies, under suitable complexity assumptions, a host of impossibility results. We also suggest a complexity-theoretic explanation for the lack of useful extensions of the Walrasian equilibrium concept: such extensions seem to require the invention of novel polynomial-time algorithms for welfare maximization.

This survey outlines a general and modular theory for proving approximation guarantees for equilibria of auctions in complex settings. This theory complements traditional economic techniques, which generally focus on exact and optimal solutions and are accordingly limited to relatively stylized settings. We highlight three user-friendly analytical tools: smoothness-type inequalities, which immediately yield approximation guarantees for many auction formats of interest in the special case of complete information and deterministic strategies; extension theorems, which extend such guarantees to randomized strategies, no-regret learning outcomes, and incomplete-information settings; and composition theorems, which extend such guarantees from simpler to more complex auctions. Combining these tools yields tight worst-case approximation guarantees for the equilibria of many widely-used auction formats.

The best algorithm for a computational problem generally depends on the “relevant inputs,” a concept that depends on the application domain and often defies formal articulation. While there is a large body of literature on empirical approaches to selecting the best algorithm for a given application domain, there has been surprisingly little theoretical analysis of the problem. This paper adapts concepts from statistical and online learning theory to reason about application-specific algorithm selection. Our models capture several state-of-the-art empirical and theoretical approaches to the problem, ranging from self-improving algorithms to empirical performance models, and our results identify conditions under which these approaches are guaranteed to perform well. We present one framework that models algorithm selection as a statistical learning problem, and our work here shows that dimension notions from statistical learning theory, historically used to measure the complexity of classes of binary- and real-valued functions, are relevant in a much broader algorithmic context. We also study the online version of the algorithm selection problem, and give possibility and impossibility results for the existence of no-regret learning algorithms.

We present a general framework for proving polynomial sample complexity bounds for the problem of learning from samples the best auction in a class of “simple” auctions. Our framework captures the most prominent examples of “simple” auctions, including anonymous and non-anonymous item and bundle pricings, with either a single or multiple buyers. The first step of the framework is to show that the set of auction allocation rules have a low-dimensional representation. The second step shows that, across the subset of auctions that share the same allocations on a given set of samples, the auction revenue varies in a low-dimensional way. Our results effectively imply that whenever it is possible to compute a near-optimal simple auction with a known prior, it is also possible to compute such an auction with an unknown prior, given a polynomial number of samples.

The k-means method is a widely used technique for clustering points in Euclidean space. While it is extremely fast in practice, its worst-case running time is exponential in the number of data points. We prove that the k-means method can implicitly solve PSPACE-complete problems, providing a complexity-theoretic explanation for its worst-case running time. Our result parallels recent work on the complexity of the simplex method for linear programming.

We study the two-party communication complexity of finding an approximate Brouwer fixed point of a composition of two Lipschitz functions g o f: [0,1]n→ [0,1]n, where Alice holds f and Bob holds g. We prove an exponential (in n) lower bound on the deterministic communication complexity of this problem. Our technical approach is to adapt the Raz-McKenzie simulation theorem (FOCS 1999) into geometric settings, thereby "smoothly lifting" the deterministic query lower bound for finding an approximate fixed point (Hirsch, Papadimitriou and Vavasis, Complexity 1989) from the oracle model to the two-party model. Our results also suggest an approach to the well-known open problem of proving strong lower bounds on the communication complexity of computing approximate Nash equilibria. Specifically, we show that a slightly "smoother" version of our fixed-point computation lower bound (by an absolute constant factor) would imply that: The deterministic two-party communication complexity of finding an ∈ = Ω(1/log2N)-approximate Nash equilibrium in an N × N bimatrix game (where each player knows only his own payoff matrix) is at least Nγfor some constant γ > 0. (In contrast, the nondeterministic communication complexity of this problem is only O(log6N)). ; The deterministic (Number-In-Hand) multiparty communication complexity of finding an ∈ = Ω(1)-Nash equilibrium in a k-player constant-action game is at least 2Ω(k/log k)(while the nondeterministic communication complexity is only O(k)).

The best algorithm for a computational problem generally depends on the "relevant inputs," a concept that depends on the application domain and often defies formal articulation. While there is a large literature on empirical approaches to selecting the best algorithm for a given application domain, there has been surprisingly little theoretical analysis of the problem.

The price of anarchy is a measure of the inefficiency of selfish behavior that has been successfully analyzed in many applications, including network routing, resource allocation, auctions, and even models of basketball. It is defined as the worst-case ratio between the welfare of a Nash equilibrium and that of an optimal (first-best) solution. Seemingly, a bound on the price of anarchy is meaningful only if players successfully reach some Nash equilibrium. The main result of this paper is that for many of the classes of games in which the price of anarchy has been studied, results are "intrinsically robust": a bound on the worst-case price of anarchy for pure Nash equilibria *necessarily* implies the exact same worst-case bound for much larger sets of outcomes, including mixed Nash equilibria, correlated equilibria, and sequences of outcomes generated by natural experimentation strategies (such as successive best responses or simultaneous regret-minimization). We also discuss subsequent developments, such as generalizations to incomplete-information games with applications to mechanism design.

This paper presents the first polynomial-time algorithm for position and matroid auction environments that learns, from samples from an unknown distribution, an auction with expected revenue arbitrarily close to the maximum possible. In contrast to most previous work, our results do not assume that the unknown distribution is regular, and require only that the distribution does not have an extremely heavy tail (a necessary assumption for any non-trivial results). Our performance guarantee is with respect to the strongest possible benchmark, the Myerson-optimal auction. Learning a near-optimal auction for an irregular distribution is technically challenging because it requires learning the appropriate "ironed intervals", a delicate global property of the distribution.

We study the problem of computing and learning non-anonymous reserve prices to maximize revenue. We first define the {\sc Maximizing Multiple Reserves (MMR)} problem in single-parameter matroid environments, where the input is $m$ valuation profiles v^1,...,v^m, indexed by the same n bidders, and the goal is to compute the vector r of (non-anonymous) reserve prices that maximizes the total revenue obtained on these profiles by the VCG mechanism with reserves r. We prove that the problem is APX-hard, even in the special case of single-item environments, and give a polynomial-time 1/2-approximation algorithm for it in arbitrary matroid environments.

The goal of this paper is to identify fundamental limitations on how efficiently algorithms implemented on platforms such as MapReduce and Hadoop can compute the central problems in the motivating application domains, such as graph connectivity problems.

We present an analysis framework for bounding the price of anarchy (POA) in games that have many players, as in many of the games most pertinent to computer science applications. We use this framework to demonstrate that, in many of the models in which the POA has been studied, the POA in large games is much smaller than the worst-case bound. Our framework also differentiates between mechanisms with similar worst-case performance, such as simultaneous uniform-price auctions and greedy combinatorial auctions, thereby providing new insights about which mechanisms are likely to perform well in realistic settings.

We design the first truthful-in-expectation, constant-factor approximation mechanisms for NP -hard cases of the welfare maximization problem in combinatorial auctions with nonidentical items and in combinatorial public projects. Our results apply to bidders with valuations that are nonnegative linear combinations of gross-substitute valuations, a class that encompasses many of the most well-studied subclasses of submodular functions, including coverage functions and weighted matroid rank functions. Our mechanisms have an expected polynomial runtime and achieve an approximation factor of 1 − 1/ e . This approximation factor is the best possible for both problems, even for known and explicitly given coverage valuations, assuming P ≠ NP . Recent impossibility results suggest that our results cannot be extended to a significantly larger valuation class. Both of our mechanisms are instantiations of a new framework for designing approximation mechanisms based on randomized rounding algorithms. The high-level idea of this framework is to optimize directly over the (random) output of the rounding algorithm , rather than the usual (and rarely truthful) approach of optimizing over the input to the rounding algorithm. This framework yields truthful-in-expectation mechanisms, which can be implemented efficiently when the corresponding objective function is concave. For bidders with valuations in the cone generated by gross-substitute valuations, we give novel randomized rounding algorithms that lead to both a concave objective function and a (1 − 1/ e )-approximation of the optimal welfare.

High triangle density---the graph property stating that a constant fraction of two-hop paths belongs to a triangle---is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius $2$ subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable $k$-median instances.

We consider a private variant of the classical allocation problem: given $k$ goods and $n$ agents with private valuation functions over bundles of goods, how can we allocate goods to agents to maximize social welfare? An important special case is when agents desire at most one good, and specify their (private) value for each good: in this case, the problem is exactly the maximum-weight matching problem in a bipartite graph. Private matching and allocation problems have not been considered in the differential privacy literature for a good reason: they are plainly impossible to solve under differential privacy. Informally, the allocation must match agents to their preferred goods in order to maximize social welfare, but this preference is exactly what agents wish to hide! Therefore, we consider the problem under the relaxed constraint of joint differential privacy: for any agent $i$, no coalition of agents excluding $i$ should be able to learn about the valuation function of agent $i$. In this setting, the full allocation is no longer published---instead, each agent is told what good to receive. We first show that if there are several identical copies of each good, it is possible to efficiently and accurately solve the matching problem while guaranteeing joint differential privacy. We then consider the more general allocation problem where bidder valuations satisfy the gross substitutes condition. Finally, we prove that the allocation problem cannot be solved to nontrivial accuracy under joint differential privacy without requiring multiple copies of each type of good.

Structured prediction tasks in machine learning involve the simultaneous prediction of multiple labels. This is often done by maximizing a score function on the space of labels, which decomposes as a sum of pairwise elements, each depending on two specific labels. The goal of this paper is to develop a theoretical explanation of the empirical effectiveness of heuristic inference algorithms for solving such structured prediction problems. We study the minimum-achievable expected Hamming error in such problems, highlighting the case of 2D grid graphs, which are common in machine vision applications. Our main theorems provide tight upper and lower bounds on this error, as well as a polynomial-time algorithm that achieves the bound.

This paper develops a general approach, rooted in statistical learning theory, to learning an approximately revenue-maximizing auction from data. We introduce t-level auctions to interpolate between simple auctions, such as welfare maximization with reserve prices, and optimal auctions, thereby balancing the competing demands of expressivity and simplicity. We prove that such auctions have small representation error, in the sense that for every product distribution F over bidders’ valuations, there exists a t-level auction with small t and expected revenue close to optimal. We show that the set of t-level auctions has modest pseudo-dimension (for polynomial t) and therefore leads to small learning error. One consequence of our results is that, in arbitrary single-parameter settings, one can learn a mechanism with expected revenue arbitrarily close to optimal from a polynomial number of samples.

We study the problem of setting a price for a potential buyer with a valuation drawn from an unknown distribution D. The seller has "data" about D in the form of m ≥ 1 i.i.d. samples, and the algorithmic challenge is to use these samples to obtain expected revenue as close as possible to what could be achieved with advance knowledge of D.

Border's theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian single-item environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border's theorem either restrict attention to relatively simple settings, or resort to approximation. This paper identifies a complexity-theoretic barrier that indicates, assuming standard complexity class separations, that Border's theorem cannot be extended significantly beyond the state-of-the-art. We also identify a surprisingly tight connection between Myerson's optimal auction theory, when applied to public project settings, and some fundamental results in the analysis of Boolean functions.

Understanding when equilibria are guaranteed to exist is a central theme in economic theory, seemingly unrelated to computation. This paper shows that the existence of pricing equilibria is inextricably connected to the computational complexity of related optimization problems: demand oracles, revenue-maximization, and welfare-maximization. This relationship implies, under suitable complexity assumptions, a host of impossibility results. We also suggest a complexity-theoretic explanation for the lack of useful extensions of the Walrasian equilibrium concept: such extensions seem to require the invention of novel polynomial-time algorithms for welfare-maximization.

The price of anarchy, defined as the ratio of the worst-case objective function value of a Nash equilibrium of a game and that of an optimal outcome, quantifies the inefficiency of selfish behavior. Remarkably good bounds on this measure are known for a wide range of application domains. However, such bounds are meaningful only if a game's participants successfully reach a Nash equilibrium. This drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash equilibria and correlated equilibria, and to sequences of outcomes generated by natural experimentation strategies, such as successive best responses and simultaneous regret-minimization. We establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives. First, we identify a “canonical sufficient condition” for an upper bound on the price of anarchy of pure Nash equilibria, which we call a smoothness argument . Second, we prove an “extension theorem”: every bound on the price of anarchy that is derived via a smoothness argument extends automatically , with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of every outcome sequence generated by no-regret learners. Smoothness arguments also have automatic implications for the inefficiency of approximate equilibria, for bicriteria bounds, and, under additional assumptions, for polynomial-length best-response sequences. Third, we prove that in congestion games, smoothness arguments are “complete” in a proof-theoretic sense: despite their automatic generality, they are guaranteed to produce optimal worst-case upper bounds on the price of anarchy.

This special section comprises eight fully refereed papers whose extended abstracts were presented at the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2012) in New Brunswick, New Jersey, October 20--23, 2012. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2012 proceedings. The regular conference program consisted of 79 papers chosen from among 248 submissions. These were selected by a program committee consisting of Susanne Albers, Nikhil Bansal, Mark Braverman, Harry Buhrman, Moses Charikar, Anupam Gupta, Adam Klivans, Swastik Kopparty, Yishay Mansour, Rasmus Pagh, Rafael Pass, Ramamohan Paturi, Sofya Raskhodnikova, Aaron Roth, Tim Roughgarden (chair), Alexander Russell, Amit Sahai, C. Seshadhri, Berthold Vöcking, Jan Vondrak, and David Woodruff. The papers invited to this special section were also selected with the input of the program committee. The eight papers in this section span a broad range of topics, including dynamical systems, cryptography, approximation algorithms, hardness of approximation, complexity theory, and combinatorics. Each paper underwent an extensive refereeing process; I thank the authors and the anonymous referees for their efforts. In addition, I would like to thank SICOMP Editor-in-Chief Leonard Schulman and SIAM Senior Publications Coordinator Heather Blythe for their help in preparing this special section.

We consider a model of user engagement in social networks, where each player incurs a cost to remain engaged but derives a benefit proportional to the number of engaged neighbors. The natural equilibrium of this model corresponds to the $k$-core of the social network---the maximal induced subgraph with minimum degree at least $k$. We introduce the problem of “anchoring” a small number of vertices to maximize the size of the corresponding anchored $k$-core---the maximal induced subgraph in which every nonanchored vertex has degree at least $k$. This problem corresponds to preventing “unraveling''---a cascade of iterated withdrawals---and it identifies the individuals whose participation is most crucial to the overall health of a social network. We classify the computational complexity of this problem as a function of $k$ and of the graph structure. We provide polynomial-time algorithms for general graphs with $k=2$ and for bounded-treewidth graphs with arbitrary $k$. We prove strong inapproximability results for general graphs and $k \ge 3$.

This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worst-case quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each motivated by a different set of problems: congestion, scheduling, and distributed welfare games, welfare-maximization in combinatorial auctions with "black-box" bidder valuations, and welfare-maximization in combinatorial auctions with succinctly described valuations. The most straightforward use of our lower bound framework is to harness an existing computational or communication lower bound to derive a lower bound on the worst-case price of anarchy (POA) in a class of games. This is a new approach to POA lower bounds, which relies on reductions in lieu of explicit constructions. More generally, the POA lower bounds implied by our framework apply to all classes of games that share the same underlying optimization problem, independent of the details of players' utility functions. For this reason, our lower bounds are particularly significant for problems of game design -- ranging from the design of simple combinatorial auctions to the computation of tolls for routing networks -- where the goal is to design a game that has only near-optimal equilibria. For example, our results imply that the simultaneous first-price auction format is optimal among all "simple combinatorial auctions" in several settings.

High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius 2 subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable k-median instances.

Deferred-acceptance auctions are auctions for binary single-parameter mechanism design problems whose allocation rule can be implemented using an adaptive reverse greedy algorithm. Milgrom and Segal [2014] recently introduced these auctions and proved that they satisfy a remarkable list of incentive guarantees: in addition to being dominant-strategy incentive-compatible, they are weakly group-strategyproof, can be implemented by ascending-clock auctions, and admit outcome-equivalent full-information pay-as-bid versions. Neither forward greedy mechanisms nor the VCG mechanism generally possess any of these additional incentive properties. The goal of this paper is to initiate the study of deferred-acceptance auctions from an approximation standpoint. We study these auctions through the lens of two canonical welfare-maximization problems, in knapsack auctions and in combinatorial auctions with single-minded bidders.

Designing double auctions is a complex problem, especially when there are restrictions on the sets of buyers and sellers that may trade with one another. The goal of this paper is to develop ``black-box reductions'' from double-auction design to the exhaustively-studied problem of designing single-sided mechanisms.

In the design and analysis of revenue-maximizing auctions, auction performance is typically measured with respect to a prior distribution over inputs. The most obvious source for such a distribution is past data. The goal of this paper is to understand how much data is necessary and sufficient to guarantee near-optimal expected revenue.

We consider a private variant of the classical allocation problem: given k goods and n agents with individual, private valuation functions over bundles of goods, how can we partition the goods amongst the agents to maximize social welfare? An important special case is when each agent desires at most one good, and specifies her (private) value for each good: in this case, the problem is exactly the maximum-weight matching problem in a bipartite graph.

We give the first black-box reduction from approximation algorithms to truthful approximation mechanisms for a non-trivial class of multi-parameter problems. Specifically, we prove that every welfare-maximization problem that admits a fully polynomial-time approximation scheme (FPTAS) and can be encoded as a packing problem also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Our reduction makes novel use of smoothed analysis by employing small perturbations as a tool in algorithmic mechanism design. We develop a “duality” between linear perturbations of the objective function of an optimization problem and of its feasible set, and we use the “primal” and “dual” viewpoints to prove the running time bound and the truthfulness guarantee, respectively, for our mechanism.

We consider a number of classical and new computational problems regarding marginal distributions, and inference in models specifying a full joint distribution. We prove general and efficient reductions between a number of these problems, which demonstrate that algorithmic progress in inference automatically yields progress for “pure data” problems. Our main technique involves formulating the problems as linear programs, and proving that the dual separation oracle for the Ellipsoid Method is provided by the target problem. This technique may be of independent interest in probabilistic inference.

We construct prior-free auctions with constant-factor approximation guarantees with ordered bidders, in both unlimited and limited supply settings. We compare the expected revenue of our auctions on a bid vector to the monotone price benchmark, the maximum revenue that can be obtained from a bid vector using supply-respecting prices that are nonincreasing in the bidder ordering and bounded above by the second-highest bid. As a consequence, our auctions are simultaneously near-optimal in a wide range of Bayesian multi-unit environments.

We study optimal and approximately-optimal mechanism design questions in the interdependent values model, which generalizes the standard setting of independent and private values. We focus our attention on ex post incentive compatible and individually rational mechanisms, and develop an analog of Myerson's optimal auction theory that applies to many interdependent settings of interest. We demonstrate two applications for specific interdependent settings: First, a parallel result to the well-known optimality of the second-price auction with reserve for i.i.d.~bidders, where the English auction replaces the second-price one. Second, we identify good prior-independent auctions --- auctions with near-optimal expected revenue across a wide range of priors --- for certain interdependent value settings.

Complements between goods--where one good takes on added value in the presence of another--have been a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymied theoretical progress in the area. For example, there are no known positive results for combinatorial auctions in which bidder valuations are multi-parameter and non-complement-free, other than the relatively weak results known for general valuations.

We define smooth games of incomplete information. We prove an "extension theorem" for such games: price of anarchy bounds for pure Nash equilibria for all induced full-information games extend automatically, without quantitative degradation, to all mixed-strategy Bayes-Nash equilibria with respect to a product prior distribution over players' preferences. We also note that, for Bayes-Nash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games.

Most results in revenue-maximizing auction design hinge on "getting the price right" --- offering goods to bidders at a price low enough to encourage a sale, but high enough to garner non-trivial revenue. Getting the price right can be hard work, especially when the seller has little or no a priori information about bidders' valuations.

Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally let f be a function mapping subsets of some ground set [n] to the non-negative real numbers. We say that f′ is an α-sketch of f if for every set S, the value f′(S) lies between f(S)/α and f(S), and f′ can be specified by poly(n) bits. We show that for every subadditive function f there exists an α-sketch where α = n1/2 · O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n1/2) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n1/3) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n1−ε. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general.

Prior-free auctions are robust auctions that assume no distribution over bidders' valuations and provide worst-case (input-by-input) approximation guarantees. In contrast to previous work on this topic, we pursue good prior-free auctions with non-identical bidders.

Abstract We study the price of anarchy of selfish routing with variable traffic rates and when the path cost is a nonadditive function of the edge costs. Nonadditive path costs are important, for example, in networking applications, where a key performance metric is the achievable throughput along a path, which is controlled by its bottleneck (most congested) edge. We prove the following results. In multicommodity networks, the worst‐case price of anarchy under the ℓp path cost with 1 < p ≤∞ can be dramatically larger than under the standard ℓ1 path cost. In single‐commodity networks, the worst‐case price of anarchy under the ℓp path cost with 1 < p < ∞ is no more than with the standard ℓ1 path norm. (A matching lower bound follows trivially from known results.) This upper bound also applies to the ℓ∞ path cost if and only if attention is restricted to the natural subclass of equilibria generated by distributed shortest path routing protocols. For a natural cost‐minimization objective function, the price of anarchy with endogenous traffic rates (and under any ℓp path cost) is no larger than that in fixed‐demand networks. Intuitively, the worst‐case inefficiency arising from the “tragedy of the commons” is no more severe than that from routing inefficiencies. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether a given database row is included. The goal of this paper is to formulate and provide strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a symmetric and monotone loss function). Our main result is the following: for each fixed count query and differential privacy level, there is a geometric mechanism $M^*$---a discrete variant of the simple and well-studied mechanism that adds random noise from a Laplace distribution---that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user $u$, no matter what its side information and preferences, derives as much utility from $M^*$ as from interacting with a differentially private mechanism $M_u$ that is optimally tailored to $u$. More precisely, for every user $u$ there is an optimal mechanism $M_u$ for it that factors into a user-independent part (the geometric mechanism $M^*$) and a user-specific postprocessing step that depends only on the output of the geometric mechanism and not on the underlying database. The first part of our proof of this result characterizes the optimal differentially private mechanism for a user as a certain basic feasible solution to a linear program with a user-specific objective function and user-independent constraints that encode differential privacy. The second part shows that all of the relevant vertices of the feasible region (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range.

Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.

We resolve the worst-case price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest non-trivial special case of affine cost functions. We make two distinct contributions. On the upper-bound side, we define the framework of “local smoothness”, which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set ℒ that satisfies mild technical conditions, the worst-case POA of pure Nash equilibria in atomic splittable congestion games with cost functions in ℒ is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games.

We design an expected polynomial time, truthful in expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions and matroid weighted-rank functions. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P ≠ NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations.

We present the first monotone randomized polynomial-time approximation scheme (PTAS) for minimizing the makespan of parallel related machines ($Q||C_{\max}$), the paradigmatic problem in single-parameter algorithmic mechanism design. This result immediately gives a polynomial-time, truthful (in expectation) mechanism whose approximation guarantee attains the best-possible one for all polynomial-time algorithms (assuming $P\neq NP$). Our algorithmic techniques are flexible and also yield a monotone deterministic quasi-PTAS for $Q||C_{\max}$ and a monotone randomized PTAS for max-min scheduling on related machines.

We give several new upper and lower bounds on the worst-case severity of Braess's paradox and the price of anarchy of selfish routing with respect to the maximum latency objective. In single-commodity networks with arbitrary continuous and nondecreasing latency functions, we prove that this worst-case price of anarchy is exactly $n-1$, where n is the number of network vertices. For Braess's paradox in such networks, we prove that removing at most c edges from a network decreases the common latency incurred by traffic at equilibrium by at most a factor of $c+1$. In particular, the worst-case severity of Braess's paradox with a single edge removal is maximized in Braess's original four-vertex network. In multicommodity networks, we exhibit an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both the worst-case severity of Braess's paradox and the price of anarchy for the maximum latency objective grow exponentially with the network size. This construction demonstrates that numerous known selfish routing results for single-commodity networks have no analogues in networks with two or more commodities. We also prove an upper bound on both of these quantities that is exponential in the network size and independent of the network latency functions, showing that our construction is close to optimal. Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability.

We give the first black-box reduction from arbitrary approximation algorithms to truthful approximation mechanisms for a non-trivial class of multi-parameter problems. Specifically, we prove that every packing problem that admits an FPTAS also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Our reduction makes novel use of smoothed analysis, by employing small perturbations as a tool in algorithmic mechanism design. We develop a “duality'' between linear perturbations of the objective function of an optimization problem and of its feasible set, and use the “primal'' and “dual'' viewpoints to prove the running time bound and the truthfulness guarantee, respectively, for our mechanism.

We design and analyze approximately revenue-maximizing auctions in general single-parameter settings. Bidders have publicly observable attributes, and we assume that the valuations of indistinguishable bidders are independent draws from a common distribution. Crucially, we assume all valuation distributions are a priori unknown to the seller. Despite this handicap, we show how to obtain approximately optimal expected revenue - nearly as large as what could be obtained if the distributions were known in advance - under quite general conditions.

We define a new interactive differentially private mechanism --- the median mechanism --- for answering arbitrary predicate queries that arrive online. Given fixed accuracy and privacy constraints, this mechanism can answer exponentially more queries than the previously best known interactive privacy mechanism (the Laplace mechanism, which independently perturbs each query result). With respect to the number of queries, our guarantee is close to the best possible, even for non-interactive privacy mechanisms. Conceptually, the median mechanism is the first privacy mechanism capable of identifying and exploiting correlations among queries in an interactive setting.

Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network cost-sharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal cost-sharing protocols for undirected and directed graphs, single-sink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the cost-sharing protocols that induce only network games with pure-strategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs and that simple priority protocols are essentially optimal in undirected graphs.

We study the distributed desynchronization problem for graphs with arbitrary topology. Motivated by the severe computational limitations of sensor networks, we present a randomized algorithm for network desynchronization that uses an extremely lightweight model of computation, while being robust to link volatility and node failure. These techniques also provide novel, ultra-lightweight randomized algorithms for quickly computing distributed vertex colorings using an asymptotically optimal number of colors.

We introduce revenue submodularity, the property that market expansion has diminishing returns on an auction's expected revenue. We prove that revenue submodularity is generally possible only in matroid markets, that Bayesian-optimal auctions are always revenue-submodular in such markets, and that the VCG mechanism is revenue-submodular in matroid markets with i.i.d bidders and "sufficient competition". We also give two applications of revenue submodularity: good approximation algorithms for novel market expansion problems, and approximate revenue guarantees for the VCG mechanism with i.i.d bidders.

The monopolist's theory of optimal single-item auctions for agents with independent private values can be summarized by two statements. The first is from Myerson [8]: the optimal auction is Vickrey with a reserve price. The second is from Bulow and Klemperer [1]: it is better to recruit one more bidder and run the Vickrey auction than to run the optimal auction. These results hold for single-item auctions under the assumption that the agents' valuations are independently and identically drawn from a distribution that satisfies a natural (and prevalent) regularity condition.

A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism M* -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user u, no matter what its side information and preferences, derives as much utility from M* as from interacting with a differentially private mechanism Mu that is optimally tailored to u. More precisely, for every user u there is an optimal mechanism Mu for it that factors into a user-independent part (the geometric mechanism M*) followed by user-specific post-processing that can be delegated to the user itself. The first part of our proof of this result characterizes the optimal differentially private mechanism for a fixed but arbitrary user in terms of a certain basic feasible solution to a linear program with constraints that encode differential privacy. The second part shows that all of the relevant vertices of this polytope (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range.

The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or "price of total anarchy"). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games --- most notably, congestion games with cost functions restricted to an arbitrary fixed set --- that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first structural characterization of atomic congestion games that are universal worst-case examples for the POA.

In a cost-sharing problem, several participants with unknown preferences vie to receive some good or service, and each possible outcome has a known cost. A cost-sharing mechanism is a protocol that decides which participants are allocated a good and at what prices. Three desirable properties of a cost-sharing mechanism are: incentive-compatibility, meaning that participants are motivated to bid their true private value for receiving the good; budget-balance, meaning that the mechanism recovers its incurred cost with the prices charged; and economic efficiency, meaning that the cost incurred and the value to the participants are traded off in an optimal way. These three goals have been known to be mutually incompatible for thirty years. Nearly all the work on cost-sharing mechanism design by the economics and computer science communities has focused on achieving two of these goals while completely ignoring the third. We introduce novel measures for quantifying efficiency loss in cost-sharing mechanisms and prove simultaneous approximate budget-balance and approximate efficiency guarantees for mechanisms for a wide range of cost-sharing problems, including all submodular and Steiner tree problems. Our key technical tool is an exact characterization of worst-case efficiency loss in Moulin mechanisms, the dominant paradigm in cost-sharing mechanism design.

We present the first monotone randomized polynomial-time approximation scheme (PTAS) for minimizing the makespan of parallel related machines (Q||Cmax), the paradigmatic problem in single-parameter algorithmic mechanism design. This result immediately gives a polynomial-time, truthful (in expectation) mechanism whose approximation guarantee attains the best-possible one for all polynomial-time algorithms (assuming P not equal to NP). Our algorithmic techniques are flexible and also yield, among other results, a monotone deterministic quasi-PTAS for Q||Cmaxand a monotone randomized PTAS for max-min scheduling on related machines.

Mechanism design is now a standard tool in computer science for aligning the incentives of self-interested agents with the objectives of a system designer. There is, however, a fundamental disconnect between the traditional application domains of mechanism design (such as auctions) and those arising in computer science (such as networks): while monetary "transfers" (i.e., payments) are essential for most of the known positive results in mechanism design, they are undesirable or even technologically infeasible in many computer systems. Classical impossibility results imply that the reach of mechanisms without transfers is severely limited. Computer systems typically do have the ability to reduce service quality--routing systems can drop or delay traffic, scheduling protocols can delay the release of jobs, and computational payment schemes can require computational payments from users (e.g., in spam-fighting systems). Service degradation is tantamount to requiring that users "burn money", and such "payments" can be used to influence the preferences of the agents at a cost of degrading the social surplus. We develop a framework for the design and analysis of "money-burning mechanisms" to maximize the residual surplus-the total value of the chosen outcome minus the payments required. Our primary contributions are the following. * We define a general template for prior-free optimal mechanism design that explicitly connects Bayesian optimal mechanism design, the dominant paradigm in economics, with worst-case analysis. In particular, we establish a general and principled way to identify appropriate performance benchmarks in prior-free mechanism design. * For general single-parameter agent settings, we characterize the Bayesian optimal money-burning mechanism. * For multi-unit auctions, we design a near-optimal prior-free money-burning mechanism: for every valuation profile, its expected residual surplus is within a constant factor of our benchmark, the residual surplus of the best Bayesian optimal mechanism for this profile. * For multi-unit auctions, we quantify the benefit of general transfers over money-burning: optimal money-burning mechanisms always obtain a logarithmic fraction of the full social surplus, and this bound is tight.

We develop polynomial-time algorithms for finding correlated equilibria—a well-studied notion of rationality that generalizes the Nash equilibrium—in a broad class of succinctly representable multiplayer games, encompassing graphical games, anonymous games, polymatrix games, congestion games, scheduling games, local effect games, as well as several generalizations. Our algorithm is based on a variant of the existence proof due to Hart and Schmeidler, and employs linear programming duality, the ellipsoid algorithm, Markov chain steady state computations, as well as application-specific methods for computing multivariate expectations over product distributions. For anonymous games and graphical games of bounded tree-width, we provide a different polynomial-time algorithm for optimizing an arbitrary linear function over the set of correlated equilibria of the game. In contrast to our sweeping positive results for computing an arbitrary correlated equilibrium, we prove that optimizing over correlated equilibria is NP-hard in all of the other classes of games that we consider.

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions—the Nash equilibria—may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context—it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widely studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is $O(\log k)$, where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

The only known general technique for designing truthful, approximatelybudget-balanced cost-sharing mechanisms is due to Moulin. While Moulin mechanisms have been successfully designed for a widerange of applications, recent negative results show that for manyfundamental cost-sharing problems, Moulin mechanisms inevitably sufferfrom poor budget-balance, poor economic efficiency, or both. We propose acyclic mechanisms, a new framework for designingtruthful, approximately budget-balanced cost-sharing mechanisms. Acyclic mechanisms strictly generalize Moulin mechanisms andoffer three important advantages. First, it is easier to design acyclic mechanisms than Moulinmechanisms: many classical primal-dual algorithms naturallyinduce a non-Moulin acyclic mechanism with good performanceguarantees. Second, for several important classes of cost-sharing problems, acyclicmechanisms have exponentially better budget-balance and economicefficiency than Moulin mechanisms.Finally, while Moulin mechanisms have found application primarily in binary demand games, we extend acyclic mechanisms to general demand games, a multi-parameter setting in which each bidder can be allocated one of several levels of service.

We present constant-factor approximation algorithms for several widely-studied NP-hard optimization problems in network design, including the multicommodity rent-or-buy, virtual private network design, and single-sink buy-at-bulk problems. Our algorithms are simple and their approximation ratios improve over those previously known, in some cases by orders of magnitude. We develop a general analysis framework to bound the approximation ratios of our algorithms. This framework is based on a novel connection between random sampling and game-theoretic cost sharing.

Keywords: NCCR-MICS ; NCCR-MICS/CL3 Reference LCA-ARTICLE-2007-012 Record created on 2007-06-06, modified on 2017-05-12

Abstract — Internet routers require buffers to hold packets during times of congestion. The buffers need to be fast, and so ideally they should be small enough to use fast memory technologies such as SRAM or all-optical buffering. Unfortunately, a widely used rule-of-thumb says we need a bandwidth-delay product of buffering at each router so as not to lose link utilization. This can be prohibitively large. In a recent paper, Appenzeller et al. challenged this rule-of-thumb and showed that for a backbone network, the buffer size can be divided by √ N without sacrificing throughput, where N is the number of flows sharing the bottleneck. In this paper, we explore how buffers in the backbone can be significantly reduced even more, to as little as a few dozen packets, if we are willing to sacrifice a small amount of link capacity. We argue that if the TCP sources are not overly bursty, then fewer than twenty packet buffers are sufficient for high throughput. Specifically, we argue that O(log W) buffers are sufficient, where W is the window size of each flow. We support our claim with analysis and a variety of simulations. The change we need to make to TCP is minimal—each sender just needs to pace packet injections from its window. Moreover, there is some evidence that such small buffers are sufficient even if we don’t modify the TCP sources so long as the access network is much slower than the backbone, which is true today and likely to remain true in the future. We conclude that buffers can be made small enough for all-optical routers with small integrated optical buffers. I.

Braess's Paradox is the counterintuitive but well-known fact that removing edges from a network with "selfish routing" can decrease the latency incurred by traffic in an equilibrium flow. Despite the large amount of research motivated by Braess's Paradox since its discovery in 1968, little is known about whether it is a common real-world phenomenon, or a mere theoretical curiosity.In this paper, we show that Braess's Paradox is likely to occur in a natural random network model. More precisely, with high probability, (as the number of vertices goes to infinity), there is a traffic rate and a set of edges whose removal improves the latency of traffic in an equilibrium flow by a constant factor. Our proof approach is robust and shows that the "global" behavior of an equilibrium flow in a large random network is similar to that in Braess's original four-node example.

We consider a model of game-theoretic network design initially studied by Anshelevich et al. [2], where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich et al. [2] proved that pure-strategy Nash equilibria always exist and that the price of stability--the ratio in costs of a minimumcost Nash equilibrium and an optimal solution--is Θ(log k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has a weight wi ≥ 1, and its cost share of an edge in its path equals wi times the edge cost, divided by the total weight of the players using the edge.This paper presents the first general results on weighted Shapley network design games. First, we give a simple example with no pure-strategy Nash equilibrium. This motivates considering the price of stability with respect to α-approximate Nash equilibria--outcomes from which no player can decrease its cost by more than an α multiplicative factor. Our first positive result is that O(log wmax)-approximate Nash equilibria exist in all weighted Shapley network design games, where wmax is the maximum player weight. More generally, we establish the following trade-off between the two objectives of good stability and low cost: for every α = Ω(log wmax), the price of stability with respect to O(α)- approximate Nash equilibria is O((log W)/α), where W is the sum of the players' weights. In particular, there is always an O(logW)-approximate Nash equilibrium with cost within a constant factor of optimal.Finally, we show that this trade-off curve is nearly optimal: we construct a family of networks without o(log wmax/ log log wmax)-approximate Nash equilibria, and show that for all α = Ω(logwmax/ log log wmax), achieving a price of stability of O(log W/α) requires relaxing equilibrium constraints by an Ω(α) factor.

A cost-sharing mechanism is a protocol that collects bids from a set of players, selects a subset of the players to receive a service (incurring a subset-dependent cost), and determines a price to charge each of these players. Three standard requirements for cost-sharing mechanisms are incentive compatibility, which states that players are motivated to bid their true valuation for the service; budget-balance, meaning that the prices charged should recover the cost incurred; and efficiency, which states that the cost incurred and the valuations of the players served should be traded off in an optimal way. These three goals have been known to be mutually incompatible for thirty years. As a result, nearly all work on cost-sharing mechanisms in the economics and theoretical computer science literatures has focused on achieving two of these goals while completely ignoring the third.We show that incentive-compatibility, budget-balance, and approximate efficiency are simultaneously achievable for a wide range of cost functions, where efficiency is measured using the social cost---the sum of the incurred service cost and the excluded valuations. In particular, we prove such guarantees for well-known mechanisms for all submodular cost functions and for the Steiner tree cost function. We also prove a generic, quantifiable trade-off between the objectives of efficiency and budget-balance in groupstrategyproof cost-sharing mechanisms.

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions - the Nash equilibria - may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context - it is the optimal solution that can be proposed from which no user will "defect". We consider the price of stability for network design with respect to one of the most widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

We study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling small jobs on a set of shared machines possessing latency functions that specify the amount of time needed to complete a job, given the machine load. We measure system performance by the total latency of the system. Assigning jobs according to the selfish interests of individual users, who wish to minimize only the latency that their own jobs experience, typically results in suboptimal system performance. However, in many systems of this type there is a mixture of "selfishly controlled" and "centrally controlled" jobs. The congestion due to centrally controlled jobs will influence the actions of selfish users, and we thus aspire to contain the degradation in system performance due to selfish behavior by scheduling the centrally controlled jobs in the best possible way. We formulate this goal as an optimization problem via Stackelberg games, games in which one player acts a leader (here, the centralized authority interested in optimizing system performance) and the rest as followers (the selfish users). The problem is then to compute a strategy for the leader (a Stackelberg strategy) that induces the followers to react in a way that (approximately) minimizes the total latency in the system. In this paper, we prove that it is NP-hard to compute an optimal Stackelberg strategy and present simple strategies with provably good performance guarantees. More precisely, we give a simple algorithm that computes a strategy inducing a job assignment with total latency no more than a constant times that of the optimal assignment of all of the jobs; in the absence of centrally controlled jobs and a Stackelberg strategy, no result of this type is possible. We also prove stronger performance guarantees in the special case where every machine latency function is linear in the machine load.

We study the multicommodity rent-or-buy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can be rented, with cost incurred on a per-unit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. The first constant-factor approximation algorithm for this problem was recently given by Kumar et al.; however, this algorithm and its analysis are both quite complicated, and its performance guarantee is extremely large. In this paper, we give a conceptually simple 12-approximation algorithm for this problem. Our analysis of this algorithm makes crucial use of cost sharing, the task of allocating the cost of an object to many users of the object in a "fair" manner. While techniques from approximation algorithms have recently yielded new progress on cost sharing problems, our work is the first to show the converse - those ideas from cost sharing can be fruitfully applied in the design and analysis of approximation algorithms.

We study economic incentives for influencing selfish behavior in networks. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimum-latency paths. The quality of a routing of traffic is historically measured by the sum of all travel times, also called the total latency.It is well known that the outcome of selfish routing (a Nash equilibrium) does not minimize the total latency and can be improved upon with coordination, and that marginal cost pricing---charging each network user for the congestion effects caused by its presence---eliminates the inefficiency of selfish routing. However, the principle of marginal cost pricing assumes that (possibly very large) taxes cause no disutility to network users; this is appropriate only when collected taxes can be feasibly returned (directly or indirectly) to the users, for example via a lump-sum refund. If this assumption does not hold and we wish to minimize the total user disutility (latency plus taxes paid)---the total cost---how should we price the network edges? Intuition may suggest that taxes should never be able to improve the cost of a Nash equilibrium, but the famous Braess's Paradox shows this intuition to be incorrect.We consider strategies for pricing network edges to reduce the cost of a Nash equilibrium. Since levying a sufficiently large tax on an edge effectively removes it from the network, our study generalizes previous work on network design citend_hard. In this paper, we prove the following results.

We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimum-latency paths. The quality of a routing of traffic is measured by the sum of travel times (the total latency).It is well known that the outcome of selfish routing (a Nash equilibrium) does not minimize the total latency. An ancient strategy for improving the selfish solution is the principle of marginal cost pricing, which asserts that on each edge of the network, each network user on the edge should pay a tax offsetting the congestion effects caused by its presence. By pricing network edges according to this principle, the inefficiency of selfish routing can always be eradicated.This result, while fundamental, assumes a very strong homogeneity property: all network users are assumed to trade off time and money in an identical way. The guarantee also ignores both the algorithmic aspects of edge pricing and the unfortunate possibility that an efficient routing of traffic might only be achieved with exorbitant taxes. Motivated by these shortcomings, we extend this classical work on edge pricing in several different directions and prove the following results.We prove that the edges of a single-commodity network can always be priced so that an optimal routing of traffic arises as a Nash equilibrium, even for very general heterogeneous populations of network users.When there are only finitely many different types of network users and all edge latency functions are convex, we show how to compute such edge prices efficiently.We prove that an easy-to-check mathematical condition on the population of heterogeneous network users is both necessary and sufficient for the existence of edge prices that induce an optimal routing while requiring only moderate taxes.

We give simple and easy-to-analyze randomized approximation algorithms for several well-studied NP-hard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following. We give a randomized 3.55-approximation algorithm for the connected facility location problem. The algorithm requires three lines to state, one page to analyze, and improves the best-known performance guarantee for the problem. We give a 5.55-approximation algorithm for virtual private network design. Previously, constant-factor approximation algorithms were known only for special cases of this problem. We give a simple constant-factor approximation algorithm for the single-sink buy-at-bulk network design problem. Our performance guarantee improves over what was previously known, and is an order of magnitude improvement over previous combinatorial approximation algorithms for the problem.

We present the first constant factor approximation algorithm for network design with multiple commodities and economies of scale. We consider the rent-or-buy problem, a type of multicommodity buy-at-bulk network design in which there are two ways to install capacity on any given edge. Capacity can be rented, with cost incurred on a per-unit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. Recent work on buy-at-bulk network design has concentrated on the special case where all sinks are identical; existing constant factor approximation algorithms for this special case make crucial use of the assumption that all commodities ship flow to the same sink vertex and do not obviously extend to the multicommodity rent-or-buy problem. Prior to our work, the best heuristics for the multicommodity rent-or-buy problem achieved only logarithmic performance guarantees and relied on the machinery of relaxed metrical task systems or of metric embeddings. By contrast, we solve the network design problem directly via a novel primal-dual algorithm.

We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a directed network in which each edge possesses a latency function describing the common latency incurred by all traffic on the edge as a function of the edge congestion. Given a rate of traffic between each pair of nodes in the network, we aspire toward an assignment of traffic to paths in which the sum of all travel times (the total latency) is minimized; however, in many settings network users are free to route their traffic in a selfish manner, without regard to the total latency. We therefore assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths (a Nash equilibrium) will not minimize the total latency; hence, selfish behavior carries the cost of decreased network performance. We quantify this degradation in network performance via the price of anarchy, defined as the worst possible ratio between the total latency of a Nash equilibrium and of a minimum-latency routing of the traffic.In this paper, we show that the underlying network topology plays no role in the determination of the price of anarchy. Specifically, we show that under weak hypotheses on the class of allowable edge latency functions, the worst-case ratio between the total latency of a Nash equilibrium and of a minimum-latency routing for any multicommodity flow network is achieved by a single-commodity instance on a set of parallel links. In the special case where the class of allowable latency functions includes all of the constant functions, we prove that a network with only two parallel links suffices to achieve the worst-possible ratio. Informally, these results imply that the inefficiency inherent in a flow at Nash equilibrium stems from the inability of selfish users to discern which of two competing routes is superior and not from the topological complexity arising from the diverse intersections of many paths belonging to different commoditie.Our proof techniques also give powerful methods for computing the price of anarchy with respect to an arbitrary class of latency functions. We apply these methods to function classes that have been well studied in the literature (such as degree-bounded polynomials and functions of the form $\ell(x) = (u—x)—1} that are used to model edges with capacity u), thereby achieving the first tight analyses of the price of anarchy for significant classes of latency functions outside the class of linear functions.

We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times---the total latency---is minimized.In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance.In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total latency of the routes chosen by unregulated selfish network users may be arbitrarily larger than the minimum possible total latency; however, we prove that it is no more than the total latency incurred by optimally routingtwiceas much traffic.

We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to s-t flows in which all flow paths have equal latency. We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n nodes and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = NP). We also prove this hardness result to be best possible by exhibiting an n/2-approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no (4/3 - /spl epsi/)-approximation algorithm for the problem (for any /spl epsi/ > 0, unless P = NP); the existence of a 4/3-approximation algorithm follows easily from existing work, proving this hardness result sharp.

AbstractWe study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with load-dependent latency functions specifying the length of time necessary to complete a job; we measure system performance by the total latency of the system. Assigning jobs according to the selfish interests of individual users (who wish to minimize only the latency that their own jobs experience) typically results in suboptimal system performance. However, in many systems of this type there is a mixture of "selfishly controlled " and "centrally controlled " jobs; as the assignment of centrally controlled jobs will influence the subsequent actions by selfish users, we aspire to contain the degradation in system performance due to selfish behavior by scheduling the centrally controlled jobs in the best possible way. We formulate this goal as an optimization problem via Stackelberg games, games in which one player acts a leader (here, the centralized authority interested in optimizing system performance) and the rest as followers (the selfish users). The problem is then to compute a strategy for the leader (a Stackelberg strategy) that induces the followers to react in a way that (at least approximately) minimizes the total latency in the system. In this paper, we prove that it is NP-hard to compute the optimal Stackelberg strategy and present simple strategies with provable performance guarantees. More precisely, we give a simple algorithm that computes a strategy inducing a job assignment with total latency no more than a constant times that of the optimal assignment of all of the jobs; in the absence of centrally controlled jobs and a Stackelberg strategy, no result of this type is possible. We also prove stronger performance guarantees in the special case where every machine latency function is linear in the machine load.

We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times-the total latency-is minimized. In many settings, including the Internet and other large-scale communication networks, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance. We quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and non-decreasing in the edge congestion.