Learning and computation of equilibria are central problems in game theory, theory of computation, and artificial intelligence. In this work, we introduce proximal regret, a new notion of regret based on proximal operators that lies strictly between external and swap regret. When every player employs a no-proximal-regret algorithm in a general convex game, the empirical distribution of play converges to proximal correlated equilibria (PCE), a refinement of coarse correlated equilibria. Our framework unifies several emerging notions in online learning and game theory—such as gradient equilibrium and semicoarse correlated equilibrium—and introduces new ones. Our main result shows that the classic Online Gradient Descent (GD) algorithm achieves an optimal O(√T) bound on proximal regret, revealing that GD, without modification, minimizes a stronger regret notion than external regret. This provides a new explanation for the empirically superior performance of gradient descent in online learning and games. We further extend our analysis to Mirror Descent in the Bregman setting and to Optimistic Gradient Descent, which yields faster convergence in smooth convex games.

In traditional algorithm design, no incentives come into play: the input is given, and your algorithm must produce a correct output. How much harder is it to solve the same problem when the input is not given directly, but instead reported by strategic agents with interests of their own? The unique challenge stems from the fact that the agents may choose to lie about the input in order to manipulate the behavior of the algorithm for their own interests, and tools from Game Theory are, therefore, required in order to predict how these agents will behave. We develop a new algorithmic framework with which to study such problems. Specifically, we provide a computationally efficient black-box reduction from solving any Bayesian optimization problem on “strategic input,” often called algorithmic mechanism design , to solving a perturbed version of that same optimization problem when the input is directly given, traditionally called algorithm design . We further demonstrate the power of our framework by making significant progress on several long-standing open problems. First, we provide an algorithmic extension of Myerson’s celebrated characterization of single item auctions [ 105 ] to multiple items, providing also a computationally efficient implementation of optimal auctions. Next, we design a computationally efficient 2-approximate mechanism for Bayesian job scheduling on unrelated machines, the original problem studied in Nisan and Ronen’s seminal article introducing the field of Algorithmic Mechanism Design [ 106 ]. This matches the guarantee of the best known computationally efficient algorithm when the input is directly given. Finally, leveraging a connection to our framework, we additionally prove hardness of approximation for Bayesian mechanism design with submodular bidders.

Most of the widely used estimators of the \emph{average treatment effect} (ATE) in causal inference rely on the assumptions of \emph{unconfoundedness} and \emph{overlap}. Unconfoundedness requires that the observed covariates account for all correlations between the outcome and treatment. Overlap requires the existence of randomness in treatment decisions for all individuals. Nevertheless, many types of studies frequently violate unconfoundedness or overlap, for instance, observational studies with deterministic treatment decisions - popularly known as Regression Discontinuity designs - violate overlap. In this paper, we initiate the study of general conditions that enable the \emph{identification} of the average treatment effect, extending beyond unconfoundedness and overlap. In particular, following the paradigm of {statistical} learning theory, we provide an interpretable condition that is sufficient and nearly necessary for the identification of ATE. Moreover, this condition characterizes the identification of the \emph{average treatment effect on the treated} (ATT) and can be used to characterize other treatment effects as well. To illustrate the utility of our condition, we present several well-studied scenarios where our condition is satisfied and, hence, we prove that ATE can be identified in regimes that prior works could not capture. For example, under mild assumptions on the data distributions, this holds for the models proposed by Tan (2006) and Rosenbaum (2002), and the Regression Discontinuity design model introduced by Thistlethwaite and Campbell (1960). For each of these scenarios, we also show that, under natural additional assumptions, ATE can be estimated from finite samples. We believe these findings open new avenues for bridging learning-theoretic insights and causal inference methodologies, particularly in observational studies with complex treatment mechanisms.

We study zero-sum games in the space of probability distributions over the Euclidean space $\mathbb{R}^d$ with entropy regularization, in the setting when the interaction function between the players is smooth and strongly convex-strongly concave. We prove an exponential convergence guarantee for the mean-field min-max Langevin dynamics to compute the equilibrium distribution of the zero-sum game. We also study the finite-particle approximation of the mean-field min-max Langevin dynamics, both in continuous and discrete times. We prove biased convergence guarantees for the continuous-time finite-particle min-max Langevin dynamics to the stationary mean-field equilibrium distribution with an explicit bias term which does not scale with the number of particles. We also prove biased convergence guarantees for the discrete-time finite-particle min-max Langevin algorithm to the stationary mean-field equilibrium distribution with an additional bias term which scales with the step size and the number of particles. This provides an explicit iteration complexity for the average particle along the finite-particle algorithm to approximately compute the equilibrium distribution of the zero-sum game.

We study last-iterate convergence properties of algorithms for solving two-player zero-sum games based on Regret Matching$^+$ (RM$^+$). Despite their widespread use for solving real games, virtually nothing is known about their last-iterate convergence. A major obstacle to analyzing RM-type dynamics is that their regret operators lack Lipschitzness and (pseudo)monotonicity. We start by showing numerically that several variants used in practice, such as RM$^+$, predictive RM$^+$ and alternating RM$^+$, all lack last-iterate convergence guarantees even on a simple $3\times 3$ matrix game. We then prove that recent variants of these algorithms based on a smoothing technique, extragradient RM$^{+}$ and smooth Predictive RM$^+$, enjoy asymptotic last-iterate convergence (without a rate), $1/\sqrt{t}$ best-iterate convergence, and when combined with restarting, linear-rate last-iterate convergence. Our analysis builds on a new characterization of the geometric structure of the limit points of our algorithms, marking a significant departure from most of the literature on last-iterate convergence. We believe that our analysis may be of independent interest and offers a fresh perspective for studying last-iterate convergence in algorithms based on non-monotone operators.

We consider the problem of maximizing a submodular function with access to a _noisy_ value oracle for the function instead of an exact value oracle. Similar to prior work, we assume that the noisy oracle is persistent in that multiple calls to the oracle for a specific set always return the same value. In this model, Hassidim and Singer (2017) design a $(1-1/e)$-approximation algorithm for monotone submodular maximization subject to a cardinality constraint, and Huang et al (2022) design a $(1-1/e)/2$-approximation algorithm for monotone submodular maximization subject to any arbitrary matroid constraint. In this paper, we design a meta-algorithm that allows us to take any "robust" algorithm for exact submodular maximization as a black box and transform it into an algorithm for the noisy setting while retaining the approximation guarantee. By using the meta-algorithm with the measured continuous greedy algorithm, we obtain a $(1-1/e)$-approximation (resp. $1/e$-approximation) for monotone (resp. non-monotone) submodular maximization subject to a matroid constraint under noise. Furthermore, by using the meta-algorithm with the double greedy algorithm, we obtain a $1/2$-approximation for unconstrained (non-monotone) submodular maximization under noise.

The convergence of online learning algorithms in games under self-play is a fundamental question in game theory and machine learning. Among various notions of convergence, last-iterate convergence is particularly desirable, as it reflects the actual decisions made by the learners and captures the day-to-day behavior of the learning dynamics. While many algorithms are known to converge in the average-iterate, achieving last-iterate convergence typically requires considerably more effort in both the design and the analysis of the algorithm. Somewhat surprisingly, we show in this paper that for a large family of games, there exists a simple black-box reduction that transforms the average iterates of an uncoupled learning dynamics into the last iterates of a new uncoupled learning dynamics, thus also providing a reduction from last-iterate convergence to average-iterate convergence. Our reduction applies to games where each player’s utility is linear in both their own strategy and the joint strategy of all opponents. This family includes two-player bimatrix games and generalizations such as multi-player polymatrix games. By applying our reduction to the Optimistic Multiplicative Weights Update algorithm, we obtain new state-of-the-art last-iterate convergence rates for uncoupled learning dynamics in multi-player zero-sum polymatrix games: (1) an $O(\frac{\log d}{T})$ last-iterate convergence rate under gradient feedback, representing an exponential improvement in the dependence on the dimension $d$ (i.e., the maximum number of actions available to either player); and (2) an $\tilde{O}(d^{\frac{1}{5}}T^{-\frac{1}{5}})$ last-iterate convergence rate under bandit feedback, improving upon the previous best rates of $\tilde{O}(\sqrt{d}T^{-\frac{1}{8}})$ and $\tilde{O}(\sqrt{d}T^{-\frac{1}{6}})$.

We study policy optimization algorithms for computing correlated equilibria in multi-player general-sum Markov Games. Previous results achieve $\tilde{O}(T^{-1/2})$ convergence rate to a correlated equilibrium and an accelerated $\tilde{O}(T^{-3/4})$ convergence rate to the weaker notion of coarse correlated equilibrium. In this paper, we improve both results significantly by providing an uncoupled policy optimization algorithm that attains a near-optimal $\tilde{O}(T^{-1})$ convergence rate for computing a correlated equilibrium. Our algorithm is constructed by combining two main elements (i) smooth value updates and (ii) the \emph{optimistic-follow-the-regularized-leader} algorithm with the log barrier regularizer.

We study constrained comonotone min-max optimization, a structured class of nonconvex-nonconcave min-max optimization problems, and their generalization to comonotone inclusion. In our first contribution, we extend the *Extra Anchored Gradient (EAG)* algorithm, originally proposed by Yoon and Ryu (2021) for unconstrained min-max optimization, to constrained comonotone min-max optimization and comonotone inclusion, achieving an optimal convergence rate of $O\left(\frac{1}{T}\right)$ among all first-order methods. Additionally, we prove that the algorithm's iterations converge to a point in the solution set. In our second contribution, we extend the *Fast Extra Gradient (FEG)* algorithm, as developed by Lee and Kim (2021), to constrained comonotone min-max optimization and comonotone inclusion, achieving the same $O\left(\frac{1}{T}\right)$ convergence rate. This rate is applicable to the broadest set of comonotone inclusion problems yet studied in the literature. Our analyses are based on simple potential function arguments, which might be useful for analyzing other accelerated algorithms.

Self play via online learning is one of the premier ways to solve large-scale zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages, including logarithmic dependence on the size of the payoff matrix and $\tilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $\delta>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/\delta$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

Partial observability of the underlying states generally presents significant challenges for reinforcement learning (RL). In practice, certain *privileged information* , e.g., the access to states from simulators, has been exploited in training and achieved prominent empirical successes. To better understand the benefits of privileged information, we revisit and examine several simple and practically used paradigms in this setting, with both computation and sample efficiency analyses. Specifically, we first formalize the empirical paradigm of *expert distillation* (also known as *teacher-student* learning), demonstrating its pitfall in finding near-optimal policies. We then identify a condition of the partially observable environment, the deterministic filter condition, under which expert distillation achieves sample and computational complexities that are *both* polynomial. Furthermore, we investigate another successful empirical paradigm of *asymmetric actor-critic*, and focus on the more challenging setting of observable partially observable Markov decision processes. We develop a belief-weighted optimistic asymmetric actor-critic algorithm with polynomial sample and quasi-polynomial computational complexities, where one key component is a new provable oracle for learning belief states that preserve *filter stability* under a misspecified model, which may be of independent interest. Finally, we also investigate the provable efficiency of partially observable multi-agent RL (MARL) with privileged information. We develop algorithms with the feature of centralized-training-with-decentralized-execution, a popular framework in empirical MARL, with polynomial sample and (quasi-)polynomial computational complexity in both paradigms above. Compared with a few recent related theoretical studies, our focus is on understanding practically inspired algorithmic paradigms, without computationally intractable oracles.

Classical auction theory typically focuses on models with exogenous signal structures, where all auction participants privately know their item valuations. However, practical scenarios often find buyers initially uninformed about their item values due to the lack of data on item quality or suitability. Instead, buyers rely on seller advertisements to learn their values. For instance, streaming services offer free trials to influence consumer perceptions before subscriptions, while real estate agencies provide property inspections for better buyer estimations.

We consider the problem of maximizing the gains from trade (GFT) in two-sided markets. The seminal impossibility result by Myerson and Satterthwaite (1983) shows that even for bilateral trade, there is no individually rational (IR), Bayesian incentive compatible (BIC) and budget balanced (BB) mechanism that can achieve the full GFT. Moreover, the optimal BIC, IR and BB mechanism that maximizes the GFT is known to be complex and heavily depends on the prior. In this paper, we pursue a Bulow-Klemperer-style question, i.e., does augmentation allow for prior-independent mechanisms to compete against the optimal mechanism? Our first main result shows that in the double auction setting with m i.i.d. buyers and n i.i.d. sellers, by augmenting O(1) buyers and sellers to the market, the GFT of a simple, dominant strategy incentive compatible (DSIC), and prior-independent mechanism in the augmented market is at least the optimal in the original market, when the buyers’ distribution first-order stochastically dominates the sellers’ distribution. The mechanism we consider is a slight variant of the standard Trade Reduction mechanism due to McAfee (1992). For comparison, Babaioff, Goldner, and Gonczarowski (2020) showed that if one is restricted to augmenting only one side of the market, then n(m + 4√m) additional agents are sufficient for their mechanism to beat the original optimal and ⌊ log2 m ⌋ additional agents are necessary for any prior-independent mechanism. Next, we go beyond the i.i.d. setting and study the power of two-sided recruitment in more general markets. Our second main result is that for any ε > 0 and any set of O(1/ε) buyers and sellers where the buyers’ value exceeds the sellers’ value with constant probability, if we add these additional agents into any market with arbitrary correlations, the Trade Reduction mechanism obtains a (1−ε)-approximation of the GFT of the augmented market. Importantly, the newly recruited agents are agnostic to the original market.

We propose a new Markov Decision Process (MDP) model for ad auctions to capture the user response to the quality of ads, with the objective of maximizing the long-term discounted revenue. By incorporating user response, our model takes into consideration all three parties involved in the auction (advertiser, auctioneer, and user). The state of the user is modeled as a user-specific click-through rate (CTR) with the CTR changing in the next round according to the set of ads shown to the user in the current round. We characterize the optimal mechanism for this MDP as a Myerson's auction with a notion of modified virtual value, which relies on the value distribution of the advertiser, the current user state, and the future impact of showing the ad to the user. Leveraging this characterization, we design a sample-efficient and computationally-efficient algorithm which outputs an approximately optimal policy that requires only sample access to the true MDP and the value distributions of the bidders. Finally, we propose a simple mechanism built upon second price auctions with personalized reserve prices and show it can achieve a constant-factor approximation to the optimal long term discounted revenue.

We study revenue maximization in multi-item auctions, where bidders have subadditive valuations over independent items [48]. Providing a simple mechanism that is approximately revenue-optimal in this setting is a major open problem in mechanism design [20]. In this paper, we present the first simple mechanism whose revenue is at least a constant fraction of the optimal revenue in multi-item auctions with subadditive bidders. Our mechanism is a simultaneous auction that incorporates either a personalized entry fee or a personalized reserve price per item. We prove that for any simultaneous auction that satisfies c-efficiency– a new property we propose, its revenue is at least an $O(c)$-approximation to the optimal revenue. We further show that both the simultaneous first-price and the simultaneous all-pay auction are $\frac{1}{2}$-efficient. Providing revenue guarantees for non-truthful simple mechanisms, e.g., simultaneous auctions, in multi-dimensional environments has been recognized by Roughgarden et al. [47] as an important open question. Prior to our result, the only such revenue guarantees are due to Daskalakis et al. [30] for bidders who have additive valuations over independent items. Our result significantly extends the revenue guarantees of these non-truthful simple auctions to settings where bidders have combinatorial valuations.

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow $\mathcal{O}(\frac{1}{\sqrt{T}})$ last-iterate convergence rate to a Nash equilibrium. While the $\mathcal{O}(\frac{1}{\sqrt{T}})$ rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal $\mathcal{O}(\sqrt{T})$ regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal $\mathcal{O}(\frac{1}{T})$ last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an $\mathcal{O}(\log T)$ individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art $\mathcal{O}(\sqrt{T})$ bound.

We revisit the problem of learning in two-player zero-sum Markov games, focusing on developing an algorithm that is *uncoupled*, *convergent*, and *rational*, with non-asymptotic convergence rates to Nash equilibrium. We start from the case of stateless matrix game with bandit feedback as a warm-up, showing an $\tilde{\mathcal{O}}(t^{-\frac{1}{8}})$ last-iterate convergence rate. To the best of our knowledge, this is the first result that obtains finite last-iterate convergence rate given access to only bandit feedback. We extend our result to the case of irreducible Markov games, providing a last-iterate convergence rate of $\tilde{\mathcal{O}}(t^{-\frac{1}{9+\varepsilon}})$ for any $\varepsilon>0$. Finally, we study Markov games without any assumptions on the dynamics, and show a *path convergence* rate, a new notion of convergence we defined, of $\tilde{\mathcal{O}}(t^{-\frac{1}{10}})$. Our algorithm removes the synchronization and prior knowledge requirement of Wei et al. (2021), which pursued the same goals as us for irreducible Markov games. Our algorithm is related to Chen et al. (2021) and Cen et al. (2021)'s and also builds on the entropy regularization technique. However, we remove their requirement of communications on the entropy values, making our algorithm entirely uncoupled.

We study the model of metric voting initially proposed by Feldman et al. [2020]. In this model, experts and candidates are located in a metric space, and each candidate possesses a quality that is independent of her location. An expert evaluates each candidate as the candidate's quality less the distance between the candidate and the expert in the metric space. The expert votes for her favorite candidate. Naturally, the expert prefers candidates that are "similar" to herself, i.e., close to her location in the metric space, thus creating bias in the vote. The goal is to select a voting rule and a committee of experts to mitigate the bias. More specifically, given m candidates, what is the minimum number of experts needed to ensure that the voting rule selects a candidate whose quality is at most ε worse than the best one?

We study the problem of social welfare maximization in bilateral trade, where two agents, a buyer and a seller, trade an indivisible item. The seminal result of Myerson and Satterthwaite shows that no incentive compatible and budget balanced (i.e., the mechanism does not run a deficit) mechanism can achieve the optimal social welfare in bilateral trade. Motivated by this impossibility result, we focus on approximating the optimal social welfare. We consider arguably the simplest form of mechanisms – the fixed-price mechanisms, where the designer offers trade at a fixed price to the seller and buyer. Besides the simple form, fixed-price mechanisms are also the only dominant strategy incentive compatible and budget balanced mechanisms in bilateral trade.

We study the question of last-iterate convergence rate of the extragradient algorithm by Korpelevich [1976] and the optimistic gradient algorithm by Popov [1980] in multi-player games. We show that both algorithms with constant step-size have last-iterate convergence rate of $O(\frac{1}{\sqrt{T}})$ to a Nash equilibrium in terms of the gap function in smooth monotone games, where each player's action set is an arbitrary convex set. Previous results only study the unconstrained setting, where each player's action set is the entire Euclidean space. Our results address an open question raised in several recent work by Hsieh et al. [2019], Golowich et al. [2020a,b], who ask for last-iterate convergence rate of either the extragradient or the optimistic gradient algorithm in the constrained setting. Our convergence rates for both algorithms are tight and match the lower bounds by Golowich et al. [2020a,b]. At the core of our results lies a new notion -- the tangent residual, which we use to measure the proximity to equilibrium. We use the tangent residual (or a slight variation of the tangent residual) as the the potential function in our analysis of the extragradient algorithm (or the optimistic gradient algorithm) and prove that it is non-increasing between two consecutive iterates.

Machine learning has developed a variety of tools for learning and representing high-dimensional distributions with structure. Recent years have also seen big advances in designing multi-item mechanisms. Akin to overfitting, however, these mechanisms can be extremely sensitive to the Bayesian prior that they target, which becomes problematic when that prior is only approximately known. At the same time, even if access to the exact Bayesian prior is given, it is known that optimal or even approximately optimal multi-item mechanisms run into sample, computational, representation and communication intractability barriers.

We study the problem of selling information to a data-buyer who faces a decision problem under uncertainty. We consider the classic Bayesian decision-theoretic model pioneered by Blackwell. Initially, the data buyer has only partial information about the payoff-relevant state of the world. A data seller offers additional information about the state of the world. The information is revealed through signaling schemes, also referred to as experiments. In the single-agent setting, any mechanism can be represented as a menu of experiments. A recent paper by Bergemann et al.[8] present a complete characterization of the revenue-optimal mechanism in a binary state and binary action environment. By contrast, no characterization is known for the case with more actions. In this paper, we consider more general environments and study arguably the simplest mechanism, which only sells the fully informative experiment. In the environment with binary state and m≥3 actions, we provide an $O(m)$-approximation to the optimal revenue by selling only the fully informative experiment and show that the approximation ratio is tight up to an absolute constant factor. An important corollary of our lower bound is that the size of the optimal menu must grow at least linearly in the number of available actions, so no universal upper bound exists for the size of the optimal menu in the general single-dimensional setting. We also provide a sufficient condition under which selling only the fully informative experiment achieves the optimal revenue.

We study revenue maximization in multi-item multi-bidder auctions under the natural item-independence assumption – a classical problem in Multi-Dimensional Bayesian Mechanism Design. One of the biggest challenges in this area is developing algorithms to compute (approximately) optimal mechanisms that are not brute-force in the size of the bidder type space, which is usually exponential in the number of items in multi-item auctions. Unfortunately, such algorithms were only known for basic settings of our problem when bidders have unit-demand or additive valuations.

We investigate the algorithmic problem of selling information to agents who face a decision-making problem under uncertainty. We adopt the model recently proposed by Bergemann et al. [BBS18], in which information is revealed through signaling schemes called experiments. In the single-agent setting, any mechanism can be represented as a menu of experiments. Our results show that the computational complexity of designing the revenue-optimal menu depends heavily on the way the model is specified. When all the parameters of the problem are given explicitly, we provide a polynomial time algorithm that computes the revenue-optimal menu. For cases where the model is specified with a succinct implicit description, we show that the tractability of the problem is tightly related to the efficient implementation of a Best Response Oracle: when it can be implemented efficiently, we provide an additive FPTAS whose running time is independent of the number of actions. On the other hand, we provide a family of problems, where it is computationally intractable to construct a best response oracle, and we show that it is NP-hard to get even a constant fraction of the optimal revenue. Moreover, we investigate a generalization of the original model by Bergemann et al. [BBS18] that allows multiple agents to compete for useful information. We leverage techniques developed in the study of auction design (see e.g. [CDW12a], [AFHHM12], [CDW12b], [CDW13a], [CDW13b]) to design a polynomial time algorithm that computes the revenue-optimal mechanism for selling information.

We study the problem of selling n heterogeneous items to a single buyer, whose values for different items are dependent. Under arbitrary dependence, Hart and Nisan[30] show that no simple mechanism can achieve a non-negligible fraction of the optimal revenue even with only two items. We consider the setting where the buyer's type is drawn from a correlated distribution that can be captured by a Markov Random Field, one of the most prominent frameworks for modeling high-dimensional distributions with structure.

We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with n heterogeneous items, where each item is owned by a different seller i, and there is a constrained-additive buyer with feasibility constraint ℱ. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market. Our first result provides an O(log(1/r))-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here r is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on r and provides an unconditional O(log n)-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another O(log n)-factor en-route. The first result is achieved using a fixed posted price mechanism, and the analysis involves a novel application of the prophet inequality or a new concentration inequality. Our second result follows from a stitching lemma that allows us to upper bound the second-best gains from trade by the first-best gains from trade from the “likely to trade” items (items with trade probability at least 1/n) and the optimal profit from selling the “unlikely to trade” items. We can obtain an O(log n)-approximation to the first term by invoking our O(log(1/r))-approximation on the “likely to trade” items. We introduce a generalization of the fixed posted price mechanism—seller adjusted posted price—to obtain an O(log n)-approximation to the optimal profit for the “unlikely to trade” items. Unlike fixed posted price mechanisms, not all seller adjusted posted price mechanisms are incentive compatible and budget balanced. We develop a new argument based on “allocation coupling” to show the seller adjusted posted price mechanism used in our approximation is indeed budget balanced and incentive-compatible.

We consider the black-box reduction from multidimensional revenue maximization to virtual welfare maximization. Cai et al. [12, 13, 14, 15] show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (∊-BIC). We provide two new polynomial time transformations that convert any ∊-BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss. Our first transformation applies to any mechanism design setting with downward-closed outcome space and only requires sample access to the agents' type distributions. Our second transformation applies to the fully general outcome space, removing the downward-closed assumption, but requires full access to the agents' type distributions. Both transformations only require query access to the original ∊-BIC mechanism. Other ∊-BIC to BIC transformations for revenue exist in the literature [23, 36, 18] but all require exponential time to run in both of the settings we consider. As an application of our transformations, we improve the reduction by Cai et al. [12, 13, 14, 15] to generate an exactly BIC mechanism.

We provide a unified view of many recent developments in Bayesian mechanism design, including the black-box reductions of Cai, Daskalakis, and Weinberg [in Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, 2013], simple auctions for additive buyers [S. Hart and N. Nisan, in Proceedings of the 13th ACM Conference on Electronic Commerce, 2012], and posted-price mechanisms for unit-demand buyers [S. Chawla, J. D. Hartline, and R. D. Kleinberg, in Proceedings of the 8th ACM Conference on Electronic Commerce, 2007, pp. 243--251]. Additionally, we show that viewing these three previously disjoint lines of work through the same lens leads to new developments as well. First, we provide a duality framework for Bayesian mechanism design, which naturally accommodates multiple agents and arbitrary objectives/feasibility constraints. Using this, we prove that either a posted-price mechanism or the Vickrey--Clarke--Groves auction with per-bidder entry fees achieves a constant factor of the optimal revenue achievable by a Bayesian Incentive Compatible mechanism whenever buyers are unit-demand or additive, unifying previous breakthroughs of Chawla et al. [in Proceedings of the 42nd ACM Symposium on Theory of Computing, 2010] and Yao [in Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 2015, pp. 92--109], and improving both approximation ratios (from 30 to 24 and 69 to 8, respectively). Finally, we show that this view also leads to improved structural characterizations in the framework of Cai, Daskalakis, and Weinberg.

We study the sample complexity of learning revenue-optimal multi-item auctions. We obtain the first set of positive results that go beyond the standard but unrealistic setting of item-independence. In particular, we consider settings where bidders' valuations are drawn from correlated distributions that can be captured by Markov Random Fields or Bayesian Networks -- two of the most prominent graphical models. We establish parametrized sample complexity bounds for learning an up-to-ε optimal mechanism in both models, which scale polynomially in the size of the model, i.e. the number of items and bidders, and only exponential in the natural complexity measure of the model, namely either the largest in-degree (for Bayesian Networks) or the size of the largest hyper-edge (for Markov Random Fields).

We propose a framework for ensuring safe behavior of a reinforcement learning agent when the reward function may be difficult to specify. In order to do this, we rely on the existence of demonstrations from expert policies, and we provide a theoretical framework for the agent to optimize in the space of rewards consistent with its existing knowledge. We propose two methods to solve the resulting optimization: an exact ellipsoid-based method and a method in the spirit of the "follow-the-perturbed-leader" algorithm. Our experiments demonstrate the behavior of our algorithm in both discrete and continuous problems. The trained agent safely avoids states with potential negative effects while imitating the behavior of the expert in the other states.

The seminal impossibility result of Myerson and Satterthwaite [1983] states that for bilateral trade, there is no mechanism that is individually rational (IR), incentive compatible (IC), weakly budget balanced, and efficient. This has led follow-up work on two-sided trade settings to weaken the efficiency requirement and consider approximately efficient simple mechanisms, while still demanding the other properties. The current state-of-the-art of such mechanisms for two-sided markets can be categorized as giving one (but not both) of the following two types of approximation guarantees on the gains from trade : a constant ex-ante guarantee, measured with respect to the second-best efficiency benchmark, or an asymptotically optimal ex-post guarantee, measured with respect to the first-best efficiency benchmark. Here the second-best efficiency benchmark refers to the highest gains from trade attainable by any IR, IC and weakly budget balanced mechanism, while the first-best efficiency benchmark refers to the maximum gains from trade (attainable by the VCG mechanism, which is not weakly budget balanced).

We provide algorithms that learn simple auctions whose revenue is approximately optimal in multi-item multi-bidder settings, for a wide range of bidder valuations including unit-demand, additive, constrained additive, XOS, and subadditive. We obtain our learning results in two settings. The first is the commonly studied setting where sample access to the bidders' distributions over valuations is given, for both regular distributions and arbitrary distributions with bounded support. Here, our algorithms require polynomially many samples in the number of items and bidders. The second is a more general max-min learning setting that we introduce, where we are given “approximate distributions,” and we seek to compute a mechanism whose revenue is approximately optimal simultaneously for all “true distributions” that are close to the ones we were given. These results are more general in that they imply the sample-based results, and are also applicable in settings where we have no sample access to the underlying distributions but have estimated them indirectly via market research or by observation of bidder behavior in previously run, potentially non-truthful auctions. All our results hold for valuation distributions satisfying the standard (and necessary) independence-across-items property. They also generalize and improve upon recent works of Goldner and Karlin [25] and Morgenstern and Roughgarden [32], which have provided algorithms that learn approximately optimal multi-item mechanisms in more restricted settings with additive, subadditive and unit-demand valuations using sample access to distributions. We generalize these results to the complete unit-demand, additive, and XOS setting, to i.i.d. subadditive bidders, and to the max-min setting. Our results are enabled by new uniform convergence bounds for hypotheses classes under product measures. Our bounds result in exponential savings in sample complexity compared to bounds derived by bounding the VC dimension and are of independent interest.

We design simple mechanisms to approximate the Gains from Trade (GFT) in two-sided markets with multiple unit-supply sellers and multiple unit-demand buyers. A classical impossibility result by Myerson and Satterthwaite showed that even with only one seller and one buyer, no Bayesian Incentive Compatible (BIC), Individually Rational (IR), and Budget-Balanced (BB) mechanism can achieve full GFT (trade whenever buyer's value is higher than the seller's cost). The same paper also proposed the ``second-best'' mechanism that maximizes the GFT subject to BIC, IR, and BB constraints, which is unfortunately rather complex for even the single-seller single-buyer case. Our mechanism is simple, BIC, IR, and BB and achieves 1/2 of the optimal GFT among all BIC, IR, and BB mechanisms. The result holds for arbitrary distributions of the buyers' and sellers' values and can accommodate any downward-closed feasibility constraints over the allocations. The analysis of our mechanism is facilitated by extending the Cai-Weinberg-Devanur duality framework to two-sided markets.

We provide simple and approximately revenue-optimal mechanisms in the multi-item multi-bidder settings. We unify and improve all previous results, as well as generalize the results to broader cases. In particular, we prove that the better of the following two simple, deterministic and Dominant Strategy Incentive Compatible mechanisms, a sequential posted price mechanism or an anonymous sequential posted price mechanism with entry fee, achieves a constant fraction of the optimal revenue among all randomized, Bayesian Incentive Compatible mechanisms, when buyers' valuations are XOS over independent items. If the buyers' valuations are subadditive over independent items, the approximation factor degrades to O(logm), where m is the number of items. We obtain our results by first extending the Cai-Devanur-Weinberg duality framework to derive an effective benchmark of the optimal revenue for subadditive bidders, and then analyzing this upper bound with new techniques.

Since the 1990s spectrum auctions have been implemented world-wide. This has provided for a practical examination of an assortment of auction mechanisms and, amongst these, two simultaneous ascending price auctions have proved to be extremely successful. These are the simultaneous multiround ascending auction (SMRA) and the combinatorial clock auction (CCA). It has long been known that, for certain classes of valuation functions, the SMRA provides good theoretical guarantees on social welfare. However, no such guarantees were known for the CCA. In this paper, we show that CCA does provide strong guarantees on social welfare provided the price increment and stopping rule are well-chosen. This is very surprising in that the choice of price increment has been used primarily to adjust auction duration and the stopping rule has attracted little attention. The main result is a polylogarithmic approximation guarantee for social welfare when the maximum number of items demanded by a bidder is fixed. Specifically, we show that either the revenue of the CCA is at least an -fraction of the optimal welfare or the welfare of the CCA is at least an -fraction of the optimal welfare, where n is the number of bidders and m is the number of items. As a corollary, the welfare ratio – the worst case ratio between the social welfare of the optimum allocation and the social welfare of the CCA allocation – is at most O( 2 · log n·· log2 m). We emphasize that this latter result requires no assumption on bidders valuation functions. Finally, we prove that such a dependence on is necessary. In particular, we show that the welfare ratio of the CCA is at least .

We provide a unified view of many recent developments in Bayesian mechanism design, including the black-box reductions of Cai et. al., simple auctions for additive buyers, and posted-price mechanisms for unit-demand buyers. Additionally, we show that viewing these three previously disjoint lines of work through the same lens leads to new developments as well. First, we provide a duality framework for Bayesian mechanism design, which naturally accommodates multiple agents and arbitrary objectives/feasibility constraints. Using this, we prove that either a posted-price mechanism or the VCG auction with per-bidder entry fees achieves a constant-factor of the optimal Bayesian IC revenue whenever buyers are unit-demand or additive, unifying previous breakthroughs of Chawla et. al. and Yao, and improving both approximation ratios (from 33.75 to 24 and 69 to 8). Finally, we show that this view also leads to improved structural characterizations in the Cai et. al. framework.

We propose an optimum mechanism for providing monetary incentives to the data sources of a statistical estimator such as linear regression, so that high quality data is provided at low cost, in the sense that the weighted sum of payments and estimation error is minimized. The mechanism applies to a broad range of estimators, including linear and polynomial regression, kernel regression, and, under some additional assumptions, ridge regression. It also generalizes to several objectives, including minimizing estimation error subject to budget constraints. Besides our concrete results for regression problems, we contribute a mechanism design framework through which to design and analyze statistical estimators whose examples are supplied by workers with cost for labeling said examples.

The simultaneous multiple-round auction (SMRA) and the combinatorial clock auction (CCA) are the two primary mechanisms used to sell bandwidth. Recently, it was shown that the CCA provides good welfare guarantees for general classes of valuation functions [7]. This motivates the question of whether similar welfare guarantees hold for the SMRA in the case of general valuation functions. We show the answer is no. But we prove that good welfare guarantees still arise if the degree of complementarities in the bidder valuations are bounded. In particular, if bidder valuations functions are $$\alpha $$ -near-submodular then, under truthful bidding, the SMRA has a welfare ratio (the worst case ratio between the social welfare of the optimal allocation and the auction allocation) of at most $$(1+\alpha )$$ . However, for $$\alpha >1$$ , this is a bicriteria guarantee, to obtain good welfare under truthful bidding requires relaxing individual rationality. We prove this bicriteria guarantee is asymptotically (almost) tight. Finally, we examine what strategies are required to ensure individual rationality in the SMRA with general valuation functions. First, we provide a weak characterization, namely secure bidding, for individual rationality. We then show that if the bidders use a profit-maximizing secure bidding strategy the welfare ratio is at most $$1+\alpha $$ . Consequently, by bidding securely, it is possible to obtain the same welfare guarantees as truthful bidding without the loss of individual rationality.

Bayesian equilibria of simultaneous auctions for individual items have been explored recently [Christodoulou et al. 2008; Bhawalkar and Roughgarden 2011; Hassidim et al. 2011; Feldman et al. 2013] as an alternative to the well-known complexity issues plaguing combinatorial auctions with incomplete information, and some strong positive results have been shown about their performance. We point out some very serious complexity obstacles to this approach: Computing a Bayesian equilibrium in such auctions is hard for PP --- a complexity class between the polynomial hierarchy and PSPACE --- and even finding an approximate such equilibrium is as hard as NP, for some small approximation ratio (additive or multiplicative); therefore, the assumption that such equilibria will be arrived at by rational agents is quite problematic. In fact, even recognizing a Bayesian Nash equilibrium is intractable. Furthermore, these results hold even if bidder valuations are quite benign: Only one bidder valuation in our construction is unit demand or monotone submodular, while all others are additive. We also explore the possibility of favorable price of anarchy results for no-regret dynamics of the Bayesian simultaneous auctions game, and identify complexity obstacles there as well.

Human–computer interactions often involve multiple tasks such as navigation, searching, conversation and data retrieving. It is a challenge to identify the user in multi-tasking situations. Here, we present the algorithm for continuous verification of pre-defined users in both text-independent and text-dependent interactions. In the system, pitch features and Mel-Frequency Cepstral Coefficients are used for voice feature extraction. Vector quantization (VQ) and Mahalanobis distance are used for voice classification. We implemented our algorithms on the Android HTC phone. Based on our experimental results, we found that our training time can be within 1 min and the test time can be within 10 s or two sentences. VQ is suitable for a quick text-dependent verification such as voice signature; Mahalanobis distance, on the other hand, is better for continuous verification such as user authentication via voice. The empirical study also shows the impact of the verification time and languages on the verification accuracy. Our empirical study suggests that speaker verification algorithms work better when a user speaks in his or her native language.

We provide a computationally efficient black-box reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximation-preserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to (not necessarily truthfully) maximizing the same objective plus virtual welfare (under the same feasibility constraints). Our reduction is based on a fundamentally new approach: we describe a mechanism's behavior indirectly only in terms of the expected value it awards bidders for certain behavior, and never directly access the allocation rule at all. Applying our new approach to revenue, we exhibit settings where our reduction holds both ways. That is, we also provide an approximation-sensitive reduction from (non-truthfully) maximizing virtual welfare to (truthfully) maximizing revenue, and therefore the two problems are computationally equivalent. With this equivalence in hand, we show that both problems are NP-hard to approximate within any polynomial factor, even for a single monotone sub modular bidder. We further demonstrate the applicability of our reduction by providing a truthful mechanism maximizing fractional max-min fairness.

We provide a Polynomial Time Approximation Scheme (PTAS) for the Bayesian optimal multi-item multi-bidder auction problem under two conditions. First, bidders are independent, have additive valuations and are from the same population. Second, every bidder's value distributions of items are independent but not necessarily identical monotone hazard rate (MHR) distributions. For non-i.i.d. bidders, we also provide a PTAS when the number of bidders is small. Prior to our work, even for a single bidder, only constant factor approximations are known. Another appealing feature of our mechanism is the simple allocation rule. Indeed, the mechanism we use is either the second-price auction with reserve price on every item individually, or VCG allocation with a few outlying items that requires additional treatments. It is surprising that such simple allocation rules suffice to obtain nearly optimal revenue.

It was recently shown in [12] that revenue optimization can be computationally efficiently reduced to welfare optimization in all multi-dimensional Bayesian auction problems with arbitrary (possibly combinatorial) feasibility constraints and independent additive bidders with arbitrary (possibly combinatorial) demand constraints. This reduction provides a poly-time solution to the optimal mechanism design problem in all auction settings where welfare optimization can be solved efficiently, but it is fragile to approximation and cannot provide solutions to settings where welfare maximization can only be tractably approximated. In this paper, we extend the reduction to accommodate approximation algorithms, providing an approximation preserving reduction from (truthful) revenue maximization to (not necessarily truthful) welfare maximization. The mechanisms output by our reduction choose allocations via black-box calls to welfare approximation on randomly selected inputs, thereby generalizing also our earlier structural results on optimal multi-dimensional mechanisms to approximately optimal mechanisms. Unlike [12], our results here are obtained through novel uses of the Ellipsoid algorithm and other optimization techniques over non-convex regions.

We provide a reduction from revenue maximization to welfare maximization in multidimensional Bayesian auctions with arbitrary - possibly combinatorial - feasibility constraints and independent bidders with arbitrary - possibly combinatorial-demand constraints, appropriately extending Myerson's single-dimensional result [21] to this setting. We also show that every feasible Bayesian auction - including in particular the revenue-optimal one - can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder's type ti is transformed into a virtual type fi(ti), via a bidder-specific function. Then, the allocation maximizing virtual welfare is chosen. Using this characterization, we show how to find and run the revenue-optimal auction given only black-box access to an implementation of the VCG allocation rule. We generalize this result to arbitrarily correlated bidders, introducing the notion of a second-order VCG allocation rule. Our results are computationally efficient for all multidimensional settings where the bidders are additive, or can be efficiently mapped to be additive, albeit the feasibility and demand constraints may still remain arbitrary combinatorial. In this case, our mechanisms run in time polynomial in the number of items and the total number of bidder types, but not type profiles. This is polynomial in the number of items, the number of bidders, and the cardinality of the support of each bidder's value distribution. For generic correlated distributions, this is the natural description complexity of the problem. The runtime can be further improved to polynomial in only the number of items and the number of bidders in itemsymmetric settings by making use of techniques from [15].

We show that every feasible, Bayesian, multi-item multi-bidder mechanism for independent, additive bidders can be implemented as a mechanism that: (a) allocates every item independently of the other items; (b) for the allocation of each item it uses a strict ordering of all bidders' types; and allocates the item using a distribution over hierarchical mechanisms that iron this ordering into a non-strict ordering, and give the item uniformly at random to the bidders whose reported types dominate all other reported types according to the non-strict ordering. Combined with cyclic-monotonicity our results provide a characterization of feasible, Bayesian Incentive Compatible mechanisms in this setting. Our characterization is enabled by a new, constructive proof of Border's theorem [Border 1991], and a new generalization of this theorem to independent (but not necessarily identically distributed) bidders, improving upon the results of [Border 2007, Che-Kim-Mierendorf 2011]. For a single item and independent bidders, we show that every feasible reduced form auction can be implemented as a distribution over hierarchical mechanisms that are consistent with the same strict ordering of all bidders' types, which every mechanism in the support of the distribution irons to a non-strict ordering. We also give a polynomial-time algorithm for determining feasibility of a reduced form auction, or providing a separation hyperplane from the set of feasible reduced forms. To complete the picture, we provide polynomial-time algorithms to find and exactly sample from a distribution over hierarchical mechanisms consistent with a given feasible reduced form. All these results generalize to multi-item reduced form auctions for independent, additive bidders. Finally, for multiple items, additive bidders with hard demand constraints, and arbitrary value correlation across items or bidders, we give a proper generalization of Border's Theorem, and characterize feasible reduced form auctions as multi-commodity flows in related multi-commodity flow instances. We also show that our generalization holds for a broader class of feasibility constraints, including the intersection of any two matroids.

We provide a Polynomial Time Approximation Scheme for the multi-dimensional unit-demand pricing problem, when the buyer's values are independent (but not necessarily identically distributed.) For all ϵ >; 0, we obtain a (1 + ϵ)-factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasi-polynomial, when sampled from regular distributions, and polynomial in npoly(log r)when sampled from general distributions supported on a set [umin,rumin]. We also provide an additive PTAS for all bounded distributions. Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all ϵ >; 0, g(1/ϵ) distinct prices suffice to obtain a (1 + ϵ)-factor approximation to the optimal revenue for MHR distributions, where g(1/ϵ) is a quasi-linear function of 1/ϵ that does not depend on the number of items. Similarly, for all ϵ >; 0 and n >; 0, g(1/ϵ · log n) distinct prices suffice for regular distributions, where n is the number of items and g(·) is a polynomial function. Finally, in the i.i.d. MHR case, we show that, as long as the number of items is a sufficiently large function of 1/ϵ, a single price suffices to achieve a (1 + ϵ)-factor approximation. Our results represent significant progress to the single-bidder case of the multidimensional optimal mechanism design problem, following Myerson's celebrated work on optimal mechanism design [Myerson 1981].

Finite automata on infinite words (omega-automata) proved to be a powerful weapon for modeling and reasoning infinite behaviors of reactive systems. Complementation of omega-automata is crucial in many of these applications. But the problem is non-trivial; even after extensive study during the past two decades, we still have an important type of omega-automata, namely Streett automata, for which the gap between the current best lower bound 2^(Omega(n lg nk)) and upper bound 2^(O (nk lg nk)) is substantial, for the Streett index size k can be exponential in the number of states n. In a previous work we showed a construction for complementing Streett automata with the upper bound 2^(O(n lg n+nk lg k)) for k = O(n) and 2^(O(n^2 lg n)) for k = omega(n). In this paper we establish a matching lower bound 2^(Omega (n lg n+nk lg k)) for k = O(n) and 2^(Omega (n^2 lg n)) for k = omega(n), and therefore showing that the construction is asymptotically optimal with respect to the ^(Theta(.)) notation.

We prove a generalization of von Neumann's minmax theorem to the class of separable multiplayer zero-sum games, introduced in [Bregman and Fokin 1998]. These games are polymatrix—that is, graphical games in which every edge is a two-player game between its endpoints—in which every outcome has zero total sum of players’ payoffs. Our generalization of the minmax theorem implies convexity of equilibria, polynomial-time tractability, and convergence of no-regret learning algorithms to Nash equilibria. Given that Nash equilibria in 3-player zero-sum games are already PPAD-complete, this class of games, i.e. with pairwise separable utility functions, defines essentially the broadest class of multi-player constant-sum games to which we can hope to push tractability results. Our result is obtained by establishing a certain game-class collapse, showing that separable constant-sum games are payoff equivalent to pairwise constant-sum polymatrix games—polymatrix games in which all edges are constant-sum games, and invoking a recent result of [Daskalakis, Papadimitriou 2009] for these games. We also explore generalizations to classes of non-constant-sum multi-player games. A natural candidate is polymatrix games with strictly competitive games on their edges. In the two player setting, such games are minmax solvable and recent work has shown that they are merely affine transformations of zero-sum games [Adler, Daskalakis, Papadimitriou 2009]. Surprisingly we show that a polymatrix game comprising of strictly competitive games on its edges is PPAD-complete to solve, proving a striking difference in the complexity of networks of zero-sum and strictly competitive games. Finally, we look at the role of coordination in networked interactions, studying the complexity of polymatrix games with a mixture of coordination and zero-sum games. We show that finding a pure Nash equilibrium in coordination-only polymatrix games is PLS-complete; hence, computing a mixed Nash equilibrium is in PLS ∩ PPAD, but it remains open whether the problem is in P. If, on the other hand, coordination and zero-sum games are combined, we show that the problem becomes PPAD-complete, establishing that coordination and zero-sum games achieve the full generality of PPAD.

Automata on infinite words (omega-automata) have wide applications in formal language theory as well as in modeling and verifying reactive systems. Complementation of omega-automata is a crucial instrument in many these applications, and hence there have been great interests in determining the state complexity of the complementation problem. However, obtaining nontrivial lower bounds has been difficult. For the complementation of Rabin automata, a significant gap exists between the state-of-the-art lower bound 2Omega(NlgN)and upper bound 2O(kNlgN), where k, the number of Rabin pairs, can be as large as 2N. In this paper we introduce multidimensional rankings to the full automata technique. Using the improved technique we establish an almost tight lower bound for the complementation of Rabin automata. We also show that the same lower bound holds for the determinization of Rabin automata.

This paper presents a tool Altair that automatically generates API function cross-references, which emphasizes reliable structural measures and does not depend on specific client code. Altair ranks related API functions for a given query according to pair-wise overlap, i.e., how they share state, and clusters tightly related ones into meaningful modules.