Competitive equilibrium is a fundamental concept in the study of markets, describing stable outcomes that can emerge from agents' trading. Proportional response is a well-established distributed algorithm which has been shown to converge to competitive equilibria in both Fisher and Arrow-Debreu markets, for various sub-families of homogeneous utilities, including linear and Constant Elasticity of Substitution (CES) utilities. However, homogeneous utilities remain a relatively restrictive subset compared to the diverse preferences that economists have considered. For instance, even the intuitive separable utilities of the form u (x) = Σj uj(xj) are generally not homogeneous. This gap motivates the open question: to what extent can the proportional response dynamics be applied to markets with non-homogeneous utility functions?

We examine two types of binary betting markets, whose primary goal is for profit (such as sports gambling) or to gain information (such as prediction markets). We articulate the interplay between belief and price-setting to analyse both types of markets, and show that the goals of maximising bookmaker profit and eliciting information are fundamentally incompatible. A key insight is that profit hinges on the deviation between (the distribution of) bettor and true beliefs, and that heavier tails in bettor belief distribution implies higher profit. Our algorithmic contribution is to introduce online learning methods for price-setting. Traditionally bookmakers update their prices rather infrequently, we present two algorithms that guide price updates upon seeing each bet, assuming very little of bettor belief distributions. The online pricing algorithm achieves stochastic regret of $\mathcal{O}(\sqrt{T})$ against the worst local maximum, or $\mathcal{O}(\sqrt{T \log T})$ with high probability against the global maximum under fair odds. More broadly, the inherent tradeoff between profit and information-seeking in binary betting may inspire new understandings of large-scale multi-agent behaviour.

This work is concerned with the dynamics of online cultural markets, namely, attention allocation of many users on a set of digital goods with infinite supply. Such dynamics are important in shaping processes and outcomes in society, from trending items in entertainment, collective knowledge creation, to election outcomes. The outcomes of online cultural markets are susceptible to intricate social influence dynamics, particularly so when the community comprises consumers with heterogeneous interests. This has made formal analysis of these markets improbable. In this paper, we remedy this by establishing robust connections between influence dynamics and optimization processes, in trial-offer markets where the consumer preferences are modelled by multinomial logit. Among other results, we show that the proportional-response-esque influence dynamic is equivalent to stochastic mirror descent on a convex objective function, thus leading to a stable and predictable outcome. When all consumers are homogeneous, the objective function has a natural interpretation as a weighted sum of efficiency and diversity of the culture market. In simulations driven by real-world preferences collected from a large-scale recommender system, we observe that ranking strategies aligned with the underlying heterogeneous preferences are more stable, and achieves higher efficiency and diversity. More broadly, we see this work as the first step in connecting computational methods for classical markets to the problem area of online attention and recommender systems. We hope this result paves the way to posing and answering a diverse set of research questions in this area.

We study the fair and efficient allocation of a set of indivisible goods among agents, where each good has several copies, and each agent has an additively separable concave valuation function with a threshold. These valuations capture the property of diminishing marginal returns, and they are more general than the well-studied case of additive valuations. We present a polynomial-time algorithm that approximates the optimal Nash social welfare (NSW) up to a factor of e1/e ≈ 1.445. This matches with the state-of-the-art approximation factor for additive valuations. The computed allocation also satisfies the popular fairness guarantee of envy-freeness up to one good (EF1) up to a factor of 2 + ε. For instances without thresholds, it is also approximately Pareto-optimal. For instances satisfying a large market property, we show an improved approximation factor. Lastly, we show that the upper bounds on the optimal NSW introduced in Cole and Gkatzelis (2018) and Barman et al. (2018) have the same value.

We present a novel control-theoretic understanding of online optimization and learning in games, via the notion of passivity. Passivity is a fundamental concept in control theory, which abstracts energy conservation and dissipation in physical systems. It has become a standard tool in analysis of general feedback systems, to which game dynamics belong. Our starting point is to show that all continuous-time Follow-the-Regularized-Leader (FTRL) dynamics, which include the well-known Replicator Dynamic, are lossless, i.e. it is passive with no energy dissipation. Interestingly, we prove that passivity implies bounded regret, connecting two fundamental primitives of control theory and online optimization. The observation of energy conservation in FTRL inspires us to present a family of lossless learning dynamics, each of which has an underlying energy function with a simple gradient structure. This family is closed under convex combination; as an immediate corollary, any convex combination of FTRL dynamics is lossless and thus has bounded regret. This allows us to extend the framework of Fox & Shamma [Games 2013] to prove not just global asymptotic stability results for game dynamics, but Poincar{é} recurrence results as well. Intuitively, when a lossless game (e.g. graphical constant-sum game) is coupled with lossless learning dynamic, their interconnection is also lossless, which results in a pendulum-like energy-preserving recurrent behavior, generalizing Piliouras & Shamma [SODA 2014] and Mertikopoulos et al. [SODA 2018].

We study learning dynamics in distributed production economies such as blockchain mining, peer-to-peer file sharing and crowdsourcing. These economies can be modelled as multi-product Cournot competitions or all-pay auctions (Tullock contests) when individual firms have market power, or as Fisher markets with quasi-linear utilities when every firm has negligible influence on market outcomes. In the former case, we provide a formal proof that Gradient Ascent (GA) can be Li-Yorke chaotic for a step size as small as Θ(1/n), where n is the number of firms. In stark contrast, for the Fisher market case, we derive a Proportional Response (PR) protocol that converges to market equilibrium. The positive results on the convergence of the PR dynamics are obtained in full generality, in the sense that they hold for Fisher markets with any quasi-linear utility functions. Conversely, the chaos results for the GA dynamics are established even in the simplest possible setting of two firms and one good, and they hold for a wide range of price functions with different demand elasticities. Our findings suggest that by considering multi-agent interactions from a market rather than a game-theoretic perspective, we can formally derive natural learning protocols which are stable and converge to effective outcomes rather than being chaotic.

We perform volume analysis of Multiplicative Weights Updates (MWU) and its optimistic variant (OMWU) in zero-sum as well as coordination games. Our analysis provides new insights into these game/dynamical systems, which seem hard to achieve via the classical techniques within Computer Science and Machine Learning. First, we examine these dynamics not in their original space (simplex of actions) but in a dual space (aggregate payoffs of actions). Second, we explore how the volume of a set of initial conditions evolves over time when it is pushed forward according to the algorithm. This is reminiscent of approaches in evolutionary game theory where replicator dynamics, the continuous-time analogue of MWU, is known to preserve volume in all games. Interestingly, when we examine discrete-time dynamics, the choices of the game and the algorithm both play a critical role. So whereas MWU expands volume in zero-sum games and is thus Lyapunov chaotic, we show that OMWU contracts volume, providing an alternative understanding for its known convergent behavior. Yet, we also prove a no-free-lunch type of theorem, in the sense that when examining coordination games the roles are reversed. Using these tools, we prove two novel, rather negative properties of MWU in zero-sum games. (1) Extremism: even in games with a unique fully-mixed Nash equilibrium, the system recurrently gets stuck near pure-strategy profiles, despite them being clearly unstable from game-theoretic perspective. (2) Unavoidability: given any set of good states (with a rather relaxed interpretation of “good” states), the system cannot avoid bad states indefinitely.

We establish that algorithmic experiments in zero-sum games “fail miserably” to confirm the unique, sharp prediction of maxmin equilibration. Contradicting nearly a century of economic thought that treats zero-sum games nearly axiomatically as the exemplar symbol of economic stability, we prove that no meaningful prediction can be made about the day-to-day behavior of online learning dynamics in zero-sum games. Concretely, Multiplicative Weights Updates (MWU) with constant step-size is \emph{Lyapunov chaotic} in the dual (payoff) space. Simply put, let’s assume that an observer asks the agents playing Matching-Pennies whether they prefer Heads or Tails (and by how much in terms of aggregate payoff so far). The range of possible answers consistent with any arbitrary small set of initial conditions blows up exponentially with time everywhere in the payoff space. This result is \emph{robust} both \emph{algorithmically} as well as \emph{game theoretically}. \textbf{Algorithmic robustness:} Chaos is robust to agents using any of a general sub-family of Follow-the-Regularized-Leader (FTRL) algorithms, the well known regret-minimizing dynamics, even when agents mix-and-match dynamics, use different or slowly decreasing step-sizes. \textbf{Game theoretic robustness:} Chaos is robust to all affine variants of zero-sum games (strictly competitive games), network variants with arbitrary large number of agents and even to competitive settings beyond these. Our result is in stark contrast with the time-average convergence of online learning to (approximate) Nash equilibrium, a result widely reported as “(weak) convergence to equilibrium”.

We extend a recently developed framework for analyzing asynchronous coordinate descent algorithms to show that an asynchronous version of tatonnement, a fundamental price dynamic widely studied in general equilibrium theory, converges toward a market equilibrium for Fisher markets with CES utilities or Leontief utilities, for which tatonnement is equivalent to coordinate descent.

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of e^{1/{e}} ~~ 1.445. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of 2 + epsilon.

We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge routing too many of the original edges. For any general connected graph with n vertices and m edges, we show that its STC is at most O(sqrt{mn}), which is asymptotically optimal since we also demonstrate graphs with STC at least Omega(sqrt{mn}). We present a polynomial-time algorithm which computes a spanning tree with congestion O(sqrt{mn}* log n). We also present another algorithm for computing a spanning tree with congestion O(sqrt{mn}); this algorithm runs in sub-exponential time when m = omega(n log^2 n). For achieving the above results, an important intermediate theorem is generalized Györi-Lovász theorem. Chen et al. [Jiangzhuo Chen et al., 2007] gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time O^*(4^n). We discuss some consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most O(n), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC Theta(n) with high probability.

Since Multiplicative Weights (MW) updates are the discrete analogue of the continuous Replicator Dynamics (RD), some researchers had expected their qualitative behaviours would be similar. We show that this is false in the context of graphical constant-sum games, which include two-person zero-sum games as special cases. In such games which have a fully-mixed Nash Equilibrium (NE), it was known that RD satisfy the permanence and Poincare recurrence properties, but we show that MW updates with any constant step-size eps > 0 converge to the boundary of the state space, and thus do not satisfy the two properties. Using this result, we show that MW updates have a regret lower bound of Omega( 1 / (eps T) ), while it was known that the regret of RD is upper bounded by O( 1 / T ). Interestingly, the regret perspective can be useful for better understanding of the behaviours of MW updates. In a two-person zero-sum game, if it has a unique NE which is fully mixed, then we show, via regret, that for any sufficiently small eps, there exist at least two probability densities and a constant Z > 0, such that for any arbitrarily small z > 0, each of the two densities fluctuates above Z and below z infinitely often.

A major goal in Algorithmic Game Theory is to justify equilibrium concepts from an algorithmic and complexity perspective. One appealing approach is to identify natural distributed algorithms that converge quickly to an equilibrium. This paper established new convergence results for two generalizations of proportional response in Fisher markets with buyers having CES utility functions. The starting points are respectively a new convex and a new convex-concave formulation of such markets. The two generalizations correspond to suitable mirror descent algorithms applied to these formulations. Several of our new results are a consequence of new notions of strong Bregman convexity and of strong Bregman convex-concave functions, and associated linear rates of convergence, which may be of independent interest. Among other results, we analyze a damped generalized proportional response and show a linear rate of convergence in a Fisher market with buyers whose utility functions cover the full spectrum of CES utilities aside the extremes of linear and Leontief utilities; when these utilities are included, we obtain an empirical $O(1/T)$ rate of convergence.

Given a weighted graph G = (V, E, w) with a set of k terminals T ⊂ V, the Steiner Point Removal problem seeks for a minor of the graph with vertex set T, such that the distance between every pair of terminals is preserved within a small multiplicative distortion. Kamma, Krauthgamer and Nguyen [13] used a ball-growing algorithm to show that the distortion is at most for general graphs. In this paper, we improve the distortion bound to . The improvement is achieved based on a known algorithm that constructs terminal-distance exact-preservation minor with (which is independent of |V|) vertices, and also two tail bounds on the sums of independent exponential random variables, which allow us to show that it is unlikely for a non-terminal being contracted to a distant terminal.

Given a graph where vertices are partitioned into $k$ terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately.The distortion of a compressed graph is the maximum multiplicative blow-up of distances between all pairs of terminals. We study the trade-off between the number of non-terminals and the distortion. This problem generalizes the Steiner Point Removal (SPR) problem, in which all non-terminals must be removed. We introduce a novel black-box reduction to convert any lower bound on distortion for the SPR problem into a super-linear lower bound on the number of non-terminals, with the same distortion, for our problem. This allows us to show that there exist graphs such that every minor with distortion less than $2~/~2.5~/~3$ must have $Ω(k^2)~/~Ω(k^{5/4})~/~Ω(k^{6/5})$ non-terminals, plus more trade-offs in between. The black-box reduction has an interesting consequence: if the tight lower bound on distortion for the SPR problem is super-constant, then allowing any $O(k)$ non-terminals will not help improving the lower bound to a constant. We also build on the existing results on spanners, distance oracles and connected 0-extensions to show a number of upper bounds for general graphs, planar graphs, graphs that exclude a fixed minor and bounded treewidth graphs. Among others, we show that any graph admits a minor with $O(\log k)$ distortion and $O(k^{2})$ non-terminals, and any planar graph admits a minor with $1+\varepsilon$ distortion and $\widetilde{O}((k/\varepsilon)^{2})$ non-terminals.

Combinatorial auctions (CA) are a well-studied area in algorithmic mechanism design. However, contrary to the standard model, empirical studies suggest that a bidder’s valuation often does not depend solely on the goods assigned to him. For instance, in adwords auctions an advertiser might not want his ads to be displayed next to his competitors’ ads. In this paper, we propose and analyze several natural graph-theoretic models that incorporate such negative externalities, in which bidders form a directed conflict graph with maximum out-degree $$\varDelta $$ . We design algorithms and truthful mechanisms for social welfare maximization that attain approximation ratios depending on $$\varDelta $$ . For CA, our results are twofold: (1) A lottery that eliminates conflicts by discarding bidders/items independent of the bids. It allows to apply any truthful $$\alpha $$ -approximation mechanism for conflict-free valuations and yields an $${\mathcal O}(\alpha \varDelta )$$ -approximation mechanism. (2) For fractionally sub-additive valuations, we design a rounding algorithm via a novel combination of a semi-definite program and a linear program, resulting in a cone program; the approximation ratio is $${\mathcal O}((\varDelta \log \log \varDelta )/\log \varDelta )$$ . The ratios are almost optimal given existing hardness results. For adwords auctions, we present several algorithms for the most relevant scenario when the number of items is small. In particular, we design a truthful mechanism with approximation ratio $$o(\varDelta )$$ when the number of items is only logarithmic in the number of bidders.

Tatonnement is a simple and natural rule for updating prices in Exchange (Arrow-Debreu) markets. In this paper we define a class of markets for which tatonnement is equivalent to gradient descent. This is the class of markets for which there is a convex potential function whose gradient is always equal to the negative of the excess demand and we call it Convex Potential Function (CPF) markets. We show the following results. CPF markets contain the class of Eisenberg Gale (EG) markets, defined previously by Jain and Vazirani. The subclass of CPF markets for which the demand is a differentiable function contains exactly those markets whose demand function has a symmetric negative semi-definite Jacobian. We define a family of continuous versions of tatonnement based on gradient descent using a Bregman divergence. As we show, all processes in this family converge to an equilibrium for any CPF market. This is analogous to the classic result for markets satisfying the Weak Gross Substitutes property. A discrete version of tatonnement converges toward the equilibrium for the following markets of complementary goods; its convergence rate for these settings is analyzed using a common potential function. Fisher markets in which all buyers have Leontief utilities. The tatonnement process reduces the distance to the equilibrium, as measured by the potential function, to an ε fraction of its initial value in O(1/ε) rounds of price updates. Fisher markets in which all buyers have complementary CES utilities. Here, the distance to the equilibrium is reduced to an ε fraction of its initial value in O(log(1/ε)) rounds of price updates.

This paper continues the study, initiated by Cole and Fleischer in [Cole and Fleischer 2008], of the behavior of a tatonnement price update rule in Ongoing Fisher Markets. The prior work showed fast convergence toward an equilibrium when the goods satisfied the weak gross substitutes property and had bounded demand and income elasticities.