We present a novel polynomial time approximation scheme for two-strategy anonymous games, in which the players' utility functions, although potentially different, do not differentiate among the identities of the other players. Our algorithm computes an $eps$-approximate Nash equilibrium of an $n$-player 2-strategy anonymous game in time $poly(n) (1/eps)^{O(1/eps^2)}$, which significantly improves upon the running time $n^{O(1/eps^2)}$ required by the algorithm of Daskalakis & Papadimitriou, 2007. The improved running time is based on a new structural understanding of approximate Nash equilibria: We show that, for any $eps$, there exists an $eps$-approximate Nash equilibrium in which either only $O(1/eps^3)$ players randomize, or all players who randomize use the same mixed strategy. To show this result we employ tools from the literature on Stein's Method.
If a game has a Nash equilibrium with probability values that are either zero or Omega(1) then this equilibrium can be found exhaustively in polynomial time. Somewhat surprisingly, we show that there is a PTAS for the games whose equilibria are guaranteed to have small-O(1/n)-values, and therefore large-Omega(n)-supports. We also point out that there is a PTAS for games with sparse payoff matrices, which are known to be PPAD-complete to solve exactly. Both algorithms are of a special kind that we call oblivious: The algorithm just samples a fixed distribution on pairs of mixed strategies, and …
We show that the problem of finding an ε-approximate Nash equilibrium in an anonymous game with seven pure strategies is complete in PPAD, when the approximation parameter ε is exponentially small in the number of players.
We study the computation of equilibria of anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is \emph{query complexity}, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. However, more positive query-comple…
We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s^2 L), where s is the number of strategies and L is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an epsilon
Log-linear learning has been extensively studied in both the game theoretic and distributed control literature. It is appealing for many applications because it often guarantees that the agents' collective behavior will converge in probability to the optimal system configuration. However, the worst case convergence time can be prohibitively long, i.e., exponential in the number of players. We formalize a modified log-linear learning algorithm whose worst case convergence time is roughly linear in the number of players. We prove this characterization for a class of potential games where agents'…
We show that there is a polynomial-time approximation scheme for computing Nash equilibria in anonymous games with any fixed number of strategies (a very broad and important class of games), extending the two-strategy result of Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a probabilistic result of more general interest: The distribution of the sum of n independent unit vectors with values ranging over {e1, e2, ...,ek}, where ei is the unit vector along dimension i of the k-dimensional Euclidean space, can be approximated by the distribution of the sum of another …
In cost sharing games, the existence and efficiency of pure Nash equilibria fundamentally depends on the method that is used to share the resources' costs. We consider a general class of resource allocation problems in which a set of resources is used by a heterogeneous set of selfish users. The cost of a resource is a (non-decreasing) function of the set of its users. Under the assumption that the costs of the resources are shared by uniform cost sharing protocols, i.e., protocols that use only local information of the resource's cost structure and its users to determine the cost shares, we e…
Network Allocation Games with Anonymous Preferences
Royal Holloway, University of London（伦敦大学皇家霍洛威学院）；AGH University of Krakow（克拉科夫AGH科技大学）；Czech Technical University in Prague（布拉格捷克理工大学）；University of Southern Denmark（南丹麦大学）
We investigate the complexity of computing approximate Nash equilibria in anonymous games. Our main algorithmic result is the following: For any $n$-player anonymous game with a bounded number of strategies and any constant $δ>0$, an $O(1/n^{1-δ})$-approximate Nash equilibrium can be computed in polynomial time. Complementing this positive result, we show that if there exists any constant $δ>0$ such that an $O(1/n^{1+δ})$-approximate equilibrium can be computed in polynomial time, then there is a fully polynomial-time approximation scheme for this problem. We also present a faster algorithm th…
In large systems, it is important for agents to learn to act effectively, but sophisticated multi-agent learning algorithms generally do not scale. An alternative approach is to find restricted classes of games where simple, efficient algorithms converge. It is shown that stage learning efficiently converges to Nash equilibria in large anonymous games if best-reply dynamics converge. Two features are identified that improve convergence. First, rather than making learning more difficult, more agents are actually beneficial in many settings. Second, providing agents with statistical information …
We add here another layer to the literature on nonatomic anonymous games started with the 1973 paper by Schmeidler. More specifically, we define a new notion of equilibrium which we call $\varepsilon$-estimated equilibrium and prove its existence for any positive $\varepsilon$. This notion encompasses and brings to nonatomic games recent concepts of equilibrium such as self-confirming, peer-confirming, and Berk--Nash. This augmented scope is our main motivation. At the same time, our approach also resolves some conceptual problems present in Schmeidler (1973), pointed out by Shapley. In that p…
The worst-case Lipschitz constant of an $n$-player $k$-action $δ$-perturbed game, $λ(n,k,δ)$, is given an explicit probabilistic description. In the case of $k\geq 3$, $λ(n,k,δ)$ is identified with the passage probability of a certain symmetric random walk on $\mathbb Z$. In the case of $k=2$ and $n$ even, $λ(n,2,δ)$ is identified with the probability that two two i.i.d.\ Binomial random variables are equal. The remaining case, $k=2$ and $n$ odd, is bounded through the adjacent (even) values of $n$. Our characterisation implies a sharp closed form asymptotic estimate of $λ(n,k,δ)$ as $δn /k\to…
We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games. For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain…
We provide the first fully polynomial time approximation scheme (FPTAS) for computing an approximate mixed-strategy Nash equilibrium in tree-structured graphical multi-hypermatrix games (GMhGs). GMhGs are generalizations of normal-form games, graphical games, graphical polymatrix games, and hypergraphical games. Computing an exact mixed-strategy Nash equilibria in graphical polymatrix games is PPAD-complete and thus generally believed to be intractable. In contrast, to the best of our knowledge, we are the first to establish an FPTAS for tree polymatrix games as well as tree graphical games wh…
This paper proposes a novel game-theoretical autonomous decision-making framework to address a task allocation problem for a swarm of multiple agents. We consider cooperation of self-interested agents, and show that our proposed decentralized algorithm guarantees convergence of agents with social inhibition to a Nash stable partition (i.e., social agreement) within polynomial time. The algorithm is simple and executable based on local interactions with neighbor agents under a strongly-connected communication network and even in asynchronous environments. We analytically present a mathematical …
The travel costs of the players (travelers) in anonymous congestion games depend on their choices of routes and also on the states of the transportation network such as incidents, weather, and road work. In this extended abstract, we consider an incomplete-information environment in which the realizations of the states are unobserved by the travelers. We study how a planner can incentivize the travelers to behave in her favor by strategically designing what and how the travelers get informed about the realizations of the states.
We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NP-complete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The two-action case can be reformulated as a natural elimination problem on a matrix, the complexity of which turns out to be surprisingly difficult to characterize and ultimately remains open. We however establish connections to a matching problem along paths in a directed graph, which is computationally hard in general but can also be used…
The notion of fault tolerant Nash equilibria has been introduced as a way of studying the robustness of Nash equilibria. Under this notion, a fixed number of players are allowed to exhibit faulty behavior in which they may deviate arbitrarily from an equilibrium strategy. A Nash equilibrium in a game with $N$ players is said to be $α$-tolerant if no non-faulty user wants to deviate from an equilibrium strategy as long as $N-α-1$ other players are playing the equilibrium strategies, i.e., it is robust to deviations from rationality by $α$ faulty players. In prior work, $α$-tolerance has been la…
We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function $g$ from the $n$-dimensional hypercube to the bounded interval $[-1,1]$, and an $n \times m$ matrix $A$ with bounded entries, maximize $g(Ax)$ over $x$ in the $m$-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. signaling). We present an approximat…
We consider the problem of computing Nash Equilibria of action-graph games (AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation of games that encapsulates both "local" dependencies as in graphical games, and partial indifference to other agents' identities as in anonymous games, which occur in many natural settings. This is achieved by specifying a graph on the set of actions, so that the payoff of an agent for selecting a strategy depends only on the number of agents playing each of the neighboring strategies in the action graph. We present a Polynomial Time Approxi…
Deciding that two network flows are essentially the same is an important problem in intrusion detection or in tracing anonymous connections. A stepping stone or an anonymity network may try to prevent flow correlation by delaying the packets, introducing chaff traffic, or even splitting the flow in several subflows. We introduce a game-theoretic framework for this problem. The framework is used to derive the Nash equilibrium under two different adversary models: the first one, when the adversary is limited to delaying packets, and the second, when the adversary also adds dummy packets and remo…
A Nash Equilibrium (NE) is a strategy profile resilient to unilateral deviations, and is predominantly used in the analysis of multiagent systems. A downside of NE is that it is not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in some cases, NE does exhibit stability against coalitional deviations, in that the benefits from a joint deviation are bounded. In this sense, NE approximates strong equilibrium. Coalition formation is a key issue in multiagent systems. We provide a framework for quantifying the stability and the performance of various assignment …
An extensive literature in economics and social science addresses contests, in which players compete to outperform each other on some measurable criterion, often referred to as a player's score, or output. Players incur costs that are an increasing function of score, but receive prizes for obtaining higher score than their competitors. In this paper we study finite games that are discretized contests, and the problems of computing exact and approximate Nash equilibria. Our motivation is the worst-case hardness of Nash equilibrium computation, and the resulting interest in important classes of …
Despite the increasing popularity and successful examples of crowdsourcing, it is stripped of aureole when collective efforts are derailed or severely hindered by elaborate sabotage. A service exchange dilemma arises when non-cooperation among self-interested users, and zero social welfare is obtained at myopic equilibrium. Traditional rating protocols are not effective to overcome the inefficiency of the socially undesirable equilibrium due to specific features of crowdsourcing: a large number of anonymous users having asymmetric service requirements, different service capabilities, and dynam…
The designs of many large-scale systems today, from traffic routing environments to smart grids, rely on game-theoretic equilibrium concepts. However, as the size of an $N$-player game typically grows exponentially with $N$, standard game theoretic analysis becomes effectively infeasible beyond a low number of players. Recent approaches have gone around this limitation by instead considering Mean-Field games, an approximation of anonymous $N$-player games, where the number of players is infinite and the population's state distribution, instead of every individual player's state, is the object …
Recent advances in multiagent learning have seen the introduction ofa family of algorithms that revolve around the population-based trainingmethod PSRO, showing convergence to Nash, correlated and coarse corre-lated equilibria. Notably, when the number of agents increases, learningbest-responses becomes exponentially more difficult, and as such ham-pers PSRO training methods. The paradigm of mean-field games pro-vides an asymptotic solution to this problem when the considered gamesare anonymous-symmetric. Unfortunately, the mean-field approximationintroduces non-linearities which prevent a str…
We present a new methodology for computing approximate Nash equilibria for two-person non-cooperative games based upon certain extensions and specializations of an existing optimization approach previously used for the derivation of fixed approximations for this problem. In particular, the general two-person problem is reduced to an indefinite quadratic programming problem of special structure involving the $n \times n$ adjacency matrix of an induced simple graph specified by the input data of the game, where $n$ is the number of players' strategies. Using this methodology and exploiting certa…
We study a static game played by a finite number of agents, in which agents are assigned independent and identically distributed random types and each agent minimizes its objective function by choosing from a set of admissible actions that depends on its type. The game is anonymous in the sense that the objective function of each agent depends on the actions of other agents only through the empirical distribution of their type-action pairs. We study the asymptotic behavior of Nash equilibria, as the number of agents tends to infinity, first by deriving laws of large numbers characterizes almos…
Studying Nash dynamics is an important approach for analyzing the outcome of games with repeated selfish behavior of self-interested agents. Sink equilibria has been introduced by Goemans, Mirrokni, and Vetta for studying social cost on Nash dynamics over pure strategies in games. However, they do not address the complexity of sink equilibria in these games. Recently, Fabrikant and Papadimitriou initiated the study of the complexity of Nash dynamics in two classes of games. In order to completely understand the complexity of Nash dynamics in a variety of games, we study the following three que…
We study a very general class of games --- multi-dimensional aggregative games --- which in particular generalize both anonymous games and weighted congestion games. For any such game that is also large, we solve the equilibrium selection problem in a strong sense. In particular, we give an efficient weak mediator: a mechanism which has only the power to listen to reported types and provide non-binding suggested actions, such that (a) it is an asymptotic Nash equilibrium for every player to truthfully report their type to the mediator, and then follow its suggested action; and (b) that when pl…
In this paper, we focus our attention on the large capacities unsplittable flow problem in a game theoretic setting. In this setting, there are selfish agents, which control some of the requests characteristics, and may be dishonest about them. It is worth noting that in game theoretic settings many standard techniques, such as randomized rounding, violate certain monotonicity properties, which are imperative for truthfulness, and therefore cannot be employed. In light of this state of affairs, we design a monotone deterministic algorithm, which is based on a primal-dual machinery, which attai…
University of Edinburgh（爱丁堡大学）；University of Southern California（南加州大学）；University of California, San Diego（加州大学圣地亚哥分校）
An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is {\em approximately sparse}, i.e., roughly speaking, its $L_1$-norm is small outside a small set. By building on this result, we obtain the following applications: {\bf Learning Theory.} We desi…
We consider the problem of resolving contention in communication networks with selfish users. In a \textit{contention game} each of $n \geq 2$ identical players has a single information packet that she wants to transmit using one of $k \geq 1$ multiple-access channels. To do that, a player chooses a slotted-time protocol that prescribes the probabilities with which at a given time-step she will attempt transmission at each channel. If more than one players try to transmit over the same channel (collision) then no transmission happens on that channel. Each player tries to minimize her own expec…
We present a unified framework for designing deterministic monotone polynomial time approximation schemes (PTAS's) for a wide class of scheduling problems on uniformly related machines. This class includes (among others) minimizing the makespan, maximizing the minimum load, and minimizing the l_p norm of the machine loads vector. Previously, this kind of result was only known for the makespan objective. Monotone algorithms have the property that an increase in the speed of a machine cannot decrease the amount of work assigned to it. The key idea of our novel method is to show that for goal fun…
