This paper is an exposition of algorithms for finding one or all equilibria of a bimatrix game (a two-player game in strategic form) in the style of a chapter in a graduate textbook. Using labeled "best-response polytopes", we present the Lemke-Howson algorithm that finds one equilibrium. We show that the path followed by this algorithm has a direction, and that the endpoints of the path have opposite index, in a canonical way using determinants. For reference, we prove that a number of notions of nondegeneracy of a bimatrix game are equivalent. The computation of all equilibria of a general bimatrix game, via a description of the maximal Nash subsets of the game, is canonically described using "complementary pairs" of faces of the best-response polytopes.
We prove that differential Nash equilibria are generic amongst local Nash equilibria in continuous zero-sum games. That is, there exists an open-dense subset of zero-sum games for which local Nash equilibria are non-degenerate differential Nash equilibria. The result extends previous results to the zero-sum setting, where we obtain even stronger results; in particular, we show that local Nash equilibria are generically hyperbolic critical points. We further show that differential Nash equilibria of zero-sum games are structurally stable. The purpose for presenting these extensions is the recen…
It is known that the set of all correlated equilibria of an n-player non-cooperative game is a convex polytope and includes all the Nash equilibria. Further, the Nash equilibria all lie on the boundary of this polytope. We study the geometry of both these equilibrium notions when the players have cumulative prospect theoretic (CPT) preferences. The set of CPT correlated equilibria includes all the CPT Nash equilibria but it need not be a convex polytope. We show that it can, in fact, be disconnected. However, all the CPT Nash equilibria continue to lie on its boundary. We also characterize the…
An heuristic approach to compute strong Nash (Aumann) equilibria is presented. The method is based on differential evolution and three variants of a generative relation for strong Nash equilibria characterization. Numerical experiments performed on the minimum effort game for up to 150 players illustrate the efficiency of the approach. The advantages and disadvantages of each variant is discussed in terms of precision and running time.
We study pure Nash equilibria in infinite-duration games on graphs, with partial visibility of actions but communication (based on a graph) among the players. We show that a simple communication mechanism consisting in reporting the deviator when seeing it and propagating this information is sufficient for characterizing Nash equilibria. We propose an epistemic game construction, which conveniently records important information about the knowledge of the players. With this abstraction, we are able to characterize Nash equilibria which follow the simple communication pattern via winning strateg…
We develop a quasi-polynomial time Las Vegas algorithm for approximating Nash equilibria in polymatrix games over trees, under a mild renormalizing assumption. Our result, in particular, leads to an expected polynomial-time algorithm for computing approximate Nash equilibria of tree polymatrix games in which the number of actions per player is a fixed constant. Further, for trees with constant degree, the running time of the algorithm matches the best known upper bound for approximating Nash equilibria in bimatrix games (Lipton, Markakis, and Mehta 2003). Notably, this work closely complements…
The rank of a bimatrix game (A,B) is the rank of the matrix A+B. We give a construction of rank-1 games with exponentially many equilibria, which answers an open problem by Kannan and Theobald (2010).
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 investigate the degree of discontinuity of several solution concepts from non-cooperative game theory. While the consideration of Nash equilibria forms the core of our work, also pure and correlated equilibria are dealt with. Formally, we restrict the treatment to two player games, but results and proofs extend to the n-player case. As a side result, the degree of discontinuity of solving systems of linear inequalities is settled.
We study the query complexity of approximate notions of Nash equilibrium in games with a large number of players $n$. Our main result states that for $n$-player binary-action games and for constant $\varepsilon$, the query complexity of an $\varepsilon$-well-supported Nash equilibrium is exponential in $n$. One of the consequences of this result is an exponential lower bound on the rate of convergence of adaptive dynamics to approxiamte Nash equilibrium.
We present a direct reduction from k-player games to 2-player games that preserves approximate Nash equilibrium. Previously, the computational equivalence of computing approximate Nash equilibrium in k-player and 2-player games was established via an indirect reduction. This included a sequence of works defining the complexity class PPAD, identifying complete problems for this class, showing that computing approximate Nash equilibrium for k-player games is in PPAD, and reducing a PPAD-complete problem to computing approximate Nash equilibrium for 2-player games. Our direct reduction makes no u…
In two player bi-matrix games with partial monitoring, actions played are not observed, only some messages are received. Those games satisfy a crucial property of usual bi-matrix games: there are only a finite number of required (mixed) best replies. This is very helpful while investigating sets of Nash equilibria: for instance, in some cases, it allows to relate it to the set of equilibria of some auxiliary game with full monitoring. In the general case, the Lemke-Howson algorithm is extended and, under some genericity assumption, its output are Nash equilibria of the original game. As a by p…
We introduce Q-Nash, a quantum annealing algorithm for the NP-complete problem of Fnding pure Nash equilibria in graphical games. The algorithm consists of two phases. The first phase determines all combinations of best response strategies for each player using classical computation. The second phase finds pure Nash equilibria using a quantum annealing device by mapping the computed combinations to a quadratic unconstrained binary optimization formulation based on the Set Cover problem. We empirically evaluate Q-Nash on D-Wave's Quantum Annealer 2000Q using different graphical game topologies.…
We investigate the complexity of bounding the uncertainty of graphical games, and we provide new insight into the intrinsic difficulty of computing Nash equilibria. In particular, we show that, if one adds very simple and natural additional requirements to a graphical game, the existence of Nash equilibria is no longer guaranteed, and computing an equilibrium is an intractable problem. Moreover, if stronger equilibrium conditions are required for the game, we get hardness results for the second level of the polynomial hierarchy. Our results offer a clear picture of the complexity of mixed Nash…
Observable games are game situations that reach one of possibly many Nash equilibria. Before an instance of the game starts, an external observer does not know, a priori, what is the exact profile of actions that will occur; thus, he assigns subjective probabilities to players' actions. However, not all probabilistic assignments are coherent with a given game. We study the decision problem of determining if a given set of probabilistic constraints assigned a priori by the observer to a given game is coherent, which we call the Coherence of Probabilistic Constraints on Equilibria, or PCE-Cohere…
We prove that in a normal form n-player game with m actions for each player, there exists an approximate Nash equilibrium where each player randomizes uniformly among a set of O(log(m) + log(n)) pure strategies. This result induces an $N^{\log \log N}$ algorithm for computing an approximate Nash equilibrium in games where the number of actions is polynomial in the number of players (m=poly(n)), where $N=nm^n$ is the size of the game (the input size). In addition, we establish an inverse connection between the entropy of Nash equilibria in the game, and the time it takes to find such an approxi…
In this paper we present a novel generic mapping between Graphical Games and Markov Random Fields so that pure Nash equilibria in the former can be found by statistical inference on the latter. Thus, the problem of deciding whether a graphical game has a pure Nash equilibrium, a well-known intractable problem, can be attacked by well-established algorithms such as Belief Propagation, Junction Trees, Markov Chain Monte Carlo and Simulated Annealing. Large classes of graphical games become thus tractable, including all classes already known, but also new classes such as the games with O(log n) t…
We study the complexity of computing a uniform Nash equilibrium on a non-win-lose bimatrix game. It is known that such a problem is NP-complete even if a bimatrix game is win-lose (Bonifaci et al., 2008). Fortunately, if a win-lose bimatrix game is planar, then uniform Nash equilibria always exist. We have a polynomial-time algorithm for finding a uniform Nash equilibrium of a planar win-lose bimatrix game (Addario-Berry et al., 2007). The following question is left: How hard to compute a uniform Nash equilibrium on a planar non-win-lose bimatrix game? This paper resolves this issue. We prove …
We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs. The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy. These games capture the idea of coordination within a local neighbourhood in the absence of globally common strategies. We study the decision problem of checking whether a given set of strategy choices for a subset of the players is consist…
In an epsilon-Nash equilibrium, a player can gain at most epsilon by changing his behaviour. Recent work has addressed the question of how best to compute epsilon-Nash equilibria, and for what values of epsilon a polynomial-time algorithm exists. An epsilon-well-supported Nash equilibrium (epsilon-WSNE) has the additional requirement that any strategy that is used with non-zero probability by a player must have payoff at most epsilon less than the best response. A recent algorithm of Kontogiannis and Spirakis shows how to compute a 2/3-WSNE in polynomial time, for bimatrix games. Here we intro…
The $\varepsilon$-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than $\varepsilon$ to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant $\varepsilon$ currently known for which there is a polynomial-time algorithm that computes an $\varepsilon$-well-supported Nash equilibrium in bimatrix games is slightly below $2/3$. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a $(1/2+δ)$-well-supported …
We study the problem of finding approximate Nash equilibria that satisfy certain conditions, such as providing good social welfare. In particular, we study the problem $ε$-NE $δ$-SW: find an $ε$-approximate Nash equilibrium ($ε$-NE) that is within $δ$ of the best social welfare achievable by an $ε$-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving $\left(\frac{1}{8} - \mathrm{O}(δ)\right)$-NE $\mathrm{O}(δ)$-SW for an $n\times n$ bimatrix game requires $n^{\mathrm{\widetilde Ω}(\log n)}$ time. Building on this result, we show similar conditional runnin…
We study the problem of computing approximate Nash equilibria of bimatrix games, in a setting where players initially know their own payoffs but not the payoffs of the other player. In order for a solution of reasonable quality to be found, some amount of communication needs to take place between the players. We are interested in algorithms where the communication is substantially less than the contents of a payoff matrix, for example logarithmic in the size of the matrix. When the communication is polylogarithmic in the number of strategies n, we show how to obtain epsilon-approximate Nash eq…
We study the deterministic and randomized query complexity of finding approximate equilibria in bimatrix games. We show that the deterministic query complexity of finding an $ε$-Nash equilibrium when $ε< \frac{1}{2}$ is $Ω(k^2)$, even in zero-one constant-sum games. In combination with previous results \cite{FGGS13}, this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a $(\frac{3 - \sqrt{5}}{2} + ε)$-Nash equilibrium using $O(\frac{k \cdot \log k}…
The problem of finding pure strategy Nash equilibria in multiagent concurrent games with finite-horizon temporal goals has received some recent attention. Earlier work solved this problem through the use of Rabin automata. In this work, we take advantage of the finite-horizon nature of the agents' goals and show that checking for and finding pure strategy Nash equilibria can be done using a combination of safety games and lasso testing in Büchi automata. To separate strategic reasoning from temporal reasoning, we model agents' goals by deterministic finite-word automata (DFAs), since finite-ho…
We discuss and solve a model for a game with many players, where a subset of truely deciding players is embedded into a hierarchy of dependent agents. These interdependencies modify the game matrix and the Nash equilibria for the deciding players. In a concrete example, we recognize the partition function of the Ising model and for high dependency we observe a phase transition to a new Nash equilibrium, which is the Pareto-efficient outcome. An example we have in mind is the game theory for major shareholders in a stock market, where intermediate companies decide according to a majority vote o…
Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute $\varepsilon$-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis, with an approximation guarantee of $(0.3393+δ)$, remains the state of the art over the last 15 years. In this paper, we propo…
The Colonel Blotto Problem proposed by Borel in 1921 has served as a widely applicable model of budget-constrained simultaneous winner-take-all competitions in the social sciences. Applications include elections, advertising, R&D and more. However, the classic Blotto problem and variants limit the study to competitions over a finite set of discrete battlefields. In this paper, we extend the classical theory to study multiplayer Blotto games over arbitrary measurable battlegrounds, provide an algorithm to efficiently sample equilibria of symmetric "equipartionable" Generalized Blotto games, and…
We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (…
We consider a track selection problem for multi-target tracking in a multifunction radar network from a game-theoretic perspective. The problem is formulated as a non-cooperative game. The radars are considered to be players in this game with utilities modeled using a proper tracking accuracy criterion and their strategies are the observed targets whose number is known. Initially, for the problem of coordination, the Nash equilibria are characterized and, in order to find equilibria points, a distributed algorithm based on the best-response dynamics is proposed. Afterwards, the analysis is ext…
We study natural improvement dynamics in weighted congestion games with polynomial latencies of maximum degree $d\geq 1$. We focus on two problems regarding the existence and efficiency of approximate pure Nash equilibria, with a reasonable small approximation factor, in these games. By exploiting a simple technique, we firstly show that such a game always admits a $d$-approximate potential function. This implies that every sequence of $d$-approximate improvement moves by the players leads to a $d$-approximate pure Nash equilibrium. As a corollary, we also obtain that, under mild assumptions o…
We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply samp…
In recent years, empirical game-theoretic analysis (EGTA) has emerged as a powerful tool for analyzing games in which an exact specification of the utilities is unavailable. Instead, EGTA assumes access to an oracle, i.e., a simulator, which can generate unbiased noisy samples of players' unknown utilities, given a strategy profile. Utilities can thus be empirically estimated by repeatedly querying the simulator. Recently, various progressive sampling (PS) algorithms have been proposed, which aim to produce PAC-style learning guarantees (e.g., approximate Nash equilibria with high probability)…
Several notions of game enjoy a Nash-like notion of equilibrium without guarantee of existence. There are different ways of weakening a definition of Nash-like equilibrium in order to guarantee the existence of a weakened equilibrium. Nash's approach to the problem for strategic games is probabilistic, \textit{i.e.} continuous, and static. CP and BR approaches for CP and BR games are discrete and dynamic. This paper proposes an approach that lies between those two different approaches: a discrete and static approach. multi strategic games are introduced as a formalism that is able to express b…
In an $ε$-Nash equilibrium, a player can gain at most $ε$ by unilaterally changing his behaviour. For two-player (bimatrix) games with payoffs in $[0,1]$, the best-known$ε$ achievable in polynomial time is 0.3393. In general, for $n$-player games an $ε$-Nash equilibrium can be computed in polynomial time for an $ε$ that is an increasing function of $n$ but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of $ε$ are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general $n$-player games where a pl…
Drawing intuition from a (physical) hydraulic system, we present a novel framework, constructively showing the existence of a strong Nash equilibrium in resource selection games (i.e., asymmetric singleton congestion games) with nonatomic players, the coincidence of strong equilibria and Nash equilibria in such games, and the uniqueness of the cost of each given resource across all Nash equilibria. Our proofs allow for explicit calculation of Nash equilibrium and for explicit and direct calculation of the resulting (unique) costs of resources, and do not hinge on any fixed-point theorem, on th…
We compute a parametric description of the totally mixed Nash equilibria of a generic game in normal form with pre-fixed structure. Using this representation, we show conditions under which a game has the maximum possible number of this kind of equilibria. Then, we present a symbolic procedure that allows us to describe and estimate the number of isolated totally mixed Nash equilibria of an arbitrary game. Under certain assumptions, the algorithm computes the exact number of these equilibria.
A bimatrix game $(A,B)$ is called a game of rank $k$ if the rank of the matrix $A+B$ is at most $k$. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1. In particular, we show that even for games of rank 1 not all equilibria can be reached by a Lemke-Howson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-degenerate game of rank 1.
Computing Nash equilibria for strategic multi-agent systems is challenging for expensive black box systems. Motivated by the ubiquity of games involving exploitation of common resources, this paper considers the above problem for potential games. We use the Bayesian optimization framework to obtain novel algorithms to solve finite (discrete action spaces) and infinite (real interval action spaces) potential games, utilizing the structure of potential games. Numerical results illustrate the efficiency of the approach in computing the Nash equilibria of static potential games and linear Nash equ…
We prove an $N^{2-o(1)}$ lower bound on the randomized communication complexity of finding an $ε$-approximate Nash equilibrium (for constant $ε>0$) in a two-player $N\times N$ game.
University of Aberdeen（阿伯丁大学）；University of Artois（阿尔图瓦大学）
In this paper we describe an argumentation-based representation of normal form games, and demonstrate how argumentation can be used to compute pure strategy Nash equilibria. Our approach builds on Modgil's Extended Argumentation Frameworks. We demonstrate its correctness, prove several theoretical properties it satisfies, and outline how it can be used to explain why certain strategies are Nash equilibria to a non-expert human user.
Designing efficient algorithms to compute Nash equilibria poses considerable challenges in Algorithmic Game Theory and Optimization. In this work, we employ integer programming techniques to compute Nash equilibria in Integer Programming Games, a class of simultaneous and non-cooperative games where each player solves a parametrized integer program. We introduce ZERO Regrets, a general and efficient cutting plane algorithm to compute, enumerate, and select Nash equilibria. Our framework leverages the concept of equilibrium inequality, an inequality valid for any Nash equilibrium, and the assoc…
MID-LIDS；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM-Research（IBM研究院）；Mila（米拉）；University of Oxford（牛津大学）
Multiagent learning settings are inherently more difficult than single-agent learning because each agent interacts with other simultaneously learning agents in a shared environment. An effective approach in multiagent reinforcement learning is to consider the learning process of agents and influence their future policies toward desirable behaviors from each agent's perspective. Importantly, if each agent maximizes its long-term rewards by accounting for the impact of its behavior on the set of convergence policies, the resulting multiagent system reaches an active equilibrium. While this new s…
If a game has a unique Nash equilibrium, then this equilibrium is arguably the solution of the game from the refinement's literature point of view. However, it might be that for almost all initial conditions, all strategies in the support of this equilibrium are eliminated by the replicator dynamics and the best-reply dynamics.
We investigate a model for representing large multiplayer games, which satisfy strong symmetry properties. This model is made of multiple copies of an arena; each player plays in his own arena, and can partially observe what the other players do. Therefore, this game has partial information and symmetry constraints, which make the computation of Nash equilibria difficult. We show several undecidability results, and for bounded-memory strategies, we precisely characterize the complexity of computing pure Nash equilibria (for qualitative objectives) in this game model.
