This paper investigates repeated win-lose coordination games (WLC-games). We analyse which protocols are optimal for these games covering both the worst case and average case scenarios, i,e., optimizing the guaranteed and expected coordination times. We begin by analysing Choice Matching Games (CM-games) which are a simple yet fundamental type of WLC-games, where the goal of the players is to pick the same choice from a finite set of initially indistinguishable choices. We give a complete classification of optimal expected and guaranteed coordination times in two-player CM-games and show that the corresponding optimal protocols are unique in every case - except in the CM-game with four choices, which we analyse separately.
Our results on CM-games are also essential for proving a more general result on the difficulty of all WLC-games: we provide a complete analysis of least upper bounds for optimal expected coordination times in all two-player WLC-games as a function of game size. We also show that CM-games can be seen as the most difficult games among all two-player WLC-games, as they turn out to have the greatest optimal expected coordination times.
This paper investigates repeated win-lose coordination games (WLC-games). We analyse which protocols are optimal for these games, covering both the worst case and average case scenarios, i,e., optimizing the guaranteed and expected coordination times. We begin by analysing Choice Matching Games (CM-games) which are a simple yet fundamental type of WLC-games, where the goal of the players is to pick the same choice from a finite set of initially indistinguishable choices. We give a fully complete classification of optimal expected and guaranteed coordination times in two-player CM-games and sho…
We introduce and motivate the study of hypergraphical clustering games of mis-coordination. For two specific variants we prove the existence of a pure Nash equilibrium and provide bounds on the price of anarchy as a function of the cardinality of the action set and the size of the hyperedges.
Motivated by understanding non-strict and strict pure strategy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs. We characterize the cases when such colorings exist and when the decision problem is NP-hard. These correspond to finding pure strategy equilibria in the anti-coordination games, whose price of anarchy we also analyze. We further consider the directed case, a generalization that captures both coordination and anti-coordination. We prove the decision problem for non-strict equilibria in directed graphs i…
This paper deals with the complexity of the problem of computing a pure Nash equilibrium for discrete preference games and network coordination games beyond $O(\log n)$-treewidth and tree metric spaces. First, we estimate the number of iterations of the best response dynamics for a discrete preference game on a discrete metric space with at least three strategies. Second, we present a sufficient condition that we have a polynomial-time algorithm to find a pure Nash equilibrium for a discrete preference game on a grid graph. Finally, we discuss the complexity of finding a pure Nash equilibrium …
Coordination mechanisms aim to mitigate the impact of selfishness when scheduling jobs to different machines. Such a mechanism defines a scheduling policy within each machine and naturally induces a game among the selfish job owners. The desirable properties of a coordination mechanism includes simplicity in its definition and efficiency of the outcomes of the induced game. We present a broad class of coordination mechanisms for unrelated machine scheduling that are simple to define and we identify one of its members (mechanism \Dcoord) that is superior to all known mechanisms. \Dcoord\ induce…
In this study, we formulate positive and negative externalities caused by changes in the supply of shared vehicles as ride sharing games. The study aims to understand the price of anarchy (PoA) and its improvement via a coordination technique in ride sharing games. A critical question is whether ride sharing games exhibit a pure Nash equilibrium (pNE) since the PoA bound assumes it. Our result shows a sufficient condition for a ride sharing game to have a finite improvement property and a pNE similar to potential games. This is the first step to analyze PoA bound and its improvement by coordin…
Whilst network coordination games and network anti-coordination games have received a considerable amount of attention in the literature, network games with coexisting coordinating and anti-coordinating players are known to exhibit more complex behaviors. In fact, depending on the network structure, such games may even fail to have pure-strategy Nash equilibria. An example is represented by the well-known matching pennies (discoordination) game. In this work, we first provide graph-theoretic conditions for the existence of pure-strategy Nash equilibria in mixed network coordination/anti-coordi…
We study natural strategic games on directed graphs, which capture the idea of coordination in the absence of globally common strategies. We show that these games do not need to have a pure Nash equilibrium and that the problem of determining their existence is NP-complete. The same holds for strong equilibria. We also exhibit some classes of games for which strong equilibria exist and prove that a strong equilibrium can then be found in linear time.
We study a novel control problem in the context of network coordination games: the individuation of the smallest set of players capable of driving the system, globally, from one Nash equilibrium to another one. Our main contribution is the design of a randomized algorithm based on a time-reversible Markov chain with provable convergence garantees.
We present a scheme for playing quantum repeated 2x2 games based on the Marinatto and Weber's approach to quantum games. As a potential application, we study twice repeated Prisoner's Dilemma game. We show that results not available in classical game can be obtained when the game is played in the quantum way. Before we present our idea, we comment on the previous scheme of playing quantum repeated games.
We study the role of local effects and finite size effects in reaching coordination and in equilibrium selection in different types of two-player coordination games. We investigate three update rules -- the replicator dynamics (RD), the best response (BR), and the unconditional imitation (UI) -- for coordination games on random graphs. Local effects turn out to me significantly more important for the UI update rule. For the pure coordination game with two equivalent strategies we find a transition from a disordered state to a state of full coordination for a critical value of the network conne…
Sharing unused vehicles is one practical solution for traffic congestion. We propose an advanced vehicle-sharing service that maximizes the sharing of vehicles and improves traffic efficiency by coordinating user trips via an information system. We formulate ride-sharing games that model externalities in vehicle sharing caused by insufficient vehicle supply. We show how Bayes correlated equilibrium can coordinate players in ride-sharing games and verify the resultant improvement in the price of anarchy.
Logit Dynamics [Blume, Games and Economic Behavior, 1993] are randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the long-term equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distrib…
We study the emergence of locally suboptimal behavior in finitely repeated games. Locally suboptimal behavior refers to players play suboptimally in some rounds of the repeated game (i.e., not maximizing their payoffs in those rounds) while maximizing their total payoffs in the whole repeated game. The central research question we aim to answer is when locally suboptimal behavior can arise from rational play in finitely repeated games. In this research, we focus on the emergence of locally suboptimal behavior in subgame-perfect equilibria (SPE) of finitely repeated games with complete informat…
Lund University（隆德大学）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）；Georgia Institute of Technology（佐治亚理工学院）
We consider the problem of distributed convergence to efficient outcomes in coordination games through dynamics based on aspiration learning. Under aspiration learning, a player continues to play an action as long as the rewards received exceed a specified aspiration level. Here, the aspiration level is a fading memory average of past rewards, and these levels also are subject to occasional random perturbations. A player becomes dissatisfied whenever a received reward is less than the aspiration level, in which case the player experiments with a probability proportional to the degree of dissat…
We prove that every repeated game with countably many players, finite action sets, and tail-measurable payoffs admits an $ε$-equilibrium, for every $ε> 0$.
Communities typically capture homophily as people of the same community share many common features. This paper is motivated by the problem of community detection in social networks, as it can help improve our understanding of the network topology and the spread of information. Given the selfish nature of humans to align with like-minded people, we employ game theoretic models and algorithms to detect communities in this paper. Specifically, we employ coordination games to represent interactions between individuals in a social network. We represent the problem of community detection as a graph …
Repeated games are difficult to analyze, especially when agents play mixed strategies. We study one-memory strategies in iterated prisoner's dilemma, then generalize the result to k-memory strategies in repeated games. Our result shows that there always exists a pure strategy best response, which can be computed with SMT or MDP solvers. However, there may not exist such pure strategy best response in multi-agent tournaments. All source code is released for verification.
Repeated games have provided an explanation how mutual cooperation can be achieved even if defection is more favorable in a one-shot game in prisoner's dilemma situation. Recently found zero-determinant strategies have substantially been investigated in evolutionary game theory. The original memory-one zero-determinant strategies unilaterally enforce linear relations between average payoffs of players. Here, we extend the concept of zero-determinant strategies to memory-two strategies in repeated games. Memory-two zero-determinant strategies unilaterally enforce linear relations between correl…
In this paper, we introduce the concept of infinitely split Nash equilibrium in repeated games in which the profile sets are chain-complete posets. Then by using a fixed point theorem on posets in [8], we prove an existence theorem. As an application, we study the repeated extended Bertrant duopoly model of price competition.
We study the problem of stochastic stability for evolutionary dynamics under the logit choice rule. We consider general classes of coordination games, symmetric or asymmetric, with an arbitrary number of strategies, which satisfies the marginal bandwagon property (i.e., there is positive feedback to coordinate). Our main result is that the most likely evolutionary escape paths from a status quo convention consist of a series of identical mistakes. As an application of our result, we show that the Nash bargaining solution arises as the long run convention for the evolutionary Nash demand game u…
We present new coordination mechanisms for scheduling selfish jobs on $m$ unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game among the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of the efficiency of schedules. The approximation ratio of a coordination mechanism quantif…
A repeated game is an effective tool to model interactions and conflicts for players aiming to achieve their objectives in a long-term basis. Contrary to static noncooperative games that model an interaction among players in only one period, in repeated games, interactions of players repeat for multiple periods; and thus the players become aware of other players' past behaviors and their future benefits, and will adapt their behavior accordingly. In wireless networks, conflicts among wireless nodes can lead to selfish behaviors, resulting in poor network performances and detrimental individual…
Stockholm University（斯德哥尔摩大学）；University of Helsinki（赫尔辛基大学）；Tampere University（坦佩雷大学）；ENS Paris-Saclay（巴黎萨克雷高等师范学校）
We study pure coordination games where in every outcome, all players have identical payoffs, 'win' or 'lose'. We identify and discuss a range of 'purely rational principles' guiding the reasoning of rational players in such games and analyze which classes of coordination games can be solved by such players with no preplay communication or conventions. We observe that it is highly nontrivial to delineate a boundary between purely rational principles and other decision methods, such as conventions, for solving such coordination games.
We study infinitely repeated games in settings of imperfect monitoring. We first prove a family of theorems that show that when the signals observed by the players satisfy a condition known as $(ε, γ)$-differential privacy, that the folk theorem has little bite: for values of $ε$ and $γ$ sufficiently small, for a fixed discount factor, any equilibrium of the repeated game involve players playing approximate equilibria of the stage game in every period. Next, we argue that in large games ($n$ player games in which unilateral deviations by single players have only a small impact on the utility o…
This paper considers repeated games in which one player has more information about the game than the other players. In particular, we investigate repeated two-player zero-sum games where only the column player knows the payoff matrix A of the game. Suppose that while repeatedly playing this game, the row player chooses her strategy at each round by using a no-regret algorithm to minimize her (pseudo) regret. We develop a no-instant-regret algorithm for the column player to exhibit last round convergence to a minimax equilibrium. We show that our algorithm is efficient against a large set of po…
The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large …
This paper aims to design a distributed coordination algorithm for solving a multi-agent decision problem with a hierarchical structure. The primary goal is to search the Nash equilibrium of a noncooperative game such that each player has no incentive to deviate from the equilibrium under its private objective. Meanwhile, the agents can coordinate to optimize the social cost within the set of Nash equilibria of the underlying game. Such an optimal Nash equilibrium problem can be modeled as a distributed optimization problem with variational inequality constraints. We consider the scenario wher…
We suggest a general method for inferring players' values from their actions in repeated games. The method extends and improves upon the recent suggestion of (Nekipelov et al., EC 2015) and is based on the assumption that players are more likely to exhibit sequences of actions that have lower regret. We evaluate this "quantal regret" method on two different datasets from experiments of repeated games with controlled player values: those of (Selten and Chmura, AER 2008) on a variety of two-player 2x2 games and our own experiment on ad-auctions (Noti et al., WWW 2014). We find that the quantal r…
One of the proposed solutions to the equilibrium selection problem for agents learning in repeated games is obtained via the notion of stochastic stability. Learning algorithms are perturbed so that the Markov chain underlying the learning dynamics is necessarily irreducible and yields a unique stable distribution. The stochastically stable distribution is the limit of these stable distributions as the perturbation rate tends to zero. We present the first exact algorithm for computing the stochastically stable distribution of a Markov chain. We use our algorithm to predict the long-term dynami…
We consider polymatrix coordination games with individual preferences where every player corresponds to a node in a graph who plays with each neighbor a separate bimatrix game with non-negative symmetric payoffs. In this paper, we study $α$-approximate $k$-equilibria of these games, i.e., outcomes where no group of at most $k$ players can deviate such that each member increases his payoff by at least a factor $α$. We prove that for $α\ge 2$ these games have the finite coalitional improvement property (and thus $α$-approximate $k$-equilibria exist), while for $α< 2$ this property does not hold.…
Zero-determinant strategies are a class of memory-one strategies in repeated games which unilaterally enforce linear relationships between payoffs. It has long been unclear for what stage games zero-determinant strategies exist. We provide a necessary and sufficient condition for the existence of zero-determinant strategies. This condition can be interpreted as the existence of two different actions which unilaterally adjust the total value of a linear combination of payoffs. A relation between the class of stage games where zero-determinant strategies exist and other class of stage games is a…
Machine learning processes, e.g. ''learning in games'', can be viewed as non-linear dynamical systems. In general, such systems exhibit a wide spectrum of behaviors, ranging from stability/recurrence to the undesirable phenomena of chaos (or ''butterfly effect''). Chaos captures sensitivity of round-off errors and can severely affect predictability and reproducibility of ML systems, but AI/ML community's understanding of it remains rudimentary. It has a lot out there that await exploration. Recently, Cheung and Piliouras employed volume-expansion argument to show that Lyapunov chaos occurs in …
The ad hoc coordination problem is to design an autonomous agent which is able to achieve optimal flexibility and efficiency in a multiagent system with no mechanisms for prior coordination. We conceptualise this problem formally using a game-theoretic model, called the stochastic Bayesian game, in which the behaviour of a player is determined by its private information, or type. Based on this model, we derive a solution, called Harsanyi-Bellman Ad Hoc Coordination (HBA), which utilises the concept of Bayesian Nash equilibrium in a planning procedure to find optimal actions in the sense of Bel…
We study the population genetics of Evolution in the important special case of weak selection, in which all fitness values are assumed to be close to one another. We show that in this regime natural selection is tantamount to the multiplicative updates game dynamics in a coordination game between genes. Importantly, the utility maximized in this game, as well as the amount by which each allele is boosted, is precisely the allele's mixability, or average fitness, a quantity recently proposed in [1] as a novel concept that is crucial in understanding natural selection under sex, thus providing a…
Strategic Delay and Coordination Efficiency in Global Games
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；University of Illinois Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）；Florida State University（佛罗里达州立大学）
It is well-known that for infinitely repeated games, there are computable strategies that have best responses, but no computable best responses. These results were originally proved for either specific games (e.g., Prisoner's dilemma), or for classes of games satisfying certain conditions not known to be both necessary and sufficient. We derive a complete characterization in the form of simple necessary and sufficient conditions for the existence of a computable strategy without a computable best response under limit-of-means payoff. We further refine the characterization by requiring the stra…
In communication systems where users share common resources, users' selfish behavior usually results in suboptimal resource utilization. There have been extensive works that model communication systems with selfish users as one-shot games and propose incentive schemes to achieve Pareto optimal action profiles as non-cooperative equilibria. However, in many communication systems, due to strong negative externalities among users, the sets of feasible payoffs in one-shot games are nonconvex. Thus, it is possible to expand the set of feasible payoffs by having users choose convex combinations of d…
How does information regarding an adversary's intentions affect optimal system design? This paper addresses this question in the context of graphical coordination games where an adversary can indirectly influence the behavior of agents by modifying their payoffs. We study a situation in which a system operator must select a graph topology in anticipation of the action of an unknown adversary. The designer can limit her worst-case losses by playing a security strategy, effectively planning for an adversary which intends maximum harm. However, fine-grained information regarding the adversary's i…
