Zero-sum games have long guided artificial intelligence research, since they possess both a rich strategy space of best-responses and a clear evaluation metric. What's more, competition is a vital mechanism in many real-world multi-agent systems capable of generating intelligent innovations: Darwinian evolution, the market economy and the AlphaZero algorithm, to name a few. In two-player zero-sum games, the challenge is usually viewed as finding Nash equilibrium strategies, safeguarding against exploitation regardless of the opponent. While this captures the intricacies of chess or Go, it avoids the notion of cooperation with co-players, a hallmark of the major transitions leading from unicellular organisms to human civilization. Beyond two players, alliance formation often confers an advantage; however this requires trust, namely the promise of mutual cooperation in the face of incentives to defect. Successful play therefore requires adaptation to co-players rather than the pursuit of non-exploitability. Here we argue that a systematic study of many-player zero-sum games is a crucial element of artificial intelligence research. Using symmetric zero-sum matrix games, we demonstrate formally that alliance formation may be seen as a social dilemma, and empirically that naïve multi-agent reinforcement learning therefore fails to form alliances. We introduce a toy model of economic competition, and show how reinforcement learning may be augmented with a peer-to-peer contract mechanism to discover and enforce alliances. Finally, we generalize our agent model to incorporate temporally-extended contracts, presenting opportunities for further work.
Reciprocity is an important feature of human social interaction and underpins our cooperative nature. What is more, simple forms of reciprocity have proved remarkably resilient in matrix game social dilemmas. Most famously, the tit-for-tat strategy performs very well in tournaments of Prisoner's Dilemma. Unfortunately this strategy is not readily applicable to the real world, in which options to cooperate or defect are temporally and spatially extended. Here, we present a general online reinforcement learning algorithm that displays reciprocal behavior towards its co-players. We show that it c…
Social dilemmas exist in various fields and give rise to the so-called free-riding problem, leading to collective fiascos. The difficulty of tracking individual behaviors makes egoistic incentives in large-scale systems a challenging task. However, the state-of-the-art mechanisms are either individual-based or state-dependent, resulting in low efficiency in large-scale networks. In this paper, we propose an egoistic incentive mechanism from a connected (network) perspective rather than an isolated (individual) perspective by taking advantage of the social nature of people. We make use of a zer…
In the last few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when non-cooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass (2013) call "translucent players". Typically, players are assumed to be "opaque", in the sense that a deviation by one player does not affect the strategies used by other players. But a player may believe that if he switches from one strategy to anoth…
Centre for Mathematics and Computer Science (CWI)（数学与计算机科学中心（CWI））；Cornell University（康奈尔大学）
In the last few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when non-cooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass call translucent players. Typically, players are assumed to be opaque, in the sense that a deviation by one player in a normal-form game does not affect the strategies used by other players. But a player may believe that if he switches from one strate…
Two-player games have had a long and fruitful history of applications stretching across the social, biological, and physical sciences. Most applications of two-player games assume synchronous decisions or moves even when the games are iterated. But different strategies may emerge as preferred when the decisions or moves are sequential, or the games are iterated. Zero-determinant strategies developed by Press and Dyson are a new class of strategies that have been developed for synchronous two-player games, most notably the iterated prisoner's dilemma. Here we apply the Press-Dyson analysis to s…
Games involving quantum strategies often yield higher payoff. Here, we study a practical realization of the three-player dilemma game using the superconductivity-based quantum processors provided by IBM Q Experience. We analyze the persistence of the quantum advantage under corruption of the input states and how this depends on parameters of the payoff table. Specifically, experimental fidelity and error are observed not to be properly anti correlated, i.e., there are instances where a class of experiments with higher fidelity yields a greater error in the payoff. Further, we find that the cla…
We study a program game version of the Prisoner's Dilemma, i.e., a two-player game in which each player submits a computer program, the programs are given read access to each other's source code and then choose whether to cooperate or defect. Prior work has introduced various programs that form cooperative equilibria against themselves in this game. For example, the $ε$-grounded Fair Bot cooperates with probability $ε$ and with the remaining probability runs its opponent's program and copies its action. If both players submit this program, then this is a Nash equilibrium in which both players …
In many cases the Nash equilibria are not predictive of the experimental players' behaviour. For some games of Game Theory it is proposed here a method to estimate the probabilities with which the different options will be actually chosen by balanced players, i.e. players that are neither too competitive, nor too cooperative. This will allow to measure the intrinsec cooperativeness degree of a game, only in function of its payoffs. The method is shaped on the Prisoner's Dilemma, then generalized for asymmetric tables, N players and N options. It is adapted to other conditions like Chicken Game…
University of Electronic Science and Technology of China（电子科技大学）；Shanghai Jiao Tong University（上海交通大学）
Repeated game has long been the touchstone model for agents' long-run relationships. Previous results suggest that it is particularly difficult for a repeated game player to exert an autocratic control on the payoffs since they are jointly determined by all participants. This work discovers that the scale of a player's capability to unilaterally influence the payoffs may have been much underestimated. Under the conventional iterated prisoner's dilemma, we develop a general framework for controlling the feasible region where the players' payoff pairs lie. A control strategy player is able to co…
For the iterated Prisoner's Dilemma, there exist Markov strategies which solve the problem when we restrict attention to the long term average payoff. When used by both players these assure the cooperative payoff for each of them. Neither player can benefit by moving unilaterally any other strategy, i.e. these are Nash equilibria. In addition, if a player uses instead an alternative which decreases the opponent's payoff below the cooperative level, then his own payoff is decreased as well. Thus, if we limit attention to the long term payoff, these \emph{good strategies} effectively stabilize c…
This paper focuses on a one person game called Indian policeman's dilemma (IPD). It represents the internal conflict between emotion and profession of a typical Indian police officer. We have 'split' the game to be played independently by different personality modules of the same player. Each module then appears as an independent individual player of the game. None of the players knows the exact payoff values of any of the others. Only greater than or less than type of inequalities among the payoff values across the players are to be inferred probabilistically. There are two Nash equilibrium (…
Purpose: We propose a model to present a possible mechanism for obtaining sizeable behavioural structures by simulating an agent based on the evolutionary public good game with available social learning. Methods: The model considered a population with a fixed number of players. For each round, the chosen players may contribute part of their value to a common pool. Then each player may imitate the strategy of another player, based on relative payoffs (whoever has the lower payoff adopts the strategy of the other player) and change strategy using different exploratory variables. Results: Relativ…
This paper examines the integration of computational complexity into game theoretic models. The example focused on is the Prisoner's Dilemma, repeated for a finite length of time. We show that a minimal bound on the players' computational ability is sufficient to enable cooperative behavior. In addition, a variant of the repeated Prisoner's Dilemma game is suggested, in which players have the choice of opting out. This modification enriches the game and suggests dominance of cooperative strategies. Competitive analysis is suggested as a tool for investigating sub-optimal (but computationally t…
A notion of pi-tolerant equilibrium is defined that takes into account that players have some tolerance regarding payoffs in a game. This solution concept generalizes Nash and refines epsilon-Nash equilibrium in a natural way. We show that pi-tolerant equilibrium can explain cooperation in social dilemmas such as Prisoner's Dilemma and the Public Good game. We then examine the structure of particularly cooperative pi-tolerant equilibria, where players are as cooperative as they can be, subject to their tolerances, in Prisoner's Dilemma. To the extent that cooperation is due to tolerance, these…
Traveler's dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between $R$ and $M$ ($R < M$) as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say $i$ and $j$ ($R \le i < j \le M$), then the payoff to the player who chooses the small value will be $i+R$ and the payoff to the other player will be $i-R$. We term the player who selec…
The problem of two companies of agents with one-step memory playing game is investigated in the context of the Iterated Prisoner's Dilemma under the partial imitation rule, where a player can imitate only those moves that he has observed in his games with his opponent. We limit our study to the special case where the players in the two groups enjoy the same conditions on a fully connected network, so that there are only two payoff matrices required: one for players playing games with members of the same company, and the other one for players playing games with members from a different company.…
How attention scarcity effects the outcomes of a game? We present our findings on a version of the Iterated Prisoners Dilemma (IPD) game in which players can accept or refuse to play with their partner. We study the memory size effect on determining the right partner to interact with. We investigate the conditions under which the cooperators are more likely to be advantageous than the defectors. This work demonstrates that, in order to beat defection, players do not need a full memorization of each action of all opponents. There exists a critical attention capacity threshold to beat defectors.…
Interactions among individuals in natural populations often occur in a dynamically changing environment. Understanding the role of environmental variation in population dynamics has long been a central topic in theoretical ecology and population biology. However, the key question of how individuals, in the middle of challenging social dilemmas (e.g., the "tragedy of the commons"), modulate their behaviors to adapt to the fluctuation of the environment has not yet been addressed satisfactorily. Utilizing evolutionary game theory and stochastic games, we develop a game-theoretical framework that…
We study a model for switching strategies in the Prisoner's Dilemma game on adaptive networks of player pairings that coevolve as players attempt to maximize their return. We use a node-based strategy model wherein each player follows one strategy at a time (cooperate or defect) across all of its neighbors, changing that strategy and possibly changing partners in response to local changes in the network of player pairing and in the strategies used by connected partners. We compare and contrast numerical simulations with existing pair approximation differential equations for describing this sys…
This work explores dynamics existing in interactions between players. The dynamic system of games is a new attitude to modeling in which an event is modeled using several games. The model allows us to analyze the interplay capabilities and the feasibility objectives of each player after a conflict with other players objectives and capabilities. As an application, we model relations between the Soviet Union and America after World War II to October 1962, by using the dynamic system of games. The dynamic system of games as an important insight clearly has significant implications for modeling st…
We propose a new class of games, called Multi-Games (MG), in which a given number of players play a fixed number of basic games simultaneously. In each round of the MG, each player will have a specific set of weights, one for each basic game, which add up to one and represent the fraction of the player's investment in each basic game. The total payoff for each player is then the convex combination, with the corresponding weights, of the payoffs it obtains in the basic games. The basic games in a MG can be regarded as different environments for the players. When the players' weights for the dif…
We investigate symmetric equilibria of mutual reinforcement learning when both players alternately learn the optimal memory-two strategies against the opponent in the repeated prisoners' dilemma game. We provide a necessary condition for memory-two deterministic strategies to form symmetric equilibria. We then provide three examples of memory-two deterministic strategies which form symmetric mutual reinforcement learning equilibria. We also prove that mutual reinforcement learning equilibria formed by memory-two strategies are also mutual reinforcement learning equilibria when both players use…
High-frequency trading (HFT) offers an excellent user case and a potential killer application of the commercially available, first generation quasi-quantum communication and computation technologies. To this end, we offer here a simple but complete game-theoretic model of HFT as the famous two player game, Prisoner's Dilemma. We explore the implementation of HFT as a game on the (quasi) quantum cloud using the Eisert, Wilkens, and Lewenstein quantum mediated communication protocol, and how this implementation can increase transaction speed and improve the lot of the players in HFT. Using coope…
Prisoner's dilemma has been heavily studied. In classical model, each player chooses to either "Cooperate" or "Defect". In this paper, we generalize the prisoner's dilemma with a new alternative which is neither defect or cooperation. The classical model is the special case under the condition that the third state is not taken into consideration.
Situations in which immediate self-interest and long-term collective interest conflict often require some form of influence to prevent them from leading to undesirable or unsustainable outcomes. Next to sanctioning, social influence and social structure, it is possible that strategic solutions can exist for these social dilemmas. However, the existence of strategies that enable a player to exert control in the long-run outcomes can be difficult to show and different situations allow for different levels of strategic influence. Here, we investigate the effect of threshold nonlinearities on the …
Cooperative behavior lies at the very basis of human societies, yet its evolutionary origin remains a key unsolved puzzle. Whereas reciprocity or conditional cooperation is one of the most prominent mechanisms proposed to explain the emergence of cooperation in social dilemmas, recent experimental findings on networked Prisoner's Dilemma games suggest that conditional cooperation also depends on the previous action of the player---namely on the `mood' in which the player currently is. Roughly, a majority of people behaves as conditional cooperators if they cooperated in the past, while they ig…
Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen an…
Pavlov, a well-known strategy in game theory, has been shown to have some advantages in the Iterated Prisoner's Dilemma (IPD) game. However, this strategy can be exploited by inveterate defectors. We modify this strategy to mitigate the exploitation. We call the resulting strategy Rational Pavlov. This has a parameter p which measures the "degree of forgiveness" of the players. We study the evolution of cooperation in the IPD game, when n players are arranged in a cycle, and all play this strategy. We examine the effect of varying p on the convergence rate and prove that the convergence rate i…
Quantum Innovation Centre (Q.InC), Agency for Science, Technology and Research (A*STAR)（量子创新中心，科学与技术研究局）；Singapore University of Technology and Design（新加坡科技设计大学）；Singapore Institute of Technology（新加坡理工大学）
Promoting Cooperation in the Public Goods Game using Artificial Intelligent Agents
Department of MicroData Analytics（微数据分析系）；BEACON Center for the Study of Evolution in Action（行为进化研究中心）；Department of Microbiology, Genetics, and Immunology（微生物学、遗传学与免疫学系）；Program in Ecology, Evolution, and Behavior（生态、进化与行为项目）；Department of Physics and Astronomy（物理与天文学系）
In two-player repeated games, Zero-Determinant (ZD) strategies enable a player to unilaterally enforce a linear payoff relation between her own and her opponent's payoff irrespective of the opponent's strategy. This manipulative nature of the ZD strategies attracted significant attention from researchers due to its close connection to controlling distributively the outcome of evolutionary games in large populations. In this paper, necessary and sufficient conditions are derived for a payoff relation to be enforceable in multiplayer social dilemmas with a finite expected number of rounds that i…
We investigate the repeated prisoner's dilemma game where both players alternately use reinforcement learning to obtain their optimal memory-one strategies. We theoretically solve the simultaneous Bellman optimality equations of reinforcement learning. We find that the Win-stay Lose-shift strategy, the Grim strategy, and the strategy which always defects can form symmetric equilibrium of the mutual reinforcement learning process amongst all deterministic memory-one strategies.
We present insights and empirical results from an extensive numerical study of the evolutionary dynamics of the iterated prisoner's dilemma. Fixation probabilities for Moran processes are obtained for all pairs of 164 different strategies including classics such as TitForTat, zero determinant strategies, and many more sophisticated strategies. Players with long memories and sophisticated behaviours outperform many strategies that perform well in a two player setting. Moreover we introduce several strategies trained with evolutionary algorithms to excel at the Moran process. These strategies ar…
In this paper, we address the much-anticipated deployment of connected and automated vehicles (CAVs) in society by modeling and analyzing the social-mobility dilemma in a game-theoretic approach. We formulate this dilemma as a normal-form game of players making a binary decision: whether to travel with a CAV (CAV travel) or not (non-CAV travel) and by constructing an intuitive payoff function inspired by the socially beneficial outcomes of a mobility system consisting of CAVs. We show that the game is equivalent to the Prisoner's dilemma, which implies that the rational collective decision is …
We have studied the impact of time-dependent learning capacities of players in the framework of spatial prisoner's dilemma game. In our model, this capacity of players may decrease or increase in time after strategy adoption according to a step-like function. We investigated both possibilities separately and observed significantly different mechanisms that form the stationary pattern of the system. The time decreasing learning activity helps cooperator domains to recover the possible intrude of defectors hence supports cooperation. In the other case the temporary restrained learning activity g…
Learning from a partner who collects higher payoff is a frequently used working hypothesis in evolutionary game theory. One of the alternative dynamical rules is when the focal player prefers to follow the strategy choice of the majority in the local neighborhood, which is often called as conformity-driven strategy update. In this work we assume that both strategy learning methods are present and compete for space within the framework of a coevolutionary model. Our results reveal that the presence of payoff-driven strategy learning method becomes exclusive for high succer's payoff and/or high …
With deeper study of the Game Theory, some conditions of Prisoner's Dilemma is no longer suitable of games in real life. So we try to develop a new model-Villager's Dilemma which has more realistic conditions to stimulate the process of game. It is emphasize that Prisoner's Dilemma is an exception which is lack of universality and the importance of rules in the game. And it puts forward that to let the rule maker take part in the game and specifies game players can stop the game as they like. This essay describes the basic model, the villager's dilemma (VD) and put some extended use of it, and…
This paper studies scenarios of cyclic dominance in a coevolutionary spatial model in which game strategies and links between agents adaptively evolve over time. The Optional Prisoner's Dilemma (OPD) game is employed. The OPD is an extended version of the traditional Prisoner's Dilemma where players have a third option to abstain from playing the game. We adopt an agent-based simulation approach and use Monte Carlo methods to perform the OPD with coevolutionary rules. The necessary conditions to break the scenarios of cyclic dominance are also investigated. This work highlights that cyclic dom…
CoopEval: Benchmarking Cooperation-Sustaining Mechanisms and LLM Agents in Social Dilemmas
Carnegie Mellon University；Foundations of Cooperative AI Lab (FOCAL)；Jinesis Lab, University of Toronto \& Vector Institute；ETH Z\" u rich；Max Planck Institute for Intelligent Systems, T\" u bingen, Germany
According to the standard imitation protocol, a less successful player adopts the strategy of the more successful one faithfully for future success. This is the cornerstone of evolutionary game theory that explores the vitality of competing strategies in different social dilemma situations. In our present work we explore the possible consequences of two slightly modified imitation protocols that are exaggerated and gradual learning rules. In the former case a learner does not only adopt, but also enlarges the strategy change for the hope of a higher income. Similarly, in the latter case a caut…
Research has shown that the addition of abstention as an option transforms social dilemmas to rock-paper-scissor type games, where defectors dominate cooperators, cooperators dominate abstainers (loners), and abstainers (loners), in turn, dominate defectors. In this way, abstention can sustain cooperation even under adverse conditions, although defection also persists due to cyclic dominance. However, to abstain or to act as a loner has, to date, always been considered as an independent, third strategy to complement traditional cooperation and defection. Here we consider probabilistic abstenti…
A generalized model of games is proposed, in which cooperative games and non-cooperative games are special cases. Some games that are neither cooperative nor non-cooperative can be expressed and analyzed. The model is based on relationships and supposed relationships between players. A relationship is a numerical value that denotes how one player cares for the payoffs of another player, while a supposed relationship is another numerical value that denotes a player's belief about the relationship between two players. The players choose their strategies by taking into consideration not only the …
A recently introduced concept of "cooperative equilibrium", based on the assumption that players have a natural attitude to cooperation, has been proven a powerful tool in predicting human behaviour in social dilemmas. In this paper, we extend this idea to more general game models, termed "Pareto solvable" games, which in particular include the Nash Bargaining Problem and the Ultimatum Game. We show that games in this class possess a unique pure cooperative equilibrium. Furthermore, for the Ultimatum Game, this notion appears to be strongly correlated with a suitably defined variant of the Dic…
The Internet has enabled the emergence of collective problem solving, also known as crowdsourcing, as a viable option for solving complex tasks. However, the openness of crowdsourcing presents a challenge because solutions obtained by it can be sabotaged, stolen, and manipulated at a low cost for the attacker. We extend a previously proposed crowdsourcing dilemma game to an iterated game to address this question. We enumerate pure evolutionarily stable strategies within the class of so-called reactive strategies, i.e., those depending on the last action of the opponent. Among the 4096 possible…
