Direct reciprocity based on the repeated prisoner's dilemma has been intensively studied. Most theoretical investigations have concentrated on memory-$1$ strategies, a class of elementary strategies just reacting to the previous-round outcomes. Though the properties of "All-or-None" strategies ($AoN_K$) have been discovered, simulations just confirmed the good performance of $AoN_K$ of very short memory lengths. It remains unclear how $AoN_K$ strategies would fare when players have access to longer rounds of history information. We construct a theoretical model to investigate the performance of the class of $AoN_K$ strategies of varying memory length $K$. We rigorously derive the payoffs and show that $AoN_K$ strategies of intermediate memory length $K$ are most prevalent, while strategies of larger memory lengths are less competent. Larger memory lengths make it hard for $AoN_K$ strategies to coordinate, and thus inhibiting their mutual reciprocity. We then propose the adaptive coordination strategy combining tolerance and $AoN_K$' coordination rule. This strategy behaves like $AoN_K$ strategy when coordination is not sufficient, and tolerates opponents' occasional deviations by still cooperating when coordination is sufficient. We found that the adaptive coordination strategy wins over other classic memory-$1$ strategies in various typical competition environments, and stabilizes the population at high levels of cooperation, suggesting the effectiveness of high level adaptability in resolving social dilemmas. Our work may offer a theoretical framework for exploring complex strategies using history information, which are different from traditional memory-$n$ strategies.
Social dilemmas are situations in which collective interests are at odds with private interests: pollution, depletion of natural resources, and intergroup conflicts, are at their core social dilemmas. Because of their multidisciplinarity and their importance, social dilemmas have been studied by economists, biologists, psychologists, sociologists, and political scientists. These studies typically explain tendency to cooperation by dividing people in proself and prosocial types, or appealing to forms of external control or, in iterated social dilemmas, to long-term strategies. But recent experi…
Disentangling trust from cooperation: Evolution of trust as reduced monitoring in social dilemmas
University of Lausanne（洛桑大学）；Teesside University（蒂斯赛德大学）；Université libre de Bruxelles（布鲁塞尔自由大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；University of Stirling（斯特灵大学）
The 2 x 2 games, in particular the Prisoner's Dilemma, have been extensively used in studies into reciprocal cooperation and, to a lesser extent, kin selection. This paper examines the suitability of the 2 x 2 games for modelling the evolution of cooperation through reciprocation and kin selection. This examination is not restricted to the Prisoner's Dilemma, but includes the other non-trivial symmetric 2 x 2 games. We show that the popularity of the Prisoner's Dilemma for modelling social and biotic interaction is justified by its superiority according to these criteria. Indeed, the Prisoner'…
We consider the one-shot Prisoner's Dilemma between algorithms with read-access to one anothers' source codes, and we use the modal logic of provability to build agents that can achieve mutual cooperation in a manner that is robust, in that cooperation does not require exact equality of the agents' source code, and unexploitable, meaning that such an agent never cooperates when its opponent defects. We construct a general framework for such "modal agents", and study their properties.
Crowdsourcing has emerged as a paradigm for leveraging human intelligence and activity to solve a wide range of tasks. However, strategic workers will find enticement in their self-interest to free-ride and attack in a crowdsourcing contest dilemma game. Hence, incentive mechanisms are of great importance to overcome the inefficiency of the socially undesirable equilibrium. Existing incentive mechanisms are not effective in providing incentives for cooperation in crowdsourcing competitions due to the following features: heterogeneous workers compete against each other in a crowdsourcing platfo…
Social dilemmas are central to human society. Depletion of natural resources, climate protection, security of energy supply, and workplace collaborations are all examples of social dilemmas. Since cooperative behaviour in a social dilemma is individually costly, Nash equilibrium predicts that humans should not cooperate. Yet experimental studies show that people do cooperate even in anonymous one-shot interactions. In spite of the large number of participants in many modern social dilemmas, little is known about the effect of group size on cooperation. Does larger group size favour or prevent …
Tianjin University（天津大学）；Washington State University（华盛顿州立大学）
The Iterated Prisoner's Dilemma has guided research on social dilemmas for decades. However, it distinguishes between only two atomic actions: cooperate and defect. In real-world prisoner's dilemmas, these choices are temporally extended and different strategies may correspond to sequences of actions, reflecting grades of cooperation. We introduce a Sequential Prisoner's Dilemma (SPD) game to better capture the aforementioned characteristics. In this work, we propose a deep multiagent reinforcement learning approach that investigates the evolution of mutual cooperation in SPD games. Our approa…
Social dilemmas are situations where individuals face a temptation to increase their payoffs at a cost to total welfare. Building artificially intelligent agents that achieve good outcomes in these situations is important because many real world interactions include a tension between selfish interests and the welfare of others. We show how to modify modern reinforcement learning methods to construct agents that act in ways that are simple to understand, nice (begin by cooperating), provokable (try to avoid being exploited), and forgiving (try to return to mutual cooperation). We show both theo…
Social dilemmas, where mutual cooperation can lead to high payoffs but participants face incentives to cheat, are ubiquitous in multi-agent interaction. We wish to construct agents that cooperate with pure cooperators, avoid exploitation by pure defectors, and incentivize cooperation from the rest. However, often the actions taken by a partner are (partially) unobserved or the consequences of individual actions are hard to predict. We show that in a large class of games good strategies can be constructed by conditioning one's behavior solely on outcomes (ie. one's past rewards). We call this c…
Exploiting others is beneficial individually but it could also be detrimental globally. The reverse is also true: a higher cooperation level may change the environment in a way that is beneficial for all competitors. To explore the possible consequence of this feedback we consider a coevolutionary model where the local cooperation level determines the payoff values of the applied prisoner's dilemma game. We observe that the coevolutionary rule provides a significantly higher cooperation level comparing to the traditional setup independently of the topology of the applied interaction graph. Int…
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 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…
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…
In many societal and industrial interactions, participants generally prefer their pure self-interest at the expense of the global welfare. Known as social dilemmas, this category of non-cooperative games offers situations where multiple actors should all cooperate to achieve the best outcome but greed and fear lead to a worst self-interested issue. Recently, the emergence of Deep Reinforcement Learning (RL) has generated revived interest in social dilemmas with the introduction of Sequential Social Dilemma (SSD). Cooperative agents mixing RL policies and Tit-for-tat (TFT) strategies have succe…
Is it rational for selfish individuals to cooperate? The conventional answer based on analysis of games such as the Prisoners Dilemma (PD) is that it is not, even though mutual cooperation results in a better outcome for all. This incompatibility between individual rationality and collective benefit lies at the heart of questions about the evolution of cooperation, as illustrated by PD and similar games. Here, we argue that this apparent incompatibility is due to an inconsistency in the standard Nash framework for analyzing non-cooperative games and propose a new paradigm, that of the co-actio…
Groups of humans are often able to find ways to cooperate with one another in complex, temporally extended social dilemmas. Models based on behavioral economics are only able to explain this phenomenon for unrealistic stateless matrix games. Recently, multi-agent reinforcement learning has been applied to generalize social dilemma problems to temporally and spatially extended Markov games. However, this has not yet generated an agent that learns to cooperate in social dilemmas as humans do. A key insight is that many, but not all, human individuals have inequity averse social preferences. This…
How cooperation emerges and persists in a population of selfish agents is a fundamental question in evolutionary game theory. Our research shows that Collective Strategies with Master-Slave Mechanism (CSMSM) defeat Tit-for-Tat and other well-known strategies in spatial iterated prisoner's dilemma. A CSMSM identifies kin members by means of a handshaking mechanism. If the opponent is identified as non-kin, a CSMSM will always defect. Once two CSMSMs meet, they play master and slave roles. A mater defects and a slave cooperates in order to maximize the master's payoff. CSMSM outperforms non-coll…
Cooperation is a very common, yet not fully-understood phenomenon in natural and human systems. The introduction of a network within the population is known to affect the outcome of cooperative dynamics, allowing for the survival of cooperation in adverse scenarios. Recently, the introduction of multiplex networks has yet again modified the expectations for the outcome of the Prisoner's Dilemma game, compared to the monoplex case. However, much remains unstudied regarding other social dilemmas on multiplex, as well as the unexplored microscopic underpinnings of it. In this paper, we systematic…
Cooperation in one-shot anonymous interactions is a widely documented aspect of human behaviour. Here we shed light on the motivations behind this behaviour by experimentally exploring cooperation in a one-shot continuous-strategy Prisoner's Dilemma (i.e. one-shot two-player Public Goods Game). We examine the distribution of cooperation amounts, and how that distribution varies based on the benefit-to-cost ratio of cooperation (b/c). Interestingly, we find a trimodal distribution at all b/c values investigated. Increasing b/c decreases the fraction of participants engaging in zero cooperation …
In this work, we develop a reinforcement learning protocol for a multiagent coordination task in a discrete state and action space: an iterated prisoner's dilemma game extended into a team based, winner-take all tournament, which forces the agents to collude in order to maximize their reward. By disallowing extra communication channels, the agents are forced to embed their coordination strategy into their actions in the prisoner's dilemma game. We develop a representation of the iterated prisoners dilemma that makes it amenable to Q-learning. We find that the reinforcement learning strategy is…
Multi-Agent, Human-Agent and Beyond: A Survey on Cooperation in Social Dilemmas
Northwestern Polytechnical University（西北工业大学）；Tsinghua University（清华大学）；University of Auckland（奥克兰大学）；Kyushu University（九州大学）；The Hong Kong Polytechnic University（香港理工大学）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）
Since the introduction of zero-determinant strategies, extortionate strategies have received considerable interest. While an interesting class of strategies, the definitions of extortionate strategies are algebraically rigid, apply only to memory-one strategies, and require complete knowledge of a strategy (memory-one cooperation probabilities). We describe a method to detect extortionate behaviour from the history of play of a strategy. When applied to a corpus of 204 strategies this method detects extortionate behaviour in well-known extortionate strategies as well others that do not fit the…
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…
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…
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…
Using simulations between pairs of $ε$-greedy q-learners with one-period memory, this article demonstrates that the potential function of the stochastic replicator dynamics (Foster and Young, 1990) allows it to predict the emergence of error-proof cooperative strategies from the underlying parameters of the repeated prisoner's dilemma. The observed cooperation rates between q-learners are related to the ratio between the kinetic energy exerted by the polar attractors of the replicator dynamics under the grim trigger strategy. The frontier separating the parameter space conducive to cooperation…
Modeling Decision-Making with Will for Cooperation in Social Dilemmas
School of Intelligence Science and Technology, Peking University（北京大学智能学院）；State Key Laboratory of General Artificial Intelligence, BIGAI（通用人工智能国家重点实验室，BIGAI）；Law School, Peking University（北京大学法学院）
We investigate the spatial distribution and the global frequency of agents who can either cooperate or defect. The agent interaction is described by a deterministic, non-iterated prisoner's dilemma game, further each agent only locally interacts with his neighbors. Based on a detailed analysis of the local payoff structures we derive critical conditions for the invasion or the spatial coexistence of cooperators and defectors. These results are concluded in a phase diagram that allows to identify five regimes, each characterized by a distinct spatiotemporal dynamics and a corresponding final sp…
In a world in which many pressing global issues require large scale cooperation, understanding the group size effect on cooperative behavior is a topic of central importance. Yet, the nature of this effect remains largely unknown, with lab experiments insisting that it is either positive or negative or null, and field experiments suggesting that it is instead curvilinear. Here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperati…
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…
In this paper, we investigate the prison's dilemma game with monte carlo rule in the view of the idea of the classic Monte Carlo method on the grid. Monte carlo rule is an organic combination of the current dynamic rules of individual policy adjustment, which not only makes full use of information but also reflects the individual's bounded rational behavior and the ambivalence between the pursuit of high returns and high risks. In addition, it also reflects the individual's behavioral execution preferences. The implementation of monte carlo rule brings an extremely good result, higher cooperat…
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…
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
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）
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…
Individual migration has been regarded as an important factor for the evolution of cooperation in mobile populations. Motivations of migration, however, can be largely divergent: one is highly frustrated by the vicinity of an exploiter or defector, while other enthusiastically searches cooperator mates. Albeit both extreme attitudes are observed in human behavior, but their specific impacts on wellbeing remained unexplored. In this work, we propose an orientation-driven migration approach for mobile individuals in combination with the mentioned migration preferences and study their roles in th…
Enforcing cooperation among substantial agents is one of the main objectives for multi-agent systems. However, due to the existence of inherent social dilemmas in many scenarios, the free-rider problem may arise during agents' long-run interactions and things become even severer when self-interested agents work in collusion with each other to get extra benefits. It is commonly accepted that in such social dilemmas, there exists no simple strategy for an agent whereby she can simultaneously manipulate on the utility of each of her opponents and further promote mutual cooperation among all agent…
Massachusetts Institute of Technology（麻省理工学院）；INESC-ID（里斯本大学高等理工学院附属信息系统与计算机工程研究所）；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等理工学院）
Designing mechanisms that leverage cooperation between agents has been a long-lasting goal in Multiagent Systems. The task is especially challenging when agents are selfish, lack common goals and face social dilemmas, i.e., situations in which individual interest conflicts with social welfare. Past works explored mechanisms that explain cooperation in biological and social systems, providing important clues for the aim of designing cooperative artificial societies. In particular, several works show that cooperation is able to emerge when specific network structures underlie agents' interaction…
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…
Heterogeneity has been studied as one of the most common explanations of the puzzle of cooperation in social dilemmas. A large number of papers have been published discussing the effects of increasing heterogeneity in structured populations of agents, where it has been established that heterogeneity may favour cooperative behaviour if it supports agents to locally coordinate their strategies. In this paper, assuming an existing model of a heterogeneous weighted network, we aim to further this analysis by exploring the relationship (if any) between heterogeneity and cooperation. We adopt a weig…
Cooperation played a significant role in the self-organization and evolution of living organisms. Both network topology and the initial position of cooperators heavily affect the cooperation of social dilemma games. We developed a novel simulation program package, called 'NetworGame', which is able to simulate any type of social dilemma games on any model, or real world networks with any assignment of initial cooperation or defection strategies to network nodes. The ability of initially defecting single nodes to break overall cooperation was called as 'game centrality'. The efficiency of this …
We outline a model to study the evolution of cooperation in a population of agents playing the prisoner's dilemma in signed networks. We highlight that if only dyadic interactions are taken into account, cooperation never evolves. However, when triadic considerations are introduced, a window of opportunity for emergence of cooperation as a stable behaviour emerges.
The Machine Psychology of Cooperation: Can GPT models operationalise prompts for altruism, cooperation, competitiveness and selfishness in economic games?
University College London（伦敦大学学院）；Middlesex University（密德萨斯大学）
