N-person Prisoner's Dilemma Research Articles

Cooperation in the Multi-Agent System with Different Types of Interactions

This paper summarizes the list of our works that contain researches about optimality principles for the "n-person prisoner's dilemma" game. The classic model is considered through the new payoff function for each player that allows to consider it without restrictions for the number of players. The new characteristic function gives an opportunity to introduce the time-consistent subset of the core of the dynamic game. In accordance with this type of game we consider some specific properties of players' payoffs and construct the new way of their interactions. Using the network representation, the classic model is modified to the wider class of games that allows to specify players' influence to each other's payoff function. These investigations can be used for the description of cooperation in the other multi-agent systems.

Cooperative n-person Prisoner's Dilemma on a Network

In the paper n-person prisoner's dilemma on the network is investigated. A cooperative game with the pairwise interaction of players is constructed. The model is a modification of the classic 2-person prisoner's dilemma problem in the game theory. Network interaction provide an ability to take into account the in uence only to the adjacent players from the whole set of players. The feature of the game is found that allows to make a decision about necessity of playing dominated strategy by a few players. This solution is based on the number of the adjacent players. The work is a continuation of the paper published earlier by Grinikh A.L. and Petrosyan L.A. in 2021.

Exit Option Can Make Cooperation Easier

In the modern society, mobility of players has increased dramatically. It is important to extend game theory to endogenize the set of players. In this paper, we extend the repeated n-person Prisoner's Dilemma by allowing some or all players to exit after each period and analyze how mobility affects endogenous cooperation in a fundamental sense: players voluntarily participate in the game and act non-myopically. We characterize the parameters (the allocation of exit options among players and the payoffs) under which eternal universal cooperation is sustained at a discount factor lower than the one for the ordinary repeated Prisoner's Dilemma. Factors that enhance endogenous long-term cooperation include payoff asymmetry in the Prisoner's Dilemma, multiple mobile players who have the exit option, and immobile players having low after-game payoffs. These are new insights which give policy implications on employment relationships and other voluntary partnership situations as well.

Group size effect on cooperation in one-shot social dilemmas

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 cooperation? We address this problem both experimentally and theoretically. Experimentally, we find that there is no general answer: it depends on the strategic situation. Specifically, we find that larger groups are more cooperative in the Public Goods game, but less cooperative in the N-person Prisoner's dilemma. Theoretically, we show that this behaviour is not consistent with either the Fehr & Schmidt model or (a one-parameter version of) the Charness & Rabin model, but it is consistent with the cooperative equilibrium model introduced by the second author.

Dangerous drivers foster social dilemma structures hidden behind a traffic flow with lane changes

Motivated by the fact that there are quite a few ill-mannered drivers who disregard traffic rules concerning lane-changing and maximum speed, we investigated an interesting question: whether or not social dilemma structures can be formed from a frequent dangerous lane-changing attitude in a typical traffic flow without any explicit bottlenecks. In our model system, two classes of driver-agents coexist: C agents (cooperative strategy) always keep to traffic regulations with respect to lane-changing and speed, while D agents (defective strategy) disregard them to move ahead. In relatively high-density flows, such as the metastable and high-density phases, we found structures that correspond to either n-person Prisoner's Dilemma (n-PD) games or to quasi-PD games. In these situations, existing ill-mannered drivers create heavy traffic jams that reduce social efficiency.

Social dilemma structures hidden behind traffic flow with lane changes

.Aiming to merge traffic flow analysis with evolutionary game theory, we investigated the question of whether such structures can be formed from frequent lane changes in usual traffic flow without any explicit bottlenecks. In our model system, two classes of driver-agents coexist: C-agents (cooperative strategy) always remain in the lane they are initially assigned, whereas D-agents (defective strategy) try to change lanes to move ahead. In relatively high-density flows, such as the metastable and high-density phases, we found structures that correspond to either n-person prisoner dilemma (n-PD) games or quasi-PD games. In these situations, lane changes by D-agents create heavy traffic jams that reduce social efficiency.

Which Cooperative Strategies are Evolutionarily Stable?

In repeated games, reciprocity helps to sustain group cooperation. Using the Markovian strategy as a touchstone, we explore the evolution of cooperation in the infinitely repeated and noisy n-person Prisoner's Dilemma. With some modified game settings, we show that a class of cooperative strategies based on reciprocity may be evolutionarily stable. When a cooperative strategy satisfies some properties in our study, that strategy becomes evolutionarily stable if it outperforms the mutant permanent defector. We also argue that evolutionary stability in a noisy n-person setting implies the Markov perfect equilibrium.

Asymmetries in Altruistic Behavior during Violent Intergroup Conflict

Recent theoretical and experimental investigations of altruistic behavior in intergroup conflict in humans frequently make use of the assumption that warfare can be modeled as a symmetrical n-person prisoner's dilemma, abstracting away the strategic differences between attack and defense. In contrast, some empirical studies on intergroup conflict in hunter-gatherer societies and chimpanzees indicate that fitness relevant risks and potential benefits of attacks and defenses might have differed substantially under ancestral conditions. Drawing on these studies, it is hypothesized that the success of defenses was much more important for individual and kin survival and that a disposition to act altruistically during intergroup conflict is thus more likely to evolve for the strategic situation of defense. It is then investigated empirically if such asymmetries in the occurrence of altruistic behavior during intergroup conflict can be found. Analyzing detailed historical case data from 20th century wars, this study finds that altruistic behavior towards members of the in-group indeed seems to occur more frequently when soldiers are defending themselves and their comrades against enemy attacks. It is proposed that this asymmetry reflects adaptive behavioral responses to the materially different strategic character of attacks and defenses under ancestral conditions. If true, this would call for a refinement of theories of the evolutionary interaction of intergroup conflict and altruism.

Bipartite Graphs as Models of Population Structures in Evolutionary Multiplayer Games

By combining evolutionary game theory and graph theory, “games on graphs” study the evolutionary dynamics of frequency-dependent selection in population structures modeled as geographical or social networks. Networks are usually represented by means of unipartite graphs, and social interactions by two-person games such as the famous prisoner’s dilemma. Unipartite graphs have also been used for modeling interactions going beyond pairwise interactions. In this paper, we argue that bipartite graphs are a better alternative to unipartite graphs for describing population structures in evolutionary multiplayer games. To illustrate this point, we make use of bipartite graphs to investigate, by means of computer simulations, the evolution of cooperation under the conventional and the distributed N-person prisoner’s dilemma. We show that several implicit assumptions arising from the standard approach based on unipartite graphs (such as the definition of replacement neighborhoods, the intertwining of individual and group diversity, and the large overlap of interaction neighborhoods) can have a large impact on the resulting evolutionary dynamics. Our work provides a clear example of the importance of construction procedures in games on graphs, of the suitability of bigraphs and hypergraphs for computational modeling, and of the importance of concepts from social network analysis such as centrality, centralization and bipartite clustering for the understanding of dynamical processes occurring on networked population structures.

Emergence of Fairness in Repeated Group Interactions

Often groups need to meet repeatedly before a decision is reached. Hence, most individual decisions will be contingent on decisions taken previously by others. In particular, the decision to cooperate or not will depend on one's own assessment of what constitutes a fair group outcome. Making use of a repeated N-person prisoner's dilemma, we show that reciprocation towards groups opens a window of opportunity for cooperation to thrive, leading populations to engage in dynamics involving both coordination and coexistence, and characterized by cycles of cooperation and defection. Furthermore, we show that this process leads to the emergence of fairness, whose level will depend on the dilemma at stake.

Review: Game theory of public goods in one-shot social dilemmas without assortment

We review the theory of public goods in biology. In the N-person prisoner's dilemma, where the public good is a linear function of the individual contributions, cooperation requires some form of assortment, for example due to kin discrimination, population viscosity or repeated interactions. In most social species ranging from bacteria to humans, however, public goods are usually a non-linear function of the contributions, which makes cooperation possible without assortment. More specifically, a polymorphic state can be stable in which cooperators and non-cooperators coexist. The existence of mixed equilibria in public goods games is a fundamental result in the study of cooperation that has been overlooked so far, because of the disproportionate attention given to the two- and N-person prisoner's dilemma. Methods and results from games with pairwise interactions or linear benefits cannot, in general, be extended to the analysis of public goods. Game theory helps explain the production of public goods in one-shot, N-person interactions without assortment, it leads to predictions that can be easily tested and allows a prescriptive approach to cooperation.

COEXISTENCE OF COOPERATION AND DEFECTION IN PUBLIC GOODS GAMES

The production of public goods by the contribution of individual volunteers is a social dilemma because an individual that does not volunteer can benefit from the public good produced by the contributions of others. Therefore it is generally believed that public goods can be produced only in the presence of repeated interactions (which allow reciprocation, reputation effects and punishment) or relatedness (kin selection). Cooperation, however, often occurs in the absence of iterations and relatedness. We show that when the production of a public good is a Volunteer's Dilemma, in which a fixed number of cooperators is necessary to produce the public good, cooperators and defectors persist in a mixed equilibrium, without iterations and without relatedness. This mixed equilibrium is absent in the N-person Prisoner's Dilemma, in which the public good is a linear function of the individual contributions. We also show that the Prisoner's Dilemma and the Volunteer's Dilemma are the two opposite extremes of a general public goods game, and that all intermediate cases can have a mixed equilibrium like the Volunteer's Dilemma. The coexistence of cooperators and defectors, therefore, is a typical outcome of most social dilemmas, which requires neither relatedness nor iterations.

Revisiting “The Evolution of Reciprocity in Sizable Groups”: Continuous reciprocity in the repeated n-person prisoner's dilemma

For many years in evolutionary science, the consensus view has been that while reciprocal altruism can evolve in dyadic interactions, it is unlikely to evolve in sizable groups. This view had been based on studies which have assumed cooperation to be discrete rather than continuous (i.e., individuals can either fully cooperate or else fully defect, but they cannot continuously vary their level of cooperation). In real world cooperation, however, cooperation is often continuous. In this paper, we re-examine the evolution of reciprocity in sizable groups by presenting a model of the n-person prisoner's dilemma that assumes continuous rather than discrete cooperation. This model shows that continuous reciprocity has a dramatically wider basin of attraction than discrete reciprocity, and that this basin's size increases with efficiency of cooperation (marginal per capita return). Further, we find that assortative interaction interacts synergistically with continuous reciprocity to a much greater extent than it does with discrete reciprocity. These results suggest that previous models may have underestimated reciprocity's adaptiveness in groups. However, we also find that the invasion of continuous reciprocators into a population of unconditional defectors becomes realistic only within a narrow parameter space in which the efficiency of cooperation is close to its maximum bound. Therefore our model suggests that continuous reciprocity can evolve in large groups more easily than discrete reciprocity only under unusual circumstances.

Cooperation as a volunteer’s dilemma and the strategy of conflict in public goods games

Conflict and cooperation for the exploitation of public goods are usually modelled as an N-person prisoner's dilemma. Many social dilemmas, however, would be described more properly as a volunteer's dilemma, in which a certain number of individuals are necessary to produce a public good. If volunteering is costly, but so is failure to produce the public good, cheaters can invade and form a stable mixed equilibrium with cooperators. The dilemma is that the benefit for the group decreases with group size because the larger the group is, the less likely it is that someone volunteers. This problem persists even in the presence of a high degree of relatedness between group members. This model provides precise, testable predictions for the stability of cooperation. It also suggests a counterintuitive but practical solution for this kind of social dilemmas: increasing the damage resulting from the failure to produce the public good increases the probability that the public good is actually produced. Adopting a strategy that entails a deliberate risk (brinkmanship), therefore, can lead to a benefit for the society without being detrimental for the individual.

Chaos, oscillation and the evolution of indirect reciprocity in n-person games

Evolution of cooperation among genetically unrelated individuals has been of considerable concern in various fields such as biology, economics, and psychology. The evolution of cooperation is often explained by reciprocity. Under reciprocity, cooperation can prevail in a society because a donor of cooperation receives reciprocation from the recipient of the cooperation, called direct reciprocity, or from someone else in the community, called indirect reciprocity. Nowak and Sigmund [1993. Chaos and the evolution of cooperation. Proc. Natl. Acad. Sci. USA 90, 5091–5094] have demonstrated that directly reciprocal cooperation in two-person prisoner's dilemma games with mutation of strategies can be maintained dynamically as periodic or chaotic oscillation. Furthermore, Eriksson and Lindgren [2005. Cooperation driven by mutations in multi-person Prisoner's Dilemma. J. Theor. Biol. 232, 399–409] have reported that directly reciprocal cooperation in n-person prisoner's dilemma games (n>2) can be maintained as periodic oscillation. Is dynamic cooperation observed only in direct reciprocity? Results of this study show that indirectly reciprocal cooperation in n-person prisoner's dilemma games can be maintained dynamically as periodic or chaotic oscillation. This is, to our knowledge, the first demonstration of chaos in indirect reciprocity. Furthermore, the results show that oscillatory dynamics are observed in common in the evolution of reciprocal cooperation whether for direct or indirect.

Punishing free riders: direct and indirect promotion of cooperation

Human cooperation in a large group of genetically unrelated people is an evolutionary puzzle. Despite its costly nature, cooperative behavior is commonly found in all human societies—a fact that has interested researchers from a wide range of disciplines, including biology, economics, and psychology, to name a few. Many behavioral experiments have demonstrated that cooperation within a group can be sustained when free riders are punished. We argue that punishment has both a direct effect and an indirect effect on promoting cooperation. The direct effect of punishment alters the consequences of cooperation and defection in such a way as to make a rational person prefer cooperation. The indirect effect of punishment promotes cooperation among conditional cooperators by providing the condition necessary for their cooperation (i.e., the expectation that other members will also cooperate). Here we present data from two one-shot n-person prisoner's dilemma games, demonstrating that the indirect effect of punishment complements the direct effect to increase cooperation in the game. Furthermore, we show that direct and indirect effects are robust across two forms of punishment technology: either when punishment is voluntarily provided by game players themselves or when it is exogenously provided by the experimenter.

The supercore for normal-form games

This paper analyzes the supercore of a system derived from a normal-form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore coincides with the set of Nash equilibria of the last game in that sequence. This result is illustrated with the characterization of the supercore for the n-person prisoner's dilemma. With regard to the mixed extension of a normal-form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria.

Cooperation driven by mutations in multi-person Prisoner's Dilemma

The n-person Prisoner's Dilemma is a widely used model for populations where individuals interact in groups. The evolutionary stability of populations has been analysed in the literature for the case where mutations in the population may be considered as isolated events. For this case, and assuming simple trigger strategies and many iterations per game, we analyse the rate of convergence to the evolutionarily stable populations. We find that for some values of the payoff parameters of the Prisoner's Dilemma this rate is so low that the assumption, that mutations in the population are infrequent on that time-scale, is unreasonable. Furthermore, the problem is compounded as the group size is increased. In order to address this issue, we derive a deterministic approximation of the evolutionary dynamics with explicit, stochastic mutation processes, valid when the population size is large. We then analyse how the evolutionary dynamics depends on the following factors: mutation rate, group size, the value of the payoff parameters, and the structure of the initial population. In order to carry out the simulations for groups of more than just a few individuals, we derive an efficient way of calculating the fitness values. We find that when the mutation rate per individual and generation is very low, the dynamics is characterized by populations which are evolutionarily stable. As the mutation rate is increased, other fixed points with a higher degree of cooperation become stable. For some values of the payoff parameters, the system is characterized by (apparently) stable limit cycles dominated by cooperative behaviour. The parameter regions corresponding to high degree of cooperation grow in size with the mutation rate, and in number with the group size. For some parameter values, we find more than one stable fixed point, corresponding to different structures of the initial population.

Attainability of International Environmental Agreements as a Social Situation

This paper applies the theory of social situations to study whether international environmental agreements (IEAs), mainly those on greenhouse gas emission reductions, can be attained. A game theoretic model is generally a black box for decision makers, where the mechanisms, which lead to solution(s) of the game, are not explicitly pointed out. This paper opens this black box by making the (institutional) move rules explicit. The usual pessimistic outcome with an ineffective and small size of stable coalitions among world regions is countered. Our model challenges conventional wisdom in the sense that large coalitions are possible outcomes of the cartel game, namely by incorporating: (1) farsightedness, and (2) coalitional moves with commitment as an alternative to myopic and individual moves which characterise the cartel game. We show that even if the international negotiations on climate change mitigation are modelled as an n-person prisoner's dilemma, one cannot rule out cooperation among world regions as a solution of the game. Indeed, in most analysed situations the grand coalition is among the solutions of the game. This shows that predictions based on cartel stability may be too pessimistic if it comes to analysing incentives to cooperate in implementing international environmental policy. Moreover, in an empirically calibrated model, we find three out of six instances where Russia (with or without the US) has an incentive to sign the Kyoto protocol.

An Atlas of interpersonal situations

Foreword Part I. Introduction and Theory: 1. Interpersonal situations: the context of social behavior 2. Outcome interdependence 3. Interaction conditions and person factors 4. Exploring the geography of the outcome patterns Part II The Situations: Preface to the Entries for the Situations Single Component Patterns: 1. Independence: we go our separate ways 2. Mutual partner control: I scratch your back, you scratch mine 3. Corresponding mutual joint control: getting in sync 4. Conflicting mutual joint control: match or mismatch Two- and three-component patterns: 5. The prisoner's dilemma: me versus we 6. Threat: trading loyalty for justice 7. Chicken: death before dishonor 8. Hero: let's do it your way 9. Conjunctive problems: together we can do it 10. Disjunctive problems: either of us can do it 11. Asymmetric dependence: you're the boss Time-extended patterns: 12. Iterated prisoner's dilemma: united we stand, divided we fall 13. Investment: building for the future 14. Delay of gratification: resisting temptation Incomplete information situations 15. Negotiation: can we agree on a deal? 16. Encounters with strangers: lack of information about a partner 17. Joint decisions under uncertainty: bird in the hand 18. Twists of fate: coping with an uncertain future N-person Situations 19. Third parties: effects of an outsider 20. N-person prisoner's dilemma: tragedy of the commons Movement from one situation to another 21. Movement among situations: where do we go from here? Part III. Epilogue.