We address the problem of how the survival of cooperation in a social system depends on the motion of the individuals. Specifically, we study a model in which Prisoner's Dilemma players are allowed to move in a two-dimensional plane. Our results show that cooperation can survive in such a system provided that both the temptation to defect and the velocity at which agents move are not too high. Moreover, we show that when these conditions are fulfilled, the only asymptotic state of the system is that in which all players are cooperators. Our results might have implications for the design of cooperative strategies in motion coordination and other applications including wireless networks.
We introduce the stochastic Network-Iterated Prisoner's Dilemma (NIPD) model, a network of players playing the Prisoner's Dilemma with their neighbours, each with a memory-one strategy which they constantly and locally update to improve their success. This process is non-deterministic, and mirrors societal interactions in many relevant aspects. We use it to assess the flexibility, noise tolerance and real-world adaptability of some well-known strategies. Furthermore, in the model a new strategy naturally emerges which proves way more successful than those. We also derive some theoretical param…
Dynamics of evolutionary games strongly depend on underlying networks. We study the coevolutionary prisoner's dilemma in which players change their local networks as well as strategies (i.e., cooperate or defect). This topic has been increasingly explored by many researchers. On the basis of active linking dynamics [J. M. Pacheco et al., J. Theor. Biol. 243, 437 (2006), J. M. Pacheco et al., Phys. Rev. Lett. 97, 258103 (2006)], we show that cooperation is enhanced fairly robustly. In particular, cooperation evolves when the payoff of the player is normalized by the number of neighbors; this is…
This manuscript explores the research topics and collaborative behaviour of authors in the field of the Prisoner's Dilemma using topic modeling and a graph theoretic analysis of the co-authorship network. The analysis identified five research topics in the Prisoner's Dilemma which have been relevant over the course of time. These are human subject research, biological studies, strategies, evolutionary dynamics on networks and modeling problems as a Prisoner's Dilemma game. Moreover, the results demonstrated the Prisoner's Dilemma is a field of continued interest, and that it is a collaborative…
Game theory is fundamental to understanding cooperation between agents. Mainly, the Prisoner's Dilemma is a well-known model that has been extensively studied in complex networks. However, although the emergence of cooperation has been investigated before, the influence of memory in its evolution is not well understood. This paper presents a detailed study of cooperation dynamics in which agents have memory. We simulate the evolutionary Prisoner's dilemma game on random, scale-free and networks presenting degree-degree correlation. Through extensive simulations, we show that assortativity can …
We study an evolutionary spatial prisoner's dilemma game where the fitness of the players is determined by both the payoffs from the current interaction and their history. We consider the situation where the selection timescale is slower than the interaction timescale. This is done by implementing probabilistic reproduction on an individual level. We observe that both too fast and too slow reproduction rates hamper the emergence of cooperation. In other words, there exists an intermediate selection timescale that maximizes cooperation. Another factor we find to promote cooperation is a diversi…
Previous studies suggest that punishment is a useful way to promote cooperation in the well-mixed public goods game, whereas it still lacks specific evidence that punishment maintains cooperation in spatial prisoner's dilemma game as well. To address this issue, we introduce a mechanism of recorded punishment, involved with memory and punishment, into spatial prisoner's dilemma game. We find that increasing punishment rate or memory length promotes the evolution of cooperation monotonously. Interestingly, compared with traditional version, recorded punishment will facilitate cooperation better…
We show that the success of cooperation in an evolutionary prisoner's dilemma on a complex network can be predicted by a simple, quantitative network analysis using mean field parameters. The criterion is shown to be accurate on a wide variety of networks with degree distributions ranging from regular to Poisson to scale-free. The network analysis uses a parameter that allows for comparisons of networks with both similar, and distinct, topologies. Furthermore, we establish the criterion here as a natural network analogue of Hamilton's classical rule for kin selection, despite arising in an ent…
We investigate an evolutionary prisoner's dilemma game among self-driven agents, where collective motion of biological flocks is imitated through averaging directions of neighbors. Depending on the temptation to defect and the velocity at which agents move, we find that cooperation can not only be maintained in such a system but there exists an optimal size of interaction neighborhood, which can induce the maximum cooperation level. When compared with the case that all agents do not move, cooperation can even be enhanced by the mobility of individuals, provided that the velocity and the size o…
We study co-evolutionary Prisoner's Dilemma games where each player can imitate both the strategy and imitation rule from a randomly chosen neighbor with a probability dependent on the payoff difference when the player's income is collected from games with the neighbors. The players, located on the sites of a two-dimensional lattice, follow unconditional cooperation or defection and use individual strategy adoption rule described by a parameter. If the system is started from a random initial state then the present co-evolutionary rule drives the system towards a state where only one evolutiona…
Prisoner's Dilemma is a game theory model used to describe altruistic behavior seen in various populations. This theoretical game is important in understanding why a seemingly selfish strategy does persist and spread throughout a population that is mixing homogeneously at random. For a population with structure determined by social interactions, Prisoner's Dilemma brings to light certain requirements for the altruistic strategy to become established. Monte Carlo simulations of Prisoner's Dilemma are carried out using both simulated social networks and a dataset of a real social network. In bot…
We investigate the win-lose relations between strategies of iterated prisoner's dilemma games by using a directed network concept to display the replicator dynamics results. In the giant strongly-connected component of the win/lose network, we find win-lose circulations similar to rock-paper-scissors and analyze the fixed point and its stability. Applying the network motif concept, we introduce dynamic motifs, which describe the population dynamics relations among the three strategies. Through exact enumeration, we find 22 dynamic motifs and display their phase portraits. Visualization using d…
The Iterated Prisoner's Dilemma (IPD) is a well studied framework for understanding direct reciprocity and cooperation in pairwise encounters. However, measuring the morality of various IPD strategies is still largely lacking. Here, we partially address this issue by proposing a suit of plausible morality metrics to quantify four aspects of justice. We focus our closed-form calculation on the class of reactive strategies because of their mathematical tractability and expressive power. We define reactive means as a tool for studying how actors in the IPD and Iterated Snowdrift Game (ISG) behave…
The conventional wisdom is that scale-free networks are prone to cooperation spreading. In this paper we investigate the cooperative behaviors on the structured scale-free network. On the contrary of the conventional wisdom that scale-free networks are prone to cooperation spreading, the evolution of cooperation is inhibited on the structured scale-free network while performing the prisoner's dilemma (PD) game. Firstly, we demonstrate that neither the scale-free property nor the high clustering coefficient is responsible for the inhibition of cooperation spreading on the structured scale-free …
Evolutionary Dynamics of Reputation-Based Voluntary Prisoner's Dilemma Games
Kyushu University（九州大学）；Teesside University（蒂斯赛德大学）；CSSC Systems Engineering Research Institute（中国船舶系统工程研究院）；CSSC Intelligent Innovation Research Institute（中国船舶智能创新研究院）；Yunnan University of Finance and Economics（云南财经大学）；University of Maribor（马里博尔大学）；Community Healthcare Center Dr. Adolf Drolc Maribor（马里博尔阿道夫·德罗尔克博士社区健康中心）；Kyung Hee University（庆熙大学）；Korea University（高丽大学）；Northwestern Polytechnical University（西北工业大学）
Understanding the role of network structure in the evolution of cooperation is a key research goal at the intersection between physics and biology. Recent studies have particularly focused on multiplex networks given that multiple social domains are interrelated and cannot be represented by single-layer networks. However, the role of network multiplexity is not fully understood when combined with another important network characteristic: network dynamics. In the present study, we investigated evolutionary prisoner's dilemma games played on dynamic two-layer multiplex networks in which the payo…
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.
Game theory formalizes certain interactions between physical particles or between living beings in biology, sociology, and economics, and quantifies the outcomes by payoffs. The prisoner's dilemma (PD) describes situations in which it is profitable if everybody cooperates rather than defects (free-rides or cheats), but as cooperation is risky and defection is tempting, the expected outcome is defection. Nevertheless, some biological and social mechanisms can support cooperation by effectively transforming the payoffs. Here, we study the related phase transitions, which can be of first order (d…
The paradox of cooperation among selfish individuals still puzzles scientific communities. Although a large amount of evidence has demonstrated that cooperator clusters in spatial games are effective to protect cooperators against the invasion of defectors, we continue to lack the condition for the formation of a giant cooperator cluster that assures the prevalence of cooperation in a system. Here, we study the dynamical organization of cooperator clusters in spatial prisoner's dilemma game to offer the condition for the dominance of cooperation, finding that a phase transition characterized b…
Promoting cooperation is an intellectual challenge in the social sciences, for which the iterated Prisoners' Dilemma (IPD) is a fundamental framework. The traditional view that there exists no simple ultimatum strategy whereby one player can unilaterally control the share of the surplus has been challenged by a new class of "zero-determinant" (ZD) strategies raised by Press and Dyson. In particular, the extortionate strategies can subdue the opponent and obtain higher scores. However, no empirical evidence has yet been found to support this theoretical finding. In a long-run laboratory experim…
The interplay of social and strategic motivations in human interactions is a largely unexplored question in collective social phenomena. Whether individuals' decisions are taken in a pure strategic basis or due to social pressure without a rational background crucially influences the model outcome. Here we study a networked Prisoner's Dilemma in which decisions are made either based on the replication of the most successful neighbor's strategy (unconditional imitation) or by pure social imitation following an update rule inspired by the voter model. The main effects of the voter dynamics are a…
We present results of the prisoner's dilemma game on complex networks that have population change. We introduce a death process with minimum requirements and show that it induces a highly cooperative society. We also study the effects of immigration on the society. We show that the acceptable number of immigrants of the society is determined by the population of the society, the ratio of cooperator among immigrants, and the immigration interval. In addition, if immigrants have a preferential attachment link, the acceptable number of immigrants increases.
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…
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…
We study the prisoner's dilemma model with a noisy imitation evolutionary dynamics on directed out-homogeneous and uncorrelated directed random networks. An heterogeneous pair mean-field approximation is presented showing good agreement with Monte Carlo simulations in the limit of weak selection (high noise) where we obtain analytical predictions for the critical temptations. We discuss the phase diagram as a function of temptation, intensity of noise and coordination number of the networks and we consider both the model with and without self-interaction. We compare our results with available …
We use replicator dynamics to study an iterated prisoners' dilemma game with memory. In this study, we investigate the characteristics of all 32 possible strategies with a single-step memory by observing the results when each strategy encounters another one. Based on these results, we define similarity measures between the 32 strategies and perform a network analysis of the relationship between the strategies by constructing a strategies network. Interestingly, we find that a win-lose circulation, like rock-paper-scissors, exists between strategies and that the circulation results from one unu…
On the existence of fair zero-determinant strategies in the periodic prisoner's dilemma game
Graduate School of Sciences and Technology for Innovation, Yamaguchi University（山口大学创新科学技术研究生院）；Department of Physics, Tokyo Metropolitan University（东京都立大学物理系）
We introduce an analytical model to study the evolution towards equilibrium in spatial games, with `memory-aware' agents, i.e., agents that accumulate their payoff over time. In particular, we focus our attention on the spatial Prisoner's Dilemma, as it constitutes an emblematic example of a game whose Nash equilibrium is defection. Previous investigations showed that, under opportune conditions, it is possible to reach, in the evolutionary Prisoner's Dilemma, an equilibrium of cooperation. Notably, it seems that mechanisms like motion may lead a population to become cooperative. In the propos…
We analyze an extended model of the Iterated Prisoner's Dilemma where agents decide to play based on the data from their limited memory or recommendations. The cooperators can decide whether to play with the matched opponent or not. The agents' decisions are directly linked to their level of optimism since they decide to play if they believe the opponent has a high probability of cooperating. Optimism is precisely tuned by parameters optimism threshold and tolerance. Our experiment showed that being optimistic is better for cooperators as it leads to more accurate exploration in the multi-agen…
The Prisoner's Dilemma game has a long history stretching across the social, biological, and physical sciences. In 2012, Press and Dyson developed a method for analyzing the mapping of the 8-dimensional strategy profile onto the 2-dimensional payoff space in an infinitely iterated Prisoner's Dilemma game, based on Markov chain analysis and memory-one strategies. We generalize this approach and introduce the concept of strategy parameter to show that linear relations among player payoffs are a ubiquitous feature of the infinitely iterated Prisoner's Dilemma game. Our extended analysis is applie…
We study the possible advantages of adopting of quantum strategies in multi-player evolutionary games. We base our study on the three-player Prisoner's Dilemma (PD) game. In order to model the simultaneous interaction between three agents we use hypergraphs and hypergraph networks. In particular, we study two types of networks: a random network and a SF-like network. The obtained results show that in the case of a three player game on a hypergraph network, quantum strategies are not necessarily stochastically stable strategies. In some cases, the defection strategy can be as good as a quantum …
The prisoner's dilemma (PD) game is a simple model for understanding cooperative patterns in complex systems consisting of selfish individuals. Here, we study a PD game problem in scale-free networks containing hierarchically organized modules and controllable shortcuts connecting separated hubs. We find that cooperator clusters exhibit a percolation transition in the parameter space (p,b), where p is the occupation probability of shortcuts and b is the temptation payoff in the PD game. The cluster size distribution follows a power law at the transition point. Such a critical behavior, resulti…
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.…
The Axelrod library is an open source Python package that allows for reproducible game theoretic research into the Iterated Prisoner's Dilemma. This area of research began in the 1980s but suffers from a lack of documentation and test code. The goal of the library is to provide such a resource, with facilities for the design of new strategies and interactions between them, as well as conducting tournaments and ecological simulations for populations of strategies. With a growing collection of 139 strategies, the library is a also a platform for an original tournament that, in itself, is of inte…
In this paper the results of a simulation of a prisoner's dilemma robin-round tournament are presented. In the tournament each participating strategy plays an iterated prisoner's dilemma against each other strategy (round-robin) and as a variant also against itself. The participants of a tournament are all strategies that are deterministic and have the same size of memory with regard to their own and their opponent's past actions: up to three most recent actions of their opponent and up to two most recent actions of their own. A focus is set on the investigation of the influence of the number …
Emergence of cooperation in self-centered individuals has been a major puzzle in the study of evolutionary ethics. Reciprocal altruism is one of explanations put forward and prisoner's dilemma has been a paradigm in this context. Emergence of cooperation was demonstrated for prisoner's dilemma on a lattice with synchronous update [Nature, 359, 826 (1992)]. However, the cooperation disappeared for asynchronous update and the general validity of the conclusions was questioned [PNAS, 90, 7716 (1993)]. Neither synchronous nor asynchronous updates are realistic for natural systems. In this paper, w…
Cooperation lies at the foundations of human societies, yet why people cooperate remains a conundrum. The issue, known as network reciprocity, of whether population structure can foster cooperative behavior in social dilemmas has been addressed by many, but theoretical studies have yielded contradictory results so far---as the problem is very sensitive to how players adapt their strategy. However, recent experiments with the prisoner's dilemma game played on different networks have shown that humans do not consider neighbors' payoffs when making their decisions, and that the network structure …
The world in which we are living is a huge network of networks and should be described by interdependent networks. The interdependence between networks significantly affects the evolutionary dynamics of cooperation on them. Meanwhile, due to the diversity and complexity of social and biological systems, players on different networks may not interact with each other by the same way, which should be described by multiple models in evolutionary game theory, such as the Prisoner's Dilemma and Snowdrift Game. We therefore study the evolutionary dynamics of cooperation on two interdependent networks…
