I present the results from a spatial model of the prisoner's dilemma, played on a toroidal lattice. Each individual has a default strategy of either cooperating ($C$) or defecting ($D$). Two strategies were tested, including ``tit-for-tat'' (TFT), in which individuals play their opponent's last play, or simply playing their default play. Each individual also has a probability of telling the truth ($0 \leq P_{truth} \leq 1$) about their last play. This parameter, which can evolve over time, allows individuals to be, for instance, a defector but present as a cooperator regarding their last play. This leads to interesting dynamics where mixed populations of defectors and cooperators with $P_{truth} \geq 0.75$ move toward populations of truth-telling cooperators. Likewise, mixed populations with $P_{truth} < 0.7$ become populations of lying defectors. Both such populations are stable because they each have higher average scores than populations with intermediate values of $P_{truth}$. Applications of this model are discussed with regards to both humans and animals.
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.
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 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…
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 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…
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…
The prisoner's dilemma game is the most known contribution of game theory into social sciences. Here we describe new implications of this game for transactional and transformative leadership. While the autocratic (Stackelberg's) leadership is inefficient for this game, we discuss a Pareto-optimal scenario, where the leader L commits to react probabilistically to pure strategies of the follower F, which is free to make the first move. Offering F to resolve the dilemma, L is able to get a larger average pay-off. The exploitation can be stabilized via repeated interaction of L and F, and turns to…
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）
A Round-Robin Tournament of the Iterated Prisoner's Dilemma with Complete Memory-Size-Three Strategies
在本文中，一个囚犯的困境罗宾轮比赛的模拟结果。在锦标赛中，每个参与策略都扮演一个迭代的囚徒困境，与其他策略（循环赛）和作为一个变体也与自己对抗。锦标赛的参与者都是确定性的策略，并且对自己和对手过去的动作具有相同大小的记忆：对手最多三个最近的动作和自己最多两个最近的动作。一个重点是设置在调查的影响，迭代次数，支付矩阵的细节，和内存大小的影响。主要结果是，对于这里进行的锦标赛，不同的策略出现作为赢家不同的收益矩阵，即使不同的收益矩阵是相似的判断，如果他们满足关系T + S = P + R或2R > T + S。由于这一结果，它建议，每当迭代囚徒困境是用来模拟一个真正的系统，不明确固定的支付矩阵，应该检查，如果结论仍然有效，当使用不同的支付矩阵。
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 …
Differences between men and women have intrigued generations of social scientists, who have found that the two sexes behave differently in settings requiring competition, risk taking, altruism, honesty, as well as many others. Yet, little is known about whether there are gender differences in cooperative behavior. Previous evidence is mixed and inconclusive. Here I shed light on this topic by analyzing the totality of studies that my research group has conducted since 2013. This is a dataset of 10,951 observations coming from 7,322 men and women living in the US, recruited through Amazon Mecha…
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…
Evolution Reinforces Cooperation with the Emergence of Self-Recognition Mechanisms: an empirical study of the Moran process for the iterated Prisoner's dilemma
Punishment and partner switching are two well-studied mechanisms that support the evolution of cooperation. Observation of human behaviour suggests that the extent to which punishment is adopted depends on the usage of alternative mechanisms, including partner switching. In this study, we investigate the combined effect of punishment and partner switching in evolutionary prisoner's dilemma games conducted on a network. In the model, agents are located on the network and participate in the prisoner's dilemma games with punishment. In addition, they can opportunistically switch interaction partn…
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…
