Strategy changing penalty promotes cooperation in spatial prisoner’s dilemma game

Many classical studies suggest that punishment is a useful way to promote cooperation in the well-mixed public goods game, whereas relative evidence in the research of spatial prisoner’s dilemma game is absent. To address this issue, we introduce a mechanism of strategy changing penalty, combining memory and penalty during the update process, into spatial prisoner’s dilemma game. We find that increasing penalty rate or memory length is able to promote the evolution of cooperation monotonously. Compared with traditional version, recorded penalty could facilitate cooperation better. Moreover, through examining the process of evolution, we provide an interpretation for this promotion phenomenon, namely, the effect of promotion can be warranted by an evolution resonance of standard deviation of fitness coefficient. Finally, we validate our results by studying the impact of uncertainty within strategy adoptions on the evolution of cooperation. We hope that our work may shed light on the understanding of the cooperative behavior in the society.     

Cooperation in spatial prisoner’s dilemma game with delayed decisions

Phenomena that time delays of information lead to delayed decisions are extensive in reality. The effect of delayed decisions on the evolution of cooperation in the spatial prisoner’s dilemma game is explored in this work. Players with memory are located on a two dimensional square lattice, and they can keep the payoff information of his neighbors and his own in every historic generation in memory. Every player uses the payoff information in some generation from his memory and the strategy information in current generation to determine which strategy to choose in next generation. The time interval between two generations is set by the parameter m. For the payoff information is used to determine the role model for the focal player when changing strategies, the focal player’s decision to learn from which neighbor is delayed by m generations. Simulations show that cooperation can be enhanced with the increase of m. In addition, just like the original evolutionary game model (m = 0), pretty dynamic fractal patterns featuring symmetry can be obtained when m > 0 if we simulate the invasion of a single defector in world of cooperators on square lattice.     

Effects of limited interactions between individuals on cooperation in spatial evolutionary prisoner’s dilemma game

We study the spatial evolutionary prisoner’s dilemma game with limited interactions by introducing two kinds of individuals, say type-A and type-B with a fraction of p and (1 − p), respectively, distributed randomly on a square lattice. Each kind of individuals can adopt two pure strategies: either to cooperate or to defect. During the evolution, the individuals can only interact with others belonging to the same kind, but they can learn from either kinds of individuals in the nearest neighborhood. Using Monte Carlo simulations, the average frequency of cooperators ρ_C is calculated as a function of p in the equilibrium state. It is shown that, compared with the case of p = 0 (only one kind of individuals existing in the system), cooperation can be evidently promoted. In particular, the cooperator density can reach a maximum level at some moderate values of p in a wide range of payoff parameters. The results imply that certain limited interactions between individuals plays an important and nontrivial role in the evolution of cooperation.     

Cooperation among mobile individuals with payoff expectations in the spatial prisoner’s dilemma game

We propose a model to address the problem how the evolution of cooperation in a social system depends on the spatial motion and the payoff expectation. In the model, if the actual payoff of an individual is smaller than its payoff expectation, the individual will either move to a new site or simply reverse its current strategy. It turns out that migration of dissatisfied individuals with relatively low expectation level leads to the aggregation of cooperators and promotion of cooperation. Moreover, under appropriate parameters migration leads to some interesting spatiotemporal patterns which seems not to have been reported in previously studied spatial games. Furthermore, it also found that a population with constant expectation can better favor cooperative behavior than a population with adaptive aspiration.     

Impact of neighborhood separation on the spatial reciprocity in the prisoner’s dilemma game

The evolutionary game theory is a very powerful tool to understand the collective cooperation behavior in many real-world systems. In the spatial game model, the payoff is often first obtained within a specific neighborhood (i.e., interaction neighborhood) and then the focal player imitates or learns the behavior of a randomly selected one inside another neighborhood which is named after the learning neighborhood. However, most studies often assume that the interaction neighborhood is identical with the learning neighborhood. Beyond this assumption, we present a spatial prisoner’s dilemma game model to discuss the impact of separation between interaction neighborhood and learning neighborhood on the cooperative behaviors among players on the square lattice. Extensive numerical simulations demonstrate that separating the interaction neighborhood from the learning neighborhood can dramatically affect the density of cooperators (ρ_C) in the population at the stationary state. In particular, compared to the standard case, we find that the medium-sized learning (interaction) neighborhood allows the cooperators to thrive and substantially favors the evolution of cooperation and ρ_C can be greatly elevated when the interaction (learning) neighborhood is fixed, that is, too little or much information is not beneficial for players to make the contributions for the collective cooperation. Current results are conducive to further analyzing and understanding the emergence of cooperation in many natural, economic and social systems.     

Cooperation in an evolutionary prisoner’s dilemma game with probabilistic strategies

In this work, we investigate an evolutionary prisoner’s dilemma game in structured populations with probabilistic strategies instead of the pure strategies of cooperation and defection. We explore the model in details by considering different strategy update rules and different population structures. We find that the distribution of probabilistic strategies patterns is dependent on both the interaction structures and the updating rules. We also find that, when an individual updates her strategy by increasing or decreasing her probabilistic strategy a certain amount towards that of her opponent, there exists an optimal increment of the probabilistic strategy at which the cooperator frequency reaches its maximum.     

The impact of the termination rule on cooperation in a prisoner’s dilemma experiment

Cooperation in prisoner’s dilemma games can usually be sustained only if the game has an infinite horizon. We analyze to what extent the theoretically crucial distinction of finite versus infinite-horizon games is reflected in the outcomes of a prisoner’s dilemma experiment. We compare three different experimental termination rules in four treatments: a known finite end, an unknown end, and two variants with a random termination rule (with a high and with a low continuation probability, where cooperation can occur in a subgame-perfect equilibrium only with the high probability). We find that the termination rules do not significantly affect average cooperation rates. Specifically, employing a random termination rule does not cause significantly more cooperation compared to a known finite horizon, and the continuation probability does not significantly affect average cooperation rates either. However, the termination rules may influence cooperation over time and end-game behavior. Further, the (expected) length of the game significantly increases cooperation rates. The results suggest that subjects may need at least some learning opportunities (like repetitions of the supergame) before significant backward induction arguments in finitely repeated game have force.     

Explaining cooperation in the finitely repeated simultaneous and sequential prisoner’s dilemma game under incomplete and complete information

Explaining cooperation in social dilemmas is a central issue in behavioral science, and the prisoner’s dilemma (PD) is the most frequently employed model. Theories assuming rationality and selfishness predict no cooperation in PDs of finite duration, but cooperation is frequently observed. We therefore build a model of how individuals in a finitely repeated PD with incomplete information about their partner’s preference for mutual cooperation decide about cooperation. We study cooperation in simultaneous and sequential PDs. Our model explains three behavioral regularities found in the literature: (i) the frequent cooperation in one-shot and finitely repeated N-shot games, (ii) cooperation rates declining over the course of the game, and (iii) cooperation being more frequent in the sequential PD than in the simultaneous PD.     

Evolutionary prisoner’s dilemma games with local interaction and best-response dynamics

This paper studies the long run behavior in evolutionary prisoner’s dilemma games. All players are assumed to sit around a circle and to interact only with their neighbors. It is known that full-defection is the unique long run equilibrium as the probability of players’ experimentation (or mutation) tends to zero in the best response dynamics. Here, it is shown that full-cooperation could emerge in the long run if one also cares for his neighbors in the bestresponse dynamics.     

Uncertainty in the traveler’s dilemma

We quantify the sensitivity of the traveler’s dilemma (Basu, Am Econ Rev 84:391–395, 1994) to perturbations from common knowledge. The perturbations entail a small uncertainty about the set of admissible actions. We show that the sensitivity scale is exponential in the range of admissible actions in the traveler’s dilemma. Such rapid growth is consistent with the intuition that a wider range makes the outcome of the traveler’s dilemma less intuitive.     

How to make cooperation the optimizing strategy in a two‐person game

This paper demonstrates the existence of a partial tit‐for‐tat (matching) strategy which, when used by one player in an iterated “Prisoner's Dilemma” game, will induce a response of pure cooperation in the other player if that player behaves optimally. The minimum matching frequency of such a strategy is shown to be monotonically related to the Rapoport‐Chammah “Cooperation Index.”     

Does cooperation in auditing matter? A comparison of a non-cooperative and a cooperative game model of auditing

A game model of auditing including internal control investigation and substantive testing is analysed as a non-cooperative game. It is shown that in order for the presumed socially desirable outcome of high and honest effort by all to be obtained, it is necessary to adjust the cost structure to ameliorate the costs of not-qualifying erroneous accounts if the auditor can prove he or she has worked hard. Comparison with a cooperative game analysis of the model shows that there is a region of parameters where both cooperative and non-cooperative versions of the game lead to this socially desirable outcome. The significance of this result is that whilst society expects an ‘independent’ auditor not to cooperate with the auditee, the practical realities of auditing require a considerable degree of cooperation. This leads to an ‘expectations gap’ between what society expects and what actually happens, except in those cost regions where both the cooperative and non-cooperative versions of the audit game lead to the same solution.     

Mathematical menopause, or, a young man’s game?

Summary  The responses were very varied. But these five statements would be generally accepted:

Modelling ‘contracts’ between a terrorist group and a government in a sequential game

In this paper, we apply a sequential game to study the possibility of ‘contracts’ (or at least mutually beneficial arrangements) between a government and a terrorist group. We find equilibrium solutions for complete and incomplete information models, where the government defends and/or provides positive rent, and the terrorist group attacks. We also study the sensitivities of equilibria as a function of both players’ target valuations and preferences for rent. The contract option, if successful, may achieve (partial) attack deterrence, and significantly increase the payoffs not only for the government, but also for some types of terrorist groups. Our work thus provides some novel insights in combating terrorism.     

An attrition game on a network ruled by Lanchester’s square law

We consider two-person zero-sum attrition games in which an attacker and a defender are in combat with each other on a network. The attacker marches from a starting node to a destination node, hoping that the initial members survive the march. The defender deploys his forces on arcs in order to intercept the attacker. If the attacker encounters the defender on an arc, the attacker incurs casualties according to Lanchester’s square law. We consider two models: a one-shot game in which the two players have no information about their opponents, and a two-stage game in which both players have some information about their opponents. For both games, the payoff is defined as the number of survivors for the attacker. The attacker’s strategy is to choose a path, and the defender’s is to deploy the defending forces on arcs. We propose a numerical algorithm, in which nonlinear programming is embedded, to derive the equilibrium of the game.     

When does inferring reputation probability countervail temptation in cooperative behaviors for the prisoners’ dilemma game?

In evolutionary games, the temptation mechanism reduces cooperation percentage while the reputation mechanism promotes it. Inferring reputation theory proposes that agent's imitating neighbors with the highest reputation takes place with a probability. Although reputation promotes cooperation, when and how it enhances cooperation is still a question. This paper investigates the condition where the inferring reputation probability promotes cooperation. Hence, the effects of reputation and temptation on cooperation are explored under the spatial prisoners’ dilemma game, utilizing the methods of simulation and statistical analysis. Results show that temptation reduces cooperation unconditionally while reputation promotes it conditionally, i.e. reputation countervails temptation conditionally. When the inferring reputation probability is less than 0.5, reputation promotes cooperation substantially and thus countervails temptation. However, when the inferring reputation probability is larger than 0.5, its contribution to cooperation is relatively weak and cannot prevent temptation from undermining cooperation. Reputation even decreases cooperation together with temptation when the probability is higher than 0.8. It should be noticed that inferring reputation does not always succeed to countervail temptation and there is a specific interval for it to promote cooperation.     

Newton’s method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation

We consider the generalized Nash equilibrium problem (GNEP), where not only the players’ cost functions but also their strategy spaces depend on the rivals’ decision variables. Existence results for GNEPs are typically shown by using a fixed point argument for a certain set-valued function. Here we use a regularization of this set-valued function in order to obtain a single-valued function that is easier to deal with from a numerical point of view. We show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria. This fixed point formulation is then used to develop a nonsmooth Newton method for computing a normalized equilibrium. The method uses a so-called computable generalized Jacobian that is much easier to compute than Clarke generalized Jacobian or B-subdifferential. We establish local superlinear/quadratic convergence of the method under the constant rank constraint qualification, which is weaker than the frequently used linear independence constraint qualification, and a suitable second-order condition. Some numerical results are presented to illustrate the performance of the method.     

Gosset’s figure in a Clifford algebra

This note describes a way to realize a “projective” version of Gosset’s 240-vertex semiregular polytope 421 using the Clifford algebra Cl(8) generated by an 8-dimensional vector space equipped with a non-degenerate quadratic form. The 120 vertices of this projective Gosset figure are also seen to coincide with a particular basis for the Lie algebra