Games — Open Access Journal

The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal-form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure, nor in mixed strategies is given. Full article

I study the path properties of adaptive heuristics that mimic the natural dynamics of play in a game and converge to the set of correlated equilibria. Despite their apparent differences, I show that these heuristics have an abstract representation as a sequence of probability distributions that satisfy a number of common properties. These properties arise due to the topological structure of the set of correlated equilibria. The characterizations that I obtain have useful applications in the study of the convergence of the heuristics. Full article

We study a model where agents face a continuum of two-player games and categorize them into a finite number of situations to make sense of their complex environment. Agents need not share the same categorization. Each agent can cooperate or defect, conditional on the perceived category. The games are fully ordered by the strength of the temptation to defect and break joint cooperation. In equilibrium agents share the same categorization, but achieve less cooperation than if they could perfectly discriminate games. All the equilibria are evolutionarily stable, but stochastic stability selects against cooperation. We model agents’ learning when they imitate successful players over similar games, but lack any information about the opponents’ categorizations. We show that imitation conditional on reaching an intermediate aspiration level leads to a shared categorization that achieves higher cooperation than under perfect discrimination. Full article

Mutation-generated variation in behavior is thought to promote the evolution of cooperation. Here, we study this by distinguishing two effects of mutation in evolutionary games of the finitely repeated Prisoner’s Dilemma in infinite asexual populations. First, we show how cooperation can evolve through the direct effect of mutation, i.e., the fitness impact that individuals experience from interactions with mutants before selection acts upon these mutants. Whereas this direct effect suffices to explain earlier findings, we question its generality because mutational variation usually generates the highest direct fitness impact on unconditional defectors (AllD). We identify special conditions (e.g., intermediate mutation rates) for which cooperation can be favored by an indirect effect of mutation, i.e., the fitness impact that individuals experience from interactions with descendants of mutants. Simulations confirm that AllD-dominated populations can be invaded by cooperative strategies despite the positive direct effect of mutation on AllD. Thus, here the indirect effect of mutation drives the evolution of cooperation. The higher level of cooperation, however, is not achieved by individuals triggering reciprocity (‘genuine cooperation’), but by individuals exploiting the willingness of others to cooperate (‘exploitative cooperation’). Our distinction between direct and indirect effects of mutation provides a new perspective on how mutation-generated variation alters frequency-dependent selection. Full article

I investigate how different dispersal patterns affect the evolution of cooperation in a spatially-structured population. I consider a finite fixed-size population of cooperators and free-riders residing on a one-dimensional lattice with periodic boundaries. Individuals interact via a multiplayer game, which is a version of a public goods game, and the population evolves via a Moran process. Individuals try to improve their interactions by evaluating the current state of the environment and moving to locations with better payoffs. I ran stochastic simulations of the evolution of this Markov process and found that if individuals disperse deterministically to locations with the best payoffs, then cooperation can still be maintained even in the worst-case scenarios, albeit at reduced levels compared to the better-case scenarios. This contrasts with an earlier investigation of probabilistic dispersal patterns, which resulted in the breakdown of cooperation in sparse populations with small interaction neighborhoods, a high mobility rate, and a large dispersal range. Full article

We study the evolution of cooperation in group interactions where players are randomly drawn from well-mixed populations of finite size to participate in a public goods game. However, due to the possibility of unforeseen circumstances, each player has a fixed probability of being unable to participate in the game, unlike previous models which assume voluntary participation. We first study how prescribed stochastic opting-out affects cooperation in finite populations, and then generalize for the limiting case of large populations. Because we use a pairwise comparison updating rule, our results apply to both genetic and behavioral evolution mechanisms. Moreover, in the model, cooperation is favored by natural selection over both neutral drift and defection if the return on investment exceeds a threshold value depending on the population size, the game size, and a player’s probability of opting-out. Our analysis further shows that, due to the stochastic nature of the opting-out in finite populations, the threshold of return on investment needed for natural selection to favor cooperation is actually greater than the one corresponding to compulsory games with the equal expected game size. We also use adaptive dynamics to study the co-evolution of cooperation and opting-out behavior. Indeed, given rare mutations minutely different from the resident population, an analysis based on adaptive dynamics suggests that over time the population will tend towards complete defection and non-participation, and subsequently cooperators abstaining from the public goods game will stand a chance to emerge by neutral drift, thereby paving the way for the rise of participating cooperators. Nevertheless, increasing the probability of non-participation decreases the rate at which the population tends towards defection when participating. Our work sheds light on understanding how stochastic opting-out emerges in the first place and on its role in the evolution of cooperation. Full article

Different strains of influenza viruses spread in human populations during every epidemic season. As the size of an infected population increases, the virus can mutate itself and grow in strength. The traditional epidemic SIR model does not capture virus mutations and, hence, the model is not sufficient to study epidemics where the virus mutates at the same time as it spreads. In this work, we establish a novel framework to study the epidemic process with mutations of influenza viruses, which couples the SIR model with replicator dynamics used for describing virus mutations. We formulated an optimal control problem to study the optimal strategies for medical treatment and quarantine decisions. We obtained structural results for the optimal strategies and used numerical examples to corroborate our results. Full article

We formally explore the idea that punishment of norm-breakers may be a vehicle for the older generation to teach youngsters about social norms. We show that this signaling role provides sufficient incentives to sustain costly punishing behavior. People punish norm-breakers to pass information about past history to the younger generation. This creates a link between past, present, and future punishment. Information about the past is important for youngsters, because the past shapes the future. Reward-based mechanisms may also work and are welfare superior to punishment-based ones. However, reward-based mechanisms are fragile, since punishment is a more compelling signaling device (in a sense that we make precise). Full article

We conducted a dynamic common pool resource experiment and found large differences among groups in the total benefits (surplus) obtained from the resource. To shed light on the factors underlying the differences, we characterized individual appropriation decisions as irresponsible, sustainable, or constructive, and defined a measure of the intensity of such actions. We then examined the relationships between group-level success and the frequency and intensity of the individual actions, finding that the average intensity of irresponsible actions was the best predictor of group success. We interpreted this as suggestive evidence that policies aimed at preserving and maximizing the benefits of renewable resources should above all else aim to reduce the intensity of irresponsible actions. Full article

Evolution of cooperation by reciprocity has been studied using two-player and n-player repeated prisoner’s dilemma games. An interesting feature specific to the n-player case is that players can vary in generosity, or how many defections they tolerate in a given round of a repeated game. Reciprocators are quicker to detect defectors to withdraw further cooperation when less generous, and better at maintaining a long-term cooperation in the presence of rare defectors when more generous. A previous analysis on a stochastic evolutionary model of the n-player repeated prisoner’s dilemma has shown that the fixation probability of a single reciprocator in a population of defectors can be maximized for a moderate level of generosity. However, the analysis is limited in that it considers only tit-for-tat-type reciprocators within the conventional linear payoff assumption. Here we extend the previous study by removing these limitations and show that, if the games are repeated sufficiently many times, considering non-tit-for-tat type strategies does not alter the previous results, while the introduction of non-linear payoffs sometimes does. In particular, under certain conditions, the fixation probability is maximized for a “paradoxical” strategy, which cooperates in the presence of fewer cooperating opponents than in other situations in which it defects. Full article

The current study aims to investigate how the presence of social norms defines belief formation on future changes in social identity (i.e., diachronic identity), and how those beliefs affect individual decisions under uncertainty. The paper proposes a theoretical model in which individuals have preferences over their own attributes and over specific information structures. The individual preferences are motivated by the presence of social norms. The norms, while establishing the socially acceptable attributes of an individual identity, also drive individuals’ preferences for information acquisition or avoidance. The model incorporates social norms as empirical expectations and provides a prior dependent theory that allows for prior-dependent information attitudes. Firstly, the model implies that decisions are mitigated by socially grounded behavioral and cognitive biases; and secondly, that it can create an incentive to avoid information, even when the latter is useful, free, and independent of strategic considerations. These biases bring out individual trade-offs between the accuracy of decision making and self-image motivated by social conformity. The two behavioral motivations are represented through a game of an intra-personal model of choice under uncertainty in which self-deception and memory manipulation mechanisms are used to overcome the individuals’ internal trade-off. Full article

We studied gender diversity and performance in endogenously formed teams in a repeated teamwork setting. In our experiment, the participants (N = 168, 84 women and 84 men) chose whether to perform a cooperative task only with members of the own gender or in a mixed-gender team. We found that independent of the choice of team, in the initial period, men contributed significantly more to the team projects than women. Men preferred the successful men-only teams in the subsequent periods, resulting in significantly higher profits for men compared to women. This endogenously emerging “gender pay gap” only closed over time. Full article

This paper analyzes the equilibrium strength of prescriptive norms to contribute to public goods. We consider three methods of establishing what an acceptable contribution to the public good is. Under the first method, the contribution of the bottom contributor is the reference point by which the comparison is being made; under the second, the median contribution is the reference point; and under the third the top contribution is the reference. The first method results in a unique equilibrium and the reference contribution is endogenously low. Each of the latter two methods allows for multiple equilibria differing in contributions made and thus in the strength of the norm to contribute. Comparing the methods we show that the median reference allows for the highest equilibrium contributions and welfare of all methods hence is the preferred method if, among the multiple equilibria, the best one can be selected. However, the bottom-reference is the maximin method, i.e., it provides safe minimal aggregate contribution and welfare that surpass the worst outcome in the other two methods. Full article

We explore how subjects with heterogeneous beliefs respond to a surprise restart in a linear public goods game played for 20 rounds using either a “partners” or a “strangers” protocol. There are two restarts: one prior to Round 11 and another prior to Round 16. We elicit subject beliefs at the outset and classify subjects into three groups—Top, Middle, and Bottom—depending on their prior beliefs about their peers’ contributions to the public good. Then, we look at how these three groups respond, in terms of their beliefs and contributions, before and after the restart. We replicate the restart effect, but find that (i) it is much more pronounced for partner matching than for stranger matching and (ii) it is less pronounced in treatments with belief elicitation compared to control treatments where beliefs are not elicited. We also find that the restart has the effect of regenerating a sense of optimism among the subjects, which is reflected in increased contributions subsequently. This increase in contribution is driven mostly by those subjects who started the game with relatively more optimistic beliefs. Our results have implications for sustaining cooperation in social dilemma games. Full article

Charities operate at different levels: national, state, or local. We test the effect of the level of the organization on charitable giving in a sample of adults in two Texas communities. Subjects make four charitable giving “dictator game” decisions from a fixed amount of money provided by the experimenter. Three decisions target different charitable organizations, all of which have a disaster-relief mission, but differ in the level of operation. The fourth targets an individual recipient, identified by the local fire department as a victim of a fire. One of the four is selected randomly for payment. Giving is significantly higher to national and local organizations compared to state. We find a higher propensity to donate and larger amount donated to the individual relative to all organizations. Subsequent analysis compares a number of demographic and attitudinal covariates with donations to specific charities. In a second decision, subjects instead indicate which of their four prior decisions they would most prefer to implement. Here we see that a majority of subjects prefer the gift to the individual. Full article

Schosser (Games2018, 9, 26) claims to have found an alternative solution to design appropriate performance measures than the State-Contingent Relative Benefit Cost Allocation (RBCA) introduced by Ortner et al. (Management Accounting Research2017, 36, 43–50), which he states is simpler and more powerful. However, this note reveals that the performance measures proposed by Schosser are, in fact, a specific subset of State-Contingent Robust RBCA performance measures and thus do not represent a new solution. Full article

We provide a game-theoretic account of endogenous intrinsic motivation within a principal–agent framework. We explore the incentives of an altruistic principal who, by exerting costly effort, can intrinsically motivate a present-biased agent to exhibit a direct preference for more far-sighted behaviour. We characterize the conditions under which this happens. We show that allowing for endogenous intrinsic motivation generates interesting interplays between exogenous economic incentives and endogenous motivation, including the possibility of crowding out. Our model can be applied in a wide variety of contexts, including public policy, self-control, and cultural transmission. Full article