A two-player iterated survival game

Evolutionarily stable payoff matrix in hawk-dove games

BACKGROUND

Classical matrix game models aim to find the endpoint of behavioural evolution for a set of fixed possible interaction outcomes. Here, we introduce an evolutionary model in which not only the players' strategies but also the payoff matrix evolves according to natural selection.

RESULTS

We start out from the hawk-dove matrix game and, in a way that is consistent with the monomorphic model setup of Maynard Smith and Price, introduce an evolving phenotypic trait that quantifies fighting ability and determines the probability of winning and the cost of losing escalated hawk-hawk fights. We define evolutionarily stable phenotypes as consisting of an evolutionarily stable strategy and an evolutionarily stable trait, which in turn describes a corresponding evolutionarily stable payoff matrix.

CONCLUSIONS

We find that the maximal possible cost of escalating fights remains constant during evolution assuming a separation in the time scales of fast behavioural and slow trait selection, despite the fact that the final evolutionarily stable phenotype maximizes the payoff of hawk-hawk fights. Our results mirror the dual nature of Darwinian evolution whereby the criteria of evolutionary success, as well as the successful phenotypes themselves, are a product of natural selection.

Best Reply Player Against Mixed Evolutionarily Stable Strategy User

We consider matrix games with two phenotypes (players): one following a mixed evolutionarily stable strategy and another one that always plays a best reply against the action played by its opponent in the previous round (best reply player, BR). We focus on iterated games and well-mixed games with repetition (that is, the mean number of repetitions is positive, but not infinite). In both interaction schemes, there are conditions on the payoff matrix guaranteeing that the best reply player can replace the mixed ESS player. This is possible because best reply players in pairs, individually following their own selfish strategies, develop cycles where the bigger payoff can compensate their disadvantage compared with the ESS players. Well-mixed interaction is one of the basic assumptions of classical evolutionary matrix game theory. However, if the players repeat the game with certain probability, then they can react to their opponents’ behavior. Our main result is that the classical mixed ESS loses its general stability in the well-mixed population games with repetition in the sense that it can happen to be overrun by the BR player.

The consequences of switching strategies in a two-player iterated survival game

We consider two-player iterated survival games in which players are able to switch from a more cooperative behavior to a less cooperative one at some step of an n-step game. Payoffs are survival probabilities and lone individuals have to finish the game on their own. We explore the potential of these games to support cooperation, focusing on the case in which each single step is a Prisoner’s Dilemma. We find that incentives for or against cooperation depend on the number of defections at the end of the game, as opposed to the number of steps in the game. Broadly, cooperation is supported when the survival prospects of lone individuals are relatively bleak. Specifically, we find three critical values or cutoffs for the loner survival probability which, in concert with other survival parameters, determine the incentives for or against cooperation. One cutoff determines the existence of an optimal number of defections against a fully cooperative partner, one determines whether additional defections eventually become disfavored as the number of defections by the partner increases, and one determines whether additional cooperations eventually become favored as the number of defections by the partner increases. We obtain expressions for these switch-points and for optimal numbers of defections against partners with various strategies. These typically involve small numbers of defections even in very long games. We show that potentially long stretches of equilibria may exist, in which there is no incentive to defect more or cooperate more. We describe how individuals find equilibria in best-response walks among n-step strategies.

Survivor's dilemma: The evolution of cooperation in volatile environments

The volatility of an environment significantly impacts cooperative behavior. In environments where viability-threatening events occur on a shorter timescale than reproduction, it is reasonable to measure the costs and benefits of cooperation in terms of their direct effect on survival probability. Then, the number of offspring increases with lifespan. With such a model, is it possible for cooperation to be evolutionarily stable, and how does cooperation depend on the benefit and cost of such interactions, and the volatility of the environment? In this paper, we develop an N-player survivor's dilemma in which prisoner's dilemma payoffs in an iteration are survival rates, and expected lifespan is the measure of reproductive fitness. We investigate cost, benefit, and volatility parameter ranges where various cooperative behaviors may occur. We observe that free-riding results in indirect punishment as the cheated partner's early death leaves the defector vulnerable. For 2- and 3-player versions of the game, we identify parameter regions where the repeated game becomes equivalent to a Harmony, Stag Hunt, or Prisoner's Dilemma static game and discuss evolutionary stability. We find that with two individuals, the initial fraction of cooperators necessary for cooperation to be selected for decreases as the benefit to cost ratio increases and as environmental volatility decreases. With the presence of a third individual, there also exists a parameter region where cooperation can invade an initially all-defecting population.

High thresholds encouraging the evolution of cooperation in threshold public-good games

For a well-mixed population, we consider a threshold public good game where group members only obtain benefits from a public good if a sufficiently large number of them cooperates. We investigate the effect of an increase in the threshold on the level of cooperation that evolves. It is shown that for sufficiently large participation costs, the level of cooperation is higher for low and for high thresholds, than it is for intermediate thresholds – where in the latter case cooperation may not evolve at all. The counterintuitive effect where an increase in the threshold from an intermediate to a high one decreases the probability of cooperation, is related to the so-called common-enemy hypothesis of the evolution of cooperation. We further apply our analysis to assess the relative weight of different game types across the parameter space, and show that game types where either a small, or a large fraction of the population evolves as cooperators, receive more weight compared to game types where an intermediate fraction of cooperators evolves.

Adversity and cooperation in heterogeneous pairs

This paper provides a game-theoretic model of the effect of higher adversity on the evolution of cooperation. The focus lies on how this effect of higher adversity is impacted when there is transient, non-genetic heterogeneity in the form of differences in the players' capabilities of contributing to the public good, in the benefits they obtain from the public good, or in their cooperation costs. A framework is provided that identifies the common mechanisms that are at work across two models of cooperation (jointly producing a public good, and jointly defending an existing public good), and across the mentioned types of heterogeneity. With relatively small heterogeneity, higher adversity generates a common-enemy effect for large cooperation costs and a deterrence effect for small cooperation costs. Yet, these results on the effect of higher adversity are completely reversed for relatively large heterogeneity.