Indirect reciprocity with simple records.

Indirect reciprocity is a foundational mechanism of human cooperation. Existing models of indirect reciprocity fail to robustly support social cooperation: Image-scoring models fail to provide robust incentives, while social-standing models are not informationally robust. Here we provide a model of indirect reciprocity based on simple, decentralized records: Each individual's record depends on the individual's own past behavior alone, and not on the individual's partners' past behavior or their partners' partners' past behavior. When social dilemmas exhibit a coordination motive (or strategic complementarity), tolerant trigger strategies based on simple records can robustly support positive social cooperation and exhibit strong stability properties. In the opposite case of strategic substitutability, positive social cooperation cannot be robustly supported. Thus, the strength of short-run coordination motives in social dilemmas determines the prospects for robust long-run cooperation.

Testing the drift-diffusion model.

The drift-diffusion model (DDM) is a model of sequential sampling with diffusion signals, where the decision maker accumulates evidence until the process hits either an upper or lower stopping boundary and then stops and chooses the alternative that corresponds to that boundary. In perceptual tasks, the drift of the process is related to which choice is objectively correct, whereas in consumption tasks, the drift is related to the relative appeal of the alternatives. The simplest version of the DDM assumes that the stopping boundaries are constant over time. More recently, a number of papers have used nonconstant boundaries to better fit the data. This paper provides a statistical test for DDMs with general, nonconstant boundaries. As a by-product, we show that the drift and the boundary are uniquely identified. We use our condition to nonparametrically estimate the drift and the boundary and construct a test statistic based on finite samples.

Bayesian posteriors for arbitrarily rare events.

We study how much data a Bayesian observer needs to correctly infer the relative likelihoods of two events when both events are arbitrarily rare. Each period, either a blue die or a red die is tossed. The two dice land on side [Formula: see text] with unknown probabilities [Formula: see text] and [Formula: see text], which can be arbitrarily low. Given a data-generating process where [Formula: see text], we are interested in how much data are required to guarantee that with high probability the observer's Bayesian posterior mean for [Formula: see text] exceeds [Formula: see text] times that for [Formula: see text] If the prior densities for the two dice are positive on the interior of the parameter space and behave like power functions at the boundary, then for every [Formula: see text] there exists a finite [Formula: see text] so that the observer obtains such an inference after [Formula: see text] periods with probability at least [Formula: see text] whenever [Formula: see text] The condition on [Formula: see text] and [Formula: see text] is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.

Recency, consistent learning, and Nash equilibrium.

We examine the long-term implication of two models of learning with recency bias: recursive weights and limited memory. We show that both models generate similar beliefs and that both have a weighted universal consistency property. Using the limited-memory model we produce learning procedures that both are weighted universally consistent and converge with probability one to strict Nash equilibrium.

Winners don't punish.

A key aspect of human behaviour is cooperation. We tend to help others even if costs are involved. We are more likely to help when the costs are small and the benefits for the other person significant. Cooperation leads to a tension between what is best for the individual and what is best for the group. A group does better if everyone cooperates, but each individual is tempted to defect. Recently there has been much interest in exploring the effect of costly punishment on human cooperation. Costly punishment means paying a cost for another individual to incur a cost. It has been suggested that costly punishment promotes cooperation even in non-repeated games and without any possibility of reputation effects. But most of our interactions are repeated and reputation is always at stake. Thus, if costly punishment is important in promoting cooperation, it must do so in a repeated setting. We have performed experiments in which, in each round of a repeated game, people choose between cooperation, defection and costly punishment. In control experiments, people could only cooperate or defect. Here we show that the option of costly punishment increases the amount of cooperation but not the average payoff of the group. Furthermore, there is a strong negative correlation between total payoff and use of costly punishment. Those people who gain the highest total payoff tend not to use costly punishment: winners don't punish. This suggests that costly punishment behaviour is maladaptive in cooperation games and might have evolved for other reasons.

Phenotype switching and mutations in random environments.

Cell populations can benefit from changing phenotype when the environment changes. One mechanism for generating these changes is stochastic phenotype switching, whereby cells switch stochastically from one phenotype to another according to genetically determined rates, irrespective of the current environment, with the matching of phenotype to environment then determined by selective pressure. This mechanism has been observed in numerous contexts, but identifying the precise connection between switching rates and environmental changes remains an open problem. Here, we introduce a simple model to study the evolution of phenotype switching in a finite population subject to random environmental shocks. We compare the successes of competing genotypes with different switching rates, and analyze how the optimal switching rates depend on the frequency of environmental changes. If environmental changes are as rare as mutations, then the optimal switching rates mimic the rates of environmental changes. If the environment changes more frequently, then the optimal genotype either maximally favors fitness in the more common environment or has the maximal switching rate to each phenotype. Our results also explain why the optimum is relatively insensitive to fitness in each environment.

Let the right one in: a microeconomic approach to partner choice in mutualisms.

One of the main problems impeding the evolution of cooperation is partner choice. When information is asymmetric (the quality of a potential partner is known only to himself), it may seem that partner choice is not possible without signaling. Many mutualisms, however, exist without signaling, and the mechanisms by which hosts might select the right partners are unclear. Here we propose a general mechanism of partner choice, "screening," that is similar to the economic theory of mechanism design. Imposing the appropriate costs and rewards may induce the informed individuals to screen themselves according to their types and therefore allow a noninformed individual to establish associations with the correct partners in the absence of signaling. Several types of biological symbioses are good candidates for screening, including bobtail squid, ant-plants, gut microbiomes, and many animal and plant species that produce reactive oxygen species. We describe a series of diagnostic tests for screening. Screening games can apply to the cases where by-products, partner fidelity feedback, or host sanctions do not apply, therefore explaining the evolution of mutualism in systems where it is impossible for potential symbionts to signal their cooperativeness beforehand and where the host does not punish symbiont misbehavior.

Emergence of cooperation and evolutionary stability in finite populations.

To explain the evolution of cooperation by natural selection has been a major goal of biologists since Darwin. Cooperators help others at a cost to themselves, while defectors receive the benefits of altruism without providing any help in return. The standard game dynamical formulation is the 'Prisoner's Dilemma', in which two players have a choice between cooperation and defection. In the repeated game, cooperators using direct reciprocity cannot be exploited by defectors, but it is unclear how such cooperators can arise in the first place. In general, defectors are stable against invasion by cooperators. This understanding is based on traditional concepts of evolutionary stability and dynamics in infinite populations. Here we study evolutionary game dynamics in finite populations. We show that a single cooperator using a strategy like 'tit-for-tat' can invade a population of defectors with a probability that corresponds to a net selective advantage. We specify the conditions required for natural selection to favour the emergence of cooperation and define evolutionary stability in finite populations.

A Dual-Self Model of Impulse Control.

We propose that a simple "dual-self" model gives a unified explanation for several empirical regularities, including the apparent time inconsistency that has motivated models of quasi-hyperbolic discounting and Rabin's paradox of risk aversion in the large and small. The model also implies that self-control costs imply excess delay, as in the O'Donoghue and Rabin models of quasi-hyperbolic utility, and it explains experimental evidence that increased cognitive load makes temptations harder to resist. The base version of our model is consistent with the Gul-Pesendorfer axioms, but we argue that these axioms must be relaxed to account for the effect of cognitive load.

Positive interactions promote public cooperation.

The public goods game is the classic laboratory paradigm for studying collective action problems. Each participant chooses how much to contribute to a common pool that returns benefits to all participants equally. The ideal outcome occurs if everybody contributes the maximum amount, but the self-interested strategy is not to contribute anything. Most previous studies have found punishment to be more effective than reward for maintaining cooperation in public goods games. The typical design of these studies, however, represses future consequences for today's actions. In an experimental setting, we compare public goods games followed by punishment, reward, or both in the setting of truly repeated games, in which player identities persist from round to round. We show that reward is as effective as punishment for maintaining public cooperation and leads to higher total earnings. Moreover, when both options are available, reward leads to increased contributions and payoff, whereas punishment has no effect on contributions and leads to lower payoff. We conclude that reward outperforms punishment in repeated public goods games and that human cooperation in such repeated settings is best supported by positive interactions with others.

Tit-for-tat or win-stay, lose-shift?

The repeated Prisoner's Dilemma is usually known as a story of tit-for-tat (TFT). This remarkable strategy has won both of Robert Axelrod's tournaments. TFT does whatever the opponent has done in the previous round. It will cooperate if the opponent has cooperated, and it will defect if the opponent has defected. But TFT has two weaknesses: (i) it cannot correct mistakes (erroneous moves) and (ii) a population of TFT players is undermined by random drift when mutant strategies appear which play always-cooperate (ALLC). Another equally simple strategy called 'win-stay, lose-shift' (WSLS) has neither of these two disadvantages. WSLS repeats the previous move if the resulting payoff has met its aspiration level and changes otherwise. Here, we use a novel approach of stochastic evolutionary game dynamics in finite populations to study mutation-selection dynamics in the presence of erroneous moves. We compare four strategies: always-defect (ALLD), ALLC, TFT and WSLS. There are two possible outcomes: if the benefit of cooperation is below a critical value then ALLD is selected; if the benefit of cooperation is above this critical value then WSLS is selected. TFT is never selected in this evolutionary process, but lowers the selection threshold for WSLS.

Evolutionary game dynamics in finite populations with strong selection and weak mutation.

We study stochastic game dynamics in finite populations. To this end we extend the classical Moran process to incorporate frequency-dependent selection and mutation. For 2 x 2 games, we give a complete analysis of the long-run behavior when mutation rates are small. For 3 x 3 coordination games, we provide a simple rule to determine which strategy will be selected in large populations. The expected motion in our model resembles the standard replicator dynamics when the population is large, but is qualitatively different when the population is small. Our analysis shows that even in large finite populations the behavior of a replicator-like system can be different from that of the standard replicator dynamics. As an application, we consider selective language dynamics. We determine which language will be spoken in finite large populations. The results have an intuitive interpretation but would not be expected from an analysis of the replicator dynamics.

Evolutionary cycles of cooperation and defection.

The main obstacle for the evolution of cooperation is that natural selection favors defection in most settings. In the repeated prisoner's dilemma, two individuals interact several times, and, in each round, they have a choice between cooperation and defection. We analyze the evolutionary dynamics of three simple strategies for the repeated prisoner's dilemma: always defect (ALLD), always cooperate (ALLC), and tit-for-tat (TFT). We study mutation-selection dynamics in finite populations. Despite ALLD being the only strict Nash equilibrium, we observe evolutionary oscillations among all three strategies. The population cycles from ALLD to TFT to ALLC and back to ALLD. Most surprisingly, the time average of these oscillations can be entirely concentrated on TFT. In contrast to the classical expectation, which is informed by deterministic evolutionary game theory of infinitely large populations, stochastic evolution of finite populations need not choose the strict Nash equilibrium and can therefore favor cooperation over defection.

Evolutionary game dynamics in finite populations.

We introduce a model of stochastic evolutionary game dynamics in finite populations which is similar to the familiar replicator dynamics for infinite populations. Our focus is on the conditions for selection favoring the invasion and/or fixation of new phenotypes. For infinite populations, there are three generic selection scenarios describing evolutionary game dynamics among two strategies. For finite populations, there are eight selection scenarios. For a fixed payoff matrix a number of these scenarios can occur for different population sizes. We discuss several examples with unexpected behavior.