Fairness versus reason in the ultimatum game.

In the Ultimatum Game, two players are offered a chance to win a certain sum of money. All they must do is divide it. The proposer suggests how to split the sum. The responder can accept or reject the deal. If the deal is rejected, neither player gets anything. The rational solution, suggested by game theory, is for the proposer to offer the smallest possible share and for the responder to accept it. If humans play the game, however, the most frequent outcome is a fair share. In this paper, we develop an evolutionary approach to the Ultimatum Game. We show that fairness will evolve if the proposer can obtain some information on what deals the responder has accepted in the past. Hence, the evolution of fairness, similarly to the evolution of cooperation, is linked to reputation.

The spatial ultimatum game.

In the ultimatum game, two players are asked to split a certain sum of money. The proposer has to make an offer. If the responder accepts the offer, the money will be shared accordingly. If the responder rejects the offer, both players receive nothing. The rational solution is for the proposer to offer the smallest possible share, and for the responder to accept it. Human players, in contrast, usually prefer fair splits. In this paper, we use evolutionary game theory to analyse the ultimatum game. We first show that in a non-spatial setting, natural selection chooses the unfair, rational solution. In a spatial setting, however, much fairer outcomes evolve.

Elementary step dynamics of catalytic hypercycles.

Two model systems for hypercyclic organisation constructed from a series of uni- and bimolecular reaction steps were studied under the condition of unlimited growth by means of qualitative analysis and numerical integration of the corresponding differential equations. It is shown that both models lead, within wide ranges of parameter and initial conditions, to the same characteristic dynamical behaviour as the elementary hypercycles and hypercycles with translation introduced in 1978 by Eigen and Schuster.

Fixation probabilities for advantageous mutants: a note on multiplication and sampling.

If the average number of gametes produced by the individual is small, as may be the case for haploid organisms, then sampling with and without replacement can lead to considerable differences in the fixation probabilities of mutant alleles. As a function of the population size N, these probabilities converge quickly to the survival probabilities given by branching processes with Poisson or Bernoulli offspring distributions.

The arithmetics of mutual help.

Computer experiments show how cooperation rather than exploitation can dominate in the Darwinian struggle for survival.

A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game.

The Prisoner's Dilemma is the leading metaphor for the evolution of cooperative behaviour in populations of selfish agents, especially since the well-known computer tournaments of Axelrod and their application to biological communities. In Axelrod's simulations, the simple strategy tit-for-tat did outstandingly well and subsequently became the major paradigm for reciprocal altruism. Here we present extended evolutionary simulations of heterogeneous ensembles of probabilistic strategies including mutation and selection, and report the unexpected success of another protagonist: Pavlov. This strategy is as simple as tit-for-tat and embodies the fundamental behavioural mechanism win-stay, lose-shift, which seems to be a widespread rule. Pavlov's success is based on two important advantages over tit-for-tat: it can correct occasional mistakes and exploit unconditional cooperators. This second feature prevents Pavlov populations from being undermined by unconditional cooperators, which in turn invite defectors. Pavlov seems to be more robust than tit-for-tat, suggesting that cooperative behaviour in natural situations may often be based on win-stay, lose-shift.

Chaos and the evolution of cooperation.

The "iterated prisoner's dilemma" is the most widely used model for the evolution of cooperation in biological societies. Here we show that a heterogeneous population consisting of simple strategies, whose behavior is totally specified by the outcome of the previous round, can lead to persistent periodic or highly irregular (chaotic) oscillations in the frequencies of the strategies and the overall level of cooperation. The levels of cooperation jump up and down in an apparently unpredictable fashion. Small recurrent and simultaneous invasion attempts (caused by mutation) can change the evolutionary dynamics from converging to an evolutionarily stable strategy to periodic oscillations and chaos. Evolution can be twisted away from defection, toward cooperation. Adding "generous tit-for-tat" greatly increases the overall level of cooperation and can lead to long periods of steady cooperation. Since May's paper [May, R. M. (1976) Nature (London) 261, 459-467], "simple mathematical models with very complicated dynamics" have been found in many biological applications, but here we provide an example of a biologically relevant evolutionary game whose dynamics display deterministic chaos. The simulations bear some resemblance to the irregular cycles displayed by the frequencies of host genotypes and specialized parasites in evolutionary "arms races" [Hamilton, W. D., Axelrod, R. & Tanese, R. (1990) Proc. Natl. Acad. Sci. USA 87, 3566-3573; Seger, J. (1988) Philos. Trans. R. Soc. London B 319, 541-555].

Gradients for the evolution of bimatrix games.

The evolutionary dynamics of bimatrix games is studied for rescaled partnership games and zero sum games. The former case leads to gradient systems. The selection equations for sexual and asexual reproduction of genotypes corresponding to mixed strategies are analysed. As examples, the origin of anisogamy and cyclic chases for predator-prey coevolution are studied.

Oscillations in the evolution of reciprocity.

A game-theoretical analysis of the Iterated Prisoner's Dilemma shows that the evolution of ensembles of stochastic strategies displays a dynamics of high complexity and unpredictability.

On the stabilizing effect of predators and competitors on ecological communities.

Ecological communities can lose their permanence if a predator or a competitor is removed: the remaining species no longer coexist. This well-known phenomenon is analysed for some low dimensional examples of Lotka-Volterra type, with special attention paid to the occurrence of hetero-clinic cycles.

Evolution of indirect reciprocity by image scoring.

Darwinian evolution has to provide an explanation for cooperative behaviour. Theories of cooperation are based on kin selection (dependent on genetic relatedness), group selection and reciprocal altruism. The idea of reciprocal altruism usually involves direct reciprocity: repeated encounters between the same individuals allow for the return of an altruistic act by the recipient. Here we present a new theoretical framework, which is based on indirect reciprocity and does not require the same two individuals ever to meet again. Individual selection can nevertheless favour cooperative strategies directed towards recipients that have helped others in the past. Cooperation pays because it confers the image of a valuable community member to the cooperating individual. We present computer simulations and analytic models that specify the conditions required for evolutionary stability of indirect reciprocity. We show that the probability of knowing the 'image' of the recipient must exceed the cost-to-benefit ratio of the altruistic act. We propose that the emergence of indirect reciprocity was a decisive step for the evolution of human societies.

A Ge-like autoantibody in the serum of a patient receiving gold therapy for rheumatoid arthritis.

A 61-year-old nulliparous woman who had never been transfused was admitted to the hospital because of severe anemia. She had been treated with gold salts for 4.5 years for rheumatoid arthritis. Because she had a negative direct antiglobulin test, an anti-Ge found in her serum was believed to be an alloantibody. Subsequent testing of her initial serum sample with autologous red cells drawn 12 months later showed them to be strongly agglutinated, indicating that she had had a transient depression of her Ge antigen and that the anti-Ge was an autoantibody. It was postulated that these transient effects may have been related to the gold therapy.

The dynamics of indirect reciprocity.

Richard Alexander has argued that moral systems derive from indirect reciprocity. We analyse a simple case of a model of indirect reciprocity based on image scoring. Discriminators provide help to those individuals who have provided help. Even if the help is never returned by the beneficiary, or by individuals who in turn have been helped by the beneficiary, discriminating altruism can be resistant against invasion by defectors. Indiscriminate altruists can invade by random drift, however, setting up a complex dynamical system. In certain situations, defectors can invade only if their invasion attempts are sufficiently rare. We also consider a model with incomplete information and obtain conditions for the stability of altruism which differ from Hamilton's rule by simply replacing relatedness with acquaintanceship.

Immune responses against multiple epitopes.

The current understanding of antigenic escape dynamics is based on models with single epitopes. The usual idea is that a mutation which enables a pathogen (virus, bacteria, etc) to escape from a given immune response confers a selective advantage. The "escape mutant" may then increase in abundance until it induces a new specific response against itself. In this paper a new picture is developed, based on mathematical models of immune responses against several epitopes; the simplest such models can have very complicated dynamics, with some surprising features. The emergence of an escape mutant can shift the immunodominant response to another epitope. Even in the absence of mutations, antigenic oscillation is found, with distinct peaks of different virus variants and fluctuations in the size and specificity of the immune responses. The model also provides a general theory for immunodominance in the presence of antigenic variation. Immunodominance is determined by the immunogenicity and by the antigenic diversity of the competing epitopes. Antigenic oscillations and fluctuations in the cytotoxic T-lymphocyte response have been observed in infections with the human immunodeficiency virus (HIV). Shifting the immune responses to weaker epitopes can represent a mechanism for disease progression based on evolutionary dynamics and antigenic diversity of the virus.

Reward and punishment.

Minigames capturing the essence of Public Goods experiments show that even in the absence of rationality assumptions, both punishment and reward will fail to bring about prosocial behavior. This result holds in particular for the well-known Ultimatum Game, which emerges as a special case. But reputation can induce fairness and cooperation in populations adapting through learning or imitation. Indeed, the inclusion of reputation effects in the corresponding dynamical models leads to the evolution of economically productive behavior, with agents contributing to the public good and either punishing those who do not or rewarding those who do. Reward and punishment correspond to two types of bifurcation with intriguing complementarity. The analysis suggests that reputation is essential for fostering social behavior among selfish agents, and that it is considerably more effective with punishment than with reward.

The logic of contrition.

A highly successful strategy for the Repeated Prisoner's Dilemma is Contrite Tit For Tat, which bases its decisions on the "standings" of the two players. This strategy is as good as Tit For Tat at invading populations of defectors, and much better at overcoming errors in implementation against players who are also using it. However, it is vulnerable to errors in perception. In this paper, we discuss the merits of Contrite Tit For Tat and compare it with other strategies, like Pavlov and the newly-introduced Remorse. We embed these strategies into an eight-dimensional space of stochastic strategies which we investigate by analytical means and numerical simulations. Finally, we show that if one replaces the conventions concerning the "standing" by other, even simpler conventions, one obtains an evolutionarily stable strategy (called Prudent Pavlov) which is immune against both mis-perception and mis-implementation.