Theft in an ultimatum game: chimpanzees and bonobos are insensitive to unfairness

Humans, but not chimpanzees, punish unfair offers in ultimatum games, suggesting that fairness concerns evolved sometime after the split between the lineages that gave rise to Homo and Pan. However, nothing is known about fairness concerns in the other Pan species, bonobos. Furthermore, apes do not typically offer food to others, but they do react against theft. We presented a novel game, the ultimatum theft game, to both of our closest living relatives. Bonobos and chimpanzee ‘proposers’ consistently stole food from the responders'' portions, but the responders did not reject any non-zero offer. These results support the interpretation that the human sense of fairness is a derived trait.     

Partner choice creates fairness in humans

Many studies demonstrate that partner choice has played an important role in the evolution of human cooperation, but little work has tested its impact on the evolution of human fairness. In experiments involving divisions of money, people become either over-generous or over-selfish when they are in competition to be chosen as cooperative partners. Hence, it is difficult to see how partner choice could result in the evolution of fair, equal divisions. Here, we show that this puzzle can be solved if we consider the outside options on which partner choice operates. We conduct a behavioural experiment, run agent-based simulations and analyse a game-theoretic model to understand how outside options affect partner choice and fairness. All support the conclusion that partner choice leads to fairness only when individuals have equal outside options. We discuss how this condition has been met in our evolutionary history, and the implications of these findings for our understanding of other aspects of fairness less specific than preferences for equal divisions of resources.     

The evolution of fairness in a biological market

Human beings universally express a concern for the fairness of social interactions, and it remains an open question that which ultimate factors led to the evolution of this preference. Here, we present a model accounting for the evolution of fairness on the basis of individual selection alone. We consider a simple social interaction based on the Dictator Game. Two individuals, a "proposer" and a "responder," have an opportunity to split a resource. When they have no choice but to interact together, the most powerful (here the proposer) reaps all the profits and fairness cannot evolve. Partner choice is the key lever to overcome this difficulty. Rather than just two individuals, we consider a population composed of two classes of individuals (either proposers or responders), and we allow the responders to choose their partner. In such a "biological market," fairness evolves as an "equilibrium price," resulting from an ecological equivalent of the law of supply and demand. If a class is disadvantaged by the chosen resource partition (i.e., if it frequently receives less than half of the resource), it is outcompeted by the other one, and automatically becomes rarer. This rarity grants it an advantage on the market, which yields in turn to the evolution of a more favorable partition. Splitting the resource into two identical halves, or more generally in a way that equalizes the payoffs of the two classes, is then the only evolutionarily stable outcome. Beyond human fairness, this mechanism also opens up new ways of explaining the distribution of benefits in many mutualistic interactions.     

Fairness evolution in the ultimatum game is a function of reward size

I formulate a simple model of the ultimatum game, in which a proposer and a responder can receive a reward if they agree on how to divide this reward between them. The model is easy to analyse and shows that strong tendencies to fair division are expected when evolution of strategy frequencies follow the traditional gradient dynamics assumed in evolutionary models. The mean stable offer is typically around 20-40% although this depends on the maximum payoff and if rejection thresholds can evolve independently from proposals. The stable proportion offered at evolutionary equilibrium increases with the maximum payoff, if proposal and acceptance thresholds are dictated by the same strategy and cannot evolve independently. If proposal and acceptance evolve independently, the stable proportion instead decreases with the maximum payoff. The stable outcome may also show substantial variation.     

Local host competition in the evolution of virulence

The evolution of parasite life histories should usually have correlated effects on host survivorship and/or reproductive success. For example, parasites that reproduce more rapidly might be expected to cause greater reductions in host fitness. Important theoretical advances have recently been made on virulence evolution, but the results are not always consistent. Here I compare two models [ Q. Rev. Biol. 71 (1996) 37 ; Q. Rev. Biol. 75 (2000) 261 ] on the evolution of virulence that get qualitatively different results with respect to the effects of coinfection. I also construct a third model that attempts to connect these two formulations. The results suggest that parasite growth rates should increase as local host competition increases, unless relatedness is at equilibrium. In addition, the qualitative effect of adding coinfections on parasite growth rates depends critically on how the number of coinfections affects transmission success.     

Statistical fluctuations in population bargaining in the ultimatum game: Static and evolutionary aspects

We explore the emergent behavior in heterogeneous populations where players negotiate via an ultimatum game: two players are offered a gift, one of them (the proposer) suggests how to divide the offer while the other player (the responder) can either accept or reject the deal. Rejection is detrimental to both players as it results in no earnings. In this context, our contribution is twofold: (i) we consider a population where the distribution of used strategies is constant over time and properties of the random payoff received by the players (average and higher moments) are reported from simple exact methods and corroborated by computer simulations; (ii) the evolution of a population is analyzed via Monte Carlo simulations where agents may change independently the proposing and accepting parameters of their strategy depending on received payoffs. Our results show that evolution leads to a stationary state in which wealth (accumulated payoff) is fairly distributed. As time evolves, an increase in average payoff and a simultaneous variance decrease is observed when we use a dynamics based on a probabilistic version of the saying: “One should not comply with small earnings, but one's greed must be limited.”     

Rock-scissors-paper game in a chaotic flow: the effect of dispersion on the cyclic competition of microorganisms

Laboratory experiments and numerical simulations have shown that the outcome of cyclic competition is significantly affected by the spatial distribution of the competitors. Short-range interaction and limited dispersion allows for coexistence of competing species that cannot coexist in a well-mixed environment. In order to elucidate the mechanisms that destroy species diversity we study the intermediate situation of imperfect mixing, typical in aquatic media, in a model of cyclic competition between toxin producing, sensitive and resistant phenotypes. It is found, that chaotic mixing, by changing the character of the spatial distribution, induces coherent oscillations in the populations. The magnitude of the oscillations increases with the strength of mixing, leading to the extinction of some species beyond a critical mixing rate. When mixing is non-uniform in space, coexistence can be sustained at much stronger mixing by the formation of partially isolated regions, that prevent global extinction. The heterogeneity of mixing may enable toxin producing and sensitive strains to coexist for very long time at strong mixing.     

公平规范与自然资源保护——基于进化博弈的理论模型

传统理论认为社区对公共自然资源的管理必然导致"公共地悲剧".但新近研究证明个人行为往往受到社会规范的影响并遵循公平互惠等"非理性"原则,进而指出社群有可能自发实现对公共资源有效的合作保护.基于这一思路,综合运用进化博弈论与行为经济学理论构建数学模型模拟了社区居民参与对自然资源的家庭承包管护的策略演化动态,并分析公平规范在该过程中能够发挥的潜在作用.结果表明在经济补偿相对管护所需劳动成本较为有限的情况下,"理性"的参与者无法长期维持高水平的保护合作;相比之下,如果公平规范对当地居民有较强的约束力,则可以实现更高的保护合作水平,其最终均衡由群体中对公平有较高要求的参与者比例决定.这一结果从理论上预测了利用公平规范推动社区参与自然资源保护合作并提高生态补偿效率的可行性.     

Root proliferation and seed yield in response to spatial heterogeneity of below-ground competition

Here, we tested the predictions of a 'tragedy of the commons' model of below-ground plant competition in annual plants that experience spatial heterogeneity in their competitive environment. Under interplant competition, the model predicts that a plant should over-proliferate roots relative to what would maximize the collective yield of the plants. We predict that a plant will tailor its root proliferation to local patch conditions, restraining root production when alone and over-proliferating in the presence of other plants. A series of experiments were conducted using pairs of pea (Pisum sativum) plants occupying two or three pots in which the presence or absence of interplant root competition was varied while nutrient availability per plant was held constant. In two-pot experiments, competing plants produced more root mass and less pod mass per individual than plants grown in isolation. In three-pot experiments, peas modulated this response to conditions at the scale of individual pots. Root proliferation in the shared pot was higher compared with the exclusively occupied pot. Plants appear to display sophisticated nutrient foraging with outcomes that permit insights into interplant competition.     

Fighting for food: a dynamic version of the Hawk-Dove game

Summary The Hawk-Dove game (Maynard Smith, 1982) has been used to analyse conflicts over resources such as food. At the evolutionarily stable strategy (ESS), either a proportionp* of animals always play Hawk, or each animal has a probabilityp* of playing Hawk. We modify the standard Hawk-Dove game to include a state variable,x, that represents the animal's level of energy reserves. A strategy is now a rule for choosing an action as a function ofx and time of day. We consider a non-reproductive period and adopt the criterion of minimizing mortality over this period. We find the ESS, which has the form play Hawk if reserves are belowc* (t) at timet, otherwise play Dove. This ESS is very different from the ESS in the standard Hawk-Dove game. It is a pure ESS that depends on the animal's state and on time. Furthermore, it is characterized by the strong condition that any single mutant that does not adopt the ESS suffers a reduction in fitness. The standard Hawk-Dove game assumes pay-offs that are related to fitness; our approach starts from a definition of fitness and derives the pay-offs in the process of finding the ESS. When the environment becomes worse (e.g. food becomes less reliable or energy expenditure increases) the ESS changes in such a way as to increase the proportion of animals that will play Hawk.     

Evolution of extraordinary female-biased sex ratios: the optimal schedule of sex ratio in local mate competition

Female-biased sex ratio in local mate competition has been well studied both theoretically and experimentally. However, some experimental data show more female-biased sex ratios than the theoretical predictions by Hamilton [1967. Science 156, 477-488] and its descendants. Here we consider the following two effects: (1) lethal male-male combat and (2) time-dependent control (or schedule) of sex ratio. The former is denoted by a male mortality being an increasing function of the number of males. The optimal schedule is analytically obtained as an evolutionarily stable strategy (ESS) by using Pontrjagin's maximum principle. As a result, an ESS is a schedule where only males are produced first, then the proportion of females are gradually increased, and finally only females are produced. Total sex ratio (sex ratio averaged over the whole reproduction period) is more female-biased than the Hamilton's result if and only if the two effects work together. The bias is stronger when lethal male combat is severer or a reproduction period is longer. When male-male combat is very severe, the sex ratio can be extraordinary female-biased (less than 5%). The model assumptions and the results generally agree with experimental data on Melittobia wasps in which extraordinary female-biased sex ratio is observed. Our study might provide a new basis for the evolution of female-biased sex ratios in local mate competition.     

Increased forest carbon storage with increased atmospheric CO2 despite nitrogen limitation: a game‐theoretic allocation model for trees in competition for nitrogen and light

Changes in resource availability often cause competitively driven changes in tree allocation to foliage, wood, and fine roots, either via plastic changes within individuals or through turnover of individuals with differing strategies. Here, we investigate how optimally competitive tree allocation should change in response to elevated atmospheric CO₂ along a gradient of nitrogen and light availability, together with how those changes should affect carbon storage in living biomass. We present a physiologically‐based forest model that includes the primary functions of wood and nitrogen. From a tree's perspective, wood is an offensive and defensive weapon used against neighbors in competition for light. From a biogeochemical perspective, wood is the primary living reservoir of stored carbon. Nitrogen constitutes a tree's photosynthetic machinery and the support systems for that machinery, and its limited availability thus reduces a tree's ability to fix carbon. This model has been previously successful in predicting allocation to foliage, wood, and fine roots along natural productivity gradients. Using game theory, we solve the model for competitively optimal foliage, wood, and fine root allocation strategies for trees in competition for nitrogen and light as a function of CO₂ and nitrogen mineralization rate. Instead of down‐regulating under nitrogen limitation, carbon storage under elevated CO₂ relative to carbon storage at ambient CO₂ is approximately independent of the nitrogen mineralization rate. This surprising prediction is a consequence of both increased competition for nitrogen driving increased fine root biomass and increased competition for light driving increased allocation to wood under elevated CO₂.     

John Maynard Smith and the importance of consistency in evolutionary game theory

John Maynard Smith was the founder of evolutionary game theory. He has also been the major influence on the direction of this field, which now pervades behavioural ecology and evolutionary biology. In its original formulation the theory had three components: a set of strategies, a payoff structure, and a concept of evolutionary stability. These three key components are still the basis of the theory, but what is assumed about each component is often different to the original assumptions. We review modern approaches to these components. We emphasis that if a game is considered in isolation, and arbitrary payoffs are assumed, then the payoffs may not be consistent with other components of the system which are not modelled. Modelling the whole system, including not only the focal game, but also the future behaviour of the players and the behaviour of other population members, allows a consistent model to be constructed. We illustrate this in the case of two models of parental care, showing how linking a focal game to other aspects of the system alters what is predicted.     

Asymmetry within social groups: division of labour and intergroup competition

Social animals vary in their ability to compete with group members over shared resources and also vary in their cooperative efforts to produce these resources. Competition among groups can promote within‐group cooperation, but many existing models of intergroup cooperation do not explicitly account for observations that group members invest differentially in cooperation and that there are often within‐group competitive or power asymmetries. We present a game theoretic model of intergroup competition that investigates how such asymmetries affect within‐group cooperation. In this model, group members adopt one of two roles, with relative competitive efficiency and the number of individuals varying between roles. Players in each role make simultaneous, coevolving decisions. The model predicts that although intergroup competition increases cooperative contributions to group resources by both roles, contributions are predominantly from individuals in the less competitively efficient role, whereas individuals in the more competitively efficient role generally gain the larger share of these resources. When asymmetry in relative competitive efficiency is greater, a group's per capita cooperation (averaged across both roles) is higher, due to increased cooperation from the competitively inferior individuals. For extreme asymmetry in relative competitive efficiency, per capita cooperation is highest in groups with a single competitively superior individual and many competitively inferior individuals, because the latter acquiesce and invest in cooperation rather than within‐group competition. These predictions are consistent with observed features of many societies, such as monogynous Hymenoptera with many workers and caste dimorphism.     

Mate choice in the face of costly competition

Studies of mate choice commonly ignore variation in preferencesand assume that all individuals should favor the highest-qualitymate available. However, individuals may differ in their matepreferences according to their own age, experience, size, orgenotype. In the present study, we highlight another simplereason why preferences may differ: if there is costly competitionfor mates, the poorest competitors might be better off avoidingthe highest-quality partners and instead targeting low-qualitypartners, so that they minimize the costs they incur. We presenta game-theoretical model of mate choice in which males of differingquality compete for access to females and try to retain themtill the time of mating. Our model predicts that high-qualitymales, who are better competitors, have a preference for thebest females that is typically several times stronger than thatof low-quality males. Early in the competitive period, the lattermay even prefer low-quality females over high-quality females.Thus, variation in competitive ability generates variation inboth the strength and direction of preferences. Differencesin competitive ability result in assortative mating with respectto quality, which is reinforced by variation in preferences.As the time of mating draws near and there is an increased riskof ending up unpaired, all males become indifferent to the qualityof potential mates. Our findings are equally applicable to femalechoice for males, and offer a new explanation for adaptive variationin mating preferences based on differing abilities to cope withthe costs of mate choice.     

Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game

Understanding the mechanisms that can lead to the evolution of cooperation through natural selection is a core problem in biology. Among the various attempts at constructing a theory of cooperation, game theory has played a central role. Here, we review models of cooperation that are based on two simple games: the Prisoner's Dilemma, and the Snowdrift game. Both games are two‐person games with two strategies, to cooperate and to defect, and both games are social dilemmas. In social dilemmas, cooperation is prone to exploitation by defectors, and the average payoff in populations at evolutionary equilibrium is lower than it would be in populations consisting of only cooperators. The difference between the games is that cooperation is not maintained in the Prisoner's Dilemma, but persists in the Snowdrift game at an intermediate frequency. As a consequence, insights gained from studying extensions of the two games differ substantially. We review the most salient results obtained from extensions such as iteration, spatial structure, continuously variable cooperative investments, and multi‐person interactions. Bridging the gap between theoretical and empirical research is one of the main challenges for future studies of cooperation, and we conclude by pointing out a number of promising natural systems in which the theory can be tested experimentally.     

Intra-plant versus Inter-plant Root Competition in Beans: avoidance,resource matching or tragedy of the commons

Root competition inhibits root proliferation. All else equal, a plant should invest roots in a nutrient patch devoid of roots rather than one already occupied by roots. Less clear is how a plant should respond to intra-plant versus inter-plant root competition. We consider three responses for how a plant may select habitats based on intra-versus inter-plant root competition: inter-plant avoidance, resource matching, or intra-plant avoidance. The first assumes that plants prefer to have their own space and preferentially proliferate roots away from neighboring plants. The second response, based on the ideal free distribution, assumes that plants invest so as to equalize average returns from roots, regardless of the identity of the neighboring roots. The third, based on game theory, assumes that the plant proliferates roots so as to maximize whole-plant fitness, in which case it is better to proliferate plants among a neighbor's roots than to continue proliferating amongst one's own roots. To test among these models we grew beans (Phaseolus varigaris, var. Kenya) in a greenhouse under two planting scenarios. Both scenario were tested under 0.5 and 0.1 strength of nutrient solution. Under scenario A (fence-sitters), two split-root plants each shared two patches by virtue of having roots in each. Under scenario B (owners) two plants each had their own patch. The results supported the game theory model of intra-plant avoidance (whole plant habitat selection). Fence-sitters produced 150% more root mass per individual than owners. Owners produced 90% more yield (dry mass of pods) than fence-sitters. Furthermore, owners had significantly higher shoot-root ratios than fence-sitters. These effects did not vary with high or low nutrient levels. The over-proliferation of roots under inter-plant competition (fence-sitters) was manifest by the tenth day after planting. In short, the fence-sitters engaged in a tragedy of the commons in which they competed with each other through root proliferation. At the ESS, the fitness maximizing strategy of the individual is to sacrifice collective yield in a quest to `steal' nutrients from its neighbor. The research has three implications. First, plants may be able to assess and respond to local opportunities in a manner that maximizes the good of the whole plant. Second, nutrient foraging as a game may provide a fresh perceptive for viewing root competition either intra-specifically or inter-specifically. Third, it may be possible to increase the yield of certain crop species by breeding more `docile' cultivars that do not overproduce roots in response to inter-plant competition.