Origins of behavioural variability: categorical and discriminative assessment in serial contests

Can social behaviour be probabilistic? Classical results in evolutionary game theory seem to suggest that recognizable asymmetries between individuals in the likelihood of winning fights (i.e. in resource holding power, RHP) rule out the stable probabilistic behaviour associated with mixed strategies. Here, using a variant of the hawk-dove game, I show that these mixed strategies can be common in asymmetric contests, because opponents of similar RHP benefit by ignoring discernable differences to form RHP equivalence categories. This process of categorizing (i.e. opting not to distinguish an opponent's discernably different RHP from one's own) can be adaptive when assessment is imprecise or expensive and mistakes can lead to dangerous combat. For large RHP differences between opponents, discriminating by acting on the discernable RHP difference is evolutionarily stable. For moderate RHP differences, typically neither categorizing nor discriminating is stable, but categorizing predominates if social interactions are frequent. More frequent interactions imply that the social status ultimately achieved is more important in accumulating fitness; this reduces the probability that a loser in combat will accept the submissive role without additional fighting, where this probability of ‘surrender’ expresses another mixed strategy. These patterns appear to fit the establishment of social relationships in some species for which combat is dangerous, and the analysis represents a step towards understanding the emergence of dominance hierarchies.     

Fighting costs stabilize aggressive behavior in intersexual conflicts

We analyze the evolution of aggressive behavior in intersexual conflicts, with a special reference to mate guarding behavior in crustaceans. An analysis of a discrete-strategy game shows that an ESS with only one of the sexes being aggressive prevail if fighting costs or fitness values of winning are asymmetric. Non-aggressiveness of both sexes is stable if fighting behavior is very costly for females and if the cost is at least partly paid independent of the strategy of the opponent. Most interestingly, the solutions of both sexes being aggressive prevails only if both sexes have some probability of winning, and if fighting costs are small. Second, we solve for the expected levels of aggressiveness in a game with continuous strategies. The form of the fighting cost function largely determines the stability of the solution. When fighting cost increases linearly with aggressiveness, mutual aggressiveness fluctuates cyclically instead of stabilizing at an ESS. However, if there is an asymmetry in fitness payoffs, a solution with only the sex having most to lose being aggressive alone is possible. With quadratically increasing fighting costs an ES combination of mutual aggressiveness may exist. It is predicted that fights between the sexes should be hardest when payoffs are symmetric, and that an overt behavioral conflict will always take place as long as there is a fitness loss to each of the sexes if losing the conflict and both sexes have a chance to win. We discuss the models in the context of fights preceding precopulatory guarding, but the models offer a general frame for analyzing any intersexual conflict. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the evolution of pure winner and loser effects: A game-theoretic model

The persistence of linear dominance hierarchies is often attributed to higher probabilities of a win after a win or a loss after a loss in agonistic interactions, yet there has been no theory on the evolution of such prior-experience effects. Here an analytic model, based on the idea that contests are determined by subjective perceptions of resource-holding potential (RHP) which animals may revise in the light of experience, demonstrates that winner and loser effects can evolve through round-robin competition among triads of animals drawn randomly from their population, and that the probability of a hierarchy increases with the strength of the combined effect. The effects are pure, in the sense that a contestant observes neither its own RHP nor its opponent’s RHP or RHP perception or win—loss record; and so the strength of an effect is unmodified by the RHPs of particular individuals, but depends on the distribution of RHP among the population at large. The greater the difference between an individual’s and its opponent’s RHP perception, the more likely it is to win a contest; however, if it overestimates its RHP, then the cost of fighting increases with the overestimate. A winner or loser effect exists only if the fitness gain of the beta individual in a hierarchy, relative to that of the alpha, is less than 0.5. Then a loser effect can exist alone, or it can coexist with a winner effect; however, there cannot exist a winner effect without a loser effect.     

Ecotypic variation in the asymmetric Hawk-Dove game: When is Bourgeois an evolutionarily stable strategy?

Summary This paper develops a mathematical model of an iterated, asymmetric Hawk-Dove game with the novel feature that not only are successive pairs of interactants — in the roles of owner and intruder contesting a site — drawn randomly from the population, but also the behaviour adopted at one interaction affects the role of a contestant in the next. Under the assumption that a site is essential for reproduction, the evolutionarily stable strategy (ESS) of the population is found to depend on the probability, w, that the game will continue for at least a further period (which is inversely related to predation risk), and five other parameters; two of them are measures of site scarcity, two are measures of fighting costs, and the last is a measure of resource holding potential (RHP). Among the four strategies — Hawk (H), Dove (D), Bourgeois (B) and anti-Bourgeois (X) — only D is incapable of being an ESS; and regions of parameter space are found in which the ESS can be only H, or only X, or only B; or either H or X; or either X or B; or either H or B; or any of the three. The scarcer the sites or the lower the costs of fighting, or the lower the value of w, the more likely it is that H is an ESS; the more abundant the sites or the higher the costs of fighting, or the higher the value of w, the more likely it is that X or B is an ESS. The different ESSs are interpreted as different ecotypes. The analysis suggests how a non-fighting population could evolve from a fighting population under decreasing risk of predation. If there were no RHP, or if RHP were low, then the ESS in the non-fighting population would be X; only if RHP were sufficiently high would the ESS be B, and the scarcer the sites, the higher the RHP would have to be. These conclusions support the thesis that if long-term territories are essential for reproduction and sites are scarce, then ownership is ruled out not only as an uncorrelated asymmetry for settling disputes in favour of owner, but also as a correlated asymmetry.     

Can transitive inference evolve in animals playing the hawk-dove game?

What should an individual do if there are no reliable cues to the strength of a competitor when fighting with it for resources? We herein examine the evolutionarily stable strategy (ESS) in the hawk-dove game, if the opponent's resource-holding potential (RHP) can only indirectly be inferred from the outcome of past interactions in the population. The strategies we examined include the classical mixed strategy in which no information on past games is utilized, the 'imprinting' strategy in which a player increases/decreases its aggressiveness if it wins/loses a game, the 'immediate inference' strategy in which a player can infer the strength of those opponents it fought before, and the 'transitive inference' strategy in which a player can infer the strength of a new opponent through a third party with which both players have fought before. Invasibility analysis for each pair of strategies revealed that (i) the transitive-inference strategy can always invade the mixed strategy and the imprinting strategy, and itself refuses invasion by these strategies; (ii) the largest advantage for transitive inference is achieved when the number of games played per individual in one generation is small and when the cost of losing an escalated game is large; (iii) the immediate inference, rather than the transitive inference, can be an ESS if the cost of fighting is small; (iv) a strong linear ranking is established in the population of transitive-inference strategists, though it does not perfectly correlate to the ranking by actual RHPs. We found that the advantage of the transitive inference is not in its ability to correct a misassessment (it is actually the worst in doing so), but in the ability of quickly lining up either incorrect or correct assessments to form a linear dominance hierarchy.     

The Hawk—Dove game revisited: Effects of continuous variation in resource-holding potential on the frequency of escalation

Summary The classic Hawk—Dove game is extended to deal with continuous variation in resource-holding potential or RHP, when RHP is observable (via any sensory modality) but RHP difference is less than perfectly reliable as a predictor of the outcome of an escalated contest. The relationship between sensory and physical magnitudes of RHP is assumed to be governed by Fechner's psychophysical law, whose effect is that contestants interact as if they had perfect information about their relative RHP (as opposed to RHP difference). Thus, an animal is aggressive if its RHP exceeds a certain fraction, called its threshold, of its opponent's RHP and otherwise is non-aggressive; and the classic Hawk and Dove strategies correspond to zero and infinite thresholds, respectively. For RHPs drawn at random from an arbitrary Gamma distribution there is a unique evolutionarily stable strategy or ESS, which depends on a parameter measuring the reliability of RHP as a predictor of the outcome of a fight, on the ratio of the valueV of winning to the costC of losing (both measured in units of reproductive fitness) and on the mean µ and variance ² of the RHP distribution. In a population at this ESS, ifV/C < 1 then the threshold is 1 and there is no fighting. AsV/C increases beyond 1 to a second critical value , however, the threshold decreases steadily from 1 to 0 and remains 0 forV/C > ; is an increasing function of , but a decreasing function of ². That a lower variance of RHP can imply a lower escalation frequencyp is a novel insight of the analysis. The prediction is at first counterintuitive, because if the aggression threshold were fixed then larger variance would imply lowerp (dispersion effect of variance). When natural selection acts on the threshold, however, increasing the variance not only reduces the probability that an animal with larger RHP will be attacked by an animal with lower RHP at the existing threshold, but also reduces the expected costs of adopting that particular threshold, so that a mutant with a somewhat lower threshold can invade the population (selection effect of variance). Forp, the selection effect dominates toward the upper end of the interval 1 V/C .     

Toward a theory of dominance hierarchies: effects of assessment, group size, and variation in fighting ability

We introduce assessment to the analysis of dominance hierarchiesby exploring the effect of an evolutionarily stable fightingrule when there is variation in resource holding potential (RHP)and RHP is not a perfectly reliable predictor of the outcomeof a fight. With assessment, the probability of a linear hierarchydecreases with group size but can remain appreciable for groupsof up to seven or eight individuals, whereas it decreases virtuallyto zero if there is no assessment. The probability of a hierarchythat correlates perfectly with RHP is low unless group sizeis small.     

Efficacy and honesty in communication between relatives

Abstract The evolution of honest communication has recently become the focus of intense theoretical attention. However, strategic models dealing with honesty have largely ignored the implications of noise and perceptual error for signal evolution (just as models dealing with signal detection in the presence of noise ignore strategic issues). Here, I analyze an extended version of Maynard Smith's strategic model of signaling of need between relatives, the Philip Sidney game, that incorporates the possibility of perceptual error. I show that even in the presence of noise, there exists over a wide range of parameter values a unique, continuously stable signaling equilibrium, at which the signaler employs a costly display when needy but refrains from doing so when healthy. For a subset of this range, there also exists a second, lower cost signaling equilibrium that is not continuously stable. At the former equilibrium, predicted signal cost is inversely related to the coefficient of relatedness (r) between signaler and receiver. Cost is not, however, predicted to drop to zero even when r = 1 and there is no conflict of interest between the two (as is the case in errofree models), because it serves to enhance the efficacy of communication as well as to discourage deceit. Equilibrium signal cost is inversely related to the probability that the signaler is needy, and tends to increase with the level of noise. If noise becomes too great (i.e., if a detectable signal is too costly to produce), signaling is no longer stable; surprisingly, it is also unstable if the level of noise is too low (i.e., if a detectable signal is too cheap to produce).     

Cyclic Game Dynamics Driven by Iterated Reasoning

Recent theories from complexity science argue that complex dynamics are ubiquitous in social and economic systems. These claims emerge from the analysis of individually simple agents whose collective behavior is surprisingly complicated. However, economists have argued that iterated reasoning–what you think I think you think–will suppress complex dynamics by stabilizing or accelerating convergence to Nash equilibrium. We report stable and efficient periodic behavior in human groups playing the Mod Game, a multi-player game similar to Rock-Paper-Scissors. The game rewards subjects for thinking exactly one step ahead of others in their group. Groups that play this game exhibit cycles that are inconsistent with any fixed-point solution concept. These cycles are driven by a “hopping” behavior that is consistent with other accounts of iterated reasoning: agents are constrained to about two steps of iterated reasoning and learn an additional one-half step with each session. If higher-order reasoning can be complicit in complex emergent dynamics, then cyclic and chaotic patterns may be endogenous features of real-world social and economic systems.     

Some game-theoretical aspects of parasitism and symbiosis

Given a two-person continuous non-constant sum game, a “solution” can be arrived at even in ignorance of the pay-off functions as each player tries to maximize his own pay-off by trial and error. The solution of the game considered here is either the intersection of two “optimal lines” (the case of the stable equilibrium) or parasitism of one player upon the other (the case of unstable equilibrium). The introduction of secondary satisfactions (linear combinations of the pay-offs) affects both the position and the stability of the equilibrium and thus the resulting primary satisfactions (the pay-offs). These effects of the matrix of transformation are investigated. In particular, it is shown that the social optimum, which is also the solution of the game where collusion is allowed, is attained if and only if the columns of the matrix are identical, and that this solution is generally stable. Some suggested connections with biological situations are discussed.     

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.     

Assessment strategy and the evolution of fighting behaviour

The view is examined that the adaptive value of conventional aspects of fighting behaviour is for assessment of relative RHP (resource holding power) of the combatants. Outcomes of aggressive disputes should be decided by each individual's fitness budget available for expenditure during a fight (determined by the fitness difference between adoption of alternative strategies, escalation or withdrawal without escalation) and on the rate of expenditure of the fitness budget if escalation occurs (determined by the RHPs of the combatants). Thus response thresholds for alternative strategies (“assessments”) will be determined by natural selection on a basis of which opponent is likely to expend its fitness budget first, should escalation occur. This “loser” should retreat (before escalation) and the winner should stay in possession of the resource. Many aggressive decisions depend on whether one is a resource holder, or an attacker. Assuming the RHP of the combatants to be equal, there are many instances of fitness pay-off imbalances between holder and attacker which should weight the dispute outcome in favour of one or other opponent by allowing it a greater expendable fitness budget. Usually the weighting favours the holder; the attacker therefore needs a correspondingly higher RHP before it may be expected to win. This is not invariably the case, and much observed data fits the predictions of this sort of model. If assessments are perfect and budget expenditure rates exactly predictable, then there would never seem to be any case for escalation. Escalation can be explained in terms of injury inflictions (expenditures) occurring as discrete events; i.e. as “bouts” won or lost during fighting. Assessment can give only a probabilistic prediction of the outcome of a bout. A simple model is developed to investigate escalation situations. Each combatant assesses relative RHP; this correlates with an absolute probability of winning the next bout (c_abs). The stake played for is infliction of loss of RHP and is determined by the fitness budgets of the opponents. (Each individual plays for the withdrawal of its opponent.) This defines a critical probability of winning (c_crit) for each combatant, above which escalation is the favourable strategy (c_abs > c_crit) and below which withdrawal is favourable (c_abs < c_crit). Escalation should occur only where c_abs-c_crit is positive for both combatants. This model gives predictions compatible with the observations, indicating that RHP loss alone can be adequate to explain withdrawal: escalation behaviour. Withdrawal tendency will be increased by low searching costs. Escalations should be restricted to closely matched RHP opponents if RHP disparity is the major imbalance. Outside the “escalation range” of a given individual, the higher RHP individual wins and the lower one loses (i.e. it should withdraw after conventional display). RHP disparity and holder: attacker imbalance should both interact to shape the observed pattern, though their relative importances will depend on species and situation. In some instances selection may favour immediate withdrawal from an occupied territory even without assessment of RHP.     

The mismeasure of animal contests

Contests between rivals placing similar value on the resource at stake are commonly won by the rival having greater ‘resource holding potential’ (RHP). Mutual assessment of RHP difference between rivals is usually expected as an economical means of resolution; weaker rivals can retreat when they detect their relative inferiority, thereby avoiding costly, futile persistence. Models of contest resolution that entail retreat decisions based on estimates of RHP difference predict that contest duration diminishes as RHP difference between rivals increases because the asymmetry is more readily detected. This prediction appears to have been fulfilled in contests of diverse taxa, generating widespread support for assessment of RHP differences in contests. But few studies have considered alternatives in which each rival simply persists in accord with its own RHP (‘own RHP-dependent persistence’). In contests decided by own RHP-dependent persistence, in which costs accrue only through each rival's own actions, weaker rivals retreat first because they are inherently less persistent, and contest duration depends primarily on the weaker (losing) rival's RHP rather than RHP difference between the rivals. We show here that the analyses most commonly used to detect effects of RHP difference cannot discriminate between these alternatives. Because RHP difference between rivals tends to be correlated with RHP of the weaker rival in a pair, a negative relation between RHP difference and contest duration may be generated even when decisions of retreat are not based on estimated RHP difference. Many studies purporting to show a negative relation between RHP difference and contest duration may actually reflect an incidental association between weaker rival RHP and RHP difference. We suggest statistical and experimental approaches that may help to discriminate between effects of weaker rival RHP and true effects of RHP difference. We also discuss whether ‘true’ negative effects of RHP difference on contest duration always reflect retreat decisions based on estimated RHP differences. Copyright 2003 Published by Elsevier Science Ltd on behalf of The Association for the Study of Animal Behaviour.      

Resource holding potential, subjective resource value, and game theoretical models of aggressiveness signalling

Empirical evidence suggests that aggressiveness (willingness to enter into, or escalate an aggressive interaction) may be more important than the ability to win fights in some species. Both empirical and theoretical traditions treat aggressiveness as a distinct property from the ability (RHP) or motivation (subjective resource value) to win a fight. I examine how these three traits are clearly distinct when modelled using a simple strategic model of escalation. I then examine game theoretical models of agonistic communication and demonstrate that models in which aggressiveness is signalled require: (1) a trait, aggressiveness, which is neither a correlate, nor consequence of RHP or motivation, (2) a handicap which negates any benefit to be gained through the use of a particular signal, and (3) the absence of any other asymmetry which could be used to assign roles to players. I conclude that it is unlikely that these assumptions are ever met, and that empirical examples of "aggressiveness" are far more likely to represent long-term differences in subjective resource value.     

Age‐dependent male mating tactics in a spider mite—A life‐history perspective

Males often fight with rival males for access to females. However, some males display nonfighting tactics such as sneaking, satellite behavior, or female mimicking. When these mating tactics comprise a conditional strategy, they are often thought to be explained by resource holding potential (RHP), that is, nonfighting tactics are displayed by less competitive males who are more likely to lose a fight. The alternative mating tactics, however, can also be explained by life‐history theory, which predicts that young males avoid fighting, regardless of their RHP, if it pays off to wait for future reproduction. Here, we test whether the sneaking tactic displayed by young males of the two‐spotted spider mite can be explained by life‐history theory. We tested whether young sneaker males survive longer than young fighter males after a bout of mild or strong competition with old fighter males. We also investigated whether old males have a more protective outer skin—a possible proxy for RHP—by measuring cuticle hardness and elasticity using nanoindentation. We found that young sneaker males survived longer than young fighter males after mild male competition. This difference was not found after strong male competition, which suggests that induction of sneaking tactic is affected by male density. Hardness and elasticity of the skin did not vary with male age. Given that earlier work could also not detect morphometric differences between fighter and sneaker males, we conclude that there is no apparent increase in RHP with age in the mite and age‐dependent male mating tactics in the mite can be explained only by life‐history theory. Because it is likely that fighting incurs a survival cost, age‐dependent alternative mating tactics may be explained by life‐history theory in many species when reproduction of old males is a significant factor in fitness.     

Persistence conditions of symmetric social hybridogenesis in haplo-diploid hymenoptera

In eusocial Hymenoptera species, females variably develop into either alate females (queens) or workers, and in most cases, caste differentiation is determined environmentally. Recently, however, female castes in two harvester ant species, Pogonomyrmex rugosus and P. barbatus, were found to be determined genetically in hybrid zones of these two species. In the hybrid populations, homozygous females (e.g. AA or BB) and heterozygous females (AB) develop into alate females and workers, respectively. This genetic caste determination system is called symmetric social hybridogenesis (SSH). It is clear that populations with SSH can persist only if all four genotypes (AA and BB females, and A and B males) coexist simultaneously. However, it is not obvious that these populations are always persistent when the four genotypes simultaneously exist. Here, we examined the stability and persistence of an SSH population using a simple mathematical model. According to the analysis of the model, the SSH population persists only when some conditions are satisfied: (1) each female mates with more than two males, and (2) male production increases less steeply than linearly with increasing numbers of workers in a colony, and alate female production increases more steeply than linearly with increasing numbers of workers, or (2') male production increases more steeply than linearly with increasing numbers of workers in a colony, and alate female production increases much more steeply than male production. Therefore, it is not obvious that SSH populations are maintained and are stable for long periods. We discuss whether these conditions are satisfied in real SSH populations.     

Evolutionary routes to stable ownership

Ownership can evolve in potentially any species. Drawing on insights from across disciplines, we distinguish between possession and ownership and present species‐neutral criteria for ownership, defined as respect for possession. We use a variant of the tug‐of‐war evolutionary game to demonstrate how ownership can evolve in the form of a new, biologically realistic strategy, Restraint With Retaliation (RWR). In our game, resource holding potential (RHP) is assumed to be equal between interactants, and resource holding asymmetry determines whether ownership is adaptive. RWR will be evolutionarily stable when the ratio of resource holdings between interactants is relatively low, but not when this ratio is sufficiently high. We offer RWR as one evolutionary route to ownership among many, and discuss how ownership unites previously described behavioural phenomena across taxa. We propose that some but not all mechanisms of territory formation and maintenance can be considered ownership, and show that territories are not the only resources that can be owned. We argue that ownership can be a powerful cooperative solution to tragedies of the commons and problems of collective action throughout the biological world. We advance recent scholarship that has begun to investigate the biological importance of ownership, and we call for a comprehensive account of its evolutionary logic and taxonomic distribution. We propose that ownership should be considered a fundamental, unifying biological phenomenon.     

Intruder Pressure Affects Territory Size and Foraging Success in Asymmetric Contests in the Water Strider Gerris lacustris

The theory of evolutionarily stable strategies (ESS) in asymmetric contests predicts that the propensity of an individual to expose itself to risk during contests depends on the individual's resource-holding-potential (RHP) and on the value of the disputed resource (V) for the individual compared with that for an opponent. If encounters of a territory owner with individuals of high RHP and high food demands (V) increase in frequency, one should expect a decrease in total aggressiveness of the territory owner, and in consequence a decrease of its territory size. Such a decrease should result in a lower amount of food consumed by the territory owner. Using natural variability in RHP and V in Gerris lacustris, I experimentally tested these predictions. The average prey item has higher value (V) for reproductive female water striders (which probably transform most of their food into eggs), than for nonreproductive females and for males. Because males are smaller, they have lower RHP than females, as RHP depends on size. Thus the reproductive females are the class of individuals of high RHP and high food demands (high V). Most nonreproductive females defend food-based territories. I observed two groups of water striders in a seminatural laboratory setting. As predicted, there was a negative correlation between the rate of encounter with reproductive females and size of the territory, and a positive correlation between territory size and number of Drosophila flies consumed by the owner. Territories were smaller in the group with high rates of encounter between territory owners and reproductive females. Territory owners caught the same number of Drosophila flies as non-territorial individuals in this group. In contrast, in the group with fewer encounters between territory owners and reproductive females, territories were larger, and territory owners gained more food than non territorial water striders.     

The cost of dishonesty

The handicap principle states that stable biological signals must be honest and costly to produce. The cost of the signal should reflect the true quality of the signaller. Here, it is argued that honest signalling may be maintained although the used signals are not handicaps. A game theoretic model in the form of a game of signalling is presented: all the existing evolutionarily stable strategies (ESSs) are found. Honest and cheap signalling of male quality is shown to be evolutionarily stable if females divorce the mate if it turns out that he has cheated about his quality. However, for this ESS to apply, the cost of lost time must not be too great. The stability of the honest signalling is based on deceivers being prevented from spreading in the population because they suffer from a cost of divorce. Under some fairly strict conditions, a mixed polymorphism of dishonesty and honesty represents another possible ESS.     

Evolutionarily stable strategy in a sex- and frequency-dependent selection model

In this paper, a sex-dependent matrix game haploid model is investigated. For this model, since the phenotypes of female and male individuals are determined by alleles located at a single locus and are sex dependent, any given genotype corresponds to a strategy pair. Thus, a strategy pair is an ESS if and only if the allele corresponding to this strategy pair cannot be invaded by any mutant allele. We show that an ESS equilibrium must be locally asymptotically stable if it exists.