Cheating as a mixed strategy in a simple model of aggressive communication

The possibility that frequency-dependent cheating can persist in an evolutionarily stable communication system has frequently been proposed. Although there is empirical evidence for this idea, however, it has not been investigated in terms of game theory. In the present paper I show for a simple symmetric game that cheating can be part of a mixed evolutionarily stable strategy (ESS). Furthermore, despite the widespread assumption that cheaters must be rare, I show that most of the population can be cheaters, while the signalling system remains evolutionarily stable. Consequences for signalling theory and experiments to detect such mixed ESS are discussed. Copyright 2000 The Association for the Study of Animal Behaviour.     

The continuous Sir Philip Sidney game: a simple model of biological signalling.

An analysis of Maynard Smith's two-player, ESS model of biological signalling, the "Sir Philip Sidney game", is presented. The stable strategies of the players in this game are shown to satisfy the conditions of Zahavi's handicap principle. At equilibrium, signals are honest, costly, and costly in a way that is related to the true quality revealed. Further analysis reveals that the level of cost required to maintain stability is inversely related to the degree of relatedness between the players. It therefore seems likely that stable biological signalling systems will feature lower signalling costs when communication occurs between relatives. A three-player, extended version of the model is investigated, in which signals are passed via an intermediate, or "messenger". It is shown that this destabilizes the signalling system, and leads to increased signalling costs. This result suggests that "kin conflict" theories of the evolution of the endosperm in flowering plants require further refinement. The introduction of a novel resource acquisition tissue, which mediates parent-offspring interaction during development, cannot be assumed to limit parent-offspring conflict simply because it carries an extra copy of the maternally inherited genes. The ability to add such complications to the Sir Philip Sidney game and still obtain solutions makes it a very useful modelling tool.     

Threat displays are not handicaps

Within a general framework of handicap signalling it was proposed that threat displays are handicaps, they can work only if they put the signaller at a disadvantage, which is only acceptable to honest signallers. The aim of the present article is to investigate this proposal with the help of a simple game-theoretical model. It was found: (1) that the use of cost-free signals is an ESS against the invasion of handicapped signals even if cheating is played as part of a mixed strategy in the population; (2) that the use of handicaps may be an ESS against cost-free signals but only if we assume that the invading cost-free signal is not accepted by weak individuals as a signal of strength; (3) that the establishment of a handicapped signal in the first place is an unresolved problem, because both cost free signals and negative-handicaps are evolutionarily stable against the invasion of handicaps; (4) that in contrast to handicaps the use of negative-handicaps can invade a population using cost-free signals (a negative-handicap is a signal which may serve other functions as well); (5) that negative-handicaps are ESS against cost-free signals as well as against handicaps; and (6) thus, the most likely evolutionary end point is that the biggest negative-handicap would be used as a threat display. This is a posture, which prepares the animal most efficiently to fight; hence, most probably it is the initial position of the fighting technique of the given species. (7) Finally, the investigation of the threat displays of well-studied taxa (great tit, cats, dogs, and hoofed mammals) confirms that threat displays are indeed negative-handicaps. They do not put the user into a disadvantaged position, instead the initial position of the species specific fighting technique is used as a threat display as predicted by the present model.     

Revisiting matrix games: the concept of neighborhood invader strategies

We extend the concept of neighborhood invader strategy (NIS) to finite-dimensional matrix games and compare this concept to the evolutionarily stable strategy (ESS) concept. We show that these two concepts are not equivalent in general. Just as ESS's may not be unique, NIS's may also not be unique. However, if there is an ESS and a NIS then these strategies must be the same. We show that an ESNIS (an ESS and NIS) for any matrix game is unique and that a mixed ESS with full support is a NIS. Thus a mixed ESS with full support is not invadable by any pure or mixed strategy and it can invade any pure or mixed strategy. An ESS which is an ESNIS, therefore, has better chance of being established evolutionarily through dynamic selection.     

Parasites and supernormal manipulation.

Social parasites may exploit their hosts by mimicking other organisms that the hosts normally benefit from investing in or responding to in some other way. Some parasites exaggerate key characters of the organisms they mimic, possibly in order to increase the response from the hosts. The huge gape and extreme begging intensity of the parasitic common cuckoo chick (Cuculus canorus) may be an example. In this paper, the evolutionary stability of manipulating hosts through exaggerated signals is analysed using game theory. Our model indicates that a parasite's signal intensity must be below a certain threshold in order to ensure acceptance and that this threshold depends directly on the rate of parasitism. The only evolutionarily stable strategy (ESS) combination is when hosts accept all signallers and parasites signal at their optimal signal intensity, which must be below the threshold. Supernormal manipulation by parasites is only evolutionarily stable under sufficiently low rates of parasitism. If the conditions for the ESS combination are not satisfied, rejector hosts can invade using signal intensity as a cue for identifying parasites. These qualitative predictions are discussed with respect to empirical evidence from parasitic mimicry systems that have been suggested to involve supernormal signalling, including evicting avian brood parasites and insect-mimicking Ophrys orchids.     

Attention-Seeking Displays

Animal communication abounds with extravagant displays. These signals are usually interpreted as costly signals of quality. However, there is another important function for these signals: to call the attention of the receiver to the signaller. While there is abundant empirical evidence to show the importance of this stage, it is not yet incorporated into standard signalling theory. Here I investigate a general model of signalling - based on a basic action-response game - that incorporates this searching stage. I show that giving attention-seeking displays and searching for them can be an ESS. This is a very general result and holds regardless whether only the high quality signallers or both high and low types give them. These signals need not be costly at the equilibrium and they need not be honest signals of any quality, as their function is not to signal quality but simply to call the attention of the potential receivers. These kind of displays are probably more common than their current weight in the literature would suggest.     

Stability in an evolutionary game

A behavior or strategy which is evolutionarily stable must be both optimal and stable. The strategy must be optimal in that it maximizes the expected fitness of all the individuals using it. In addition, the strategy must be resistant to invasion by a mutant. The difference between the Nash solution of game theory and the ESS used in ecology is that the Nash solution only satisfies an optimality criterion and not an evolutionary stability criterion. We extend the ESS definition of Maynard Smith and Price so that it can be applied directly to two-strategy evolutionary games. The concept of a balanced game is introduced, and necessary conditions are derived which are similar to the Nash necessary conditions. The balanced game necessary conditions may be used for direct calculation of ESS candidates. These results are used to examine the optimal flowering time of an annual plant experiencing competition from neighboring plants. The plant competition model is general, and the results may be applied to a wide range of interference competition problems.     

Error-prone signalling.

The handicap principle of Zahavi is potentially of great importance to the study of biological communication. Existing models of the handicap principle, however, make the unrealistic assumption that communication is error free. It seems possible, therefore, that Zahavi's arguments do not apply to real signalling systems, in which some degree of error is inevitable. Here, we present a general evolutionarily stable strategy (ESS) model of the handicap principle which incorporates perceptual error. We show that, for a wide range of error functions, error-prone signalling systems must be honest at equilibrium. Perceptual error is thus unlikely to threaten the validity of the handicap principle. Our model represents a step towards greater realism, and also opens up new possibilities for biological signalling theory. Concurrent displays, direct perception of quality, and the evolution of 'amplifiers' and 'attenuators' are all probable features of real signalling systems, yet handicap models based on the assumption of error-free communication cannot accommodate these possibilities.     

From reciprocity to unconditional altruism through signalling benefits

Cooperation among genetically unrelated individuals is commonly explained by the potential for future reciprocity or by the risk of being punished by group members. However, unconditional altruism is more difficult to explain. We demonstrate that unconditional altruism can evolve as a costly signal of individual quality (i.e. a handicap) as a consequence of reciprocal altruism. This is because the emergent correlation between altruism and individual quality in reciprocity games can facilitate the use of altruism as a quality indicator in a much wider context, outside the reciprocity game, thus affecting its further evolution through signalling benefits. Our model, based on multitype evolutionary game theory shows that, when the additive signalling benefit of donating help exceeds the cost for only some individuals (of high-quality state) but not for others (of low-quality state), the population possesses an evolutionarily stable strategy (ESS) profile wherein high-quality individuals cooperate unconditionally while low-quality individuals defect or play tit-for-tat (TfT). Hence, as predicted by Zahavi's handicap model, signalling benefits of altruistic acts can establish a stable generosity by high-quality individuals that no longer depends on the probability of future reciprocation or punishment.     

Evolutionarily stable communication between kin: a general model

At present, the most general evolutionary theory of honest communication is Grafen''s model of Zahavi''s ''handicap'' signalling system, in which honesty of signals about the signaller''s quality (e.g. mate suitability or fighting ability) is maintained by the differentially high cost of signals to signallers having lower quality. The latter model is here further generalized to include any communication between signallers and receivers that are genetically related (e.g. parents and begging offspring, cooperative or competing siblings). Signalling systems involving relatives are shown to be evolutionarily stable, despite a potential pay-off for false signalling, if the Zahavian assumption of differential signal costs holds and there are diminishing reproductive returns to the signaller as the receiver''s assessed value of its attribute increases, or if, regardless of whether the Zahavian assumption holds, signallers with high values of the attribute benefit more from a given receiver assessment than signallers with low values (e.g. begging chicks that are hungrier benefit more from being fed). In stable systems of signalling among kin, it is also shown to be generally true that (i) levels of signalling and thus observed signal costs will decline as relatedness increases or as the receiver''s reproductive penalty for erroneous assessment increases, and (ii) receivers will consistently, altruistically overestimate the true value of the signalled attribute.     

Signalling among relatives. I. Is costly signalling too costly?

Zahavi''s handicap principle,originally proposed as an explanation for sexual selection ofelaborate male traits, suggests that a sufficient cost to dishonest signals can outweigh the rewards of deception and allow individuals to communicate honestly. Maynard Smith (1991) and Johnstone and Grafen (1992) introduce the Sir Philip Sidney game in order to extend the handicap principle to interactions among related individuals, and to demonstrate that stable costly signalling systems can exist among relatives.In this paper we demonstrate that despite the benefits associated with honest information transfer, the costs incurred in a stable costly signalling system may leave all participants worse off than they would be in a system with no signalling at all. In both the discrete and continuous forms of the Sir Philip Sidney game, there exist conditions under which costly signalling among relatives, while stable, is so costly that it is disadvantageous compared with no signalling at all. We determine the factors which dictate signal cost and signal benefit in a generalized version of this game, and explain how signal cost can exceed signal value. Such results raise concerns about theevolutionary pathways which could have led to the existence of signalling equilibria in nature. The paper stresses the importance of comparing signalling equilibria with other possible strategies, beforedrawing conclusions regarding the optimality of signalling.     

Sexual Signals as Advertisers of Resistance to Mistakes

To be evolutionarily stable, sexual signals need to convey information regarding the quality or motivation of individuals. This can be achieved through direct or indirect signal costs that guarantee honest communication or through signal designs that facilitate the assessment of quality. Here, I present the case for an alternative hypothesis: that signalling exposes informative mistakes. Signalling mistakes range from occasional to frequent and from slight distortions of typical signals to grossly atypical signals. Their occurrence may be enhanced by disease or stress, thus revealing individual quality or motivation, and receivers typically respond negatively to them. By this mechanism, honest communication is due to costs of developing resistance to mistakes. Therefore, the hypothesis can function independently of signal design costs, although it can also be enhanced by signal design costs when those increase the occurrence of mistakes. This hypothesis widens the scope of signals expected to be sexually selected and creates new approaches to research in sexual selection and animal communication.     

On evolutionarily stable sets

As an extension of the concept of an evolutionarily stable strategy (ESS) evolutionarily stable sets are introduced, i.e. sets of equilibrium strategies (EQS) which have much of the properties of an ESS. They are primarily used with evolutionary game models that allow a continuum of EQSs, none of which can be an ESS, but also include common ESSs as a special case. For a large class even of nonlinear models it can be shown that the standard dynamics converge towards some equilibrium point in an ES set if started within a neighbourhood of the set. Important applications of ES sets include e.g. mixed-strategist models and evolutionary game models in sexual populations.     

Cheap talk when interests conflict

Most evolutionary analyses of animal communication suggest that low-cost signals can evolve only when both the signaller and the recipient rank outcomes in the same order. When there is a conflict of interest between sender and receiver, honest signals must be costly. However, recent work suggests that low-cost signals can be evolutionarily stable, even when the sender and the receiver rank outcomes in different orders, as long as the interest in achieving coordination is sufficiently great. In this paper, we extend this body of work by analysing a game theory model that shows that low-cost signals can evolve when there are conflicts of interest and no interest in coordination, as long as individuals interact repeatedly. We also present an empirical example indicating that female rhesus macaques, Macaca mulatta, use honest, low-cost, vocal signals to facilitate interactions when conflicts of interest exist. Copyright 2000 The Association for the Study of Animal Behaviour.     

When will evolution lead to deceptive signaling in the Sir Philip Sidney game?

Sir Philip Sidney games are a widely used model of simple signaling. Johnstone and Grafen [Johnstone, R.A., Grafen, A., 1993. Dishonesty and the handicap principle. Animal Behaviour 46, 759–764] present a version in which the Evolutionarily Stable Strategy (ESS) is for most signalers to “honestly” signal, with a small minority of signalers who “cheat”. This model is among the most frequently cited papers on the topic of “dishonest” signaling and supports the view that signals may be “dishonest” as long as they are “honest on average”. Using genetic algorithms, we demonstrate that another solution exists to the game, an evolutionarily stable set of Nash equilibria in which members of the set never signal and all donors give their resource. Payoffs to players using this set of strategies is greater those when playing the “dishonest” signaling ESS. We demonstrate that a random population is far more likely to evolve to this non-communicating strategy set than the “dishonest” signaling ESS. We also discuss the dynamics of biological game theory models and the advances of genetic algorithms as a heuristic solution method for these models.     

Evolutionary matrix games and optimization theory

An evolutionarily stable strategy (ESS) is only required to be capable of resisting invasion by rare mutant strategies. In contrast, an absolute invader strategy (AIS) is a rare mutant strategy that can invade any established strategy. We show that the predictions of the outcome of evolution made by optimization models are compatible with those made by the classical expected payoff comparisons in matrix games. We also show that if a matrix game has an AIS that AIS is unique and is also an ESS. But an ESS need not be an AIS. In pure-strategy submodels, an AIS need not be unique. An AIS of a matrix game has global asymptotic stability property in the game dynamics which involve only pure strategies including the AIS.     

An ESS maximum principle for matrix games

Previous work has demonstrated that for games defined by differential or difference equations with a continuum of strategies, there exists a G-function, related to individual fitness, that must take on a maximum with respect to a virtual variable v whenever v is one of the vectors in the coalition of vectors which make up the evolutionarily stable strategy (ESS). This result, called the ESS maximum principle, is quite useful in determining candidates for an ESS. This principle is reformulated here, so that it may be conveniently applied to matrix games. In particular, we define a matrix game to be one in which fitness is expressed in terms of strategy frequencies and a matrix of expected payoffs. It is shown that the G-function in the matrix game setting must again take on a maximum value at all the strategies which make up the ESS coalition vector. The reformulated maximum principle is applicable to both bilinear and nonlinear matrix games. One advantage in employing this principle to solve the traditional bilinear matrix game is that the same G-function is used to find both pure and mixed strategy solutions by simply specifying an appropriate strategy space. Furthermore we show how the theory may be used to solve matrix games which are not in the usual bilinear form. We examine in detail two nonlinear matrix games: the game between relatives and the sex ratio game. In both of these games an ESS solution is determined. These examples not only illustrate the usefulness of this approach to finding solutions to an expanded class of matrix games, but aids in understanding the nature of the ESS as well.     

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.     

Biological signals as handicaps

An ESS model of Zahavi's handicap principle is constructed. This allows a formal exposition of how the handicap principle works, and shows that its essential elements are strategic. The handicap model is about signalling, and it is proved under fairly general conditions that if the handicap principle's conditions are met, then an evolutionarily stable signalling equilibrium exists in a biological signalling system, and that any signalling equilibrium satisfies the conditions of the handicap principle. Zahavi's major claims for the handicap principle are thus vindicated. The place of cheating is discussed in view of the honesty that follows from the handicap principle. Parallel signalling models in economics are discussed. Interpretations of the handicap principle are compared. The models are not fully explicit about how females use information about male quality, and, less seriously, have no genetics. A companion paper remedies both defects in a model of the handicap principle at work in sexual selection.     

Properties of a mixed ESS candidate in continuous strategy sets

An evolutionarily stable strategy (ESS) is a strategy that if almost all members of the population adopt, then this population cannot be invaded by any mutant strategy. An ESS is not necessarily a possible end point of the evolutionary process. Moreover, there are cases where the population evolves towards a strategy that is not an ESS. This paper studies the properties of a unique mixed ESS candidate in a continuous time animal conflict. A member of a group sized three finds itself at risk and needs the assistance of another group member to be saved. In this conflict, a player's strategy is to choose the probability distribution of the interval between the beginning of the game and the moment it assists the player which is at risk. We first assume that a player is only allowed to choose an exponential distribution, and show that in this case the ESS candidate is an attracting ESS; the population will always evolve towards this strategy, and once it is adopted by most members of the population it cannot be invaded by mutant strategies. Then, we extend the strategy sets and allow a player to choose any continuous distribution. We show that although this ESS candidate may no longer be an ESS, under fairly general conditions the population will tend towards it. This is done by characterizing types of strategies that if established in the population, can be invaded by this ESS candidate, and by presenting possible paths of transition from other types of common strategies to this ESS candidate.