Hawk-Dove Game Research Articles

The Stability of Two-Community Replicator Dynamics with Discrete Multi-Delays

This article investigates the stability of evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. In the real environment, players interact simultaneously while the return of their choices may not be observed immediately, which implies one or more time-delays exists. In addition to using the method of classic characteristic equations, we also apply linear matrix inequality (i.e., LMI) to discuss the stability of the mixed evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. We derive a delay-dependent stability and a delay-independent stability sufficient conditions of the evolutionarily stable strategy in the two-community replicator dynamics with two delays, and manage to extend the sufficient condition to n time delays. Lastly, numerical trials of the Hawk–Dove game are given to verify the effectiveness of the theoretical consequences.

Replicator dynamics for the game theoretic selection models based on state

The paper presents an attempt to integrate the classical evolutionary game theory based on replicator dynamics and the state-based approach of Houston and McNamara. In the new approach, individuals have different heritable strategies; however, individuals carrying the same strategy can differ in terms of state, role or the situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect, the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths), can lead to the change in state of the competing individuals. Special cases of reversible and irreversible incremental stage-structured models, where the state changes can describeenergy accumulation, developmental steps or aging, are derived for discrete and continuous versions. The new approach is illustrated using the example of the Owner-Intruder game with explicit dynamics of the role changes. The new model presents a generalization of the demographic version of the Hawk-Dove game,with the difference being that the opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Here, the Intruders check random nest sites and play the Hawk-Dove game with the Owner if they are occupied. Meanwhile, the Owners produce newborns that become Intruders, since they must find a free nest site to reproduce. An interesting feedback mechanism is produced via the fluxes of individuals between the different subpopulations. In addition, the population growth suppression mechanism resulting from the fixation Bourgeois strategy is analyzed.

Possession is Nine-Tenths of the Law: possession, property, and coordination in a Hawk–Dove Experiment

AbstractIn all legal systems, possession and property are inextricably linked. Game theory captures this relationship in the Hawk–Dove game: players competing for an asset are better off when the possessor plays Hawk and the intruder plays Dove (the bourgeois strategy) so that property can emerge as a spontaneous convention. This theory has been supported by large experimental evidence with animals. This paper presents a lab experiment where possession is manipulated to study the emergence of the property convention with human subjects. We show that the highest coordination emerges when possession is achieved meritoriously and that possession induces only bourgeois coordination (never antibourgeois).

Thermodynamic susceptibility as a measure of cooperative behavior in social dilemmas.

The emergence of cooperation in the thermodynamic limit of social dilemmas is an emerging field of research. While numerical approaches (using replicator dynamics) are dime a dozen, analytical approaches are rare. A particularly useful analytical approach is to utilize a mapping between the spin-1/2 Ising model in 1D and the social dilemma game and calculate the magnetization, which is the net difference between the fraction of cooperators and defectors in a social dilemma. In this paper, we look at the susceptibility, which probes the net change in the fraction of players adopting a certain strategy, for both classical and quantum social dilemmas. The reason being, in statistical mechanics problems, the thermodynamic susceptibility as compared to magnetization is a more sensitive probe for microscopic behavior, e.g., observing small changes in a population adopting a certain strategy. In this paper, we find the thermodynamic susceptibility for reward, sucker's payoff, and temptation in classical Prisoner's Dilemma to be positive, implying that the turnover from defect to cooperate is greater than vice versa, although the Nash equilibrium for the two-player game is to defect. In the classical hawk-dove game, the thermodynamic susceptibility for resource suggests that the number of players switching to hawk from dove strategy is dominant. Entanglement in Quantum Prisoner's Dilemma has a non-trivial role in determining the behavior of thermodynamic susceptibility. At maximal entanglement, we find that sucker's payoff and temptation increase the number of players switching to defect. In the zero-temperature limit, we find that there are two second-order phase transitions in the game, marked by a divergence in the susceptibility. This behavior is similar to that seen in type-II superconductors wherein also two second-order phase transitions are seen.

Analysis of Multilevel Replicator Dynamics for General Two-Strategy Social Dilemma.

Here, we consider a game-theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on multilevel social dilemmas: games in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. While past work on such multilevel dynamics has restricted attention to scenarios with exactly solvable within-group dynamics, we use comparison principles and an invariant property of the tail of the population distribution to extend our analysis to all possible two-player, two-strategy social dilemmas. In the Stag-Hunt and similar games with coordination thresholds, we show that any amount of between-group competition allows for fixation of cooperation in the population. For the prisoners' dilemma and Hawk-Dove game, we characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: If groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups.

The evolution of taking roles

Individuals are randomly matched to play an ex-ante symmetric hawk-dove game. Individuals assume one of a finite set of observable labels and condition their action choice on their opponent’s label. We study the evolutionary stability of chosen labels and their social interaction structure. Evolutionarily stable social structures differ for games in which a dove player prefers the opponent to play hawk (anti-coordination games), and those in which everyone prefers their opponent to play dove (conflict games). Non-trivial hierarchical social structures can only emerge in anti-coordination games. Egalitarian social structures can emerge in both, but are more fragile in conflict games.

Replicator based on imitation for finite and arbitrary networked communities

This paper introduces a novel replicator equations to cover evolutionary games. This replicator is applied on a finite set of agent communities organized on arbitrary graphs. The communities located at the nodes of the graph compete with their neighbours according to the weights of the links that connect them. The communities replicate by imitation probabilities those neighbourhood’s strategies with higher utility. The communities also execute a best response addressed to maximize the entropy associated to imitation probabilities. We explore possible connexions between our replicator equations and The Second Law of Thermodynamics, and prove that populations reach consensus equilibria as expressions of maximum entropy states. We also explore connexions with learning dynamics, and prove that under suitable assumptions and conditions, the communities carry out and intelligent learning process. We illustrate results with an example of the classical hawk-dove game applied on fully-connected and arbitrary populations.

Stability of Replicator Dynamics with Bounded Continuously Distributed Time Delay

In this paper, we consider evolutionary games and construct a model of replicator dynamics with bounded continuously distributed time delay. In many circumstances, players interact simultaneously while impacts of their choices take place after some time, which implies a time delay exists. We consider the time delay as bounded continuously distributed other than some given constant. Then, we investigate the stability of the evolutionarily stable strategy in the replicator dynamics with bounded continuously distributed time delay in two-player game contexts. Some stability conditions of the unique interior Nash equilibrium are obtained. Finally, the simple but important Hawk–Dove game is used to verify our results.

Analysis of direct and indirect genetic effects in fighting sea anemones.

Theoretical models of animal contests such as the Hawk-Dove game predict that variation in fighting behavior will persist due to mixed evolutionarily stable strategies (ESS) under certain conditions. However, the genetic basis for this variation is poorly understood and a mixed ESS for fighting can be interpreted in more than one way. Specifically, we do not know whether variation in aggression within a population arises from among-individual differences in fixed strategy (determined by an individual’s genotype—direct genetic effects [DGEs]), or from within-individual variation in strategy across contests. Furthermore, as suggested by developments of the original Hawk-Dove model, within-individual variation in strategy may be dependent on the phenotype and thus genotype of the opponent (indirect genetic effects—IGEs). Here we test for the effect of DGEs and IGEs during fights in the beadlet sea anemone Actinia equina. By exploiting the unusual reproductive system of sea anemones, combined with new molecular data, we investigate the role of both additive (DGE + IGE) and non-additive (DGE × IGE) genetic effects on fighting parameters, the latter of which have been hypothesized but never tested for explicitly. We find evidence for heritable variation in fighting ability and that fight duration increases with relatedness. Fighting success is influenced additively by DGEs and IGEs but we found no evidence for non-additive IGEs. These results indicate that variation in fighting behavior is driven by additive indirect genetic effects (DGE + IGE), and support a core assumption of contest theory that strategies are fixed by DGEs.

Dominance, sharing, and assessment in an iterated Hawk–Dove game

Animals use a wide variety of strategies to reduce or avoid aggression in conflicts over resources. These strategies range from sharing resources without outward signs of conflict to the development of dominance hierarchies, in which initial fighting is followed by the submission of subordinates. Although models have been developed to analyse specific strategies for resolving conflicts over resources, little work has focused on trying to understand why particular strategies are more likely to arise in certain situations. In this paper, we use a model based on an iterated Hawk–Dove game to analyse how resource holding potentials (RHPs) and other factors affect whether sharing, dominance relationships, or other behaviours are evolutionarily stable. We find through extensive numerical simulations that sharing is stable only when the cost of fighting is low and the animals in a contest have similar RHPs, whereas dominance relationships are stable in most other situations. We also explore what happens when animals are unable to assess each other’s RHPs without fighting, and we compare a range of strategies for contestants using simulations. We find (1) that the most successful strategies involve a limited period of assessment followed by a stable relationship in which fights are avoided and (2) that the duration of assessment depends both on the costliness of fighting and on the difference between the animals’ RHPs. Along with our direct work on modelling and simulations, we develop extensive software to facilitate further testing. It is available at https://bitbucket.org/CameronLHall/dominancesharingassessmentmatlab/.

A single 'weight-lifting' game covers all kinds of games.

Game theory has been studied extensively to answer why cooperation is promoted in human and animal societies. All games are classified into five games: the Prisoner's Dilemma, chicken game (including hawk–dove game), stag hunt game and two trivial games of either all cooperation or all defect, which are studied separately. Here, we propose a new game that covers all five game categories: the weight-lifting game. The player choose either to (1) carry a weight (cooperate: pay a cost) or (2) pretend to carry it (defect: pay no cost). The probability of success in carrying the weight depends on the number of cooperators, and the players either gain the success reward or pay the failure penalty. All five game categories appear in this game depending on the success probabilities for the number of cooperators. We prove that this game is exactly equivalent to the combination of all five games in terms of a pay-off matrix. This game thus provides a unified framework for studying all five types of games.

Analysis of Fallow Farming Decision-Making Behavior of Farmers Based on Hawk-Dove Game Theory: The Case of Guizhou Province

Implementing a land fallow policy is of great significance for ensuring China’s food security and promoting the improvement of the ecological environment. The implementation of a fallow project involves different stakeholders. Farmers are the main participants in the fallow project. The decision of farmers to practice fallow is the key factor for the successful development of the fallow project. Therefore, this study theoretically reveals the decision-making mechanism of farmers’ participation in cultivated land fallow by utilizing the hawk-dove evolutionary game theory among farmers and explains some challenges in the implementation of fallow in Guizhou Province. We drew the following conclusions: (1) The behavior of farmers will be affected by other farmers in the same situation, and the effects of mutual incentives and imitations between the groups of farmers are affected by their interests; (2) in the fallow project, the rate of choosing either fallow or unfallow depends on the ratio of fallow income to planting income. If the income of participating in fallow is higher, the demonstration effect of farmers participating in fallow is stronger, and the strategy of continued cultivation is adopted. The fewer unfallow farmers there are, the more consolidated the results of fallow will be; and (3) the government should protect the income of farmers after fallow as much as possible, implement flexible subsidy policies, and formulate corresponding policies to successfully consolidate the fallow results.

Rapid environmental change in games: complications and counter-intuitive outcomes

Human-induced rapid environmental change (HIREC) has recently led to alterations in the fitness and behavior of many organisms. Game theory is an important tool of behavioral ecology for analyzing evolutionary situations involving multiple individuals. However, game theory bypasses the details by which behavioral phenotypes are determined, taking the functional perspective straight from expected payoffs to predicted frequencies of behaviors. In contrast with optimization approaches, we identify that to use existing game theoretic models to predict HIREC effects, additional mechanistic details (or assumptions) will often be required. We illustrate this in relation to the hawk-dove game by showing that three different mechanisms, each of which support the same ESS prior to HIREC (fixed polymorphism, probabilistic choice, or cue dependency), can have a substantial effect on behavior (and success) following HIREC. Surprisingly, an increase in the value of resources can lead to a reduction in payoffs (and vice versa), both in the immediate- and long-term following HIREC. An increase in expected costs also increases expected payoffs. Along with these counter-intuitive findings, this work shows that simply understanding the behavioral payoffs of existing games is insufficient to make predictions about the effects of HIREC.

Persistence and extinction for stochastic ecological models with internal and external variables.

The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. To understand this interplay, we prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities i.e. stochastic persistence, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, in the case of persistence, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight several challenges for future research.

Possession Is Nine-Tenths of the Law: Bourgeois Behavior, Property and Inequality in a Hawk-Dove Experiment

In all legal systems, possession and property are inextricably linked. Game theory captures this relationship in the Hawk-Dove game. When two contenders compete to appropriate an asset, one of the two most efficient outcomes is achieved when both play the bourgeois strategy: they coordinate in such a way that the incumbent possessor plays Hawk and the intruder plays Dove. In our Hawk-Dove experiment, we test the role of possession with respect to the emergence of such a bourgeois convention. Our treatment variation concerns the way the initial claim to the asset is established. We manipulate the type of information provided and the process of acquisition. We show that i) the highest level of bourgeois coordination emerges whenever the information is based on possession but also when the process is meritorious; ii) possession, both arbitrary and meritorious, induces only bourgeois coordination and never antibourgeois (this is the equally efficient opposite strategy of playing Hawk when the player is the intruder and playing Dove when he is the possessor); and iii) without merit or possession, coordination on a “bullying” equilibrium emerges, generating very inequitable outcomes. Possession thus triggers the establishment of property, and this prevents inequality from arising.

Coexistence of Microbial Species in Structured Communities by Forming a Hawk-Dove Game Like Interactive Relationship.

Microorganisms evolve kinds of elaborate interaction models that can form relatively stable communities in a wide range of ecosystems. It is recognized that the spatial genetic structure of microbes in surface-attached environments lays a good foundation for the persistence of polymicrobial communities in adverse conditions. However, the interacting dynamics of microbes in facilitating the formation and stabilization of community structure still remains elusive. In this study, we identify a hawk-dove game like interspecific relationship between the two Gram-negative opportunistic pathogens Pseudomonas aeruginosa and Klebsiella pneumoniae, which naturally coexist in insect gut and can cocolonize human tissues. Specifically, although P. aeruginosa had significant competitive advantage over cocultured K. pneumoniae on solid medium with rich nutrient factors, K. pneumoniae could resist the suppression of P. aeruginosa by enhancing the expression of membrane transporters induced by the extracellular metabolites of P. aeruginosa. By contrast, under the condition that K. pneumoniae had a growth advantage but P. aeruginosa met a metabolic burden in producing quorum-sensing-controlled extracellular products, the frequency of K. pneumoniae would be slightly higher than P. aeruginosa during the coexistence because K. pneumoniae was also capable of exploiting the extracellular metabolite from P. aeruginosa. In addition, P. aeruginosa quorum-sensing variant could reap benefits from K. pneumoniae in turn and reach a relatively stable two species equilibrium. These findings provide an explanation for the formation and maintenance of polymicrobial communities in different spatially structured environments, and thus may contribute to understanding the complex interspecific interactions of microbes in local communities and shed new light on the development of social microbiology.

Within- and between-person variation in morning testosterone is associated with economic risk-related decisions in athletic women across the menstrual cycle

Literature suggests that women experience ovulatory shifts in risk-taking behaviours across different domains, which might be partly attributed to changes in testosterone (T). Thus, we investigated associations between menstrual variability in T concentrations and economic risk-related decisions among athletic women. Thirty-five women were monitored across three consecutive menstrual cycles. Testing occurred on day seven (D7), 14 (D14) and 21 (D21) following the onset of menses. The morning (7 to 8 am) assessment of salivary T (sal-T) and cortisol (sal-C) was followed by the economic Hawk-Dove game (11 am to 12 pm) played in pairs, where hawk decisions were used to index risk. Morning sal-T concentration increased from D7 to D14, before decreasing on D21 (p < 0.001), representing moderate effect size (ES) changes of 0.6 to 0.8. Morning sal-C did not vary over time. Hawk choices paralleled the sal-T results, being elevated on D14 (p < 0.001) with large ES changes of 1.8. Regression analyses revealed that morning sal-T concentration was positively related (p ≤ 0.01) to the number of hawks chosen between- (beta = 0.47) and within-participants (beta = 0.10) when controlling for training hours and menstrual day. In summary, the risk-related choices of athletic women during a dyadic contest covaried with morning sal-T concentrations across the menstrual cycle. Both outcomes were positively correlated on a within- and between-person level. Confirming the major sources of T variation across the menstrual cycle, whilst discerning its relationship with other risk-related behaviours, would be worthwhile avenues for research.

The replicator dynamics for multilevel selection in evolutionary games.

We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner's Dilemma or the Hawk-Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner's Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot be sustained at all in the case where between-group competition favors an even mix of cooperators and defectors.

Capuchin monkeys (Sapajus [Cebus] apella) play Nash equilibria in dynamic games, but their decisions are likely not influenced by oxytocin.

Comparative approaches to experimental economics have shed light on the evolution of social decision-making across a range of primate species, including humans. Here we replicate our previous work looking at six pairs of capuchin monkeys' (Sapajus [Cebus] apella) responses to scenarios requiring both coordination (Assurance Game) and anti-coordination (Hawk-Dove Game). This then provides a foundation for assessing their responses to two additional games, one with a scenario of beneficial cooperation with a temptation to defect (Prisoner's Dilemma) and one with an environment requiring changing strategies within short temporal proximity (Alternating Economic Game). We additionally explored the effects of exogenous oxytocin on decision-making. Oxytocin did not affect decisions in any of our games. Results from the first two games largely replicated our previous findings. Responses to the Prisoner's Dilemma were more varied than was seen in previous games, with pairs respectively cooperating, defecting, and failing to establish stable strategies. Such variability indicates that this game may be a good assay for individual differences in social decision-making. Finally, capuchins were able to flexibly switch between their previously established strategies within each of the different games, even when the games were presented within the same session, requiring strategy adjustments within short temporal proximity. These results build on earlier findings showing that capuchins can alter decision-making strategies as the context demands, which is likely essential for decision-making in naturally occurring contexts.

Dynamics of multiplayer games on complex networks using territorial interactions.

The modeling of evolution in structured populations has been significantly advanced by evolutionary graph theory, which incorporates pairwise relationships between individuals on a network. More recently, a new framework has been developed to allow for multiplayer interactions of variable size in more flexible and potentially changing population structures. While the theory within this framework has been developed and simple structures considered, there has been no systematic consideration of a large range of different population structures, which is the subject of this paper. We consider a large range of underlying graphical structures for the territorial raider model, the most commonly used model in the new structure, and consider a variety of important properties of our structures with the aim of finding factors that determine the fixation probability of mutants. We find that the graphical temperature and the average group size, as previously defined, are strong predictors of fixation probability, while all other properties considered are poor predictors, although the clustering coefficient is a useful secondary predictor when combined with either temperature or group size. The relationship between temperature or average group size and fixation probability is sometimes, however, nonmonotonic, with a directional reverse occurring around the temperature associated with what we term "completely mixed" populations in the case of the hawk-dove game, but not the public goods game.