Stag Hunt Game Research Articles

Competition and partnership between conformity and payoff-based imitations in social dilemmas

Learning from a partner who collects a higher payoff is a frequently used working hypothesis in evolutionary game theory. One of the alternative dynamical rules is when the focal player prefers to follow the strategy choice of the majority in the local neighborhood, which is often called a conformity-driven strategy update. In this work we assume that both strategy learning methods are present and compete for space within the framework of a coevolutionary model. Our results reveal that the presence of a payoff-driven strategy learning method becomes exclusive for high sucker’s payoff and/or high temptation values that represent a snowdrift game dilemma situation. In general, however, the competition of the mentioned strategy learning methods could be useful to enlarge the parameter space where only cooperators prevail. The success of cooperation is based on the enforced coordination of cooperator players which reveals the benefit of the latter strategy. Interestingly, the payoff-based and the conformity-based cooperator players can form an effective alliance against defectors that can also extend the parameter space of full cooperator solution in the stag-hunt game region. Our work highlights that the coevolution of strategies and individual features such as the learning method can provide a novel type of pattern formation mechanism that cannot be observed in a static model, and hence remains hidden in traditional models.

Conditional behavior and learning in similar stag hunt games

This paper reports an experiment using Stag Hunt games with different payoff ranges. Specifically, in one treatment for half of the games the payoff dominant and the risk dominant equilibrium coincide and for the other half of the games they conflict; in the second treatment they always conflict. The experiment provides evidence that the payoff range experienced by the participant influences the likelihood of efficient conventions emerging. In particular, experiencing games where payoff dominance and risk dominance coincide appears to make payoff dominance more attractive in games in which they conflict. In the experiment, we also observe conditional behavior emerging with experience. We develop a model of conditional expectations to explain these stylized facts that depends crucially on the assumption that after a brief learning period participants categorize their experience using the same relative bandwidth in both treatments even though the range of experience is twice as large in treatment 1 as it is in treatment 2. The assumption cannot be rejected by the data. The analysis provides a formal example in which increasing experienced diversity by changing the way similar experiences are categorized increases the likelihood of efficient conventions emerging in communities playing similar Stag Hunt games.

Stag hunt and trust emergence in social networks

Trust plays an important role in interactions within complex social systems. In this paper, we use multi-agent learning to study trust emergence in social networks. In our setting, agents play an iterative game of Stag Hunt with their neighbors. An agent adopts the learning rule imitate-the-best to learn how to play the game. Trust emerges if all agents choose to hunt stag after repeated play of the Stag Hunt game. We study the probability of the emergence of trust among agents living in different social networks. Using experiments, we reveal critical points of trust emergence in lattice network, ring network and small world model. The probability of trust emergence is relatively low if the quotient of the value of trust and the basic utility is smaller than the critical point. The probability, however, grows quickly when the quotient is larger than the critical point. In scale-free networks, there is no such critical point. Our findings also demonstrate that on scale-free networks, trust emerges only if the value of trust is several times larger than the basic utility.

When less information is good enough: experiments with global stag hunt games

There is mixed evidence on whether subjects coordinate on the efficient equilibrium in experimental stag hunt games under complete information. A design that generates an anomalously high level of coordination, Rankin et al. (Games Econo Behav 32(2):315–337, 2000), varies payoffs each period in repeated play rather than holding them constant. These payoff “perturbations” are eerily similar to those used to motivate the theory of global games, except the theory operates under incomplete information. Interestingly, that equilibrium selection concept is known to coincide with risk dominance, rather than payoff dominance. Thus, in theory, a small change in experimental design should produce a different equilibrium outcome. We examine this prediction in two treatments. In one, we use public signals to match Rankin et al. (2000)’s design; in the other, we use private signals to match the canonical example of global games theory. We find little difference between treatments, in both cases, subject play approaches payoff dominance. Our literature review reveals this result may have more to do with the idiosyncrasies of our complete information framework than the superiority of payoff dominance as an equilibrium selection principle.

How Should the E.U. Respond to Brexit and Trump?: The Lessons from Trade Wars

The U.K.’s decision to exit the E.U. (popularly known as “Brexit”) sets the stage for a potential retaliatory trade war. Similarly, the aggressive nationalism of U.S. President Donald Trump does also. In both cases, game theory suggests how such a conflict might be resolved. This paper first examines three standard game theory models—the Chicken Game, the Prisoner’s Dilemma, and the Stag-Hunt Game—and then turns to the strong incentives for rent-seeking by special interests and considers how rent-seeking likely affects how these games might play out. Although the conventional wisdom expects that the U.K. will suffer retaliation for Brexit, this paper argues that retaliation will produce counter-retaliation in a mutually destructive cycle. Thus, it proposes a strategy to produce a more clearly iterated game and hence increase the potential for cooperation and a compromise aimed at enhancing public welfare, rather than private interests.

Restricted and free-form cheap-talk and the scope for efficient coordination

This paper is the first to compare the efficiency-enhancing capacity of two communication protocols in experimental stag-hunt games. The traditional restricted protocol that allows communicating intentions only fails to improve efficiency over the no-communication conditions. When players are allowed to send any messages (free-form), the majority of them send messages that underscore a reason for which players should choose the efficient action. To explore further whether the absence of such a richer message in the traditional restricted communication protocol can be the cause of its ineffectiveness, we ran an additional treatment that included the intention-based message as well as a reason-based message. The data show that the richer restricted communication is as effective as free-form communication, and that reason-based messages are an effective efficiency-enhancing device in this class of coordination games regardless of whether the protocol is restricted or free-form.

Lead Us (Not) into Temptation: Testing the Motivational Mechanisms Linking Honesty–Humility to Cooperation

Over the past decades, there has been considerable interest in individual differences in cooperative behaviour and how these can be explained. Whereas the Honesty–Humility dimension from the HEXACO model of personality has been identified as a consistent predictor of cooperation, the underlying motivational mechanisms of this association have remained unclear—especially given the confound between the temptation to exploit others and the fear of being exploited as motivational drivers of defection in social dilemmas. In a reanalysis and a new experiment, we tease apart these mechanisms by manipulating the rank order of pay–offs in a symmetric two–person game paradigm, essentially implementing the classic prisoner's dilemma, stag hunt, and chicken games. Results revealed that Honesty–Humility predicted cooperation specifically in the games in which temptation was a potential motivator of defection, whereas it did not account for cooperation in those games in which only fear implied defection. Our findings thereby shed light on the underlying motivational mechanisms of the Honesty–Humility–cooperation link and, more generally, demonstrate how economic games can be used to disentangle such mechanisms. Copyright © 2018 European Association of Personality Psychology

A Social Heuristics Hypothesis for the Stag Hunt: Fast- and Slow-Thinking Hunters in the Lab

In this paper, we analyze the role of intuitive versus deliberative thinking in stag hunt games. To do so we, first, provide a conceptual framework predicting that, under the assumption that stag is the ruling social convention in real life interactions, players who make their choices fast and intuitively, relying on social heuristics, choose stag more often than other players. Second, we run a lab experiment and use a time pressure treatment to induce fast and intuitive thinking. We find that: (i) players under the time pressure treatment are more likely to choose stag than individuals in the control group; (ii) individual choices under the time pressure treatment are less sensitive to the size of the basin of attraction of stag; (iii) these results are largely driven by less experienced participants. Overall, our findings provide support to the Social Heuristics Hypothesis applied to stag hunt interactions.

Hunting the Stag in Harry Potter's Games

AbstractThis paper uses Game Theory as a theoretical framework to analyse the war strategies of Albus Dumbledore and Harry Potter during the Harry Potter series (Rowling 1997–2007). Over the course of the series, Dumbledore's Prisoner's Dilemma creates a zero-sum Game that relies on the ignorance of the people his Game is built around; namely, Harry Potter. Harry's knowledge of himself is deliberately limited by Dumbledore, which establishes Harry as the embodiment of Dumbledore's equilibria rather than as an active player with agency. Harry Potter reclaims his agency and personhood after his death and subsequent return to life, and he does this by adopting a Stag Hunt Game strategy, which empowers both himself and his opponent, Lord Voldemort. This paper argues that the Harry Potter series constructs the limitation of agency and the ignorance of young characters negatively, regardless of the potential good it could serve society. By contrast, Harry Potter's cooperative Game strategy is empowering both to society and to other people because it provides them with the means to make informed decisions and exercise productive agency.

Stress Induces Contextual Blindness in Lotteries and Coordination Games.

In this paper, we study how stress affects risk taking in three tasks: individual lotteries, Stag Hunt (coordination) games, and Hawk-Dove (anti-coordination) games. Both control and stressed subjects take more risks in all three tasks when the value of the safe option is decreased and in lotteries when the expected gain is increased. Also, subjects take longer to take decisions when stakes are high, when the safe option is less attractive and in the conceptually more difficult Hawk-Dove game. Stress (weakly) increases reaction times in those cases. Finally, our main result is that the behavior of stressed subjects in lotteries, Stag Hunt and Hawk-Dove are all highly predictive of each other (p-value < 0.001 for all three pairwise correlations). Such strong relationship is not present in our control group. Our results illustrate a “contextual blindness” caused by stress. The mathematical and behavioral tensions of Stag Hunt and Hawk-Dove games are axiomatically different, and we should expect different behavior across these games, and also with respect to the individual task. A possible explanation for the highly significant connection across tasks in the stress condition is that stressed subjects habitually rely on one mechanism to make a decision in all contexts whereas unstressed subjects utilize a more cognitively flexible approach.

Evolutionary games with self-questioning adaptive mechanism and the Ising model

A class of evolutionary games using a self-questioning strategy switching mechanism played in a population of connected agents is shown to behave as an Ising model Hamiltonian of spins connected in the same way. The payoff parameters combine to give the coupling between spins and an external magnetic field. The mapping covers the prisoner's dilemma, snowdrift and stag hunt games in structured populations. A well-mixed system is used to illustrate the equivalence. In a chain of agents/spins, the mapping to Ising model leads to an exact solution to the games effortlessly. The accuracy of standard approximations on the games can then be quantified. The site approximation is found to show varied accuracies depending on the payoff parameters, and the link approximation is shown to give the exact result in a chain but not in a closed form. The mapping established here connects two research areas, with each having much to offer to the other.

A Dynamic Evolution Model of Multiplayers Stag-Hunt Game in a Cycle Network

Social contracts and norms can be defined as the selection of Nash-equilibrium in coordination game. However, existing studies on group coordination game are lack of a theoretical process which simulates the evolution of behaviors of agents, especially in a cycle network. In this paper, by means of establishing Cellular Automata Model (CAM) based on the Stag Hunt Game as the bottom model in cycle network, we simulate the dynamically evolutionary Group Stag-Hunt Game in which assumed that an individual has complete rationality. The conclusion suggested that the final results were related to the size of the payoff factor and the risk factor and the structure features of the network. This model provides dynamically evolutionary Group Stag Hunt Game researches in a cycle network with a theoretical support.

Stags, Hawks, and Doves: Social Evolution Theory and Individual Variation in Cooperation.

One of the triumphs of evolutionary biology is the discovery of robust mechanisms that promote the evolution of cooperative behaviors even when cooperation reduces the fertility or survival of cooperators. These mechanisms include, kin selection, reciprocity, and direct benefits to cooperation that are often nonlinear. Though they have been extensively studied separately, investigating the joint action of these mechanisms has been more difficult. Moreover, how these mechanisms shape variation in cooperation is not well known. Such variation is crucial for understanding the evolution of behavioral syndromes and animal personality. Here, I use the tools of kin selection and evolutionary game theory to build a framework that integrates these mechanisms for pairwise social interactions. Using relatedness as a measure of the strength of kin selection, responsiveness as a measure of reciprocity, and synergy as a measure of payoff nonlinearity, I show how different combinations of these three parameters produce directional selection for or against cooperation or variation in levels of cooperation via stabilizing or diversifying selection. Moreover, each of these outcomes maps uniquely to one of four classic games from evolutionary game theory, which means that modulating relatedness, responsiveness, and synergy effectively transforms the payoff matrix from one the evolutionary game to another. Assuming that cooperation exacts a fertility cost on cooperators and provides a fertility benefit to social partners, a prisoner's dilemma game and directional selection against cooperation occur when relatedness and responsiveness are low and synergy is not too positive. Enough positive synergy in these conditions generates a stag-hunt game and diversifying selection. High levels of relatedness or responsiveness turn cooperation from a fitness cost into a fitness benefit, which produces a mutualism game and directional selection for cooperation when synergy is not too negative. Sufficiently negative synergy in this case creates a hawk-dove game and stabilizing selection for cooperation. I extend the results with relatedness and synergy to social groups and show that how group size changes the effect of relatedness and synergy on selection for cooperation depends on how the per capita benefit of cooperation changes with group size. Together, these results provide a general framework with which to generate comparative predictions that can be tested using quantitative genetic techniques and experimental techniques that manipulate investment in cooperation. These predictions will help us understand both interspecific variation in cooperation as well as within-population and within-group variation in cooperation related to behavioral syndromes.

Equilibrium selection in similar repeated games: experimental evidence on the role of precedents

We report on an experiment examining behavior and equilibrium selection in two similar, infinitely repeated games, Stag Hunt and Prisoner’s Dilemma under anonymous random matching. We are interested in the role that historical precedents may play for equilibrium selection between these two repeated games. We find that a precedent for efficient play in the repeated Stag Hunt game does not carry over to the repeated Prisoner’s Dilemma game despite the possibility that efficient play can be sustained as an equilibrium of the indefinitely repeated game. Similarly, a precedent for inefficient play in the repeated Prisoner’s Dilemma game does not extend to the repeated Stag Hunt game. We conclude that equilibrium selection between similar repeated games may have less to do with historical precedents and might instead depend more on strategic considerations associated with the different payoffs of these similar repeated games.

Topology-dependent rationality and quantal response equilibria in structured populations.

Given that the assumption of perfect rationality is rarely met in the real world, we explore a graded notion of rationality in socioecological systems of networked actors. We parametrize an actors' rationality via their place in a social network and quantify system rationality via the average Jensen-Shannon divergence between the games Nash and logit quantal response equilibria. Previous work has argued that scale-free topologies maximize a system's overall rationality in this setup. Here we show that while, for certain games, it is true that increasing degree heterogeneity of complex networks enhances rationality, rationality-optimal configurations are not scale-free. For the Prisoner's Dilemma and Stag Hunt games, we provide analytic arguments complemented by numerical optimization experiments to demonstrate that core-periphery networks composed of a few dominant hub nodes surrounded by a periphery of very low degree nodes give strikingly smaller overall deviations from rationality than scale-free networks. Similarly, for the Battle of the Sexes and the Matching Pennies games, we find that the optimal network structure is also a core-periphery graph but with a smaller difference in the average degrees of the core and the periphery. These results provide insight on the interplay between the topological structure of socioecological systems and their collective cognitive behavior, with potential applications to understanding wealth inequality and the structural features of the network of global corporate control.

Equilibrium selection in the stag hunt game under generalized reinforcement learning

We apply the generalized reinforcement (GR) learning protocol to the stag hunt game. GR learning combines positive and negative reinforcement. The GR learning rule generates the GR dynamic, which governs the evolution of the mixed strategy of agents in the population. We identify conditions under which the GR dynamic converges globally to one of the two pure strategy Nash equilibria of the game.

Greed and fear in multiperson social dilemmas

We consider greed and fear in social dilemmas, represented by multiplayer games with two strategies, cooperation and defection. The dilemmas are defined by relevant axioms. The N-person Prisoner’s Dilemma, Public Goods, Tragedy of the Commons, Volunteer’s Dilemma, and Assurance Game, are included in the considered axiomatization. For two-player interactions the scheme leads to three types of social dilemmas, the PD (and the Weak PD), the Chicken and the Stag Hunt game. We define greed and fear for multiplayer social dilemmas, and observe that all the dilemmas have greed or fear build in.

Risk preferences, responsibility, and self-monitoring in a Stag Hunt

Previous experimental research suggests that people often switch their strategy in games of uncertainty when faced with responsibility over another individual's outcomes. We extend this literature by assessing which personal traits may lead to changes in decisions. Specifically, we investigate how risk preferences and self-monitoring are related to decision changes in the Stag Hunt game under different levels of responsibility. As expected, we find that risk-averse individuals tend to make riskier choices less often than risk-loving individuals do in a series of economic games. Having responsibility for another player in the Stag Hunt seems to amplify risk attitudes as risk-averse individuals are more likely to make cautious choices and risk loving participants are more likely to make risky choices with responsibility than without it. High self-monitors are more likely to switch their strategy than low self-monitors when faced with responsibility over another's outcomes, and when they change strategies they tend to change in a manner consistent with their own preferences for risk.

Reference-Dependent Preferences in Stag-Hunt Coordination Games

We show that a number of previously isolated experimental results for behavior in stag-hunt coordination games can all be described by means of reference-dependent preferences. This includes an aversion to negative payoffs (loss avoidance) and a sensitivity to payoff differences. In addition, we also connect other-regarding preferences to this framework. Specifically, we show that an exogenous reference point can induce cooperative behavior, provided that the players are sufficiently risk-loving in the loss domain. Finally, we comment on the relevance of these findings to theories of equilibrium selection.

Coordination and transfer

We study the ability of subjects to transfer principles between related coordination games. Subjects play a class of order statistic coordination games closely related to the well-known minimum (or weak-link) and median games (Van Huyck et al. in Am Econ Rev 80:234–248, 1990, Q J Econ 106(3):885–910, 1991). When subjects play a random sequence of games with differing order statistics, play is less sensitive to the order statistic than when a fixed order statistic is used throughout. This is consistent with the prediction of a simple learning model with transfer. If subjects play a series of similar stag hunt games, play converges to the payoff dominant equilibrium when a convention emerges, replicating the main result of Rankin et al. (Games Econ Behav 32:315–337, 2000). When these subjects subsequently play a random sequence of order statistic games, play is shifted towards the payoff dominant equilibrium relative to subjects without previous experience. The data is consistent with subjects absorbing a general principle, play of the payoff dominant equilibrium, and applying it in a new related setting.