Stag Hunt Game Research Articles

Interdependent Self-Organizing Mechanisms for Cooperative Survival.

Cooperative survival "games" are situations in which, during a sequence of catastrophic events, no one survives unless everyone survives. Such situations can be further exacerbated by uncertainty over the timing and scale of the recurring catastrophes, while the resource management required for survival may depend on several interdependent subgames of resource extraction, distribution, and investment with conflicting priorities and preferences between survivors. In social systems, self-organization has been a critical feature of sustainability and survival; therefore, in this article we use the lens of artificial societies to investigate the effectiveness of socially constructed self-organization for cooperative survival games. We imagine a cooperative survival scenario with four parameters: scale, that is, n in an n-player game; uncertainty, with regard to the occurrence and magnitude of each catastrophe; complexity, concerning the number of subgames to be simultaneously "solved"; and opportunity, with respect to the number of self-organizing mechanisms available to the players. We design and implement a multiagent system for a situation composed of three entangled subgames-a stag hunt game, a common-pool resource management problem, and a collective risk dilemma-and specify algorithms for three self-organizing mechanisms for governance, trading, and forecasting. A series of experiments shows, as perhaps expected, a threshold for a critical mass of survivors and also that increasing dimensions of uncertainty and complexity require increasing opportunity for self-organization. Perhaps less expected are the ways in which self-organizing mechanisms may interact in pernicious but also self-reinforcing ways, highlighting the need for some reflection as a process in collective self-governance for cooperative survival.

Which Expectations to Follow: The Impact of First- and Second-Order Beliefs on Strategy Choices in a Stag Hunt Game.

Many situations require coordinated actions of individuals to achieve common goals. Such situations include organizing mass protests or adjusting behavior to new behavioral recommendations that aim to slow down the spread of a contagious disease. However, there is a risk of coordination failure in such situations that can lead to a worse outcome for those who acted in a coordinated manner than for those who chose not to. In this paper, we investigate the main determinant of individuals' decisions in these situations to determine whether beliefs regarding the action of others (empirical expectations), beliefs regarding others' beliefs (normative expectations), or risk attitudes are dominant determinants. To this end, we conducted an experiment analyzing the relationship between an individual's choices in a stag hunt game, their probabilistic empirical and normative expectations (i.e., first-order and second-order beliefs, respectively), and their risk attitudes. Our central finding is that expectations, not risk attitudes, explain individuals' strategy selection. In addition, we found evidence that normative expectations are a better predictor of strategy selection than empirical expectations. This could have implications for developing more targeted strategies intended to promote new behavioral standards and to guide individuals' behavior toward a welfare-maximizing equilibrium.

The influence of own historical information and environmental historical information on the evolution of cooperation

This article has studied the influence of own historical information and environmental historical information on the evolution of cooperation in the three common social dilemma games. The results show that only the past two-step information can effectively promote cooperation. In the prisoner's dilemma game, cooperation can be improved by using only the individual own information. In the stag hunt game, using only environmental information can promote cooperation. In the snowdrift game, when the parameter values are different, the results are different. The cooperation ratio can be maximized by using either its own information or environmental information only. This article provides a new way to promote and understand cooperation.

Evolution of cooperation in multigame with environmental space and delay

Replicator dynamics is widely used in evolutionary game theory, however, most previous studies on replicator dynamics focus on single games and ignore multiple social dilemmas encountered by individuals in a population. This paper uses replicator dynamics to construct a multigame system with environmental space and delay based on three social dilemmas. For the non-delayed and delayed multigame systems, rich dynamics for stability, bistability, transcritical bifurcation, Hopf bifurcation, and the direction, stability and periodic variation of periodic solutions are comprehensively investigated. Accordingly, we use numerical simulations to assist in exploring the effects of multigame, environmental space, and time delay on strategic dynamics. The results show that large proportions of snowdrift game and stag hunt game are conducive to the prosperity of cooperators, and defectors are easy to survive when the proportion of prisoner’s dilemma is large. The cooperator gains the advantage of benefit distribution from environmental space, or the defector gets less benefit distribution as punishment, which will make pure cooperation the dominant strategy. Furthermore, environmental space can allow cooperation and defection to coexist oscillatingly. Interestingly, large delays reverse the coexistence of cooperation and defection to a situation dominated by the purely cooperative strategy.

Limiting Dynamics for Q-Learning with Memory One in Symmetric Two-Player, Two-Action Games

We develop a method based on computer algebra systems to represent the mutual pure strategy best-response dynamics of symmetric two-player, two-action repeated games played by players with a one-period memory. We apply this method to the iterated prisoner’s dilemma, stag hunt, and hawk-dove games and identify all possible equilibrium strategy pairs and the conditions for their existence. The only equilibrium strategy pair that is possible in all three games is the win-stay, lose-shift strategy. Lastly, we show that the mutual best-response dynamics are realized by a sample batch Q-learning algorithm in the infinite batch size limit.

Congestion aware and game based odd even adaptive routing in network on chip many-core architecture

The era of single processors had almost reached a saturation state, and the industry had moved to multi-core processors for the newer generation of many-core architecture. Interconnections between multiple cores with network on chip (NoC) surpass traditional bus architecture for its quality of service (QoS) and other additional services. Seamless communication among the cores is more significant for better performance and the proper utilization of the cores. The rise in the cores count in a semiconductor chip adds the complexity of the communication among cores. Cache misses request and packet transmission’s traffic possibly will reduce the performance of the architecture. A theoretical game-based methodology is proposed to improvise the performance and communication by routing the request packets in the NoC of the many core architectures and the throughput is maximized with reduced latency by using the stag-hunt game (SHG) model. The proposed communication algorithm routes the packets in an adaptive way by detecting the congestion in routers. The SHG based odd-even routing algorithm is adaptive and can divert the packets towards less congested routers using the information gathered about congestion in the system, so that the overall performance of the system in terms of latency and throughput is improved.

Reward and Punishment Mechanism with weighting enhances cooperation in evolutionary games

The essential of the human cooperative behavior is to find an effective mechanism to promote cooperation. In reality, there is a class of professionals who punish evil and promote good, such as police officers, judges, and volunteers. Their actions and responsibilities can profoundly affect the actions of other individuals by fines and bonuses. Motivated by this phenomenon, we propose a Reward and Punishment Mechanism with weighting (RPMW) for promoting cooperation. The core of RPMW lies in three aspects: (i) A subset of players are tagged as supervisors, who are generated by a random algorithm. If the supervisor chooses to cooperate, she/he acts as the doer of rewards and punishments, otherwise she/he is a general player. (ii) Each supervisor asks their neighbors in turn as well as rewards cooperators and punishes defectors. Bonuses (penalties) are calculated according to a weighting which is the ratio of the number of current evolutionary round cooperators (defectors) to the total of players. (iii) A factor α is added to control the intensity of rewards and punishments, and the β is denoted as the percentage of supervisors in the group. Furthermore, we do a detailed ablation study on the Reward Mechanism with Weighting (RMW) and Punishment Mechanism with Weighting (PMW) in three social dilemmas. The simulation results show that RPMW can effectively promote cooperation in prisoner’s dilemma (PDG), snowdrift game (SDG) and stag-hunt game (SHG). We find that the frequency of cooperation reaches highest when the α is at its maximum and the β is in a positive impact on evolutionary cooperation. The results prove that RPMW is more effective than RMW and PMW on enhancing cooperation in PDG, SDG.

Coevolution of vaccination behavior and perceived vaccination risk can lead to a stag-hunt-like game.

Voluntary vaccination is effective to prevent infectious diseases from spreading. Both vaccination behavior and cognition of the vaccination risk play important roles in individual vaccination decision making. However, it is not clear how the coevolution of the two shapes population-wide vaccination behavior. We establish a coupled dynamics of epidemic, vaccination behavior, and perceived vaccination risk with three different time scales. We assume that the increase of vaccination level inhibits the rise of perceived vaccination risk, and the increase of perceived vaccination risk inhibits the rise of vaccination level. It is shown that the resulting vaccination behavior is similar to the stag-hunt game, provided that the basic reproductive ratio is moderate and that the epidemic dynamics evolves sufficiently fast. This is in contrast with the previous view that vaccination is a snowdriftlike game. And we find that epidemic breaks out repeatedly and eventually leads to vaccine scares if these three dynamics evolve on a similar time scale. Furthermore, we propose some ways to promote vaccination behavior, such as controlling side-effect bias and perceived vaccination costs. Our work sheds light on epidemic control via vaccination by taking into account the coevolutionary dynamics of cognition and behavior.

Average abundancy of cooperation in multi-player games with random payoffs

We consider interactions between players in groups of size [Formula: see text] with payoffs that not only depend on the strategies used in the group but also fluctuate at random over time. An individual can adopt either cooperation or defection as strategy and the population is updated from one time step to the next by a birth-death event according to a Moran model. Assuming recurrent symmetric mutation and payoffs to cooperators and defectors according to the composition of the group whose expected values, variances, and covariances are of the same small order, we derive a first-order approximation for the average abundance of cooperation in the selection-mutation equilibrium. In general, we show that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation. As for the effect of the covariance between any payoff for cooperation and any payoff for defection, we show that it depends on the number of cooperators in the group associated with these payoffs. We study in particular the public goods game, the stag hunt game, and the snowdrift game, all social dilemmas based on random benefit b and random cost c for cooperation, which lead to correlated payoffs to cooperators and defectors within groups. We show that a decrease in the scaled variance of b or c, or an increase in their scaled covariance, makes it easier for weak selection to favor the abundance of cooperation in the stag hunt game and the snowdrift game. The same conclusion holds for the public goods game except that the variance of b has no effect on the average abundance of C. Moreover, while the mutation rate has little effect on which strategy is more abundant at equilibrium, the group size may change it at least in the stag hunt game with a larger group size making it more difficult for cooperation to be more abundant than defection under weak selection.

Environmental heterogeneity unifies the effect of spatial structure on the altruistic cooperation in game-theory paradigms

As vivid metaphors for social dilemmas, the prisoner’s dilemma game, snowdrift game, and stag hunt game have been the focus of game theory research. Incorporating spatial structure into game models has been proven to have the opposite impact on the evolution of cooperation. Nevertheless, it is demonstrated here that spatial structure always fosters cooperation provided the environmental heterogeneity is included, regardless of the social dilemma. We consider the static and dynamic heterogeneous environment categories. The results exhibit that, in a static environment with two state types (poor and rich), a higher quality level of the rich state can bring a more pronounced improvement effect, and a moderate proportion of the rich-type environment will enable the density of cooperators to reach its maximum. To portray a more realistic scenario, the state types are further increased to fulfill the uniform distribution and the normal distribution, respectively, meanwhile they are allowed to fluctuate. It is shown that the enhancement effect is invariably verified. Additionally, the uniform distribution performs better in the static environment while two distributions present virtually consistent impact in the dynamic environment. The robustness analysis reveals that our claims are supported under various initial distributions, environment compositions, population configurations, and even the heat maps of parameter combination T-S. Our conclusions imply that environmental heterogeneity is crucial to the development of cooperation in social dilemmas.

The coevolving behavior of games and strategies under different network structures

In this paper, we study the coevolving behavior of mixed games when agents have a relationship represented by a fully connected network or a square lattice. Under the imitation update rule, whether the system will evolve to a state of pure game or mixed games and what the level of cooperation of the population will finally be are dependent on the initial fraction of mixed games, the game parameters and the network structures. We find that agents prefer to afford the prisoner’s dilemma (PD) game than the snowdrift game in the full connected network or in the square lattice and thus the cooperation is greatly suppressed. When the PD game mixes with the stag hunt game initially, they will coexist during evolution and a bistable phenomenon is observed. Meanwhile, the fraction of cooperation is enhanced when agents compete in a square lattice by comparison with the case of a fully connected network. If the PD game mixes with the harmony game (HG) initially, which one will dominate the other is related to the game parameters. The cooperation prevails in the population if the HG dominates the PD game. We also analyze the case of a fully connected network by a theory and the theoretical results are in good agreement with the simulation data.

Decomposing coordination failure in stag hunt games

Decomposing coordination failure in stag hunt games

The disconnection-reconnection-elite mechanism enhances cooperation of evolutionary game on lattice

How cooperative behavior evolves is the primary research topic in evolutionary game theory. In recent years, a growing number of researchers have been investigating mechanisms that enhance cooperation behavior. In the present paper, we propose a disconnection-reconnection-elite mechanism for prisoner’s dilemma, snowdrift game, and stag-hunt game models on spatial lattice networks. The core of the mechanism lies in: player i decides whether to keep or delete its neighbors from the set of interactive neighbors by comparing its payoff with those of its neighbors; and if there is no more neighbor in player i’s interactive neighbor set, the neighbor with the highest payoff in player i’s potential neighbor set is chosen and added to the interactive neighbor set. The proposed mechanism can significantly promote cooperation in the three games mentioned above. In addition, increasing population stochasticity significantly promotes cooperative evolution in prisoner’s dilemma and snowdrift game.

The choice-decision based on memory and payoff favors cooperation in stag hunt game on interdependent networks

The choice-decision based on memory and payoff favors cooperation in stag hunt game on interdependent networks

Dying to cooperate: the role of environmental harshness in human collaboration.

It has been proposed that environmental stress acted as a selection pressure on the evolution of human cooperation. Through agent-based evolutionary modelling, mathematical analysis, and human experimental data we illuminate the mechanisms by which the environment influences cooperative success and decision making in a Stag Hunt game. The modelling and mathematical results show that only cooperative foraging phenotypes survive the harshest of environments but pay a penalty for miscoordination in favourable environments. When agents are allowed to coordinate their hunting intentions by communicating, cooperative phenotypes outcompete those who pursue individual strategies in almost all environmental and payoff scenarios examined. Data from human participants show flexible decision-making in face of cooperative uncertainty, favouring high-risk, high-reward strategy when environments are harsher and starvation is imminent. Converging lines of evidence from the three approaches indicate a significant role for environmental variability in human cooperative dynamics and the species-unique cognition designed to support it.

The determinants of efficient behavior in coordination games

We study the determinants of efficient behavior in stag hunt games (2×2 symmetric coordination games with Pareto ranked equilibria) using data from eight previous experiments on stag hunt games and a new experiment that allows for a more systematic variation of parameters. We find that subjects do not necessarily play the efficient action (stag), stressing the importance of strategic uncertainty in coordination games. While the frequency of playing stag is greater when stag is risk dominant, there is still large variation in behavior that cannot be explained by risk dominance. Part of this variation is explained by the risk arising from strategic uncertainty that we measure with the size of the basin of attraction of stag. We also explore the importance of other determinants of efficient behavior and we show that the results are robust to paying subjects using the lottery method in an attempt to induce risk neutral preferences.

Evolution of cooperation in heterogeneously stochastic interactions

• We propose a novel game model with heterogeneously stochastic interactions. • Our model shows that heterogeneously stochastic interactions promote cooperation. • The probability distribution meets the power-law, exponential and uniform distributions. Heterogeneity is omnipresent in the real world, which plays a vital role in affecting the evolution of cooperation. Within this paper, we will discard the hypothesis of deterministic interactions among individuals and further propose a novel game model with heterogeneously stochastic interactions to mainly investigate the evolution of cooperation on a 2D square lattice, in which individuals play games with their neighbors in accordance with the probability of different heterogeneous distributions, such as power-law, exponential and uniform distribution, respectively. Through abundant numerical simulations, we find that such a simple setup has different effects on the evolution of cooperation in different social dilemmas. For the prisoner’s dilemma game and snowdrift game, the evolution of cooperation can be significantly facilitated. While for the stag hunt game, heterogeneously stochastic interactions can even impede the evolution of cooperation. Thus, we mainly focus on the impact of heterogeneously stochastic interaction on the evolution of cooperation under the prisoner’s dilemma and snowdrift game. On the whole, the stronger heterogeneity in the interaction probability can bring more benefit to the evolution of cooperation. We also verify the robustness of the proposed model with other potential structures and strategy update pattern, and the results are almost qualitatively consistent. Our exploration further clarifies the positive role of heterogeneity in the evolution of cooperation, which can help us to deeply comprehend the persistence or emergence of collective cooperation occurring on top of many natural and social systems.

Anything for a cheerio: Brown capuchins (Sapajus [Cebus] apella) consistently coordinate in an Assurance Game for unequal payoffs.

Unequal outcomes disrupt cooperation in some situations, but this has not been tested in the context of coordination in economic games. To explore this, we tested brown capuchins (Sapajus [Cebus] apella) on a manual version of the Stag Hunt (or Assurance) Game, in which individuals sequentially chose between two options, Stag or Hare, and were rewarded according to their choices and that of their partner. Typically, coordination on Stag results in an equal highest payout, whereas coordinating on Hare results in a guaranteed equal but lower payoff and uncoordinated play results in the lowest payoff when playing Stag. We varied this structure such that one capuchin received double the rewards for the coordinated Stag outcome; thus, it was still both animals' best option, but no longer equally rewarding. Despite the inequality, capuchins coordinated on Stag in 78% of trials, and neither payoff structure nor their partner's choice impacted their decision. Additionally, there was no relationship between self-scratching, a measure of stress in capuchins, and choices. After completing the study, we discovered our reward, cheerios, was sufficiently valuable that in another study, capuchins never refused it, so post hoc we repeated the study using a lower value reward, banana flavored pellets. Capuchins completed only 26% of the pellet trials (compared to 98% with cheerios), constraining our ability to interpret the results, but nonetheless the monkeys showed a decrease in preference for Stag, particularly when they received fewer rewards for the coordinated Stag outcome. These results reinforce capuchins' ability to find coordinated outcomes in the Stag Hunt game, but more work is needed to determine whether the monkeys did not mind the inequality or were unwilling to sacrifice a highly preferred food to rectify it. In either case, researchers should carefully consider the impact of their chosen rewards on subjects' choices.

Consensus towards Partially Cooperative Strategies in Self-Regulated Evolutionary Games on Networks

Cooperation is widely recognized to be fundamental for the well-balanced development of human societies. Several different approaches have been proposed to explain the emergence of cooperation in populations of individuals playing the Prisoner’s Dilemma game, characterized by two concurrent natural mechanisms: the temptation to defect and the fear to be betrayed by others. Few results are available for analyzing situations where only the temptation to defect (Chicken game) or the fear to be betrayed (Stag-Hunt game) is present. In this paper, we analyze the emergence of full and partial cooperation for these classes of games. We find the conditions for which these Nash equilibria are asymptotically stable, and we show that the partial one is also globally stable. Furthermore, in the Chicken and Stag-Hunt games, partial cooperation has been found to be more rewarding than the full one of the Prisoner’s Dilemma game. This result highlights the importance of such games for understanding and sustaining different levels of cooperation in social networks.

Coordination problems triggered by sunspots in the laboratory

A sunspot variable is any random variable that is not related to fundamental factors of the economy but a potential coordination device. The coordination power of sunspots has been analysed in theory and in experiments. However, some have discussed whether sunspots, e.g., public announcements such as financial market ratings, can create coordination problems. That discussion reached a new peak during the European sovereign debt crisis. We ask: can a sunspot variable, in form of a random forecast, trigger coordination problems? To answer that, we use a repeated three-player stag hunt game with fixed groups. In our experiment, a sunspot variable points randomly at the risk-dominant or the payoff-dominant choice. We find out-of-equilibrium behaviour caused by the sunspot variable in the short run. In the long run, the sunspot variable can lead to coordination on payoff-dominated equilibria. Only if the sunspot variable points more often to the payoff-dominated alternative, some groups use the sunspot variable consistently as a coordination device.