Stag Hunt Game Research Articles

Solidarity. . . Forever? Teaching Labor and Race at a Predominantly White and Emerging Hispanic Serving Institution with Games

In the past year, I have utilized two series of games to teach a predominantly white classroom the barriers that race plays to labor organizing. Students play Prisoner’s Dilemma and Stag-Hunt games, without and with player inequality. Player inequality is designed to mirror power asymmetry in the abstract but also white supremacy specifically by mirroring scenarios in Noel Ignatiev’s posthumous memoir of his time at the Gary Works— Acceptable Men. Though players are predisposed to cooperation, inequality between players strains solidaristic play. This cooperation exceeds that found in Acceptable Men, and students are encouraged to reflect upon why this may be the case. JEL Classification: A22, J15, J51, Z13

Computational joint action: Dynamical models to understand the development of joint coordination.

Coordinating with others is part of our everyday experience. Previous studies using sensorimotor coordination games suggest that human dyads develop coordination strategies that can be interpreted as Nash equilibria. However, if the players are uncertain about what their partner is doing, they develop coordination strategies which are robust to the actual partner's actions. This has suggested that humans select their actions based on an explicit prediction of what the partner will be doing-a partner model-which is probabilistic by nature. However, the mechanisms underlying the development of a joint coordination over repeated trials remain unknown. Very much like sensorimotor adaptation of individuals to external perturbations (eg force fields or visual rotations), dynamical models may help to understand how joint coordination develops over repeated trials. Here we present a general computational model-based on game theory and Bayesian estimation-designed to understand the mechanisms underlying the development of a joint coordination over repeated trials. Joint tasks are modeled as quadratic games, where each participant's task is expressed as a quadratic cost function. Each participant predicts their partner's next move (partner model) by optimally combining predictions and sensory observations, and selects their actions through a stochastic optimization of its expected cost, given the partner model. The model parameters include perceptual uncertainty (sensory noise), partner representation (retention rate and internale noise), uncertainty in action selection and its rate of decay (which can be interpreted as the action's learning rate). The model can be used in two ways: (i) to simulate interactive behaviors, thus helping to make specific predictions in the context of a given joint action scenario; and (ii) to analyze the action time series in actual experiments, thus providing quantitative metrics that describe individual behaviors during an actual joint action. We demonstrate the model in a variety of joint action scenarios. In a sensorimotor version of the Stag Hunt game, the model predicts that different representations of the partner lead to different Nash equilibria. In a joint two via-point (2-VP) reaching task, in which the actions consist of complex trajectories, the model captures well the observed temporal evolution of performance. For this task we also estimated the model parameters from experimental observations, which provided a comprehensive characterization of individual dyad participants. Computational models of joint action may help identifying the factors preventing or facilitating the development of coordination. They can be used in clinical settings, to interpret the observed behaviors in individuals with impaired interaction capabilities. They may also provide a theoretical basis to devise artificial agents that establish forms of coordination that facilitate neuromotor recovery.

A complete classification of evolutionary games with environmental feedback

Abstract A tragedy of the commons, in which rational behavior of individuals to maximize their own payoffs depletes common resources, is one of the most important research topics in game theory. To better understand the social dilemma problem, recent studies have developed a theoretical framework of feedback-evolving game where individual behavior affects an environmental (renewable) resource and the environmental resource changes individual payoffs. While previous studies assumed that the frequency of defectors increases (prisoner's dilemma [PD] game) when the environmental resource is abundant to investigate an oscillating tragedy of the commons, it is also possible for other types of game to produce the social dilemma. In this paper, we extend the feedback-evolving game by considering not only PD game, but also the other three game structures when the environmental resource is replete for a reasonably complete classification. The three games are Chicken game where defectors and cooperators coexist through minority advantage, Stag-Hunt (SH) game with minority disadvantage, and Trivial game where the frequency of cooperators increases. In addition, we utilize a dilemma phase plane to visually track (transient) dynamics of game structure changes. We found that an emergent initial condition dependence (i.e. bistability) is pervasive in the feedback-evolving game when the three games are involved. We also showed that persistent oscillation dynamics arise even with Chicken or SH games in replete environments. Our generalized analysis will be an important step to further extend the theoretical framework of feedback-evolving game to various game situations with environmental feedback.

Communication in stag hunt games: When does it really help?

This paper examines whether the effectiveness of cheap-talk communication in Stag-Hunt-like games is influenced by payoff asymmetry between players. We find that communication improves efficiency when both players benefit from the Pareto-dominant equilibrium. Otherwise, its impact is limited, regardless of whether one or both players communicate their intended actions.

Coordination in stag hunt games

Stag hunt games display a tension between a payoff superior option (stag) and a less risky but payoff inferior alternative (hare). We explore that tension by proposing a selection criterion (which we denote as relative salience) where subjects choose to coordinate in one option by comparing the salience of stag’s main aspect (its payoff) to the salience of hare’s main aspect (its risk) by accounting for both payoff-relevant factors and unobservable individual-specific (idiosyncratic) preferences. Using data from 10 experiments, we find that this criterion is a significant determinant of individual choices in stag-hunt games, outperforming other selection methods.

Stability of pairwise social dilemma games: Destructive agents, constructive agents, and their joint effects.

Destructive agents, who opt out of the game and indiscriminately harm others, paradoxically foster cooperation, representing an intriguing variant of the voluntary participation strategy. Yet, their impact on cooperation remains inadequately understood, particularly in the context of pairwise social dilemma games and in comparison to their counterparts, constructive agents, who opt out of the game but indiscriminately benefit others. Furthermore, little is known about the combined effects of both agent types on cooperation dynamics. Using replicator dynamics in infinite and well-mixed populations, we find that contrary to their role in facilitating cooperation in multiplayer games, destructive agents fail to encourage cooperation in pairwise social dilemmas. Instead, they replace defection in the prisoners' dilemma and stag-hunt games. Similarly, in the chicken game, they can destabilize or replace the mixed equilibrium of cooperation and defection, undermining cooperation in the harmony (trivial) game. Conversely, constructive agents, when their payoffs exceed their contributions to opponents, can exhibit effects similar to destructive agents. However, if their payoffs are lower, while they destabilize defection in prisoners' dilemma and stag-hunt games, they do not disrupt the cooperation equilibrium in harmony games and have a negligible impact on the coexistence of cooperation in chicken games. The combination of destructive and constructive agents does not facilitate cooperation, but instead generates complex evolutionary dynamics, including bistable, tristable, and quadstable states, with outcomes contingent on their relative payoffs and game types. These results, taken together, enhance our understanding of the impact of the voluntary participation mechanism on cooperation, contributing to a more comprehensive understanding of its influence.

"Fight alone" to "win–win cooperation": a quantum stag hunt game model for analyzing cooperative R&D between enterprises

"Fight alone" to "win–win cooperation": a quantum stag hunt game model for analyzing cooperative R&D between enterprises

Cooperation in primary school children: The role of cardiac vagal tone and cognitive control

Cooperation is crucial in classrooms for positive interactions among pupils, but the factors influencing children’s cooperative behaviors remain debated. This study explores the impact of cardiac vagal tone (CVT), a measure of physiological self-regulation, and cognitive control on cooperation among 109 primary school students. Heart rate variability was used to assess self-regulation, and cognitive control was evaluated through an AX-Continuous Performance Task. Cooperative behavior was measured with a child-friendly Stag Hunt game. Results showed that moderate CVT is related to increased cooperative actions, but no direct relationship between cognitive control types and cooperation was found. However, an interaction effect indicated that moderate CVT particularly boosts cooperation in children who employ proactive cognitive control. These findings emphasize the need to develop physiological self-regulation skills to enhance cooperative behavior in educational settings, supporting better social, and emotional well-being, and learning outcomes.

Risk-dominant Equilibrium in Chicken and Stag-hunt Games with Different Dilemma Strengths

Objective: This study investigates how the strength of risk-averting and gamble-intending dilemmas can affect the risk-dominant equilibrium in chicken and stag-hunt games. The focus is on deriving conditions under which the risk-dominant equilibrium can maximize group benefit. Methods: Basic concepts of game theory were used to determine the Nash equilibria of each game.The Harsanyi-Selten theory was applied to determine the risk-dominant equilibrium. By manipulating the parameters related to risk-averting and gamble-intending dilemmas, different game scenarios are sim-ulated to observe the resulting equilibria. Results: The study found that for a given strength of the risk-averting dilemma, the risk-dominant equilibrium in the chicken game continuously shifts towards mutual cooperation as the strength of the gamble-intending dilemma decreases. In the SH game, the degree of cooperation of the risk-dominant equilibrium for a given strength of the gamble-intending dilemma decreases with increasing strength of the risk-averting dilemma and takes the discrete values: 1, 0.5 and 0. The results show the importance of understanding individual risk preferences to predict strategic decisions and equilibrium outcomes in these game settings. Conclusion: The risk-dominant equilibrium in chicken and stag-hunt games depends on the relative strength of the risk-averting and gamble-intending dilemmas. By taking these factors into account, researchers and policy makers can better predict the likely outcomes of strategic interactions in scenarios with conflicting interests and different risk tolerances. The risk-dominant equilibrium can provide reasonable solutions to real-world conflict situations, such as the Iran-Iraq conflict over shared oil and gas resources.

A Review of the U.S.-China Science and Technology Cooperation Agreement and Its Prospect–Based on the Perspective of Game Theory

《中美科技合作协定》的签署历程与发展历史解释了中美两国签订该协定的动因与维持双边科技合作关系的考量，并深刻揭示了两国科技合作关系背后的经济与政治博弈。现阶段，美国对华战略认知的调整与其将科技问题安全化的倾向导致了《中美科技合作协定》遭遇续签危机，其未来走向成为中美各界尤其关心的问题。博弈论为中美科技关系的战略转型提供有效的分析框架。其中，中美猎鹿博弈模型揭示了美国对华科技脱钩力度较弱的情况下双方进行合作的进路，而重复囚徒困境博弈模型则指出长期合作与互信的建立才是中美实现双赢的关键。中美两国应努力消除分歧，构建合作与竞争并存的新型关系，避免零和游戏，共同推动科技进步和全球发展。 The signing process and development of the US-China Science and Technology Cooperation Agreement (US-China S&T Cooperation Agreement) explain the reasons for the signing of the agreement and the considerations for maintaining the bilateral S&T cooperation relationship between China and the US, and deeply reveals the economic and political games behind the S&T cooperation relationship between the two countries. At this stage, the adjustment of the U.S. strategic perception of China and its tendency to securitize science and technology issues have led to the renewal crisis of the US-China S&T Cooperation Agreement, and its future direction has become an issue of particular concern to all sectors in the U.S. and China. Game theory provides an effective analytical framework for the strategic transformation of Sino-US S&T relations. The Stag Hunt Game Model reveals the way forward for cooperation between the two sides in the case of weak US S&T decoupling from China, while the Repeated Prisoner’s Dilemma Game Model points out that the establishment of long-term cooperation and mutual trust is the key to achieving a win-win situation for both China and the US. China and the United States should strive to eliminate differences, build a new type of relationship where cooperation and competition coexist, avoid zero-sum games, and jointly promote scientific and technological progress and global development.

Dogs understand the role of a human partner in a cooperative task

Humans are exceptionally flexible in cooperation, partly due to our ability to recognize the roles of cooperative partners. While some non-human animals understand the need for a partner in such interactions, it is unclear whether they grasp the consequences of their partner’s actions and adjust accordingly. Previous studies utilizing economic games with non-human animals yielded mixed results. We investigated dogs, known for their close cooperation with humans, in a stag hunt game. Dogs could cooperate for better rewards or defect for lower ones, while their human partners would either cooperate, never cooperate, or act randomly. We control for attraction to food, side bias, and local enhancement. Dogs were more likely to coordinate with their partners when it led to better rewards, suggesting that they understood their partner’s actions. By highlighting this cognitive skill in dogs, we advance our knowledge of the intricate mechanisms driving cooperative behavior in non-human animals.

Coupling injunctive social norms with evolutionary games

Human behaviors are strongly influenced by social norms. This work couples injunctive social norms with evolutionary games. Injunctive norms refer to behaviors which individuals should conform to and thereby entail social pressure to adhere to a particular action. Here we investigate impact of such social norms on both pairwise and multi-player games. It is shown that injunctive norms can enforce the system to either extreme, meaning that they give rise to the possibility of bi-stability (two stable equilibrium states) in evolutionary games. We derive the condition for such bi-stability in the mean-field limit. While investigating pairwise games with social norms, we observe that —besides other possibilities—Prisoner's dilemma (PD), snow-drift (SD), and Trivial (TR) games may experience the bi-stability like the stag-hunt (SH) game. Interestingly, SD game with social norms exhibits four possible asymptotic outcomes: all-cooperation (AllC), all-defection (AllD), either AllC or AllD (bi-stable), and coexistence. However, its multi-player variant does not manifest a unique coexistence equilibrium. Moreover, it demonstrates another bi-stable equilibrium, comprising either coexistence or AllC, besides other outcomes observed in the two-player case. Conversely, PD and its multi-player variant, i.e., public goods game with injunctive norms show identical evolutionary outcomes (AllD or bi-stable).

Facilitating cooperation in human-agent hybrid populations through autonomous agents

Cooperative AI has shown its effectiveness in solving the conundrum of cooperation. Understanding how cooperation emerges in human-agent hybrid populations is a topic of significant interest, particularly in the realm of evolutionary game theory. In this article, we scrutinize how cooperative and defective Autonomous Agents (AAs) influence human cooperation in social dilemma games with a one-shot setting. Focusing on well-mixed populations, we find that cooperative AAs have a limited impact in the prisoner's dilemma games but facilitate cooperation in the stag hunt games. Surprisingly, defective AAs can promote complete dominance of cooperation in the snowdrift games. As the proportion of AAs increases, both cooperative and defective AAs have the potential to cause human cooperation to disappear. We then extend our investigation to consider the pairwise comparison rule and complex networks, elucidating that imitation strength and population structure are critical for the emergence of human cooperation in human-agent hybrid populations.

China’s Policy Toward Afghanistan and Its Impact on India

The South Asia subcontinent is witnessing new security and political dynamics after the United States withdrew its troops in mid-2021. Even as the Taliban-led government is stepping up engagement with the outside world to earn wider diplomatic recognition and exhibit its governing credentials, Beijing and New Delhi are adjusting their regional policies in a bid to acquire more influence in a changing geopolitical landscape. Assuming nation states to be rational actors, this paper uses the stag hunt game to analyze the potential maximum payoffs for Beijing, Kabul, and New Delhi, respectively, if they could make the right strategic calculations about their short-term interests and long-term goals. Even though the subcontinent is not immune to the effect of great power rivalry that is unfolding on the world stage, regional stakeholders could still choose to cooperate on issues of common concern while limiting the fallout of competition and confrontation on other more contentious challenges.

Game-theoretical description of the go-or-grow dichotomy in tumor development for various settings and parameter constellations

A medically important feature of several types of tumors is their ability to “decide” between staying at a primary site in the body or leaving it and forming metastases. The present theoretical study aims to provide a better understanding of the ultimate reasons for this so-called “go-or-grow” dichotomy. To that end, we use game theory, which has proven to be useful in analyzing the competition between tumors and healthy tissues or among different tumor cells. We begin by determining the game types in the Basanta–Hatzikirou–Deutsch model, depending on the parameter values. Thereafter, we suggest and analyze five modified variants of the model. For example, in the basic model, the deadlock game, Prisoner’s Dilemma, and hawk-dove game can occur. The modified versions lead to several additional game types, such as battle of the sexes, route-choice, and stag-hunt games. For some game types, all cells are predicted to stay on their original site (“grow phenotype”), while for other types, only a certain fraction stay and the other cells migrate away (“go phenotype”). If the nutrient supply at a distant site is high, all the cells are predicted to go. We discuss our predictions in terms of the pros and cons of caloric restriction and limitations of the supply of vitamins or methionine. Our results may help devise treatments to prevent metastasis.

Structured heterogeneity of network in evolutionary social dilemmas under assortative linking and learning

We use assortativity in linking and learning by studying social dilemmas on a dynamic network, where assortativity in linking runs faster than assortativity in learning, so that both can work together. The emergence of cooperation is based on the structured heterogeneity of a network, which includes heterogeneous interactions and a heterogeneous population structure made by assortativity. In an infinite population, cooperation is more likely to happen in the Snowdrift game when there is heterogeneous interaction, in the Stag-hunt game when there is heterogeneous population structure, and in the Prisoner’s Dilemma game when there are two different kinds of settings. In a finite population size, the heterogeneous interaction plays an important role in emergence of cooperation in all dilemmas.

Weighted-elite-memory mechanism enhances cooperation in social dilemmas

The issue of how to enhance cooperation has been a hot topic of research in evolutionary games for a long time. A mechanism is proposed to facilitate the cooperation behavior of evolutionary groups on networks in three game models, including prisoner's dilemma, snowdrift game, and stag hunt game. The core of the mechanism lies in: 1) Each player has a length of memory and uses the information of the elite in the memory span to update its strategy. 2) Each player has the chance to game with a certain neighbor more than once in each round. 3) The accumulative payoff of a player consists of two parts, one from playing with elites in memory length and another from playing with current neighbors, and a weight is introduced to adjust these two parts. The findings of the simulation demonstrate that a small weight can significantly enhance cooperation in three typical social dilemmas. Furthermore, the level of cooperation increases at first and then stays stable as the memory length increases.

The conflict between self-interaction and updating passivity in the evolution of cooperation

In social dilemmas under weak selection, the capacity for a player to exhibit updating passivity or interact with its own strategy can lead to conflicting outcomes. The central question is which effect is stronger and how their simultaneous presence influences the evolution of cooperation. We introduce a model that considers both effects using different weight factors. We derive theoretical solutions for the conditions of cooperation success and the cooperation level under weak selection, scanning the complete parameter space. When the weight factors are equally strong, the promoting effect of self-interaction to cooperation surpasses the inhibitory effect of updating passivity. Intriguingly, however, we identify non-monotonous cooperation-supporting effects when the weight of updating passivity increases more rapidly. Our findings are corroborated by Monte Carlo simulations and demonstrate robustness across various game types, including the prisoner’s dilemma, stag-hunt, and snowdrift games.

Social Learning and the Exploration-Exploitation Tradeoff

Cultures around the world show varying levels of conservatism. While maintaining traditional ideas prevents wrong ones from being embraced, it also slows or prevents adaptation to new times. Without exploration there can be no improvement, but often this effort is wasted as it fails to produce better results, making it better to exploit the best known option. This tension is known as the exploration/exploitation issue, and it occurs at the individual and group levels, whenever decisions are made. As such, it is has been investigated across many disciplines. We extend previous work by approximating a continuum of traits under local exploration, employing the method of adaptive dynamics, and studying multiple fitness functions. In this work, we ask how nature would solve the exploration/exploitation issue, by allowing natural selection to operate on an exploration parameter in a variety of contexts, thinking of exploration as mutation in a trait space with a varying fitness function. Specifically, we study how exploration rates evolve by applying adaptive dynamics to the replicator-mutator equation, under two types of fitness functions. For the first, payoffs are accrued from playing a two-player, two-action symmetric game, we consider representatives of all games in this class, including the Prisoner’s Dilemma, Hawk-Dove, and Stag Hunt games, finding exploration rates often evolve downwards, but can also undergo neutral selection as well depending on the games parameters or initial conditions. Second, we study time dependent fitness with a function having a single oscillating peak. By increasing the period, we see a jump in the optimal exploration rate, which then decreases towards zero as the frequency of environmental change increases. These results establish several possible evolutionary scenarios for exploration rates, providing insight into many applications, including why we can see such diversity in rates of cultural change.

Invasion of Optimal Social Contracts

The stag-hunt game is a prototype for social contracts. Adopting a new and better social contract is usually challenging because the current one is already well established and stable due to sanctions imposed on non-conforming members. Thus, how does a population shift from the current social contract to a better one? In other words, how can a social system leave a locally optimum configuration to achieve a globally optimum state? Here, we investigate the effect of promoting diversity on the evolution of social contracts. We consider group-structured populations where individuals play the stag-hunt game in all groups. We model the diversity incentive as a snowdrift game played in a single focus group where the individual is more prone to adopting a deviant norm. We show that a moderate diversity incentive is sufficient to change the system dynamics, driving the population over the stag-hunt invasion barrier that prevents the global optimum being reached. Thus, an initial fraction of adopters of the new, better norm can drive the system toward the optimum social contract. If the diversity incentive is not too large, the better social contract is the new equilibrium and remains stable even if the incentive is turned off. However, if the incentive is large, the population is trapped in a mixed equilibrium and the better social norm can only be reached if the incentive is turned off after the equilibrium is reached. The results are obtained using Monte Carlo simulations and analytical approximation methods.