Spatial Game Research Articles

The spatial public goods game on hypergraphs with heterogeneous investment

The emergence of cooperation in the multi-person social dilemma has been extensive investigated under the pairwise interactions-based spatial public goods game (PGG). However, higher-order interactions are more suitable to describe the group relationships in real society and have attracted intensive interests. In this work, we propose an extension of the spatial PGG on hypergraphs, where cooperators are allowed to allocate their investments unevenly. A tunable parameter, α, is introduced to characterize the investment preference. α>0 means that cooperators will allocate more investment to the PGG group from which they obtain higher payoff, while for α<0, cooperators will allocate more investment to the group from which they gain lower payoff. Results from numerical simulations indicate that compared with the results of homogeneous case, allocating more resources to profitable groups can effectively promote cooperation. Additionally, for a relative high synergy factor r, there exists a moderate negative value of α that will induce the lowest cooperation level, larger or smaller values of α can enable a higher cooperation level relative to the worst result. Moreover, the nontrivial results are robust against hypergraphs with different orders.

Ecological Compensation Assessment from the Perspective of Spatial Game: a Case Study of the Watershed Ecological Compensation in Anhui Section of the Huaihe River Basin, China

Ecological Compensation Assessment from the Perspective of Spatial Game: a Case Study of the Watershed Ecological Compensation in Anhui Section of the Huaihe River Basin, China

Interaction state Q-learning promotes cooperation in the spatial prisoner's dilemma game

Many recent studies have used reinforcement learning methods to investigate the behavior of agents in evolutionary games. Q-learning, in particular, has become a mainstream method during this development. Here we introduce Q-learning agents into the evolutionary prisoner's dilemma game on a square lattice. Specifically, we associate the state space of Q-learning agents with the strategies of their neighbors, and we introduce a neighboring reward information sharing mechanism. We thus provide Q-learning agents with the payoff information of their neighbors, in addition to their strategies, which has not been done in previous studies. Through simulations, we show that considering neighborhood payoff information can significantly promote cooperation in the population. Moreover, we show that for an appropriate strength of neighborhood payoff information sharing, a chessboard pattern emerges on the lattice. We analyze in detail the reasons for the emergence of the chessboard pattern and the increase in cooperation frequency, and we also provide a theoretical analysis based on the pair approximation method. We hope that our research will inspire effective approaches for resolving social dilemmas by means of sharing more information among reinforcement learning agents during evolutionary games.

Heterogeneous reputation promotes cooperation in spatial public goods game

Many properties of things are heterogeneous, which promotes the reproduction and development of human society. Naturally, the reputation has heterogeneity, and the heterogeneity of reputation can effectively promote cooperation in evolutionary systems. However, how to measure the heterogeneity of reputation and reveal the evolutionary mechanisms by which it fosters cooperation is not fully understood. In this paper, a modified public goods game (PGG) model is proposed based on co-evolution mode, in which a player's reputation is measured by the number of times the player adheres to the same strategy. To better understand the influence of heterogeneous reputation on individual's decision, we propose two evolutionary mechanisms, namely the absolute mechanism and the relative mechanism based on reputation. Simulation verifies that two proposed mechanisms are easier to promote cooperation compared with the traditional PGG model, and the relative mechanism is easier to promote the evolution of cooperation than the absolute mechanism.

Evolution of cooperation in spatial public goods game based on heterogenous social preferences and dynamic punishment

The mechanism of promoting cooperation in the public goods game has always been concerned by scholars. However, most of the existing studies are based on the premise that participants are self-interested. In order to explore why some sellers on e-commerce platforms voluntarily maintain the platform’s reputation, we incorporate heterogeneous social preferences of sellers into the spatial public goods game. We find that heterogeneous social preferences can enhance cooperation by improving collective rationality. Specifically, the altruistic preference of sellers can greatly reduce free-riding behavior, while the inequality aversion preference has a little inhibitory effect. Interestingly, when the benefit of maintaining the platform’s reputation is relatively high, the reciprocal preference can inhibit cooperation, but it can promote cooperation when the benefit is relatively small. This is due to the existence of some loosely connected but stable cooperative or defective clusters of sellers in e-commerce platforms. Furthermore, we propose a dynamic punishment mechanism to punish free riders. We observe that the dynamic punishment mechanism is more effective than the static punishment mechanism in solving the second-order free-riding problem faced by punishers. Increasing the enhancement factor of public goods is identified as a fundamental approach to mitigating this problem.

Measuring the evolutionary game process among three functional space types at the county scale in Henan Province, China

Conflicts among various spatial planning policies have created a spatial game over land function and utilization due to the limitation of land resources. We quantitatively assess the evolutionary game process among three functional space types (agricultural production, urban–rural living, and blue–green ecological) using the concept of game theory during the period of 1980–2020 in Henan Province, China, as a case study. A research framework for assessing the spatial game process was established, and a measurement method based on spatial game type (SGT) and spatial game intensity (SGI) is proposed. The results demonstrate that the evolutionary game process among three functional space types exhibited significant spatiotemporal variation and spatial agglomeration. SGI increased from low to high values and then decreased again, and exhibited spatial autocorrelation. SGT supported classification into positive-sum and zero-sum games. The period of 1990–2000 had the largest SGI and the most counties with high-intensity zero-sum games. A highly urbanized urban center showed a high-intensity zero-sum game at the expense of agricultural production spaces. Balancing the areas devoted to various uses, enhancing spatial connectivity and improving functional mixing degree of land use may be effective ways to alleviate noncooperative games and spatial conflicts among these three functional space types.

The emergence of rich complex dynamics in a spatial dyadic game with resource storage, participation cost, and agent interaction propensity

We present an evolutionary game model combining subordinate elements from several viewpoints to address whether a resource-storing mechanism promotes a society where the wealthy engage in or refrain from conflict. The model is based on the pairwise game, which incorporates the accumulation of payoffs over time and introduces the concept of participation probability based on wealthiness. Our study encompasses four distinct game classes: Prisoner's Dilemma, Trivial, Stag Hunt, and Chicken. By incorporating these diverse social dilemma structures, we strive to comprehensively understand the dynamics within different game scenarios. Additionally, we broaden the scope of our analysis by considering two network types: a regular lattice network and a Barabasi-Albert scale-free (BA-SF) graph. Through simulation results, we have discovered that the commonly held belief or human philosophical wisdom that “the wealthy do not fight” leads to the emergence of a cooperative society, depending on the intensity of the dilemma. In contrast, our findings strongly suggest that the prevailing notion of “the wealthy do fight” fosters an imbalanced exploitation-based society where defectors who exploit the poor cooperators thrive. Further analysis shows variations in beliefs and dynamics between cooperators and defectors, highlighting the emergence of social dilemmas and the impact of payoff storage. Our result reported here proves that the proposed model based on a minimal spatial game setting by 2-player & 2-strategy game just added several subordinate components can reproduce rich, complex scenarios likely observed in a real human society.

The impact of current and historical reputation with non-uniform change on cooperation in spatial public goods game

In the complex human social environment, reputation plays an important role in interpersonal communication. Normally, individuals pay attention to their neighbors’ current and historical reputations during strategy learning, and the variation of individual reputation over time is often non-uniform. This paper proposes a new spatial public goods game model that involves current and historical reputation. An exponential function of the rounds in which the individual continuously maintains the cooperative or defective strategy unchanged is defined to reflect the change of the individual reputation. And two discount coefficients involving five parameters are introduced to the imitation probability of strategy learning. Simulation results indicate that high sensitivities of individuals to her/his own reputation, to the reputation of her/his neighbors, and to changes in neighbors’ historical reputations are all beneficial for the cooperative behavior of the system. And individual behavior of paying attention to changes in neighbors’ historical reputation is also beneficial for the cooperation, but attention to more historical reputation changes in neighbors does not further promote cooperative behavior. The sensitivity of the population to individual historical information about strategy has little impact on the cooperative behavior of the system.

Asymmetry of individual activity promotes cooperation in the spatial prisoner's dilemma game.

We consider an aspiration-based asymmetric individual activity co-evolutionary prisoner's dilemma game model on the square lattice. In detail, each player has an individual weight that evolves with its strategy. We introduce an asymmetric setting that only μ fractions of players in the network chosen to be active can update their individual weights according to whether their payoffs satisfy their aspirations. Therefore, our individual weights can be regarded as a type of intrinsic motivation satisfaction. Through Monte Carlo simulations on the square lattice, our co-evolution mechanism has been proven to significantly promote cooperation. Furthermore, we find that the moderate μ leads to the highest cooperation level. By dividing the players into four types according to their strategies and weights, we prove that the inversion of the dominant relationship between cooperators and defectors with low weights leads to a conversion cycle of the four types of players, which enhances spatial reciprocity to promote cooperation. Moderate μ results in an appropriate number of low-weight players in this cycle. Interestingly, our results show that higher weight heterogeneity does not lead to higher levels of cooperation, which is contrary to the intuition formed in the previous work.

The finagle point is close to the yolk

The yolk and the finagle point are two well-known solution concepts in spatial voting games without a core. No relationship between them has been rigorously established in the literature. We prove that every finagle point must be within 2.32472 yolk radii of every yolk center, in all dimensions. This result tightens and generalizes a claimed bound of 2.5 for two dimensions. The bound applies to all points with finagle radius less than the yolk radius. If the finagle radius is half the yolk radius, the bound decreases to 1.7607.

Role polarization and its effects in the spatial ultimatum game.

Human society is believed to be becoming increasingly polarized, yet it remains unclear how role polarization influences the evolution of fairness. In addition, little is known about role adaptation, despite the fact that altering the roles of players can often change the outcome of the ultimatum game. Unlike earlier static, random, symmetric role assignment, here I suggest a succeed-reinforce-fail-slacken role adaptation rule that encourages successful proposers in the present round to propose again in the next round and vice versa. The results demonstrate that this simple rule can tip the scales in favor of fair strategies when it comes to the proposer advantage, and therein lies the key to promoting fairness. Depending on its pace, notably, role adaptation can direct the system to equilibrium states that bear variable degrees of role polarization, with two consequences incidentally. Not only does it favor fairness, it also fosters empathy. Noise associated with role adaptation often reduces role polarization and thus has a negative impact on fairness and empathy. The comparison of experiments with various networks validates the substantial resilience of role polarization to structural changes. These findings add to the evidence for role polarization and highlight the centrality of role adaptation in the evolution of fairness.

Evolution of cooperation with tag-based expulsion in spatial public goods game

We introduce tag-based expulsion and explore its dynamic influence on cooperative behavior within a spatial public goods game. Our model employs a multi-agent system where agents, characterized by their strategies and tags, participate in public goods games and expel neighbors with dissimilar tags. The expulsion process effectively reshapes the interaction structure, providing a protective shield for cooperative clusters against defector infiltration. We demonstrate that expulsion, tag mutability, and tolerance are pivotal in driving cooperation levels and the optimal interaction structure. Even in less favorable conditions characterized by low enhancement factors, cooperation can emerge given appropriate tolerance levels. Our research underscores the importance of moderate population density, tag-based expulsion, and well-calibrated expulsion intensity in fostering cooperation, introducing fresh perspectives on the interplay between social dynamics and evolutionary processes. Additionally, we provide evidence of the dynamic adaptability of expulsion mechanisms, suggesting they may be integral to understanding cooperation in various societal contexts. These findings open up new avenues for further exploration of the complexity of cooperative behaviors and the roles of expulsion mechanisms in the evolution of cooperation.

A model for herd behaviour based on a spatial public goods game

In groups and crowds, individuals have a propensity to replicate the actions of their immediate neighbours, often disregarding rational considerations. This phenomenon is known as herd behaviour, which has been extensively investigated in biological and economic contexts to comprehend the dynamics of fragile systems. Examples of herd behaviour scenarios include the bubble effect in the stock market, animal coordination, and individuals congregating together during emergency situations to avert peril. The understanding of the underlying causes and consequences of individual decision-making in crowd dynamics can mitigate the negative outcomes of these situations. Consequently, models are utilised to propose effective stock market controls and design emergency exits that enable swift responses to crises. This paper presents a spatial game that induces a level of herding in the population, whereby a multiplayer public good game serves as the platform for interaction among individuals. Herding behaviour arises when individuals shift from emulating the most successful strategy to copying the predominant strategy in their neighbourhood. The objective of the game is to investigate how the induced level of herding and the return on investment affect the total wealth, wealth distribution, and level of cooperation in the population. In summary, when herd behaviour is more prevalent, it leads to reduced cooperation and increased wealth inequality. Nonetheless, the outcome is contingent on the initial concentration of states in the simulation.

Evolutionary public good games based on the long-term payoff mechanism in heterogeneous networks

Over the past two decades, it has been of great concern to solve the puzzle of social dilemmas in spatial public goods games. It has been demonstrated that the introduction of memory effects can effectively promote cooperative emergence to some extent. In this paper, we focus on the impact of scale-free and high aggregation properties on the public goods game of memory heterogeneous groups. Here, we propose an optimal strategy selection mechanism based on long-term historical payoffs. In our model, a memory loss function ω is introduced to characterize the memory length of players, and accordingly, pairs determine the optimal strategy by comparing historical gains for the next game cycle. Experiments conducted on scale-free networks and small world networks show that the strategy selection mechanism based on long-term payoff can keep the proportion of cooperators at a high level. In addition, this study demonstrates that the heterogeneity of memory length in the network promotes the enhancement of network reciprocity and robustness, which is beneficial to further reveal the mechanism of cooperative behaviour emergence in complex networks.

Antisocial behavior-based environmental feedback in spatial prisoner's dilemma game

The environment, people, and behavior are interdependent and always interactive influence. In reality, a higher cooperation level may change the environment in a way that is beneficial for all competitors. Herein, this work proposed a coevolutionary model in the prisoner's dilemma game where the community environment determines the anti-social revenue structure. Moreover, to capture this phenomenon that behavioral changes induced by the individual's perception of the external environment, the memory effect is also taken into account in this work. It suggests that individuals' stored information about community cooperative level (referred to as indirect experience) can trigger antisocial behavior by influencing their perceptions of antisocial revenue, and that their own past strategic information (referred to as direct experience) also plays a similar role. Numerous simulations show that the proposed mechanism significantly improves cooperation, especially when sensitivity to the environment is taken into account. Furthermore, when memory is considered, a lack of experience and insufficient environmental information may result in different optimal cooperative levels corresponding to different b. In addition, the impact of the sufficient indirect experience and the immediate environmental feedback on cooperation are nearly identical. When individuals have sufficient direct and indirect experiences, their roles can be interchangeably in decision-making. Finally, we discovered that the proposed mechanism exhibits good robustness. Overall, this study demonstrates an innovative use of quantitative calculations to describe the conclusions of qualitative analysis, which have theoretical and practical implications.

Evolution of cooperation in spatial public goods game with interactive diversity

Evolution of cooperation in spatial public goods game with interactive diversity

Inertia in spatial public goods games under weak selection

Due to limited cognitive skills for perceptual error or other emotional reasons, players may keep their current strategies even if there is a more promising choice. Such behavior inertia has already been studied, but its consequences remained unexplored in the weak selection limit. To fill this gap, we consider a spatial public goods game model where inertia is considered during the imitation process. By using the identity-by-descent method, we present analytical forms of the critical synergy factor $r^\star$, which determines when cooperation is favored. We find that inertia hinders cooperation, which can be explained by the decelerated coarsening process under weak selection. Interestingly, the critical synergy conditions for different updating protocols, including death-birth and birth-death rules, can be formally linked by the extreme limits of the inertia factor. To explore the robustness of our observations, calculations are made for different lattices and group sizes. Monte Carlo simulations also confirm the results.

The advantage of cooperation for moderate switching between group exclusion and group reward in the spatial public goods game

Group exclusion and group reward are prevalent in institutional management, which are often used to maintain public cooperation and common resources. However, it remains theoretically unclear how the combination of both mechanisms affects social cooperation. Here we propose a conditional switching between group exclusion and group reward based on an assessment threshold in the spatial public goods game. This mechanism implies that once the frequency of cooperators falls below the assessment threshold, all members in the group are excluded, otherwise, they are all rewarded equally. Using Monte Carlo numerical analysis, we find that a moderate switching between group exclusion and group reward (MSER) can quickly establish high level of cooperation even in harsh social dilemmas, while smaller or excessively large assessment thresholds can weaken the cooperative advantage. Furthermore, we reveal that the MSER mechanism has significant advantages over the pure group exclusion and the pure group reward in terms of the evolution of cooperation and the propagation of mechanism. Moreover, the pure group reward clearly outperforms the pure group exclusion in sustaining cooperation only when the synergy factor is relatively small and the reward intensity is high, while the pure group exclusion significantly performs better than the pure group reward only when the synergy factor is relatively large and the reward intensity is small.

Zealous cooperation does not always promote cooperation in public goods games.

There is a conventional belief that prosocial behaviors cannot arise through selfish human nature, because defection always exploits cooperation to achieve a higher payoff at an individual level. Unyieldingly, some people hope to move society to cooperation through their zealous cooperation, regardless of payoffs. From the perspective of spatial evolutionary games, however, such zealous behavior is unnecessary because cooperation can emerge from selfish human nature by aggregating in evolution. Yet, to what extent can zealous cooperation induce others to cooperate? We assume a fraction of zealous agents in spatial public goods games who always cooperate. The results show that a moderate proportion of these zealous cooperators can diminish the cooperation level in the system, and cooperation is only promoted when zealots are many. Regarding spatial behaviors, the areas of zealous cooperation in a medium density can prevent evolutionary cooperation from passing through and aggregating. The phenomenon of zealous cooperation impeding cooperation becomes more pronounced when agents become less random and more selfish. This is because dotted zealous cooperation provides significant payoffs to neighboring defection, making them more solid in fitness. In this way, we also find that when zealous cooperators have low productivity, the neighbors receive fewer benefits by exploitation, thus allowing cooperation to spread. We also study replicator dynamics in unstructured populations where zealous cooperation always promotes cooperation, agreeing that zealous cooperation hindering cooperation is a spatial effect.

An Examination of the Effects of Virtual Reality Training on Spatial Visualization and Transfer of Learning.

Spatial visualization ability (SVA) has been identified as a potential key factor for academic achievement and student retention in Science, Technology, Engineering, and Mathematics (STEM) in higher education, especially for engineering and related disciplines. Prior studies have shown that training using virtual reality (VR) has the potential to enhance learning through the use of more realistic and/or immersive experiences. The aim of this study was to investigate the effect of VR-based training using spatial visualization tasks on participant performance and mental workload using behavioral (i.e., time spent) and functional near infrared spectroscopy (fNIRS) brain-imaging-technology-derived measures. Data were collected from 10 first-year biomedical engineering students, who engaged with a custom-designed spatial visualization gaming application over a six-week training protocol consisting of tasks and procedures that varied in task load and spatial characteristics. Findings revealed significant small (Cohen's d: 0.10) to large (Cohen's d: 2.40) effects of task load and changes in the spatial characteristics of the task, such as orientation or position changes, on time spent and oxygenated hemoglobin (HbO) measures from all the prefrontal cortex (PFC) areas. Transfer had a large (d = 1.37) significant effect on time spent and HbO measures from right anterior medial PFC (AMPFC); while training had a moderate (d = 0.48) significant effect on time spent and HbR measures from left AMPFC. The findings from this study have important implications for VR training, research, and instructional design focusing on enhancing the learning, retention, and transfer of spatial skills within and across various VR-based training scenarios.