Games, Volume 12, Issue 3 (September 2021) – 20 articles

In hedonic games, coalitions are created as a result of the strategic interaction of independent players. In particular, in additively separable hedonic games, every player has valuations for all other ones, and the utility for belonging to a coalition is given by the sum of the valuations for all other players belonging to it. So far, non-cooperative hedonic games have been considered in the literature only with respect to totally selfish players. Starting from the fundamental class of additively separable hedonic games, we define and study a new model in which, given a social graph, players also care about the happiness of their friends: we call this class of games social context additively separable hedonic games (SCASHGs). We focus on the fundamental stability notion of Nash equilibrium, and study the existence, convergence and performance of stable outcomes (with respect to the classical notions of price of anarchy and price of stability) in SCASHGs. In particular, we show that SCASHGs are potential games, and therefore Nash equilibria always exist and can be reached after a sequence of Nash moves of the players. Finally, we provide tight or asymptotically tight bounds on the price of anarchy and the price of stability of SCASHGs. Full article

In simple dyadic games such as rock, paper, scissors (RPS), people exhibit peculiar sequential dependencies across repeated interactions with a stable opponent. These regularities seem to arise from a mutually adversarial process of trying to outwit their opponent. What underlies this process, and what are its limits? Here, we offer a novel framework for formally describing and quantifying human adversarial reasoning in the rock, paper, scissors game. We first show that this framework enables a precise characterization of the complexity of patterned behaviors that people exhibit themselves, and appear to exploit in others. This combination allows for a quantitative understanding of human opponent modeling abilities. We apply these tools to an experiment in which people played 300 rounds of RPS in stable dyads. We find that although people exhibit very complex move dependencies, they cannot exploit these dependencies in their opponents, indicating a fundamental limitation in people’s capacity for adversarial reasoning. Taken together, the results presented here show how the rock, paper, scissors game allows for precise formalization of human adaptive reasoning abilities. Full article

The present paper provides theoretical insights regarding the determinants of firms’ incentives to invest in a Circular Economy. The analysis relies on a Cournot model disaggregating the disposal cost in the production function. In a non-simultaneous sequential game, two risk-neutral firms are endowed with a green innovation project that, if successful, would reduce the overall production costs and implement a Circular Economy. Firms are plagued by asymmetric information about the exact value of the other firm’s innovation. In this setting, the R&D investment in a Circular Economy, by affecting the distribution of production and disposal costs, influences the production decisions of both the innovating and the rival firms. The sign of the impact depends on the firms’ strategy in the product market. Furthermore, the analysis points out that cooperation in R&D of firms competing in the product market reinforces incentives to invest in green innovation. This suggests that governments aimed to advance a Circular Economy should encourage firms’ cooperation. Full article

While the ontogeny of prosociality during infancy, childhood, and adolescence has received substantial attention over the last decades, little is known about how prosocial preferences develop beyond emerging adulthood. Recent evidence suggests that the previously observed positive association between age and prosocial preferences is less robust than assumed. This study reports results on the association between social preferences, age, gender, and education from an Austrian representative sample (N = 777, aged 16–94 years) in which incentivized social value orientations (SVO) were measured along with various other sociodemographic characteristics. The analyses confirm that men are less prosocial than women, however, mainly during emerging adulthood (16–25 years). At the same time, the decline of prosociality is stronger among women leading to a convergence of prosociality between men and women as they age. Overall, we find that a prosocial value orientation is negatively correlated with people’s age. We suspect that the susceptibility of peoples’ social preferences to the preferences of others in their social environment is a critical factor unifying these different observations in the development of prosociality. We hypothesize that the opposite associations between age and SVO observed in two previous studies using unincentivized measures of social preferences are explained in parts by an age-related change in social desirability, measurement inaccuracy (continuous vs. categorical), and cross-cultural differences promoting competitive preferences among emerging adults in Japan. Moreover, we find that political orientations towards right-wing populists are consistently associated with less prosocial preferences, while education seems to be positively associated with prosociality. Overall, our study highlights the importance of conducting representative studies using incentivized measurements across cultures. Full article

Using a nationally representative sample of 1052 respondents from the United Kingdom, we systematically tested the associations between the experimental trust game and a range of popular self-reported measures for trust, such as the General Social Survey (GSS) and the Rosenberg scale for self-reported trust. We find that, in our UK representative sample, the experimental trust game significantly and positively predicts generalised self-reported trust in the GSS. This association is robust across a number of alternative empirical specifications, which account for multiple hypotheses corrections and control for other social preferences as measured by the dictator game and the public good game, as well as for a broad range of individual characteristics, such as gender, age, education, and personal income. We discuss how these results generalise to nationally representative samples from six other Organisation for Economic Co-operation and Development (OECD) countries (France, Germany, Italy, Korea, Slovenia, and the US). Full article

This paper studies fishery strategies in lakes, seas, and shallow rivers subject to agricultural and industrial pollution. The flowing pollutants are modeled by a nonlinear differential equation in a general manner. The logistic growth model for the fish population is modified to cover the pollution impact on the fish growth rate. We start by presenting the stability analysis of the dynamical system to discern the different types of the evolution of the fish population according to human actions. A cooperative game is formulated to design strategies for preserving the fish population by controlling the pollution as well as the fish stock for harvesting. The sufficient conditions for implementing the cooperative strategy are investigated through an incentive design approach with an adaptive taxation policy for the players. Numerical results are presented to illustrate the benefit of the cooperative for fish population preservation but also for the players’ rewards. Full article

This paper studies how to form an efficient coalition—a group of people. More specifically, we compare two mechanisms for forming a coalition by running a laboratory experiment and reveal which mechanism leads to higher social surplus. In one setting, we invite the subjects to join a meeting simultaneously, so they cannot know the other subjects’ decisions. In the other setting, we ask them sequentially, which allows each subject to know his or her predecessor’s choice. Those who decide to join the meeting form a coalition and earn payoffs according to their actions and individual preferences. As a result, we obtain the following findings. First, the sequential mechanism induces higher social surplus than the simultaneous mechanism. Second, most subjects make choices consistent with the subgame-perfect Nash equilibrium in the sequential setting and choose the dominant strategy in the simultaneous setting, when a dominant strategy exists. Finally, when the subjects need to look further ahead to make a theoretically rational choice, they are more likely to fail to choose rationally. Full article

Experiments using the public goods game have repeatedly shown that in cooperative social environments, punishment makes cooperation flourish, and withholding punishment makes cooperation collapse. In less cooperative social environments, where antisocial punishment has been detected, punishment was detrimental to cooperation. The success of punishment in enhancing cooperation was explained as deterrence of free riders by cooperative strong reciprocators, who were willing to pay the cost of punishing them, whereas in environments in which punishment diminished cooperation, antisocial punishment was explained as revenge by low cooperators against high cooperators suspected of punishing them in previous rounds. The present paper reconsiders the generality of both explanations. Using data from a public goods experiment with punishment, conducted by the authors on Israeli subjects (Study 1), and from a study published in Science using sixteen participant pools from cities around the world (Study 2), we found that: 1. The effect of punishment on the emergence of cooperation was mainly due to contributors increasing their cooperation, rather than from free riders being deterred. 2. Participants adhered to different contribution and punishment strategies. Some cooperated and did not punish (‘cooperators’); others cooperated and punished free riders (‘strong reciprocators’); a third subgroup punished upward and downward relative to their own contribution (‘norm-keepers’); and a small sub-group punished only cooperators (‘antisocial punishers’). 3. Clear societal differences emerged in the mix of the four participant types, with high-contributing pools characterized by higher ratios of ‘strong reciprocators’, and ‘cooperators’, and low-contributing pools characterized by a higher ratio of ‘norm keepers’. 4. The fraction of ‘strong reciprocators’ out of the total punishers emerged as a strong predictor of the groups’ level of cooperation and success in providing the public goods. Full article

When we play competitive games, the opponents that we face act as predictors of the outcome of the game. For instance, if you are an average chess player and you face a Grandmaster, you anticipate a loss. Framed in a reinforcement learning perspective, our opponents can be thought of as predictors of rewards and punishments. The present study investigates whether facing an opponent would be processed as a reward or punishment depending on the level of difficulty the opponent poses. Participants played Rock, Paper, Scissors against three computer opponents while electroencephalographic (EEG) data was recorded. In a key manipulation, one opponent (HARD) was programmed to win most often, another (EASY) was made to lose most often, and the third (AVERAGE) had equiprobable outcomes of wins, losses, and ties. Through practice, participants learned to anticipate the relative challenge of a game based on the opponent they were facing that round. An analysis of our EEG data revealed that winning outcomes elicited a reward positivity relative to losing outcomes. Interestingly, our analysis of the predictive cues (i.e., the opponents’ faces) demonstrated that attentional engagement (P3a) was contextually sensitive to anticipated game difficulty. As such, our results for the predictive cue are contrary to what one might expect for a reinforcement model associated with predicted reward, but rather demonstrate that the neural response to the predictive cue was encoding the level of engagement with the opponent as opposed to value relative to the anticipated outcome. Full article

We consider a demand management problem in an energy community, in which several users obtain energy from an external organization such as an energy company and pay for the energy according to pre-specified prices that consist of a time-dependent price per unit of energy as well as a separate price for peak demand. Since users’ utilities are their private information, which they may not be willing to share, a mediator, known as the planner, is introduced to help optimize the overall satisfaction of the community (total utility minus total payments) by mechanism design. A mechanism consists of a message space, a tax/subsidy, and an allocation function for each user. Each user reports a message chosen from her own message space, then receives some amount of energy determined by the allocation function, and pays the tax specified by the tax function. A desirable mechanism induces a game, the Nash equilibria (NE), of which results in an allocation that coincides with the optimal allocation for the community. As a starting point, we design a mechanism for the energy community with desirable properties such as full implementation, strong budget balance and individual rationality for both users and the planner. We then modify this baseline mechanism for communities where message exchanges are allowed only within neighborhoods, and consequently, the tax/subsidy and allocation functions of each user are only determined by the messages from their neighbors. All of the desirable properties of the baseline mechanism are preserved in the distributed mechanism. Finally, we present a learning algorithm for the baseline mechanism, based on projected gradient descent, that is guaranteed to converge to the NE of the induced game. Full article

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. Full article

A dynamic Bertrand-duopoly model where price leadership emerges in equilibrium is developed. In the price leadership equilibrium, a firm leads price changes and its competitor always matches in the next period. The firms produce a homogeneous product and are identical except for the information they possess about demand. The market size follows a two-state Markov process. Market size realizations are observed by one of the firms but not the other. Without explicit communication, price leadership allows firms to jointly approximate monopolistic profits in equilibrium as the market size becomes more persistent provided that firms are patient. In the presence of persistent market dynamics, the informed firm’s price serves as a signal of current and therefore future market conditions. In the proposed price leadership equilibrium, the informed firm could cut prices without being detected, but it does not do so because it would lead the uninformed to also lower their price in the following period. Full article

This paper investigates the importance of concerns about intentions and outcomes in a sequential prisoner’s dilemma game with nature. In the game, there is a chance that the first mover’s choice is reversed. This allows the separation of intended actions from the resulting outcomes. Equilibrium predictions from theoretical models of fairness are tested experimentally by varying the chance the first mover’s choice is reversed and whether the second mover observes the first mover’s choice. The results show that second mover cooperation is higher when the first mover has little control over their choice and when the second mover is not told what the first mover chose. While subject behavior is consistent with concerns for both intentions and outcomes, the results indicate that these concerns work in ways not predicted by current theoretical models. In addition, I find that psychometric measures of empathic concern and perspective taking are correlated with second mover cooperation and provide potential explanations for the experimental results. Full article

This note provides two numerical representations of a nested system of strict partial orders. The first representation is based on utility and threshold functions. We generalize the threshold representation of menu-dependent preferences by allowing the threshold to depend not only on the menu but also on the pair of alternatives under comparison. The threshold function can be interpreted as the distance between alternatives. The second representation is based on the aggregation of multi-dimensional preference. This representation describes a decision-making procedure where multiple criteria are gradually aggregated into an overall assessment. Full article

We study the criminal networks that will emerge in the long run when criminals are neither myopic nor completely farsighted but have some limited degree of farsightedness. We adopt the horizon-K farsighted set to answer this question. We find that in criminal networks with n criminals, the set consisting of the complete network is a horizon-K farsighted set whenever the degree of farsightedness of the criminals is larger than or equal to $(n-1)$. Moreover, the complete network is the unique horizon-$(n-1)$ farsighted set. Hence, the predictions obtained in case of completely farsighted criminals still hold when criminals are much less farsighted. Full article

In this paper, we propose a game in which each player decides with whom to establish a costly connection and how much local public good is provided when benefits are shared among neighbors. We show that, when agents are homogeneous, Nash equilibrium networks are nested split graphs. Additionally, we show that the game is a potential game, even when we introduce heterogeneity along several dimensions. Using this result, we introduce stochastic best reply dynamics and show that this admits a unique and stationary steady state distribution expressed in terms of the potential function of the game. Hence, even if the set of Nash equilibria is potentially very large, the long run predictions are sharp. Full article

There are many competing game-theoretic analyses of terrorism. Most of these models suggest nonlinear relationships between terror attacks and some variable of interest. However, to date, there have been very few attempts to empirically sift between competing models of terrorism or identify nonlinear patterns. We suggest that machine learning can be an effective way of undertaking both. This feature can help build more salient game-theoretic models to help us understand and prevent terrorism. Full article

We consider an evolutionary model of social coordination in a 2 × 2 game where two groups of players prefer to coordinate on different actions. Players can pay a cost to learn their opponent’s group: if they pay it, they can condition their actions concerning the groups. We assess the stability of outcomes in the long run using stochastic stability analysis. We find that three elements matter for the equilibrium selection: the group size, the strength of preferences, and the information’s cost. If the cost is too high, players never learn the group of their opponents in the long run. If one group is stronger in preferences for its favorite action than the other, or its size is sufficiently large compared to the other group, every player plays that group’s favorite action. If both groups are strong enough in preferences, or if none of the groups’ sizes is large enough, players play their favorite actions and miscoordinate in inter-group interactions. Lower levels of the cost favor coordination. Indeed, when the cost is low, in inside-group interactions, players always coordinate on their favorite action, while in inter-group interactions, they coordinate on the favorite action of the group that is stronger in preferences or large enough. Full article

This research studied the strategies that players use in sequential adversarial games. We took the Rock-Paper-Scissors (RPS) game as an example and ran players in two experiments. The first experiment involved two humans, who played the RPS together for 100 times. Importantly, our payoff design in the RPS allowed us to differentiate between participants who used a random strategy from those who used a Nash strategy. We found that participants did not play in agreement with the Nash strategy, but rather, their behavior was closer to random. Moreover, the analyses of the participants’ sequential actions indicated heterogeneous cycle-based behaviors: some participants’ actions were independent of their past outcomes, some followed a well-known win-stay/lose-change strategy, and others exhibited the win-change/lose-stay behavior. To understand the sequential patterns of outcome-dependent actions, we designed probabilistic computer algorithms involving specific change actions (i.e., to downgrade or upgrade according to the immediate past outcome): the Win-Downgrade/Lose-Stay (WDLS) or Win-Stay/Lose-Upgrade (WSLU) strategies. Experiment 2 used these strategies against a human player. Our findings show that participants followed a win-stay strategy against the WDLS algorithm and a lose-change strategy against the WSLU algorithm, while they had difficulty in using an upgrade/downgrade direction, suggesting humans’ limited ability to detect and counter the actions of the algorithm. Taken together, our two experiments showed a large diversity of sequential strategies, where the win-stay/lose-change strategy did not describe the majority of human players’ dynamic behaviors in this adversarial situation. Full article