Repeated Game Research Articles

Modelling and Research on Intuitionistic Fuzzy Goal-Based Attack and Defence Game for Infrastructure Networks.

Network attack and defence games are gradually becoming a new approach through which to study the protection of infrastructure networks such as power grids and transportation networks. Uncertainty factors, such as the subjective decision preferences of attackers and defenders, are not considered in existing attack and defence game studies for infrastructure networks. In this paper, we introduce, respectively, the attacker's and defender's expectation value, rejection value, and hesitation degree of the target, as well as construct an intuitionistic fuzzy goal-based attack and defence game model for infrastructure networks that are based on the maximum connectivity slice size, which is a network performance index. The intuitionistic fuzzy two-player, zero-sum game model is converted into a linear programming problem for solving, and the results are analysed to verify the applicability and feasibility of the model proposed in this paper. Furthermore, different situations, such as single-round games and multi-round repeated games, are also considered. The experimental results show that, when attacking the network, the attacker rarely attacks the nodes with higher importance in the network, but instead pays more attention to the nodes that are not prominent in the network neutrality and median; meanwhile, the defender is more inclined to protect the more important nodes in the network to ensure the normal performance of the network.

Short- and long-term repeated game behaviours of two parallel supply chains based on government subsidy in the vehicle market

In response to environmental protection policies issued by the United Nations Environment Program, battery electric vehicles (BEVs) have been developed. This paper studies the short- and long-term repeated game behaviours of two parallel supply chains involving duopoly vehicle manufacturers producing BEVs and fuel vehicles (FVs). Taking consumers’ low-carbon preference and government subsidy scheme into account, optimal decisions and comparative analyses under the three game strategies of non-cooperative game and cooperative game and cost-sharing contract under supply chain coordination have been conducted. In the short-term game supply chain, three game models (non-cooperative, cooperative and cost-sharing contract) are formulated and three optimal decisions are compared. In the long-term repeated game supply chain, the dynamic behaviours of two parallel supply chains under non-cooperative game are analysed. We uncovered that the prices of BEVs are more dependent on the reduction of carbon emissions and government subsidy than FVs. In addition to supply chain coordination, consumers’ low-carbon preference also improves the total profits of the three game models. Furthermore, the adjustment speed of the eco-friendly level of BEVs has the most significant impact on the vehicle market.

Heuristics in experiments with infinitely large strategy spaces

We introduce a new methodology that enables detection of the onset of convergence towards Nash equilibria in simple repeated games with infinitely large strategy spaces, thereby revealing the heuristics used in decision-making. The method works by constraining on a special finite subset of strategies, called decoupled strategies. We show how the technique can be applied to understand price formation in financial market experiments by introducing a predictive measure ΔD: the different between positive decoupled strategies (recommending to buy) and negative decoupled strategies (recommending to sell). Using ΔD we illustrate how the method can predict (at certain special times) participants' actions with a high success rate in a series of experiments.

A distributed algorithm to obtain repeated games equilibria with discounting

We introduce a distributed algorithm to negotiate equilibria on repeated games with discounting. It is based on the Folk Theorem, which allows obtaining better payoffs for all players by enforcing cooperation among players when possible. Our algorithm works on incomplete information games: each player needs not knowing the payoff function of the rest of the players. Also, it allows obtaining Pareto-efficient payoffs for all players using either Nash or correlated equilibrium concepts. We explain the main ideas behind the algorithm, explain the two key procedures on which algorithm relies on, provide a theoretical bound on the error introduced and show empirically the performance of the algorithm on four well-known repeated games.

Power Rationing for Tradeoff Between Energy Consumption and Profit in Multimedia Heterogeneous Networks

With the explosive growth of multimedia applications, heterogeneous cellular networks (HetNets) are widely deployed to meet the increasingly impressing demands on communication capacity. However, supporting the massive real-time traffic generated by always-on multimedia applications in HetNets causes enormous energy consumptions. The tradeoff between energy consumptions and profits of service providers while maintaining satisfied quality-of-service (QoS) has become a significant objective. To this end, this paper proposes a novel framework of power rationing in HetNets to achieve maximal profit and guaranteed service performance. A dynamic power rationing strategy that employs Tullock contest is developed to model the power control of multiple cells into a game and solve the contradiction of profits and energy consumption as the tradeoff between QoS and cost. Two principal challenges in gaming, incomplete information, and the curse of dimensionality are resolved by the designed virtual repeated game that adopts the Monte-Carlo method and particle swarm optimization (PSO) to obtain the Nash equilibrium. The equilibrium of power rationing balances the energy consumptions and profits of cells, which ensures the optimal solution for the multimedia service providers and HetNets operators. The experimental results demonstrate that the developed model can serve as an efficient tool for power rationing in multimedia HetNets.

Analysis of Training Plans in Basketball: Gender and Formation Stage Differences.

Scientific literature has stated the presence of various stages in athletes sportive development, with different objectives in each one of them. This should lead coaches to different training plans according to the athlete’s formation stage. The aim of this study was to analyse training plans and identify differences in basketball objectives according to formative stages (U’12 and U’14) in boys and girls. A total of 1,976 training tasks were collected and analysed, for a total of four teams (girls and boys of U’12 and U’14 categories) during an entire season. Pedagogical variables, game phases, game situations, training means and content were studied. The results showed significant differences between genders. Girls’ teams performed more tasks on offense and technical skills. By contrast, boys’ teams performed more defensive tasks and tactical contents. The 1-on-0 and 1-on-1 were the most repeated game situations in all teams. Coaches used different training tasks according to gender and age. In male U’12 teams, drills predominated, whereas in the other categories, games predominated. For boys’ teams, the contents were tactical oriented, and for girls’ teams, the contents were oriented toward skill acquisition. Studying the pedagogical variables of the training process allowed for identification of the utility of training, assessment, and modification of this process.

Constructing strategies in the indefinitely repeated prisoner’s dilemma game

We propose a new approach for running lab experiments on indefinitely repeated games with high continuation probability. This approach has two main advantages. First, it allows us to run multiple long repeated games per session. Second, it allows us to incorporate the strategy method with minimal restrictions on the set of pure strategies that can be implemented. This gives us insight into what happens in long repeated games and into the types of strategies that subjects construct. We report results obtained from the indefinitely repeated prisoner’s dilemma with a continuation probability of δ=.95. We find that during such long repeated prisoner’s dilemma games, cooperation drops from the first period of a supergame to the last period of a supergame. When analyzing strategies, we find that subjects rely on strategies similar to those found in the literature on shorter repeated games—specifically Tit-For-Tat, Grim Trigger, and Always Defect. However, we also identify features of strategies that depend on more than just the previous period that are responsible for the drop in cooperation within supergames, but that may be overlooked when using the common strategy frequency estimation approach.

The method of mutual influence object getting along with two parties based on repetitive game control

To meet the goal of promoting the effectiveness of game analysis towards the governmental regulations and the rules and regulations for enterprise safety, a game analysis method of governmental regulations and the rules and regulations for enterprise safety is proposed on the basis of distributed power and controlled by repeated game. First and foremost, based on the Nash negotiation and the limited repeated game, the distributed repeated game control scheme is established, and the game strategy is effectively promoted. Secondly, on the basis of proposed game analysis, the game analysis model for governmental regulations and the rules and regulations for enterprise safety is established, and the analysis of the relations between governmental regulations and the rules and regulations for enterprise safety is carried out. Lastly, the through the analytical investigation carried out towards the practical cases, it is indicated that the proposed method can be more precise, and this method can be applied for guiding the perfection of enterprises’ regulation and promoting the enterprise benefit.

Crosstalk in concurrent repeated games impedes direct reciprocity and requires stronger levels of forgiveness

Direct reciprocity is a mechanism for cooperation among humans. Many of our daily interactions are repeated. We interact repeatedly with our family, friends, colleagues, members of the local and even global community. In the theory of repeated games, it is a tacit assumption that the various games that a person plays simultaneously have no effect on each other. Here we introduce a general framework that allows us to analyze “crosstalk” between a player’s concurrent games. In the presence of crosstalk, the action a person experiences in one game can alter the person’s decision in another. We find that crosstalk impedes the maintenance of cooperation and requires stronger levels of forgiveness. The magnitude of the effect depends on the population structure. In more densely connected social groups, crosstalk has a stronger effect. A harsh retaliator, such as Tit-for-Tat, is unable to counteract crosstalk. The crosstalk framework provides a unified interpretation of direct and upstream reciprocity in the context of repeated games.

Bounding payoffs in repeated games with private monitoring: n-player games

We provide a simple upper bound on the Nash equilibrium payoff set at a fixed discount factor in repeated games with imperfect private monitoring. The bound admits a tractable recursive characterization and can thus be applied “off-the-shelf” to any repeated game. The bound is not tight in general, but it is tight if the stage game is “concave” or if a certain form of observable mixed actions is allowed. We illustrate our results with applications to the repeated prisoners' dilemma and to canonical public goods and oligopoly games.

Heuristics in Experiments with Infinitely Large Strategy Spaces

We introduce a new methodology that enables detection of the onset of convergence towards Nash equilibria in simple repeated games with infinitely large strategy spaces, thereby revealing the heuristics used in decision-making. The method works by constraining on a special finite subset of strategies. We illustrate how the method can predict (at certain special times) participants' actions with a high success rate in a series of experiments.

The Determinants of Response Time in a Repeated Constant-Sum Game: A Robust Bayesian Hierarchical Model

The investigation of response time and behavior has a long tradition in cognitive psychology, particularly for non-strategic decision making. Recently, experimental economists have also studied response time in strategic interactions, but mostly for one-shot games. This is the first study to investigate the determinants of response time in a repeated game with fixed partner matching, specifically a constant-sum game with a unique mixed strategy Nash equilibrium. I specify and test hypotheses of the relationship between response time and the history of play, such as patterns in past joint actions and payoffs, the degree of randomization and the difference in expected payoffs of the available actions. The main finding is that actions that are consistent with the win-stay/lose-shift heuristic are markedly faster than inconsistent actions. This was found in virtually the whole population supporting previous findings based on action data only that this heuristic is widespread as an intuitive strategy in both humans and animals. Furthermore, I relate the magnitude of response time differences to the degree of regret and disappointment experienced (or avoided) in the prior round, calculated relative to the appropriate counterfactuals. The higher the experienced regret and disappointment from a loss in the prior round, the faster the subsequent response time when subjects shifted away from the previous action. Furthermore, disappointment had a stronger effect on response times than regret. These results highlight the important role of emotions arising from the history of game play, and how they may drive the implementation of strategic behavior.

Commitments and Contests

This article offers a contribution to the broader theoretical goal of increasing our systematic understanding of the social mechanisms that underlie collective behavior. Specifically, we ask how it is possible that equity can arise in a market situation. Even though fair outcomes can be observed across a wide range of social situations that are not centrally coordinated, we argue that there is scant understanding of how these outcomes are facilitated and what impact they have on interaction patterns. We use experimental game theory to test two mechanisms of preferential association, which is defined as a situation where there is competition for partners and in which two actors have a preference for each other. The first mechanism, committed partnership, is abundant in social life and is exemplified by friendships. The second mechanism, competitive altruism, is the intuitive idea that to secure a desirable matching, one must display desirable behavior. To test the mechanisms, we use a laboratory experiment in which participants play a version of the well-known repeated Ultimatum Game. In our version of the game, players either (a) play each other randomly, (b) according to reputation, or (c) according to preferential association in which players are matched according to a preferred ranking of past player histories. The results suggest that fairness can arise from the reduced risk associated with committing to a desirable partner. This is a study of micro-to-micro mechanisms, investigating the micro-level conditions that produce a particular type of social action, namely, a sustainable equitable division of offerings.

Constructing Strategies in the Indefinitely Repeated Prisoner's Dilemma Game

We propose a new approach for running lab experiments on indefinitely repeated games with high continuation probability. This approach has two main advantages. First, it allows us to run multiple long repeated games per session. Second, it allows us to incorporate the strategy method with minimal restrictions on the set of pure strategies that can be implemented. This gives us insight into what happens in long repeated games and into the types of strategies that subjects construct. We report results obtained from the indefinitely repeated prisoner's dilemma with a continuation probability of δ =0.95. We find that during such long repeated prisoner's dilemma games, cooperation drops from the first period of a supergame to the last period of a supergame. When analyzing strategies, we find that subjects rely on strategies similar to those found in the literature on shorter repeated games --- specifically Tit-For-Tat, Grim Trigger, and Always Defect. However, we also identify features of strategies that depend on more than just the previous period that are responsible for the drop in cooperation within supergames, but that may be overlooked when using the common strategy frequency estimation approach.

Cross-layer distributed power control: a repeated game formulation to improve the sum energy efficiency

The main objective of this work is to improve the energy-efficiency (EE) of a multiple access channel (MAC) system, through power control, in a distributed manner. In contrast with many existing works on energy-efficient power control, which ignore the possible presence of a queue at the transmitter, we consider a new generalized cross-layer EE metric. This approach is relevant when the transmitters have a non-zero energy cost even when the radiated power is zero and takes into account the presence of a finite packet buffer and packet arrival at the transmitter. As the Nash equilibrium (NE) is an energy-inefficient solution, the present work aims at overcoming this deficit by improving the global energy-efficiency. Indeed, as the considered system has multiple agencies each with their own interest, the performance metric reflecting the individual interest of each decision maker is the global energy-efficiency defined then as the sum over individual energy-efficiencies. Repeated games (RG) are investigated through the study of two dynamic games (finite RG and discounted RG), whose equilibrium is defined when introducing a new operating point (OP), Pareto-dominating the NE and relying only on individual channel state information (CSI). Accordingly, closed-form expressions of the minimum number of stages of the game for finite RG (FRG) and the maximum discount factor of the discounted RG (DRG) were established. The cross-layer model in the RG formulation leads to achieving a shorter minimum number of stages in the FRG even for higher number of users. In addition, the social welfare (sum of utilities) in the DRG decreases slightly with the cross-layer model when the number of users increases while it is reduced considerably with the Goodman model. Finally, we show that in real systems with random packet arrivals, the cross-layer power control algorithm outperforms the Goodman algorithm.

History Dependence in an Experimental Labor Market

We examine the possibility that other regarding preferences may change in response to past observations. We use a repeated gift exchange game, (an experimental labor market), where an employer decides on a wage and an employee decides on an effort level. In our treatments, low or high wages are imposed by the computer for ten periods, followed by subject-determined wages for ten periods. We find that the computer-imposed wages affect subjects’ subsequent play in terms of both the wages offered and the wage-effort relationship.

History Dependence in an Experimental Labor Market

We examine the possibility that other regarding preferences may change in response to past observations. We use a repeated gift exchange game (an experimental labor market), where an employer decides on a wage and an employee decides on an effort level. In our treatments, low or high wages are imposed by the computer for 10 periods, followed by subject-determined wages for 10 periods. We find that the computer-imposed wages affect subjects’ subsequent play in terms of both the wages offered and the wage–effort relationship.

Governance, trade, and investment

Imperfections in the rule of law create new problems of contractibility, in addition to the familiar one of unverifiability of information. Alternative social institutions for governance of property and contract arise but are also imperfect, and can interact well or poorly with the formal state institutions. Security of property and contract is especially problematic for foreign traders and investors. This paper considers some theoretical analysis of such situations. The game-theoretic ideas and methods range from simple two-stage games with strategic moves to bilateral and multilateral repeated games with added issues of asymmetric information and imperfect communication.

Non Negative Benefits of the Commons in Infinitely Repeated Game

In this paper, two groups use the commons as a resource with its near-sightedness and far-sightedness. We study the game with majority of the group and unanimity of the group in its decision making. And we look into the game with the individuals’ utility in making use of the commons with near-sightedness and farsightedness in infinitely repeated game. SPE exists for nearsightedness which tends to destroy the commons in near large period of the game but farsightedness makes safe and useful for infinite period of the survivility of the humanity.

Between Challenge and Defeat: Repeated Player-Death and Game Enjoyment

This article contributes to research on the complex relation between dying in a digital game and player experience. In this article, we extend recent findings (Ravaja, Turpeinen, Saari, Kelti-kangas-Järvinen, & Puttonen, 2008) indicating that player death may induce positive affect. The study explored whether smiling upon dying in a game should be attributed to transient relief experiences or, instead, related to being challenged to try harder. Facial expressions were recorded while players experienced frequent deaths and occasionally reached safe zones in a shooter game. Players, indeed, smiled after dying, as testified by deactivation of the corrugator supercilii and activation of the zygomaticus major. Although retrospective subjective evaluations showed player-death events were appraised negatively, importantly, zygomaticus major activation decreased with repeated player-deaths. Moreover, safe zones did not result in patterns of facial electromyography activity indicating pleasure, countering a transient relief explanation. We propose that, as part of the ongoing game, player death reinforces the sense of challenge that the game offers. While dying may not be fun in itself, being challenged may initially be experienced as enjoyable and, therefore, evoke a smile. Without clear progress in the game and a lost sense of challenge, however, dying is no longer a laughing matter.