Payoffs Of Players Research Articles

On the Transmission of Continuous Cultural Traits

This paper generalizes the discrete cultural transmission model proposed by Bisin and Verdier (2001) to continuous trait space. The resulting cultural evolutionary dynamic can be characterized by a continuous imitative dynamic in a population game in which a player's payoff is equal to the aggregate cultural intolerance he has towards other agents. We show that cultural heterogeneity is always preserved. In addition, we model each agent's cultural intolerance towards another agent as an increasing function of cultural distance --- the distance between that other agent's trait and his own trait in the trait space. This captures people's general tendencies of evaluating culturally more distant people with stronger biases, and it is most easily modeled on a continuous trait space. We find that the curvature of the cultural intolerance function plays an important role in determining the long-run cultural phenomena. In particular, when cultural intolerance is a convex function of cultural distance, only the most extremely polarized state is a stable limit point.

Computing Approximate Nash Equilibria in Polymatrix Games

In an $$\epsilon $$∈-Nash equilibrium, a player can gain at most $$\epsilon $$∈ by unilaterally changing his behavior. For two-player (bimatrix) games with payoffs in [0, 1], the best-known $$\epsilon $$∈ achievable in polynomial time is 0.3393 (Tsaknakis and Spirakis in Internet Math 5(4):365---382, 2008). In general, for n-player games an $$\epsilon $$∈-Nash equilibrium can be computed in polynomial time for an $$\epsilon $$∈ that is an increasing function of n but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of $$\epsilon $$∈ are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general n-player games where a player's payoff is the sum of payoffs from a number of bimatrix games. There exists a very small but constant $$\epsilon $$∈ such that computing an $$\epsilon $$∈-Nash equilibrium of a polymatrix game is $$\mathtt {PPAD}$$PPAD-hard. Our main result is that a $$(0.5+\delta )$$(0.5+ź)-Nash equilibrium of an n-player polymatrix game can be computed in time polynomial in the input size and $$\frac{1}{\delta }$$1ź. Inspired by the algorithm of Tsaknakis and Spirakis [28], our algorithm uses gradient descent style approach on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a $$(0.5+\delta )$$(0.5+ź)-Nash equilibrium in a two-player Bayesian game.

The value of technology improvements in games with externalities: A fresh look at offsetting behavior

We model the effect of safety technology improvements in a symmetric game in which each player's payoff depends on his own precaution and the other players' average precaution. We derive conditions under which an improved technology increases or decreases players' equilibrium utilities.For mandatory safety technologies, the direction of the welfare effect depends on whether the externality between players is positive or negative, and on whether the technology improvement is a complement or substitute for individual precaution. For safety technologies that individuals can choose whether or not to purchase, individuals expend too much on reducing the loss size but may spend either too much or too little on features that reduce an individual's loss probability.

From Hierarchies to Levels: New Solutions for Games

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many of these problems, players are organized according to either a hierarchical structure or a levels structure that restrict players’ possibilities to cooperate. In this paper, we propose three new solutions for games with hierarchical structure and characterize them by properties that relate a player’s payoff to the payoffs of other players located in specific positions in the structure relative to that player. To define each of these solutions, we consider a certain mapping that transforms any hierarchical structure into a levels structure, and then we apply the standard generalization of the Shapley Value to the class of games with levels structure. The transformations that map the set of hierarchical structures to the set of levels structures are also studied from an axiomatic viewpoint by means of properties that relate a player’s position in both types of structure.

The Farsighted Stable Set

Harsanyi (1974) criticized the von Neumann–Morgenstern (vNM) stable set for its presumption that coalitions are myopic about their prospects. He proposed a new dominance relation incorporating farsightedness, but retained another feature of the stable set: that a coalition S can impose any imputation as long as its restriction to S is feasible for it. This implicitly gives an objecting coalition complete power to arrange the payoffs of players elsewhere, which is clearly unsatisfactory. While this assumption is largely innocuous for myopic dominance, it is of crucial significance for its farsighted counterpart. Our modification of the Harsanyi set respects “coalitional sovereignty.” The resulting farsighted stable set is very different from both the Harsanyi and the vNM sets. We provide a necessary and sufficient condition for the existence of a farsighted stable set containing just a single-payoff allocation. This condition roughly establishes an equivalence between core allocations and the union of allocations over all single-payoff farsighted stable sets. We then conduct a comprehensive analysis of the existence and structure of farsighted stable sets in simple games. This last exercise throws light on both single-payoff and multi-payoff stable sets, and suggests that they do not coexist.

Development of a secure routing protocol using game theory model in mobile ad hoc networks

In mobile ad-hoc networks (MANETs), nodes are mobile in nature. Collaboration between mobile nodes is more significant in MANETs, which have as their greatest challenges vulnerabilities to various security attacks and an inability to operate securely while preserving its resources and performing secure routing among nodes. Therefore, it is essential to develop an effective secure routing protocol to protect the nodes from anonymous behaviors. Currently, game theory is a tool that analyzes, formulates and solves selfishness issues. It is seldom applied to detect malicious behavior in networks. It deals, instead, with the strategic and rational behavior of each node. In our study, we used the dynamic Bayesian signaling game to analyze the strategy profile for regular and malicious nodes. This game also revealed the best actions of individual strategies for each node. Perfect Bayesian equilibrium (PBE) provides a prominent solution for signaling games to solve incomplete information by combining strategies and payoff of players that constitute equilibrium. Using PBE strategies of nodes are private information of regular and malicious nodes. Regular nodes should be cooperative during routing and update their payoff, while malicious nodes take sophisticated risks by evaluating their risk of being identified to decide when to decline. This approach minimizes the utility of malicious nodes and it motivates better cooperation between nodes by using the reputation system. Regular nodes monitor continuously to evaluate their neighbors using belief updating systems of the Bayes rule.

Nonlocality and conflicting interest games.

Nonlocality enables two parties to win specific games with probabilities strictly higher than allowed by any classical theory. Nevertheless, all known such examples consider games where the two parties have a common interest, since they jointly win or lose the game. The main question we ask here is whether the nonlocal feature of quantum mechanics can offer an advantage in a scenario where the two parties have conflicting interests. We answer this in the affirmative by presenting a simple conflicting interest game, where quantum strategies outperform classical ones. Moreover, we show that our game has a fair quantum equilibrium with higher payoffs for both players than in any fair classical equilibrium. Finally, we play the game using a commercial entangled photon source and demonstrate experimentally the quantum advantage.

From Hierarchies to Levels: New Solutions for Games with Hierarchical Structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many of these problems, players are organized according to either a hierarchical structure or a levels structure that restrict players’ possibilities to cooperate. In this paper, we propose three new solutions for games with hierarchical structure and characterize them by properties that relate a player’s payoff to the payoffs of other players located in specific positions in the structure relative to that player. To define each of these solutions, we consider a certain mapping that transforms any hierarchical structure into a levels structure, and then we apply the standard generalization of the Shapley Value to the class of games with levels structure. The transformations that map the set of hierarchical structures to the set of levels structures are also studied from an axiomatic viewpoint by means of properties that relate a player’s position in both types of structure.

Risk consideration and cooperation in the iterated prisoner’s dilemma

This paper investigates the cooperative behavior in the two-player iterated prisoner's dilemma (IPD) game with the consideration of income stream risk. The standard deviation of one-move payoffs for players is defined for measuring the income stream risk, and thus the risk effect on the cooperation in the two-player IPD game is examined. A two-population coevolutionary learning model, embedded with a niching technique, is developed to search optimal strategies for two players to play the IPD game. As experimental results illustrate, risk-averse players perform better than risk-seeking players in cooperating with opponents. In particular, in the case with short game encounters, in which cooperation has been demonstrated to be difficult to achieve in previous work, a high level of cooperation can be obtained in the IPD if both players are risk-averse. The reason is that risk consideration induces players to negotiate for stable gains, which lead to steady mutual cooperation in the IPD. This cooperative pattern is found to be quite robust against low levels of noise. However, with increasingly higher levels of noise, only intermediate levels of cooperation can be achieved in games between two risk-averse players. Games with risk-seeking players get to even lower cooperation levels. By comparing the players' strategies coevolved with and without a high level of noise, the main reason for the reduction in the extent of cooperation can be explained as the lack of contrition and forgiveness of players in the high-noise interactions. Moreover, although increasing encounter length is helpful in improving cooperation in the noiseless and low-noise IPD, we find that it may enforce the absence of contrition and forgiveness, and thus make cooperation even more difficult in the high-noise games.

Bayesian Games with Baire Class 1 Payoff Functions

This paper considers Bayesian games where a player's payoff function is a Baire class 1 function. A Baire class 1 function (Baire (1905) is defined as a pointwise limit of a sequence of continuous functions and include semicontinuous functions, step functions, functions of bounded variations, and monotone functions. Bayesian games with Baire class 1 payoff's generalize the canonical model of Milgrom and Weber (1985) and encompass a large class of auction games and mechanism design problems. A Bayesian game with Baire class 1 payoff's has a Bayesian equilibrium via continuous approximations when the better reply security condition of Reny (1999) is satisfied. This continuous approximation approach can be regarded as a generalization of the path-following method to games with discontinuous payoff's. Using the continuous approximation method, we characterize the optimal auction mechanism with heterogeneous objects and multidimensional types with continuous distributions by unifying Myerson (1981), Mussa and Rosen (1978), and Rochet and Chone (1998). We illustrate the method by solving an example where the buyers private values are uniformly distributed on a square.

Weakly monotonic solutions for cooperative games

The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some player's marginal contribution to all coalitions increase or stay the same, then this player's payoff should not decrease. We investigate the class of values that satisfy efficiency, symmetry, and weak monotonicity. It turns out that this class coincides with the class of egalitarian Shapley values. Thus, weak monotonicity reflects the nature of the egalitarian Shapley values in the same vein as strong monotonicity reflects the nature of the Shapley value. An egalitarian Shapley value redistributes the Shapley payoffs as follows: First, the Shapley payoffs are taxed proportionally at a fixed rate. Second, the total tax revenue is distributed equally among all players.

Games on graphs: A minor modification of payoff scheme makes a big difference

Many techniques developed in simulations of physical models have been adopted in studies of game theory by researchers including physicists and mathematicians. In this work, we show that a seemingly non-essential mechanism —what we refer to as a “payoff scheme”— has a large impact on strategic outcomes of some games. The payoff scheme here refers to how each player's payoff is calculated in each round after the states of all of the players are determined. Conventionally either the accumulated or the average payoff of a player is used, where its payoff is calculated from pairing up the player with all of its neighboring players. Here we consider to calculate the payoff from pairing up with only one random player from the neighboring players. The average payoff scheme that involves averaging over all of the neighbors should, in a sense, be equivalent to repeatedly randomly pairing up with one neighbor a time, which we refer to as the stochastic payoff scheme. However, our simulation of games on graphs shows that, in many cases, the two payoff schemes lead to qualitatively different levels of cooperation: Seemingly non-essential modifications might have large impact on behavioral outcomes. We have also observed that results from the stochastic scheme are more robust than the average scheme: different updating rules and initial states of the players do not have a large impact on the final level of cooperation in the former case when compared with those in the latter case.

Strategies generalization and payoff fluctuation optimization in the iterated ultimatum game

An iterated version of ultimatum game, based on generalized probabilistic strategies, which are mathematically modeled by accepting proposal functions is presented. These strategies account for the behavior of the players by mixing levels of altruism and greed. We obtained analytically the moments of the payoff of the players under such a generalization. Our analysis is divided into two cases: (i) no memory players, where players do not remember previous decisions, and (ii) one-step memory players, where the offers depend on players’ last decision. We start considering the former case. We show that when the combination of the proposer’s altruism and responder’s greed levels balances the proposer’s greedy and responder’s altruism levels, the average and variance of the payoff of both players are the same. Our analysis is carried out considering that the acceptance of an offer depends on: (a) a fixed probability p or (b) the value offered. The combination of cases (i) and (a) shows that there exists a p value that maximizes the cumulative gain after n iterations. Moreover, we show n×p diagrams with ïso-average” and ïso-variance” of the cumulative payoff. Our analytical results are validated by Monte Carlo simulations. For the latter case, we show that when players have no memory (i), there are cutoff values, which the variance of the proposer’s cumulative payoff presents local maximum and minimum values, while for the responder, the same amount presents a global maximum. Case (b) combined with one-step memory players (ii), we verified, via MC simulations that, for the same number of iterations, the responder obtains different cumulative payoffs by setting different cutoff values. This result composes an interesting pattern of stripes in the cutoff per n diagrams. Simultaneously, looking at variance of this amount, for the responder player in a similar diagram, we observe regions of iso-variance in non trivial patterns which depend on initial value of the proposal. Our contributions detailed by analytical and MC simulations are useful to design new experiments in the ultimatum game in stochastic scenarios.

Dynamic Stackelberg game model for water rationalization in drought emergency

Summary In water resource management, in case of a limited resource, there is a conflict situation between different consumers. In this paper, a dynamic game-theoretical model is suggested for the solution of such conflict. Let us suppose that in a region, water supply is based on a given aquifer, from which a quantity of effective reserve can be used without damaging the aquifer, and a long drought is foreseen. The use of water is divided between the social sector represented by the local authority, and the production sector, in our case, simplified to a single agricultural producer using water for irrigation; they are the players in the game. For a fixed time period, every day, a given amount is available, from which first the authority, then the producer takes a proportion, which corresponds to the strategy choices of the players. A price function is given, which depends on the total available reserve, the payoffs of both players are quantified as their net incomes for the whole period: for the producer: profit from selling the product minus price of water and tax paid, for the authority: tax received plus the gain for the authority from selling the water bought to the social sector minus price of water purchased. A solution (equilibrium) of the game consists of such strategy choices of both players, with which each player maximizes her/his total payoff (over the whole time horizon of the game) provided that the other player also maximizes her/his own payoff. In the paper, in a mathematical model for the above conflict situation, a deterministic continuum-strategy two-player discrete-time dynamic Stackelberg game with fixed finite time duration and closed-loop information structure is proposed, where the authority is “leader” and the producer is “follower”. The algorithms for the solution of the game are based on recent theoretical results of the authors. Illustrative numerical examples are also given.

Communication in repeated network games with imperfect monitoring

I consider repeated games with private monitoring played on a network. Each player has a set of neighbors with whom he interacts: a player's payoff depends on his own and his neighbors' actions only. Monitoring is private and imperfect: each player observes his stage payoff but not the actions of his neighbors. Players can communicate costlessly at each stage: communication can be public, private or a mixture of both. Payoffs are assumed to be sensitive to unilateral deviations. First, for any network, a folk theorem holds if some Joint Pairwise Identifiability condition regarding payoff functions is satisfied. Second, a necessary and sufficient condition on the network topology for a folk theorem to hold for all payoff functions is that no two players have the same set of neighbors not counting each other.

The Impact of Free Agency on Players’ Compensation

This study examines whether players benefit from free agency or not. In professional sports, free agency has been a hot topic in recent years. Based on a bargaining model with arbitrary bargaining power, we found that players who have a relative smaller bargaining power to their owners still get lower payoffs after the implement of free agency, and vice versa. Moreover, the expected payoffs of players and sports teams are both not influenced by free agency.

Cooperative Mean-Field Type Games

In the standard formulation of a game, a player's payoff function depends on the states and actions of all the players. Yet, real world applications suggest to consider also a functional of the probability measure of states and actions of all the players. In this paper, we consider cooperative mean-field type games in which the state dynamics and the payoffs depend not only on the state and actions but also on their probability measure. We establish stochastic maximum principle and provide a time-dependent payoff allocation procedure for coalitions. The allocated payoff considers not only fairness property but also the cost of making the coalition. Finally, time consistency and subgame perfectness solution concept equations are established.

A Tree Formulation for Signaling Games with Noise

The paper provides an analysis of a sender-receiver sequential signaling game. The private information of the sender is transmitted with noise by a Machine, i.e. does not always correctly reflect the state of nature. Hence, a truthful revelation by the sender of his information does not necessarily imply that the signal he sends is correct. Also, the receiver can take a correct action even if the sender transmits an incorrect signal. The payoffs of the two players depend on their combined actions. Perfect Bayesian Equilibria which can result from different degrees of noise is analysed. The Bayesian updating of probabilities is explained. The fixed point theorem which makes the connection with the idea of rational expectations in economics is calculated. Given a number of equilibria, we comment on the most credible one on the basis of the implied payoffs for both players. The equilibrium signals are an example of the formation of a language convention discussed by D. Lewis.

When Ignorance is Bliss - Information Asymmetries Enhance Prosocial Behavior in Dictator Games

In most laboratory experiments concerning prosocial behavior subjects are fully informed how their decision influences the payoff of other players. Outside the laboratory, however, individuals typically have to decide without such detailed knowledge. To assess the effect of information asymmetries on prosocial behavior, we conduct a laboratory experiment with a simple non-strategic interaction. A dictator has only limited knowledge about the benefits his prosocial action generates for a recipient. We observe subjects with heterogenous social preferences. While under symmetric information only individuals with the same type of preferences transfer, under asymmetric information different types transfer at the same time. As a consequence and the main finding of our experiment, uninformed dictators behave more prosocially than informed dictators.

Strategic real options with stochastic volatility in a duopoly model

The investment-timing problem has been considered by many authors under the assumption that the instantaneous volatility of the demand shock is constant. Recently, Ting et al. (2013) [12] carried out an asymptotic approach in a monopoly model by letting the volatility parameter follow a stochastic process. In this paper, we consider a strategic game in which two firms compete for a new market under an uncertain demand, and extend the analysis of Ting et al. to duopoly models under different strategic game structures. In particular, we investigate how the additional uncertainty in the volatility affects the investment thresholds and payoffs of players. Several numerical examples and comparison of the results are provided to confirm our analysis.