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Showing papers on "Stochastic game published in 2007"


Posted Content
TL;DR: For example, Selten et al. as mentioned in this paper showed that in the guessing game, players engage in a finite depth of reasoning on players' beliefs about one another, where a player selects a strategy at random without forming beliefs or picks a number that is salient to him.
Abstract: Consider the following game: a large number of players have to state simultaneously a number in the closed interval [0, 100]. The winner is the person whose chosen number is closest to the mean of all chosen numbers multiplied by a parameter p, where p is a predetermined positive parameter of the game; p is common knowledge. The payoff to the winner is a fixed amount, which is independent of the stated number and p. If there is a tie, the prize is divided equally among the winners. The other players whose chosen numbers are further away receive nothing.' The game is played for four rounds by the same group of players. After each round, all chosen numbers, the mean, p times the mean, the winning numbers, and the payoffs are presented to the subjects. For 0 c p < 1, there exists only one Nash equilibrium: all players announce zero. Also for the repeated supergame, all Nash equilibria induce the same announcements and payoffs as in the one-shot game. Thus, game theory predicts an unambiguous outcome. The structure of the game is favorable for investigating whether and how a player's mental process incorporates the behavior of the other players in conscious reasoning. An explanation proposed, for out-of-equilibrium behavior involves subjects engaging in a finite depth of reasoning on players' beliefs about one another. In the simplest case, a player selects a strategy at random without forming beliefs or picks a number that is salient to him (zero-order belief). A somewhat more sophisticated player forms first-order beliefs on the behavior of the other players. He thinks that others select a number at random, and he chooses his best response to this belief. Or he forms second-order beliefs on the first-order beliefs of the others and maybe nth order beliefs about the (n I )th order beliefs of the others, but only up to a finite n, called the ndepth of reasoning. The idea that players employ finite depths of reasoning has been studied by various theorists (see e.g., Kenneth Binmore, 1987, 1988; Reinhard Selten, 1991; Robert Aumann, 1992; Michael Bacharach, 1992; Cristina Bicchieri, 1993; Dale 0. Stahl, 1993). There is also the famous discussion of newspaper competitions by John M. Keynes (1936 p. 156) who describes the mental process of competitors confronted with picking the face that is closest to the mean preference of all competitors.2 Keynes's game, which he considered a Gedankenexperiment, has p = 1. However, with p = 1, one cannot distinguish between different steps of reasoning by actual subjects in an experiment. There are some experimental studies in which reasoning processes have been analyzed in ways similar to the analysis in this paper. Judith Mehta et al. (1994), who studied behavior in two-person coordination games, suggest that players coordinate by either applying depth of reasoning of order I or by picking a focal point (Thomas C. Schelling, 1964), which they call "Schelling salience." Stahl and Paul W. Wilson (1994) analyzed behavior in symmetric 3 x 3 games and concluded that subjects were using depths of reasoning of orders 1 or 2 or a Nash-equilibrium strategy. * Department of Economics, Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, Spain. Financial support from Deutsche Forschungsgemeinschaft (DFG) through Sonderforschungsbereich 303 and a postdoctoral fellowship from the University of Pittsburgh are gratefully acknowledged. I thank Reinhard Selten, Dieter Balkenborg, Ken Binmore, John Duffy, Michael Mitzkewitz, Alvin Roth, Karim Sadrieh, Chris Starmer, and two anonymous referees for helpful discussions and comments. I learned about the guessing game in a game-theory class given by Roger Guesnerie, who used the game as a demonstration experiment. 'The game is mentioned, for example, by Herve Moulin (1986), as an example to explain rationalizability, and by Mario H. Simonsen (1988). 2 This metaphor is frequently mentioned in the macroeconomic literature (see e.g., Roman Frydman, 1982).

1,221 citations


Journal ArticleDOI
01 Feb 2007-EPL
TL;DR: The results highlight the importance of asymmetry characterizing the exchange of master-follower role during the strategy adoptions in evolutionary Prisoner's Dilemma games with quenched inhomogeneities in the spatial dynamical rules.
Abstract: Evolutionary Prisoner's Dilemma games with quenched inhomogeneities in the spatial dynamical rules are considered. The players following one of the two pure strategies (cooperation or defection) are distributed on a two-dimensional lattice. The rate of strategy adoption from randomly chosen neighbors is controlled by the payoff difference and a two-value pre-factor w characterizing the players whom the strategy learned from. The reduced teaching activity of players is distributed randomly with concentrations ν at the beginning and fixed further on. Numerical and analytical calculations are performed to study the concentration of cooperators as a function of w and ν for different noise levels and connectivity structures. Significant increase of cooperation is found within a wide range of parameters for this dynamics. The results highlight the importance of asymmetry characterizing the exchange of master-follower role during the strategy adoptions.

399 citations


Journal ArticleDOI
TL;DR: This work introduces a temperature (of selection) to account for stochastic effects and calculates the fixation probabilities and fixation times for any symmetric 2 x 2 game, for any intensity of selection and any initial number of mutants.

343 citations


Proceedings ArticleDOI
01 Mar 2007
TL;DR: A dynamic Cournot game is formulated in which the strategy of one secondary user is selected solely based on the pricing information obtained from the primary user, and the stability condition of the dynamic behavior for this spectrum sharing scheme is investigated.
Abstract: "Cognitive radio" is an emerging technique to improve the utilization of radio frequency spectrum in wireless networks. In this paper, we consider the problem of spectrum sharing among a primary user and multiple secondary users. We formulate this problem as an oligopoly market competition and use a Cournot game to obtain the spectrum allocation for secondary users. Nash equilibrium is considered as the solution of this game. We first present the formulation of a static Cournot game for the case when all secondary users can observe the adopted strategies and the payoff of each other. However, this assumption may not be realistic in some cognitive radio systems. Therefore, we formulate a dynamic Cournot game in which the strategy of one secondary user is selected solely based on the pricing information obtained from the primary user. The stability condition of the dynamic behavior for this spectrum sharing scheme is investigated.

212 citations


Journal ArticleDOI
TL;DR: It is shown that for finite state spaces, all three sets of winning states can be computed in polynomial time: type-1 states in linear time, andType-2 and type-3 states in quadratic time, which enable the construction of the winning and spoiling strategies.

199 citations


Journal ArticleDOI
TL;DR: For Bayesian games of strategic complementarities, this work provides a constructive proof of the existence of a greatest and a least Bayes-Nash equilibrium - each one in strategies monotone in type - if the payoff to a player displays increasing differences in own action and the profile of types.

186 citations


Journal ArticleDOI
TL;DR: This paper presents an approach to single-product dynamic revenue management that accounts for errors in the underlying model at the optimization stage and obtains an optimal pricing policy through a version of the so-called Isaacs' equation for stochastic differential games.
Abstract: In the area of dynamic revenue management, optimal pricing policies are typically computed on the basis of an underlying demand rate model. From the perspective of applications, this approach implicitly assumes that the model is an accurate representation of the real-world demand process and that the parameters characterizing this model can be accurately calibrated using data. In many situations, neither of these conditions are satisfied. Indeed, models are usually simplified for the purpose of tractability and may be difficult to calibrate because of a lack of data. Moreover, pricing policies that are computed under the assumption that the model is correct may perform badly when this is not the case. This paper presents an approach to single-product dynamic revenue management that accounts for errors in the underlying model at the optimization stage. Uncertainty in the demand rate model is represented using the notion of relative entropy, and a tractable reformulation of the “robust pricing problem” is obtained using results concerning the change of probability measure for point processes. The optimal pricing policy is obtained through a version of the so-called Isaacs' equation for stochastic differential games, and the structural properties of the optimal solution are obtained through an analysis of this equation. In particular, (i) closed-form solutions for the special case of an exponential nominal demand rate model, (ii) general conditions for the exchange of the “max” and the “min” in the differential game, and (iii) the equivalence between the robust pricing problem and that of single-product revenue management with an exponential utility function without model uncertainty, are established through the analysis of this equation.

176 citations


Book ChapterDOI
01 Sep 2007
TL;DR: This chapter presents algorithms for repeated play of a matrix game with the guarantee that against any opponent, they will perform nearly as well as the best fixed action in hindsight, and presents a general reduction showing how to convert any algorithm for minimizing external regret to one that minimizes this stronger form of regret as well.
Abstract: Many situations involve repeatedly making decisions in an uncertain environment: for instance, deciding what route to drive to work each day, or repeated play of a game against an opponent with an unknown strategy. In this chapter we describe learning algorithms with strong guarantees for settings of this type, along with connections to game-theoretic equilibria when all players in a system are simultaneously adapting in such a manner. We begin by presenting algorithms for repeated play of a matrix game with the guarantee that against any opponent, they will perform nearly as well as the best fixed action in hindsight (also called the problem of combining expert advice or minimizing external regret). In a zero-sum game, such algorithms are guaranteed to approach or exceed the minimax value of the game, and even provide a simple proof of the minimax theorem. We then turn to algorithms that minimize an even stronger form of regret, known as internal or swap regret. We present a general reduction showing how to convert any algorithm for minimizing external regret to one that minimizes this stronger form of regret as well. Internal regret is important because when all players in a game minimize this stronger type of regret, the empirical distribution of play is known to converge to correlated equilibrium. The third part of this chapter explains a different reduction: how to convert from the full information setting in which the action chosen by the opponent is revealed after each time step, to the partial information (bandit) setting, where at each time step only the payoff of the selected action is observed (such as in routing), and still maintain a small external regret. Finally, we end by discussing routing games in the Wardrop model, where one can show that if all participants minimize their own external regret, then

176 citations


Journal ArticleDOI
TL;DR: It is proved that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games.

175 citations


Patent
14 Mar 2007
TL;DR: In this article, the authors present a system that allows a player to wager varying amounts on whether a player will have multiple consecutive wins, losses, or ties of a game of chance.
Abstract: A gaming method and system which allows a player to wager varying amounts on whether that player will have multiple consecutive wins, losses, or ties of a game of chance and having the option of withdrawing from a consecutive wins, loss or tie bet before reaching the number of consecutive wins, losses or ties originally selected and select a reduced payoff or to make a consecutive wins, losses or ties bet without selecting a predetermined number of consecutive wins, losses or ties and which provides a simplified arrangement for monitoring the progress of such bets made by one or more players, as well as providing a multiple outcome bet tracking system and mark matching system to protect against bet tampering.

169 citations


Journal ArticleDOI
TL;DR: This article studied external regret in sequential prediction games with both positive and negative payoffs and derived new and sharper regret bounds for the well-known exponentially weighted average forecaster and for a second forecaster with a different multiplicative update rule.
Abstract: This work studies external regret in sequential prediction games with both positive and negative payoffs. External regret measures the difference between the payoff obtained by the forecasting strategy and the payoff of the best action. In this setting, we derive new and sharper regret bounds for the well-known exponentially weighted average forecaster and for a second forecaster with a different multiplicative update rule. Our analysis has two main advantages: first, no preliminary knowledge about the payoff sequence is needed, not even its range; second, our bounds are expressed in terms of sums of squared payoffs, replacing larger first-order quantities appearing in previous bounds. In addition, our most refined bounds have the natural and desirable property of being stable under rescalings and general translations of the payoff sequence.

Journal ArticleDOI
TL;DR: In this paper, the authors provide an agent-based model of the Walrasian economy, which is a computer simulation of the repeated play of a game in which a large number of agents are endowed with software-encoded strategies governing both how they play the game and how they gather information and update their behaviour.
Abstract: In Walras original description of general equilibrium (Walras, 1954 [1874]), market clearing was effected by a central authority. This authority, which has come to be known as the auctioneer, remains today because no one has succeeded in producing a plausible decentralised dynamic model of producers and consumers engaged in market interaction in which prices and quantities move towards market-clearing levels. Only under implausible assumptions can the continuous auctioneer dynamic be shown to be stable (Fisher, 1983), and in a discrete model, even these assumptions (gross substitutability, for instance) do not preclude instability and chaos in price movements (Saari, 1985; Bala and Majumdar, 1992). 1 Moreover, contemporary analysis of excess demand functions suggests that restrictions on preferences are unlikely to entail the stability of t^ (Sonnenschein, 1972, 1973; Debreu, 1974; Kirman and Koch, 1986). It has been a half century since Debreu (1952) and Arrow and Debreu (1954) provided a satisfactory analysis of the equilibrium properties market economies, yet we know virtually nothing systematic about Walrasian dynamics. This suggests that we lack understanding of one or more fundamental properties of market exchange. This article provides an agent-based model of the Walrasian economy. An agentbased model is a computer simulation of the repeated play of a game in which a large number of agents are endowed with software-encoded strategies governing both how they play the game and how they gather information and update their behaviour. The disequilibrium behaviour of agents in our agent-based models is governed by a replicator dynamic (Taylor and Jonker, 1978) in which, over time, successful agents tend in Darwinian fashion to increase in frequency at the expense of unsuccessful agents. We describe the process of shifting from lower to higher payoff strategies as imitation,

Journal ArticleDOI
TL;DR: Five mechanisms for the evolution of cooperation are discussed: direct reciprocity, indirect reciprocities, kin selection, group selection, Group selection, and network reciprocity (or graph selection).
Abstract: How does natural selection lead to cooperation between competing individuals? The Prisoner's Dilemma captures the essence of this problem. Two players can either cooperate or defect. The payoff for mutual cooperation, R, is greater than the payoff for mutual defection, P. But a defector versus a cooperator receives the highest payoff, T, where as the cooperator obtains the lowest payoff, S. Hence, the Prisoner's Dilemma is defined by the payoff ranking T > R > P > S. In a well-mixed population, defectors always have a higher expected payoff than cooperators, and therefore natural selection favors defectors. The evolution of cooperation requires specific mechanisms. Here we discuss five mechanisms for the evolution of cooperation: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity (or graph selection). Each mechanism leads to a transformation of the Prisoner's Dilemma payoff matrix. From the transformed matrices, we derive the fundamental conditions for the evolution of cooperation. The transformed matrices can be used in standard frameworks of evolutionary dynamics such as the replicator equation or stochastic processes of game dynamics in finite populations.

Book ChapterDOI
12 Dec 2007
TL;DR: An efficient algorithm is provided that computes 0.3393- approximate equilibria, the best approximation till now, based on the formulation of an appropriate function of pairs of mixed strategies reflecting the maximum deviation of the players' payoffs from the best payoff each player could achieve given the strategy chosen by the other.
Abstract: In this paper we propose a new methodology for determining approximate Nash equilibria of non-cooperative bimatrix games and, based on that, we provide an efficient algorithm that computes 0.3393- approximate equilibria, the best approximation till now. The methodology is based on the formulation of an appropriate function of pairs of mixed strategies reflecting the maximum deviation of the players' payoffs from the best payoff each player could achieve given the strategy chosen by the other. We then seek to minimize such a function using descent procedures. As it is unlikely to be able to find global minima in polynomial time, given the recently proven intractability of the problem, we concentrate on the computation of stationary points and prove that they can be approximated arbitrarily close in polynomial time and that they have the above mentioned approximation property. Our result provides the best Ɛ till now for polynomially computable Ɛ-approximate Nash equilibria of bimatrix games. Furthermore, our methodology for computing approximate Nash equilibria has not been used by others.

Posted Content
Naoki Masuda1
TL;DR: In replicator-type evolutionary dynamics, it is shown that even a relatively small participation cost extinguishes the merit of heterogeneous networks in terms of cooperation.
Abstract: Real social interactions occur on networks in which each individual is connected to some, but not all, of others. In social dilemma games with a fixed population size, heterogeneity in the number of contacts per player is known to promote evolution of cooperation. Under a common assumption of positively biased payoff structure, well-connected players earn much by playing frequently, and cooperation once adopted by well-connected players is unbeatable and spreads to others. However, maintaining a social contact can be costly, which would prevent local payoffs from being positively biased. In replicator-type evolutionary dynamics, it is shown that even a relatively small participation cost extinguishes the merit of heterogeneous networks in terms of cooperation. In this situation, more connected players earn less so that they are no longer spreaders of cooperation. Instead, those with fewer contacts win and guide the evolution. The participation cost, or the baseline payoff, is irrelevant in homogeneous populations but is essential for evolutionary games on heterogeneous networks.

Journal ArticleDOI
TL;DR: It is pointed out the difference between both limits of weak selection and the condition under which the differences vanish and it turns out that this condition is fulfilled by the popular parametrization of the prisoner's dilemma in benefits and costs.

Journal ArticleDOI
TL;DR: In this paper, the authors study the impact of stochastic payoff variations with different distributions on the evolution of cooperation in the spatial prisoner's dilemma game and find that Gaussian-distributed payoff variations are most successful in promoting cooperation irrespective of the temptation to defect.
Abstract: We study the impact of stochastic payoff variations with different distributions on the evolution of cooperation in the spatial prisoner's dilemma game. We find that Gaussian-distributed payoff variations are most successful in promoting cooperation irrespective of the temptation to defect. In particular, the facilitative effect of noise on the evolution of cooperation decreases steadily as the frequency of rare events increases. Findings are explained via an analysis of local payoff ranking violations. The relevance of results for economics and sociology is discussed.

Journal ArticleDOI
TL;DR: In this article, a new class of two-player continuous-time games with imperfect monitoring is investigated, where the players' observations of each other's actions are distorted by Brownian motions.
Abstract: This paper investigates a new class of two-player games in continuous time, in which the players' observations of each other's actions are distorted by Brownian motions. These games are analogous to repeated games with imperfect monitoring in which the players take actions frequently. Using a differential equation, we find the set E(r) of payoff pairs achievable by all public perfect equilibria of the continuous-time game, where r is the discount rate. The same differential equation allows us to find public perfect equilibria that achieve any value pair on the boundary of the set 9(r). These public perfect equilibria are based on a pair of continuation values as a state variable, which moves along the boundary of E(r) during the course of the game. In order to give players incentives to take actions that are not static best responses, the pair of continuation values is stochastically driven by the players' observations of each other's actions along the boundary of the set S(r).

Journal ArticleDOI
22 Jul 2007
TL;DR: This dissertation explores the problem of constructing an effective general game-playing program, with an emphasis on techniques for automatically constructing effective heuristic evaluation functions from game descriptions, and presents a technique based on abstract models of games.
Abstract: A general game playing program plays games that it has not previously encountered. A game manager program sends the game playing programs a description of a game's rules and objectives in a well-defined game description language. A central challenge in creating effective general game playing programs is that of constructing heuristic evaluation functions from game descriptions. This paper describes a method for constructing evaluation functions that represent exact values of simplified games. The simplified games are abstract models that incorporate the most essential aspects of the original game, namely payoff, control, and termination. Results of applying this method to a sampling of games suggest that heuristic evaluation functions based on our method are both comprehensible and effective.

Journal ArticleDOI
TL;DR: A simple mean-field approximation is derived that captures the average effect of the payoff stochasticity, which reduces the intensity of selection and therefore increases the temperature of selection in populations of finite size.

Journal ArticleDOI
TL;DR: A discrete velocity mathematical model for vehicular traffic along a one-way road using the kinetic scale to capture the probabilistic essence of the interactions among the vehicles and offers the opportunity of a profitable analytical investigation of the relevant global features of the system.
Abstract: Following some general ideas on the discrete kinetic and stochastic game theory proposed by one of the authors in a previous work, this paper develops a discrete velocity mathematical model for vehicular traffic along a one-way road. The kinetic scale is chosen because, unlike the macroscopic one, it allows to capture the probabilistic essence of the interactions among the vehicles, and offers at the same time, unlike the microscopic one, the opportunity of a profitable analytical investigation of the relevant global features of the system. The discretization of the velocity variable, rather than being a pure mathematical technicality, plays a role in including the intrinsic granular nature of the flow of vehicles in the mathematical theory of traffic. Other important characteristics of the model concern the gain and loss terms of the kinetic equations, namely the construction of a density-dependent table of games to model velocity transitions and the introduction of a visibility length to account for non...

Proceedings ArticleDOI
01 Dec 2007
TL;DR: This work introduces three different payoff based processes for increasingly general scenarios and proves that after a sufficiently large number of stages, player actions constitute a Nash equilibrium at any stage with arbitrarily high probability.
Abstract: We consider repeated multi-player games in which players repeatedly and simultaneously choose strategies from a finite set of available strategies according to some strategy adjustment process. We focus on the specific class of weakly acyclic games, which is particularly relevant for multi-agent cooperative control problems. A strategy adjustment process determines how players select their strategies at any stage as a function of the information gathered over previous stages. Of particular interest are "payoff based" processes, in which at any stage, players only know their own actions and (noise corrupted) payoffs from previous stages. In particular, players do not know the actions taken by other players and do not know the structural form of payoff functions. We introduce three different payoff based processes for increasingly general scenarios and prove that after a sufficiently large number of stages, player actions constitute a Nash equilibrium at any stage with arbitrarily high probability. We also show how to modify player utility functions through tolls and incentives in so-called congestion games, a special class of weakly acyclic games, to guarantee that a centralized objective can be realized as a Nash equilibrium. We illustrate the methods with a simulation of distributed routing over a network.

Journal ArticleDOI
TL;DR: In this paper, the authors presented a cooperative differential game of transboundary industrial pollution, in which the industrial sectors remain competitive among themselves while the governments cooperate in pollution abatement.
Abstract: This paper presents a cooperative differential game of transboundary industrial pollution. A noted feature of the game model is that the industrial sectors remain competitive among themselves while the governments cooperate in pollution abatement. It is the first time that time consistent solutions are derived in a cooperative differential game on pollution control with industries and governments being separate entities. A stochastic version of the model is presented and a subgame-consistent cooperative solution is provided. This is the first study of pollution management in a stochastic differential game framework.

Journal ArticleDOI
TL;DR: It is proved that finding a Nash equilibrium that minimizes the potential function is NP-hard and an upper bound of O(radicnlog2 n) on the price of anarchy, and a lower bound of Omega(log n/log log n) are established.
Abstract: We consider a multicast game with selfish non- cooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NP-hard. We focus on the price of anarchy of a Nash equilibrium resulting from the best-response dynamics of a game course, where the players join the game sequentially. For a game with in players, we establish an upper bound of O(radicnlog2 n) on the price of anarchy, and a lower bound of Omega(log n/log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium.

Journal ArticleDOI
TL;DR: Flexibility of the suppliers' capability is also found to affect the PPD decisions and the use of platform commonality and modularity has been found generally beneficial not only to the supply chain as a whole but also to individual players that are eventually configured into the game.
Abstract: This paper is concerned with optimizing the configuration of a set of platform products and the associated supply chain consisting of one manufacturer and multiple suppliers using a three-move dynamic game-theoretic approach. The variants in the product family share a common platform for developing/configuring variant modules which are substitutable in the sense that high-end module options can functionally replace low-end ones at higher prices. As the customer in the supply chain, the manufacturer takes its leading role by making the first move to give decisions on platform products development (PPD) and supplier selection. The concerned suppliers make the second move to optimize their decisions including price discounts and their ordering policies. The manufacturer finishes the game by taking the last move to make his ordering decisions. The ranges of the rational reactions for the players are derived from the analyses of their payoff models, and an enumerative algorithm is developed to find the subgame perfect equilibrium of the game through the technique of backward induction. The game model and the proposed solution procedure are illustrated through a series of simulation experiments and sensitivity analyses using a numerical example. The results have allowed us to draw some meaningful interpretations and useful managerial insights. The use of platform commonality and modularity has been found generally beneficial not only to the supply chain as a whole but also to individual players that are eventually configured into the game. Flexibility of the suppliers' capability is also found to affect the PPD decisions

Journal ArticleDOI
TL;DR: In contrast to the predictions, the increase in contributions in the dynamic game does not depend critically on the existence of a completion benefit jump or on whether players can condition their decisions on the behavior of other members of their group as mentioned in this paper.

Journal ArticleDOI
Jun Tanimoto1
TL;DR: One reason for PMN's effectiveness is the local strategy adaptation mechanism, which helps both the preservation and fixation of C agents, and not that the payoff matrix noise makes a dilemma game into a Trivial (dilemma-free) game.
Abstract: A series of numerical simulations of a $2\ifmmode\times\else\texttimes\fi{}2$ symmetric game on a network examined whether payoff matrix noise promotes cooperation, as reported initially by Perc [New J. Phys. 8, 22 (2006)]. Agents have no memory (they offer cooperation, $C$, or defection, $D$). We assume that the network is time invariable. The effect of payoff matrix noise (PMN) is measured by a simulated payoff difference between a normal network game and a network game with PMN. The effect of PMN appears only when a local strategy adaptation is implemented (for example, a network game with imitation dynamics). The influence of PMN becomes more significant with a larger stochastic deviation, and less significant in a larger degree network. One reason for PMN's effectiveness is the local strategy adaptation mechanism, which helps both the preservation and fixation of $C$ agents, and not that the payoff matrix noise makes a dilemma game into a Trivial (dilemma-free) game.

Journal ArticleDOI
TL;DR: In this paper, Ansolabehere et al. examine the implications of inequity aversion for bargaining games in which unanimity is not required and show that it may lead to a more inequitable outcome than would occur with selfish preferences.
Abstract: Inequity aversion models have been used to explain equitable payoff divisions in bargaining games. I show that inequity aversion can actually increase the asymmetry of payoff division if unanimity is not required. This is because responders may be willing to accept a lower share rather than risk being left out. Inequity aversion may also affect comparative statics: the advantage of being the proposer can decrease as players become more impatient. Game theory usually assumes that players care only about their own material payoffs. This hypothesis is clearly refuted by the experimental evidence in the ultimatum and related games; see Camerer (2003) for a recent survey. Inequity aversion theories have been developed in order to account for the stylised facts observed in the laboratory; see Fehr and Schmidt (1999) and Bolton and Ockenfels (2000). Inequity aversion means that people are willing to give up some material payoffs in order to achieve more equitable outcomes. Inequity averse responders prefer to reject small offers in the ultimatum game, and the proposers, anticipating this, make higher offers. In this article I examine the implications of inequity aversion for bargaining games in which unanimity is not required (e. g., legislative bargaining games) and show that it may lead to a more inequitable outcome than would occur with selfish preferences. The leading model of legislative bargaining is due to Baron and Ferejohn (1989). In this model, n symmetric players must divide a budget by simple majority. Each player has an equal chance of being chosen to propose a division of the budget. Once a proposal is made, the remaining players vote 'yes' or 'no'; if a majority of the players supports the proposal it is implemented and the game ends; otherwise the procedure is repeated. This model predicts that minimal winning coalitions will form and that the proposer will receive a disproportionate share of the proceedings. Thus, the equilibrium of the Baron-Ferejohn model with selfish preferences exhibits a substantial amount of inequity: some players are excluded (almost half of them if the decision rule is simple majority), and the proposer receives a substantial share (more than half of the total payoff if the decision rule is simple majority). The advantage of the proposer increases as players become more impatient or more risk averse. The Baron-Ferejohn model has led to many applications and extensions.1 In its simplest form, it assumes that parties are selfish, risk neutral and only concerned with their share of cabinet posts as opposed to policy. The predictions of the model under these assumptions have been tested by Ansolabehere et al. (2005) using data on the

Journal ArticleDOI
TL;DR: In this paper, the authors consider a two-person intertemporal bargaining problem where players choose actions and offers each period, and collect payoffs (as a function of that period's actions) while bargaining proceeds.
Abstract: Consider a two-person intertemporal bargaining problem in which players choose actions and offers each period, and collect payoffs (as a function of that period's actions) while bargaining proceeds. This can alternatively be viewed as an infinitely repeated game wherein players can offer one another enforceable contracts that govern play for the rest of the game. Theory is silent with regard to how the surplus is likely to be split, because a folk theorem applies. Perturbing such a game with a rich set of behavioral types for each player yields a specific asymptotic prediction for how the surplus will be divided, as the perturbation probabilities approach zero. Behavioral types may follow nonstationary strategies and respond to the opponent's play. In equilibrium, rational players initially choose a behavioral type to imitate and a war of attrition ensues. How much should a player try to get and how should she behave while waiting for the resolution of bargaining? In both respects she should build her strategy around the advice given by the “Nash bargaining with threats” (NBWT) theory developed for two-stage games. In any perfect Bayesian equilibrium, she can guarantee herself virtually her NBWT payoff by imitating a behavioral type with the following simple strategy: in every period, ask for (and accept nothing less than) that player's NBWT share and, while waiting for the other side to concede, take the action Nash recommends as a threat in his two-stage game. The results suggest that there are forces at work in some dynamic games that favor certain payoffs over all others. This is in stark contrast to the classic folk theorems, to the further folk theorems established for repeated games with two-sided reputational perturbations, and to the permissive results obtained in the literature on bargaining with payoffs as you go.

Journal ArticleDOI
TL;DR: A condition, uniform payoff security, is introduced for games with compact Hausdorff strategy spaces and payoffs bounded and measurable in players’ strategies and it is established that if any compact game G is uniformly payoff secure, then its mixed extension G¯ is payoff secure.