Showing papers in "Dynamic Games and Applications in 2011"
TL;DR: The authors provide an overview of recent research on belief and opinion dynamics in social networks and discuss both Bayesian and non-Bayesian models of social learning and focus on the implications of the form of learning.
Abstract: We provide an overview of recent research on belief and opinion dynamics in social networks. We discuss both Bayesian and non-Bayesian models of social learning and focus on the implications of the form of learning (e.g., Bayesian vs. non-Bayesian), the sources of information (e.g., observation vs. communication), and the structure of social networks in which individuals are situated on three key questions: (1) whether social learning will lead to consensus, i.e., to agreement among individuals starting with different views; (2) whether social learning will effectively aggregate dispersed information and thus weed out incorrect beliefs; (3) whether media sources, prominent agents, politicians and the state will be able to manipulate beliefs and spread misinformation in a society.
500 citations
TL;DR: Various extensions are considered, such as modeling the effects of the concern for relative performance on the over-exploitation of resources, and applying the tragedy of the commons paradigm to the declining effectiveness of antibiotics and pesticides.
Abstract: This article provides a comprehensive survey of models of dynamic games in the exploitation of renewable and exhaustible resources. It includes dynamic games at the industry level (oligopoly, cartel versus fringe, tragedy of the commons) and at the international level (tariffs on exhaustible resources, fish wars, entry deterrence). Among more recent topics are international strategic issues involving the link between resource uses and transboundary pollution, the design of taxation to ensure efficient outcomes under symmetric and asymmetric information, the rivalry among factions in countries where property rights on natural resources are not well established. Various extensions are considered, such as (i) modeling the effects of the concern for relative performance (relative income, relative consumption, and social status) on the over-exploitation of resources, (ii) applying the tragedy of the commons paradigm to the declining effectiveness of antibiotics and pesticides. Outcomes under Nash equilibria and Stackelberg equilibria are compared. The paper ends with some suggestions for future research.
123 citations
TL;DR: This survey paper presents some new advances in theoretical aspects of differential game theory, and particular focus on three topics: differential games with state constraints; backward stochastic differential equations approach to stochastics differential games; differentialGames with incomplete information.
Abstract: This survey paper presents some new advances in theoretical aspects of differential game theory. We particular focus on three topics: differential games with state constraints; backward stochastic differential equations approach to stochastic differential games; differential games with incomplete information. We also address some recent development in nonzero-sum differential games (analysis of systems of Hamilton–Jacobi equations by conservation laws methods; differential games with a large number of players, i.e., mean-field games) and long-time average of zero-sum differential games.
83 citations
TL;DR: Analysis of an evolutionary game between two strategies interacting on an extreme heterogeneous graph, the star graph, finds explicit expressions for the fixation probability of mutants, and the time to absorption and fixation and appropriate conditions under which one strategy is favoured over the other.
Abstract: Evolutionary game dynamics have been traditionally studied in well-mixed populations where each individual is equally likely to interact with every other individual. Recent studies have shown that the outcome of the evolutionary process might be significantly affected if the population has a non-homogeneous structure. In this paper we study analytically an evolutionary game between two strategies interacting on an extreme heterogeneous graph, the star graph. We find explicit expressions for the fixation probability of mutants, and the time to absorption (elimination or fixation of mutants) and fixation (absorption conditional on fixation occurring). We investigate the evolutionary process considering four important update rules. For each of the update rules, we find appropriate conditions under which one strategy is favoured over the other. The process is considered in four different scenarios: the fixed fitness case, the Hawk–Dove game, the Prisoner’s dilemma and a coordination game. It is shown that in contrast with homogeneous populations, the choice of the update rule might be crucial for the evolution of a non-homogeneous population.
59 citations
TL;DR: Under relatively weak conditions, it is shown that there is a solution to the optimality equation for the maxmin control problem as well as an optimal strategy for the controller and these results are applied to the theory of optimal growth and the Hansen–Sargent robust control model in macroeconomics.
Abstract: We study a discounted maxmin control problem with general state space. The controller is unsure about his model in the sense that he also considers a class of approximate models as possibly true. The objective is to choose a maxmin strategy that will work under a range of different model specifications. This is done by dynamic programming techniques. Under relatively weak conditions, we show that there is a solution to the optimality equation for the maxmin control problem as well as an optimal strategy for the controller. These results are applied to the theory of optimal growth and the Hansen–Sargent robust control model in macroeconomics. We also study a class of zero-sum discounted stochastic games with unbounded payoffs and simultaneous moves and give a brief overview of recent results on stochastic games with weakly continuous transitions and the limiting average payoffs.
56 citations
TL;DR: This paper studies linear quadratic games with set up costs monotonic on the number of active players, namely, players whose action is non-null and the influence of an additional disturbance in the spirit of the literature on H∞ control.
Abstract: This paper studies linear quadratic games with set up costs monotonic on the number of active players, namely, players whose action is non-zero. Such games arise naturally in joint replenishment inventory systems. Building upon a preliminary analysis of the properties of the best response strategies and Nash equilibria for the given game, the main contribution is the study of the same game under large population. Numerical illustrations are provided.
43 citations
TL;DR: In this article, the authors extend the concept of the price of anarchy to a general differential game framework, and introduce the Price of Information (PoI) to characterize comparative game performances under different information structures, as well as the Price Of Cooperation to capture the extent of benefit or loss a player accrues as a result of altruistic behavior.
Abstract: The price of anarchy (PoA) has been widely used in static games to quantify the loss of efficiency due to noncooperation. Here, we extend this concept to a general differential games framework. In addition, we introduce the price of information (PoI) to characterize comparative game performances under different information structures, as well as the price of cooperation to capture the extent of benefit or loss a player accrues as a result of altruistic behavior. We further characterize PoA and PoI for a class of scalar linear quadratic differential games under open-loop and closed-loop feedback information structures. We also obtain some explicit bounds on these indices in a large population regime.
40 citations
TL;DR: It is shown that the extension of the one-third law of evolution from the 2- player game to the d-player game is the same for all exchangeable models in the domain of application of the Kingman coalescent in the limit of a large neutral population.
Abstract: We show that the extension of the one-third law of evolution from the 2-player game to the d-player game is the same for all exchangeable models in the domain of application of the Kingman coalescent in the limit of a large neutral population. The extension relies on an approximation of the probability of fixation of a single mutant in terms of expected times in ancestral sample states that are calculated by induction. An interpretation based on the concept of projected average excess in payoff is discussed.
38 citations
TL;DR: The role of voluntary participation in establishing and upholding cooperation with or without punishment is reviewed, and two distinct forms of punishment are dealt with, namely peer punishment and pool punishment, and their stability and their efficiency are compared.
Abstract: Punishment of free-riders is generally viewed as an important factor in promoting cooperation. But since it is often costly to sanction exploiters, the emergence of such a behavior and its stability raise interesting problems. Players who do not contribute to the sanctions, but profit from the increased level of cooperation caused by them, act as “second-order exploiters” and threaten the joint enterprise. In this paper, we review the role of voluntary participation in establishing and upholding cooperation with or without punishment. In particular, we deal with two distinct forms of punishment, namely peer punishment and pool punishment, and compare their stability and their efficiency. The emergence and upkeep of collaborative undertakings can strongly depend on whether participation is voluntary or mandatory. The possibility to opt out of a joint enterprise often helps in curbing exploiters and boosting pro-social behavior.
37 citations
TL;DR: This work sheds light on the emergence and maintenance of cooperation on dynamically changing social networks, where reputation plays a decisive role in the formation of social ties.
Abstract: Coevolution of individual strategies and social ties, in which individuals not only adjust their strategies by social learning but also switch their adverse partners to search for potential beneficial ones, has attracted increasing attention very recently. It is found that the interplay of strategic updating and partner network adaptation can facilitate the escape from the stalemate of cooperation in social dilemmas. But the question how individual preferential partner choice shapes the dynamical and topological organization of cooperation has yet to be fully answered. Here we propose a simple evolutionary game model to address this problem. In our model, when severing a current disadvantageous partnership, individuals can choose a new partner, either among their friends of friends preferentially according to their reputation scores or randomly from the remaining population. In addition to partner switching, individuals also update their strategies by imitating social neighbors. The interplay between these two processes gives rise to rich evolutionary dynamics. We focus on both strategic and topological evolution. We find that reputation-based partner selection leads to highly heterogeneous and often disassortative partner networks. During the coevolutionary process, a few successful individuals who attain a large number of partners emerge as social hubs and thus directly influence periphery individuals of small degree, forming leader–follower hierarchical structures. Cooperation prevails because of the positive feedback effects: good guys attract more partnerships and “the rich get richer.” Our work sheds light on the emergence and maintenance of cooperation on dynamically changing social networks, where reputation plays a decisive role in the formation of social ties.
31 citations
TL;DR: Three models of time-delay stochastic dynamics of finite well-mixed or random-matching populations with a unique mixed evolutionarily stable strategy are presented.
Abstract: We discuss combined effects of stochasticity and time delays in simple evolutionary games with a unique mixed evolutionarily stable strategy. We present three models of time-delay stochastic dynamics of finite well-mixed or random-matching populations. We show that in the first two models the evolutionarily stable strategy loses its stability and there appears a stable cycle around it with the time period and the amplitude proportional to the delay. In the third model, only one randomly chosen individual can update his strategy at a time. This slows down the dynamics and makes the evolutionarily stable strategy stable with respect to both time delay and stochastic perturbations.
TL;DR: In this paper, the authors analyze the dynamics of evolutionary games in which fitness is defined as an affine function of the expected payoff and a constant contribution, and apply the affine fitness function in a model for tumor-normal cell interactions to determine which are the most successful tumor strategies.
Abstract: We analyze the dynamics of evolutionary games in which fitness is defined as an affine function of the expected payoff and a constant contribution. The resulting inhomogeneous replicator equation has an homogeneous equivalent with modified payoffs. The affine terms also influence the stochastic dynamics of a two-strategy Moran model of a finite population. We then apply the affine fitness function in a model for tumor–normal cell interactions to determine which are the most successful tumor strategies. In order to analyze the dynamics of concurrent strategies within a tumor population, we extend the model to a three-strategy game involving distinct tumor cell types as well as normal cells. In this model, interaction with normal cells, in combination with an increased constant fitness, is the most effective way of establishing a population of tumor cells in normal tissue.
TL;DR: The main results show that in two-strategy games with strict Nash equilibria A and B, if selection favors strategy A when a single individual is using it initially, then one-third of the opponents one meets before fixation are A-individuals.
Abstract: The one-third law introduced by Nowak et al (Nature 428:646–650, 2004) for the Moran stochastic process has proven to be a robust criterion to predict when weak selection will favor a strategy invading a finite population In this paper, we investigate fixation probability, mean effective fixation time, and average and expected fitnesses in the diffusion approximation of the stochastic evolutionary game Our main results show that in two-strategy games with strict Nash equilibria A and B: (i) the one-third law means that, if selection favors strategy A when a single individual is using it initially, then one-third of the opponents one meets before fixation are A-individuals; and (ii) the average fitness of strategy A about the mean effective fixation time is larger than that of strategy B The analysis reinforces the universal nature of the one-third law as of fundamental importance in models of selection We also connect risk dominance of strategy A to its larger expected fitness with respect to the stationary distribution of the diffusion approximation that includes a small mutation rate between the two strategies
TL;DR: It is shown that a matrix game within groups in a finite-island model is effectively equivalent to a matrixgame in a well-mixed population because hard selection reduces competition within groups compared to soft selection.
Abstract: We show that a matrix game within groups in a finite-island model is effectively equivalent to a matrix game in a well-mixed population. The effective game matrix is a sum of interaction effects minus competition effects, weighted by identity measures involving up to three individuals. These identity measures are computed in the absence of selection but depend on the selection regime and the dispersal pattern: differential viability or fertility, hard selection or soft selection, uniform dispersal or local extinction followed by uniform recolonization. Hard selection, which allows for group selection, understood as differential contributions of groups, reduces competition within groups compared to soft selection. Moreover, the reduction is more pronounced in the case of uniform dispersal than in the case of local extinction. Fertility differences add competition effects between an individual and itself. A personal inclusive payoff can be defined from the effective game matrix and used to predict the increase or decrease in frequency of a mutant strategy. However, this personal inclusive payoff is generally frequency-dependent and its mean does not necessarily increase over time.
TL;DR: The purpose of this survey is to describe some recent advances in zero-sum repeated games and in particular new connections to differential games.
Abstract: The purpose of this survey is to describe some recent advances in zero-sum repeated games and in particular new connections to differential games. Topics include: approachability, asymptotic analysis: recursive formula and operator approach, dual game and incomplete information, uniform approach.
TL;DR: In this paper, the authors study the welfare implications of Stackelberg leaderships for an individual country and the world and overcome the problem of time-inconsistency by imposing a "credibility condition" on the Markovian strategy of the leader.
Abstract: Formulating a dynamic game model of a world exhaustible resource market, in this paper, we study welfare implications of Stackelberg leaderships for an individual country and the world. We overcome the problem of time-inconsistency by imposing a “credibility condition” on the Markovian strategy of the Stackelberg leader. Under this condition, we show that the presence of a global Stackelberg leader leaves the follower worse off relative to the Nash equilibrium. Moreover, the world welfare is highest in the Nash equilibrium as compared with the two Stackelberg equilibria.
TL;DR: The main properties that are valid over this whole class of games are studied and both surveys known results and provides new ones.
Abstract: Supergames are repeated games in which a fixed known finite one-shot game is repeated over and over. Information about the actions chosen at each stage is provided by a signalling technology. This paper studies the main properties that are valid over this whole class of games and both surveys known results and provides new ones.
TL;DR: The aim is to study intertemporal coordination and contracting in the supply chain, using a franchise contract, which includes as special cases a wholesale-price contract, a two-part tariff, and a revenue-sharing contract.
Abstract: The paper studies a stylized supply chain, consisting of a manufacturer who sells a particular product to an independent retailer. The aim is to study intertemporal coordination and contracting in the supply chain, using a franchise contract. A franchise contract includes as special cases a wholesale-price contract, a two-part tariff, and a revenue-sharing contract. The retailer determines her ordering rate and the consumer price while the manufacturer determines the production rate and the parameters of the franchise contract. Contract parameters are time-dependent and determined under two, alternative, objectives. First, the manufacturer wishes to achieve an outcome in which total supply chain profits are maximized. Second, the manufacturer seeks an outcome which maximizes its individual profits. The setup is a differential game played over a fixed and finite horizon.
TL;DR: A new approach to the Gal theory is presented, based on ideas borrowed from the author's recent extension of Gal’s theory to networks which can be searched at speeds depending on the location and direction in the network.
Abstract: A network is called weakly Eulerian if it consists of a finite number of disjoint Eulerian networks which are connected in a tree-like fashion. S. Gal and others developed a theory of (zero-sum) hide-and-seek games on such networks. The minimax search time for a network is called its search value. A network is called simply searchable if its search value is half the minimum time to tour it. A celebrated result of Gal is that a network is simply searchable if and only if it is weakly Eulerian. This expository article presents a new approach to the Gal theory, based on ideas borrowed from the author's recent extension of Gal's theory to networks which can be searched at speeds depending on the location and direction in the network. Most of the proofs are new.
TL;DR: This research addresses the problem of the optimal interception of an optimally evasive orbital target by a pursuing spacecraft or missile and incorporates a recently developed method, termed “semi-direct collocation with nonlinear programming”, for the numerical solution of dynamic games.
Abstract: This research addresses the problem of the optimal interception of an optimally evasive orbital target by a pursuing spacecraft or missile. The time for interception is to be minimized by the pursuing space vehicle and maximized by the evading target. This problem is modeled as a two-sided optimization problem, i.e. as a two-player zero-sum differential game. The work incorporates a recently developed method, termed “semi-direct collocation with nonlinear programming”, for the numerical solution of dynamic games. The method is based on the formal conversion of the two-sided optimization problem into a single-objective one, by employing the analytical necessary conditions for optimality related to one of the two players. An approximate, first attempt solution for the method is provided through the use of a genetic algorithm in a “preprocessing” phase. Three qualitatively different cases are considered. In the first example the pursuer and the evader are represented by two spacecraft orbiting the Earth in two distinct orbits. The second and the third case involve two missiles, and a missile that pursues an orbiting spacecraft, respectively. The numerical results achieved in this work testify to the robustness and effectiveness of the method also in solving large, complex, three-dimensional problems.
TL;DR: The introduction of mutation to the system dramatically alters evolutionary outcomes and leads to a fair split of resources between two agents, and the effect of asymmetry in selection intensity on the resulting pattern of resource division is studied.
Abstract: Cooperation is fundamental in animal societies including humans, yet how to divide the resources obtained through cooperation is not a trivial question. The Nash demand game provides an excellent framework to study resource division between selfish agents. We herein study evolutionary dynamics of strategies in the Nash demand game. Our evolutionary model confirms the traditional prediction based on a Nash-equilibrium analysis that any possible resource division can be a stable outcome. Next, we study the effect of mutation (or exploration in cultural evolution). We model mutation as diffusion in the strategy space and analyze a pair of reaction diffusion equations. We find that the introduction of mutation to the system dramatically alters evolutionary outcomes and leads to a fair split of resources between two agents. We also study the effect of asymmetry in selection intensity on the resulting pattern of resource division.
TL;DR: The results indicate that the effect of RHP variation on the tendency of individuals to resolve conflicts without fighting is more subtle than previously anticipated.
Abstract: There is increasing awareness that inter-individual variation can affect the propensity of individuals to cooperate. Here we formally examine the role of variation in the context of animal conflict, assuming that individuals vary in resource holding potential (RHP) and are capable of adopting strategies (Hawk if strong, Dove if weak) based on this trait. Our models allow winners and/or losers of contests to pay costs of fighting that vary with the difference in RHP between contestants. In contrast to earlier studies, we identify a broad range of conditions under which increasing variation in RHP serves to reduce rather than to enhance the proportion of conflicts resolved without fighting. Moreover, we show that in many cases ignorance of one’s RHP can lead to a higher tendency for individuals to cooperate than awareness of one’s RHP. Equally counterintuitively, precluding fights between individuals strong enough to want to fight and individuals not strong enough to want to fight decreases the overall extent to which conflicts are settled without conflict, by enhancing the incentive to act like a Hawk. Collectively, our results indicate that the effect of RHP variation on the tendency of individuals to resolve conflicts without fighting is more subtle than previously anticipated.
TL;DR: This paper introduces simple conditionally stationary strategies and related stationary penal codes for dynamic games with perfect monitoring, and the main results are necessary and sufficient conditions for extremal strategies, or corresponding penal codes, which give the minimal payoffs to players among conditionALLY stationary equilibria.
Abstract: This paper introduces simple conditionally stationary strategies and related stationary penal codes for dynamic games with perfect monitoring. The class of equilibria that these strategies support is defined—they are called conditionally stationary equilibria. The main results are necessary and sufficient conditions for extremal strategies, or corresponding penal codes, which give the minimal payoffs to players among conditionally stationary equilibria. Results are applied to resource extraction games.
TL;DR: This paper model the case of an international non-renewable resource monopolist as a dynamic game between a monopolist and n importing countries governments, and investigates whether a tariff on resource imports can be advantageous for consumers in importing countries.
Abstract: In this paper we model the case of an international non-renewable resource monopolist as a dynamic game between a monopolist and n importing countries governments, and investigate whether a tariff on resource imports can be advantageous for consumers in importing countries. We analyse both the case of a price-setting monopolist and the case of a quantity-setting monopolist. We find that a tariff is advantageous for consumers, even when there is no commitment to the trade policy and importing countries do not coordinate their policies. Using a numerical example, we find that a tariff is more advantageous for the importing countries if the monopolist chooses the quantity instead of the price and that the optimal temporal path when the monopolist chooses the price is consistently below the optimal temporal path when the monopolist chooses the quantity for the entire period of exploitation of the resource. Nevertheless, the variation in total welfare between the two regimens is small.
TL;DR: A non-standard evolutionary game approach is taken, in which the starting point of the modeling is the dynamic evolution of the population as a function of the strategies used, instead of a fitness-based model (in which the fitness functions determine the evolution).
Abstract: We revisit in this paper the well-studied Hawk and Dove game within a dynamic framework. A non-standard evolutionary game approach is taken, in which the starting point of the modelling is the dynamic evolution of the populations as a function of the strategies used, instead of a fitness based model (in which the fitness functions determine the evolution). This work is motivated by the discussion in the book of Thomas L. Vincent co-authored with J. Brown [4] in which they raise (on page 73) the puzzling question of whom should one consider to be the players: the individuals or the populations?
TL;DR: In this paper, the quest for hegemony is modeled as a game, with countries being differentiated initially only by some endowment which yields a pollution free flow of income, as measured by its level of pollution is directly related to its economic strength.
Abstract: This paper builds on the assumption that countries behave in such a way as to improve, via their economic strength, the probability that they will attain the hegemonic position on the world stage. The quest for hegemony is modeled as a game, with countries being differentiated initially only by some endowment which yields a pollution free flow of income. A country’s level of pollution is assumed directly related to its economic strength, as measured by its level of production. Two types of countries are distinguished: richly endowed countries, for which the return on their endowment is greater than the return they can expect from winning the hegemony race, and poorly endowed countries, who can expect a greater return from winning the race than from their endowment. The paper analyzes the effect on global pollution of the distribution between the two types of countries and of various measures that change the relative or absolute lot of the poorly endowed.
TL;DR: In this article, the authors study duopolistic competition in a dierentiated market with rms setting prices and quantities, without explicitly imposing market clearing, and they assume that rms behavior to be shaped by a Darwinian dynamic.
Abstract: We study duopolistic competition in a dierentiated market with rms setting prices and quantities, without explicitly imposing market clearing. Unlike the commonly adopted assumption of prot maximizing rms, we assume rm behavior to be shaped by a Darwinian dynamic: the less
TL;DR: A trade model relating stock price to the demand strategies of these traders who wish to maximize their payoffs is formulated and the Nash equilibrium concept is used to compute the optimal value functions for these large traders and study the stability of the state process under equilibrium strategies of the large traders.
Abstract: We study the influence of large traders in the stock market in the presence of a fringe of marginal “noise traders”. We formulate a trade model relating stock price to the demand strategies of these traders who wish to maximize their payoffs. Using the Nash equilibrium concept, we compute the optimal value functions for the large traders and study the stability of the state process (log price) under equilibrium strategies of the large traders. In the process, we propose two measures. The first one is to measure the big traders’ total faith on the market’s valuation (φ0), and the second one is to measure the big traders’ interaction between themselves (φ1). We discuss what values of the measures might indicate a collusion of the big traders to corner the market for their benefit and illustrate this with numerical examples. We also illustrate, with diagrams, the historical and instantaneous correlation among the value processes for these large traders to highlight certain interesting features that influence the market.