Stable Conventions in Hawk-Dove Games with Many Players
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Stable Conventions in Hawk-Dove Games with Many Players Daniel H. Wood∗ January, 2012 Abstract This paper investigates the evolution of conventions in hawk-dove games between more than two players when multiple players could share the same payoff-irrelevant label, such as “blue” or “green”. Asymmetric conventions where one role is more aggressive develop; which con- vention is more likely depends on how many players in the contest share each label. Conven- tions closer to a pure strategy equilibrium of the game are stochastically stable. This logic offers one reason for the emergence of informal property rights. In disputes over property, in- dividuals naturally separate into two roles: the possessor, who is unique, and non-possessors, who can be numerous. If the value of objects is low relative to the cost of conflict over them, this asymmetry favors the development of informal property rights conventions. JEL Classification: C73, D23 Keywords: n-player games, hawk-dove, anti-coordination, stochastic stability, evolution, con- vention, informal property rights ∗Department of Economics, Clemson University. email: [email protected]. I would like to thank Jerry Green, Michael Kremer, and Al Roth for their suggestions and support. I also thank Attila Ambrus, Drew Fudenberg, Chuck Thomas, Patrick Warren, and seminar audiences at Harvard and Clemson for helpful questions and comments. 1 1 Introduction The two-player hawk-dove game, shown in Figure 1, has a unique equilibrium if players are iden- tical ex ante, in which hawk is played V=C fraction of the time. When the players have distinct identities such as “row” and “column”, there are also pure strategy equilibria in which one role plays hawk and the other role plays dove. Evolutionary dynamics generally predict that a pop- ulation with asymmetries will universally adopt one of the pure strategy equilibria as the shared convention (Selten 1980).1 If these asymmetries are “nominal” – such as colors or labels assigned to each role but not affecting players’ payoff functions – then either convention is equally likely to develop. Hawk Dove V − C V − C Hawk ; V; 0 2 2 V V Dove 0;V ; 2 2 Figure 1: Normal Form of the Hawk-Dove Game with Two Players. 0 < V < C. In this paper I extend hawk-dove games to consider several players competing for an indivisible resource.2 With several players, there can be payoff-irrelevant asymmetries between roles that are not purely nominal: some roles may be more numerous than other roles. For example, a single contest could be between two “green” players and three “blue” players. Consider a population whose members participate continually in these conflicts, where indi- 1“Convention” means a shared pure strategy Nash equilibrium a la Young (1993) if one exists. However, sometimes a pure strategy Nash equilibria does not exist in the class of games I analyze and in that case “convention” refers to whatever shared equilibrium is reached in the long run. 2Hauert et al. (2006) develop an n-player version of the hawk-dove game. Their aim is to provide a general classification of the evolutionary dynamics of social dilemmas with n players. They do not consider asymmetries between players, which is the major focus of this paper. An older paper, Schaffer (1988), analyzes evolutionarily stable strategies in hawk-dove games with more than two players in a finite population, but also assumes players are undifferentiated. 2 vidual members follow role-contingent pure strategies. Members of the population slowly update their strategies, usually by switching to a better pure strategy, but sometimes by switching to a worse strategy. A strategy is stochastically stable if the entire population is likely to eventually adopt it when mistakes in updating are rare but not unheard of. I show that if a pure strategy Nash equilibrium exists in these games, it is the unique stochastically stable equilibrium. If such an equilibrium does not exist, generally the mixed strategy equilibrium that is “closer” to a pure strategy equilibrium – in the sense of more a homogenous set of strategy choices by the players – is uniquely stable.3 In contrast to the two-player case, populations that repeatedly engage in mul- tiplayer hawk-dove games will develop conventions that favor particular payoff-irrelevant labels. Those conventions involve less numerous roles being more aggressive if V=C is low, and the op- posite if V=C is high. Section 2 illustrates these results with the case of repeated contests between three randomly chosen players who before each contest draw straws from a set of two short and one long straw. Section 3 defines multiplayer hawk-dove games more generally. Section 4 defines the evolutionary modeling framework, which is similar to Kandori et al. (1993). Section 5 partially characterizes stochastically stable equilibria in multiplayer hawk-dove games where each player has a role and some players share each role. The selection of particular conventions based on payoff-irrelevant characteristics sheds light on likely conventions in the two player game. When there are nominal asymmetries between players that can serve as coordination devices, equilibria that ignore those asymmetries are not evolu- tionarily stable (Maynard Smith and Parker 1976; Selten 1980). However evolutionary stability does not predict which convention will evolve in cases with payoff-irrelevant and sometimes even payoff-relevant labels.4 “Paradoxical” equilibria are theoretically stable, such as animals abandon- 3Existence of pure strategy equilibria, where all players in a role play the same strategy, is not guaranteed for general V and C. If V < C, no pure strategy equilibrium exists in the blue / green example. 4A large biology literature on evolutionary stability with different asymmetries exists, beginning with Maynard Smith and Parker (1976) and Hammerstein (1981), who examined single possibly nominal asymmetries and multiple 3 ing their territory to intruding rivals or larger animals losing conventionally to smaller animals. If the two player model is an abstraction from occasional conflicts with several participants, the logic of conflicts with many players would rule out the owner-intruder sort of paradoxical equilibrium.5 This logic is especially relevant for economics, since human social conflicts often occur be- tween more than two players. Many types of de facto property rights exist without legal protec- tion.6 Section 6 applies the results of Section 5 to understanding informal property rights. I show that even in the absence of legal protection or other advantage, ownership will be stable if the value of possessing an object is low enough relative to expected cost of fighting over it. The exclusive nature of the possession relationship favors the development of informal property rights. 2 Example: Drawing Straws This section analyzes the evolution of conventions for a series of three-player hawk-dove contests in each of which the players are given a nominal asymmetry as possible coordination device. It illustrates the more general results of the paper. Consider an ex ante homogenous population of players with access to a sequence of discrete valuable prizes. Each round a randomly selected group of three players compete for that round’s asymmetries, respectively. One branch of the literature continues the analysis of payoff-revelant asymmetries (for example, Enquist and Leimar (1987) or Mesterton-Gibbons (1994)). Another branch in this literature does not focus on payoff-relevant asymmetries but instead how a player’s behavior in a particular hawk-dove contest affects his future role. These include Grafen (1987), which argues that coordination on asymmetries which leaves players in a particular role permanently disadvantaged in reproduction would leave that role with no real cost to fighting – what Grafen terms the “desperado” effect. Mesterton-Gibbons (1992) examines which strategies are evolutionarily stable when a player’s success in one round leads to him being the owner in a future round, and finds different sets of parameters where any convention might be an evolutionarily stable strategy. Finally, Kokko et al. (2006) endogenizes the value of territories and cost of fighting and shows that coordination on ownership is favored by these feedback loops. 5Straightforward calculations show that the big-small sort of paradoxical equilibrium will not be stochastically stable when size affects players’ payoffs, although this sort will be evolutionarily stable for small payoff-relevant asymmetries. 6Ellickson (1991) is a classic study. 4 prize. Each player draws a straw from a set of one long and two short straws, and these players then play a hawk-dove game. Players may condition their action on which straw they have drawn. The prize winner is determined through the players’ simultaneous choice of either hawk or dove actions. If no player chooses hawk, one randomly wins the prize and receives V , while the others receive 0. If one or more plays hawk, a hawk randomly wins the prize, receiving V , while other hawks suffer a cost of fighting C, and any doves receive 0. Expected payoffs are given in Figure 2. Note that unlike the two player hawk-dove game, where hawk is a dominant strategy when V > C, in the three player game, hawk is only dominant when V > 2C. When the value of the prize relative to the cost of losing a fight is high enough, the contest could have the “capacity” for two pure-strategy hawks in equilibrium. Two hawks One hawk and one dove Two doves Hawk V=3 − 2C=3 V=2 − C=2 V Dove 0 0 V=3 Figure 2: Three Player Hawk-Dove Payoffs. Expected payoff from playing hawk or dove as a function of the action choice of the other two players. Players in the population follow pure strategies that they update slowly. After each round, a randomly selected player from the population switches her strategy, with probability π, to the best response to the population’s distribution of strategies, or, with probability , to a random other strategy.