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Conjectural Variations in Aggregative Games: an Evolutionary Perspective

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Conjectural Variations in Aggregative Games: an Evolutionary Perspective Conjectural Variations in Aggregative Games: An Evolutionary Perspective Alex Possajennikov∗ University of Nottingham Final version accepted for Mathematical Social Sciences, July 2015 Published version available at http://dx.dox.org/10.1016/j.mathsocsci.2015.07.003 Abstract Suppose that in symmetric aggregative games, in which payoffs depend only on a player's strategy and on an aggregate of all players' strategies, players have conjectures about the reaction of the aggregate to marginal changes in their strategy. The players play a conjectural variation equilibrium, which determines their fitness payoffs. The paper shows that only consistent conjectures can be evolutionarily stable in an infinite population, where a conjecture is consistent if it is equal to the marginal change in the aggregate determined by the actual best responses. In the finite population case, only zero conjectures representing aggregate-taking behavior can be evolutionarily stable. Keywords: conjectural variations, aggregative games, indirect evolution, evolution- ary stability JEL Codes: C72, D84 ∗School of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom. Tel: +44 115 9515461, fax: +44 115 9514159, email: [email protected], http://www.nottingham.ac.uk/~lezap. 1 1 Introduction This paper shows that in symmetric aggregative games there is a link between certain properties of conjectures and the evolutionary success of those conjectures. An aggregative game is a game in which the payoff of a player depends only on the player's own strategy and on an aggregate of the strategies of all players. A typical example of an aggregative game is the Cournot oligopoly; aggregative games were analyzed e.g. in Corch´on(1994) and more recently in Cornes and Hartley (2012) and Acemoglu and Jensen (2013), where further examples of such games are discussed. Conjectures, also called conjectural variations, describe players' beliefs about the reac- tion of other players to a change in a player's strategy. Theoretically, they allow for an extended range of behaviors (see Figui`ereset al., 2004, for a book-length discussion of the theory of conjectures and for further references).1 In aggregative games, conjectures can be seen as beliefs about the reaction of the aggregate. Such beliefs determine players' strategies via a (conjectural variations) equilibrium, in which each player's strategy is a best response given the player's conjecture about the reaction of the aggregate to a deviation. Generally, players with different conjectures will play different strategies in equilibrium and thus get different payoffs. If there is an evolutionary process selecting players on the basis of these payoffs, then some conjectures will perform better than others. Suppose that in a large (infinite) population agents are randomly matched to play a given finite-player aggregative game. This paper shows that in well-behaved games the evolutionarily stable constant conjectures must be consistent at equilibrium: the beliefs about the reaction to a (small) deviation from equilibrium coincide with the marginal change in the aggregate, derived from the players' best response functions. On the other hand, evolutionary stability for finite populations (where all players interact in the same game) selects zero conjectural variations: players believe that the aggregate does not change if their strategy changes. Such behavior is akin to the price-taking behavior in the standard perfect competition model (as noted e.g. in Kamien and Schwartz, 1983). Players with consistent conjecture correctly anticipate the reaction of the aggregate. Nevertheless, it is not obvious why they should get a higher payoff in evolutionary terms, since other players may adjust their behavior to such a conjecture. A consistent conjecture, however, does not have a detrimental strategic effect; indeed, consistency takes this strategic effect into account. In a finite population, relative payoffs are important in evolutionary terms. In symmetric games, the effect of the aggregate is the same on any player and thus 1Conjectural variations are also used in empirical industrial organization (see e.g. Belleflamme and Peitz, 2010, pp. 70-71, and references there) and in policy analysis (see a report for the former main competition authority in the UK, Office of Fair Trading, 2011). 2 cancels out from the relative payoff evaluation. Players with zero conjectures behave as if the aggregate does not change thus mimicking the condition for maximizing relative payoffs. The results generalize and combine several previous observations in the literature about evolutionary justifications of conjectures. For the infinite population case, Dixon and Somma (2003) and M¨ullerand Normann (2005) showed that consistent conjectures have evolutionary foundations in certain duopoly games, while Possajennikov (2009) generalized the result to arbitrary well-behaved two-player games. Reddy Rachapalli and Kulshreshtha (2013) obtained the same result in a linear-quadratic oligopoly model. The present paper extends the result to arbitrary well-behaved n-player aggregative games. For the finite population case Possajennikov (2003) and Al´os-Ferrer and Ania (2005) provided evolution- ary background for aggregate-taking behavior in aggregative games. The present paper shows that this result can be reinterpreted via conjectures, since aggregate-taking behavior is equivalent to zero conjectural variation about the change in the aggregate.2 2 Aggregative Games and Conjectures A symmetric aggregative game on the real line is given by G = (N; X; u), where N = f1; : : : ; ng is the set of players, X ⊂ R is the strategy set, common across players, and u(xi;A) = ui(xi;A): X × R ! R is the payoff function for each player i. Here, A = n g(x1; : : : ; xn): X ! R is the aggregate of players' strategies, such that for any permutation π on the set of players g(π(x1); : : : ; π(xn)) = g(x1; : : : ; xn). The game is assumed to be well behaved: the set X is convex and the functions u and g are twice continuously differentiable. e A player i's conjecture ri = (dA=dxi) 2 R, where R is a convex subset of the real line R, is a number representing the player's belief, or expectation, about the change in the aggregate in response to a marginal change in the player's own strategy. It is assumed that players entertain constant conjectures: ri does not vary with x1; : : : ; xn. Given the conjecture ri, player i maximizes the payoff u(xi;A). If the solution of the player's maximization problem is interior, then the first-order condition @u=@xi(xi;A) + e @u=@A(xi;A) · (dA=dxi) = 0 holds. Thus @u @u Fi(xi;A; ri) := (xi;A) + (xi;A) · ri = 0. (1) @xi @A For each player i, it is assumed that the optimal choice is determined by this equation. Suppose that all n players have some conjectures. If each player's solution of the payoff maximization problem is interior, then a (conjectural variations) equilibrium is characterized 2M¨ullerand Normann (2007) showed that in a duopoly, finite population evolutionary stability indeed leads to a result that would be obtained with zero conjectures about the aggregate. 3 by the following equations: F1(x1;A; r1) = 0 ··· (2) Fn(xn;A; rn) = 0 A − g(x1; : : : ; xn) = 0. ∗ ∗ ∗ Definition 1 Strategy vector x = (x1(r); : : : ; xn(r)), together with the value of the aggre- ∗ ∗ ∗ ∗ gate A = A (r) = g(x1(r); : : : ; xn(r)), is a conjectural variation equilibrium (CVE) for a ∗ ∗ conjecture vector r = (r1; : : : ; rn) if x ;A satisfy the system of equations (2) for this r. In general a CVE may not exist or there may be multiple equilibria. An equilibrium selection assigns for each vector r a unique equilibrium. It is assumed in the sequel that a locally well behaved (i.e. differentiable) equilibrium selection x∗(r), A∗(r) exists for the relevant values of conjectures. In a symmetric game, if conjectures are symmetric (ri = rj for all ∗ ∗ i; j), then, whenever a CVE exists, there exists a symmetric CVE with xi (r) = xj (r) for all i; j. The equilibrium selection is assumed to select this symmetric CVE. Certain values of conjectures represent well-known solution concepts. In a standard e Nash equilibrium analysis, the actions of the other players are fixed, thus ri = (dA=dxi) = @g=@xi. The current setting allows for many more conjectures, some of which turn out to be relevant in terms of evolution. For example, a player with zero conjectural variation e ri = (dA=dxi) = 0 believes that a change in his or her own action does not affect the aggregate (aggregate-taking behavior, Possajennikov, 2003, and Al´os-Ferrer and Ania, 2005). To define a (symmetric) consistent conjecture, suppose that players' conjectures are ri = rj for all i; j and imagine that player i's strategy xi is fixed (the definition is based on the definition for the Cournot oligopoly context in Perry, 1982, which generalized the definitions for duopoly in Laitner, 1980, and Bresnahan, 1981). The remaining n−1 players play a CVE of the reduced game with player i's strategy fixed at xi. Then there are n equations characterizing the CVE of the reduced game: Fj(xj;A; rj) = 0, j 6= i (3) A − g(x1; : : : ; xn) = 0. ∗∗ For various xi, this system implicitly defines reaction functions of the other players xj (xi) ∗∗ ∗∗ ∗∗ and A (xi) = g(x1 (xi); : : : ; xi; : : : ; xn (xi)). It is assumed that the selection of the reaction functions is such that for a symmetric conjecture vector r and a corresponding symmetric ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ CVE x ;A , if xi = xi , then xj (xi ) = xj for all j 6= i and A (xi ) = A , since xj ;A are C then a solution of system (3). For a symmetric r with ri = r for all i define 4 Definition 2 Conjecture rC is consistent if for any i ∗∗ C dA ∗ dg ∗∗ ∗ ∗ ∗∗ ∗ r = (xi ) = (x1 (xi ); : : : ; xi ; : : : ; xn (xi )), dxi dxi ∗∗ ∗∗ ∗∗ ∗ ∗ where xj (xi);A (xi) is a differentiable selection of reaction functions with xj (xi ) = xj ∗∗ ∗ ∗ ∗ ∗ ∗ for j 6= i, A (xi ) = A , where (x1; : : : ; xn);A is a symmetric CVE for r.
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