Identification and Estimation of Dynamic Games when Players Beliefs are not in Equilibrium

Transcription

1 and of Dynamic Games when Players Beliefs are not in Equilibrium Victor Aguirregabiria and Arvind Magesan Presented by Hanqing Institute, Renmin University of China

2 Outline General Views 1 General Views 2 Basic assumption DP problem given belief B jt 3 Testable null hypothesis of equilibrium beliefs identify payoff and belief functions 4 with nonparametric payoff function with parametric payoff function and of Dynamic Games when Players Be

3 Cases when player s belief is not in the equilibrium Competition in oligopoly industries: Firm managers have incentives to misrepresent their own strategies and face significant uncertainty about the strategies of their competitors. Policy change in a strategic environment: Firms need to take time to learn about strategies of competitors after the policy change. In laboratory experiments, there exists significant heterogeneity in agents elicited beliefs, and that this heterogeneity is often one of the most important factors in explaining heterogeneity in observed behavior. and of Dynamic Games when Players Be

4 Main Problem we want to deal with Relax the assumption of equilibrium beliefs. When players beliefs are not in equilibrium they are different from the actual distribution of players actions. This paper concentrates on the identification of players payoff functions and beliefs and does not want to make any arbitrary assumption on beliefs. However, without other restrictions, beliefs cannot be identified and estimated by simply using a nonparametric estimator of the distribution of players actions. (Order condition for identification is not satisfied, the number of restrictions is less than the number of parameters). and of Dynamic Games when Players Be

5 Main Steps General Views First, a standard exclusion restriction can provide testable nonparametric restrictions of the null hypothesis of equilibrium beliefs. Second, new results on the nonparametric point-identification of payoff functions and beliefs. (no strategic uncertainty at two extreme points should be imposed). Third, a simple two-step estimation method of structural parameters and beliefs is proposed. Fourth, an empirical application of a dynamic game of store location by retail chains is illustrated. (Omitted because of time limitation) and of Dynamic Games when Players Be

6 Basic assumption DP problem given belief B jt Two player,i,j. Finite period, T. Y it 0, 1 represents the choice of player i in period t. Optimal expected utility E t ( T s=0 βs i i,t+s ). One period payoff function i,t = π it(y jt, X t ) ε it if Y it = 1, i,t = 0 if Y it = 0. Y jt represents the current action of the other player. X t is a vector of state variables which are common knowledge for both players. and of Dynamic Games when Players Be

7 Basic assumption DP problem given belief B jt Common knowledge X it has three parts: X t (W t, S it, S jt ). W t is a vector of state variables that evolve exogenously according to a Markov process with transition probability function. ( Market Size ) S it, S jt are endogenous state variables. They evolve over time according to a transition probability function f St (S t+1 Y it, Y jt, X t ). ( the number of consecutive years of player i in market ). Y jt represents the current action of the other player. X t is a vector of state variables which are common knowledge for both players. and of Dynamic Games when Players Be

8 An example General Views Basic assumption DP problem given belief B jt Deterministic transition rule: S it+1 = Y it (S it + Y it ). Current payoff function it = π(y jt, S it )W t ε it if Y it = 1, and it = 0 ify it = 0. we got π(y jt, s it ) = ((1 Y jt )θi M + Y jt θi D ) θi0 FC θfc i1 exp( S it) 1(S it = 0)θi1 EC and of Dynamic Games when Players Be

9 Basic assumption are structural parameters. and represent the per capita variable profit offirm when DP problem given belief B jt the firm is a monopolist and when it is a duopolist, respectively. is a parameter that represents market entry cost. And 0 and 1 are parameters that represent the fixed operating costs and how they depend on firm s experience. Markov Perfect Equilibrium (MPE) Most previous literature on estimation of dynamic discrete games assumes that the data comes from a Markov Perfect Equilibrium (MPE). This equilibrium concept incorporates three main assumptions. ASSUMPTION 1 (Payoff relevant state variables): Players strategy functions depend only on payoff relevant state variables: X and. ASSUMPTION 2 (Rational beliefs on own future behavior): Players are forward looking, maximize expected intertemporal payoffs, and have rational expectations on their own behavior in the future. ASSUMPTION EQUIL : (Rational or equilibrium beliefs on other players actions): Strategy functions are common knowledge, and players have rational expectations on the current and future behavior of other players. That is, players beliefs about other players behavior are consistent with the actual behavior of other players. First, let us examine the implications of imposing only Assumption 1. The payoff-relevant information set of player is {X }. The space of X is X W S 2. At period, players observe X and choose their respective actions. Let (X ) be a strategy function for player at period. This is a function from the support of (X ) into the binary set {0 1}, i.e., and of Dynamic Games when Players Be

10 Basic assumption DP problem given belief B jt Given the strategy function σ it, we have the Conditional Choice Probability function P it (X t ) = 1{σ it (X t, ε it ) = 1}dΛ i (ε it ). When assumption Equil not holds, the belief of choice probability B jt (X t ) = 1{b jt (X t, ε it ) = 1}dΛ i (ε it ) will be not equal to the actual choice probability P jt. and of Dynamic Games when Players Be

11 Basic assumption DP problem given belief B jt Given the expected one-period payoff function π B it (X t) = (1 B jt (X t ))π it (0, X t ) + B jt (X t )π it (1, X t ) Using backwards induction in the following Bellman equation: B We denote v it ( X t ) a threshold value function because it represents the threshold value that makes player i indifferent between the choice of alternatives 0 and 1. and of Dynamic Games when Players Be

12 Basic assumption DP problem given belief B jt Given the threshold, the optimal response function is Y it = 1 iff {ε it v B it (X t)}. Under assumption 1 and 2 the actual behavior of player i, satisfying P it (X t ) = Λ i (v B it (X t)). When player belief are in equilibrium, we have that B jt (X t ) = P jt (X t ). and of Dynamic Games when Players Be

13 Basic Assumption General Views Testable null hypothesis of equilibrium beliefs identify payoff and belief functions We concentrate on the identification of the payoff functions π it and belief function B it and assume that {f St, f Wt, Λ i, β i : i = 1, 2} are known. The transition probability base on belief is the combination between belief function and actual transition function fit B(S t+1 Y it, X t ) = (1 B jt (X t ))f it (S t+1 Y it, 0, X t ) + B jt (X t )f it (S t+1 Y it, 1, X t ) A common restriction that has been used to obtain nonparametric identification of payoff functions in games with equilibrium beliefs is a particular kind of exclusion restriction (see Bajari and Hong, 2005, and Bajari et al., 2010). We follow this approach. ASSUMPTION 4 (Exclusion Restriction): The one-period payoff function of player depends on the actions of both players, and, the common state variables W,andtheownstockvariable,, but it does not depend on the stock variable of the other player,. ( X )= ( W ) This type of exclusion restriction Weilong appears Zhangnaturally in someand dynamic games. of Dynamic For instance, Games when thisplayers Be

14 Testable null hypothesis of equilibrium beliefs identify payoff and belief functions Under the assumption, the null hypothesis of equilibrium belief is testable. Best response condition: P i (X) = Λ i ((1 B jt (X t ))π it (0, X t ) + B jt (X t )π it (1, X t )) define q i (X) = Λ 1 i (P i (X)) = π i (0, X) + [π i (1, X) π i (0, X)]B j (X) given four values of vector X, say X a, X b, X c, X d, with same (S i, W ) but different S j, then they will have same payoff π i. Then q i (X a ) q i (X b ) q i (X c ) q i (X d ) = B j (X a ) B j (X b ) B j (X c ) B j (X d ) and of Dynamic Games when Players Be

15 Testable null hypothesis of equilibrium beliefs identify payoff and belief functions A nonparametric test for the null hypothesis of equilibrium beliefs. δ(x a, X b, X c, X d ) = q i(x a ) q i (X b ) q i (X c ) q i (X d ) P j(x a ) P j (X b ) P j (X c ) P j (X d ) and of Dynamic Games when Players Be

16 Additional Assumption We Need Testable null hypothesis of equilibrium beliefs identify payoff and belief functions and of Dynamic Games when Players Be

17 Proof General Views Testable null hypothesis of equilibrium beliefs identify payoff and belief functions Given we have known the belief in two extreme points, we have that for any value of (Sj, W): and of Dynamic Games when Players Be

18 Proof continuing General Views Testable null hypothesis of equilibrium beliefs identify payoff and belief functions and of Dynamic Games when Players Be

19 with nonparametric payoff function with parametric payoff function with nonparametric payoff function and of Dynamic Games when Players Be

20 with nonparametric payoff function with parametric payoff function with parametric payoff function and of Dynamic Games when Players Be

21 Iteration to Converge General Views with nonparametric payoff function with parametric payoff function and of Dynamic Games when Players Be