Global Games I. Mehdi Shadmehr Department of Economics University of Miami. August 25, 2011

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1 Global Games I Mehdi Shadmehr Department of Economics University of Miami August 25, 2011

2 Definition What is a global game? 1 A game of incomplete information. 2 There is an unknown and uncertain parameter. 3 There is strategic uncertainty: players are uncertain about each others behavior because they do not know each other s information (signal) about the uncertain parameter. Example: investment game: player i Invest No Invest player i Invest θ, θ θ 1, 0 No Invest 0, θ 1 0, 0

3 The Investment Game player i player i Invest No Invest Invest θ θ 1 No Invest 0 0 What are the equilibria of the complete information game? 1 θ > 1: Invest is a strictly dominant strategy. 2 θ < 0: No Invest is a strictly dominant strategy. 3 0 < θ < 1: Do whatever the other player do, i.e., invest if she invests, and does not invest if she does not. When θ (0, 1), there are multiple equilibria as there are in other coordination games. Which equilibrium is played in the real world? Harsanyi and Selten (88): Payoff-dominance vs. risk-dominance. Risk-dominant equilibrium: is the equilibrium that has the largest basin of attraction, i.e., the more strategic uncertainty, the more likely that the risk-dominant equilibrium is played.

4 Risk-Dominant Equilibrium of the Investment Game player i player i Invest No Invest Invest θ θ 1 No Invest 0 0 If i plays Invest with probability p, then i plays Invest iff p θ + (1 p) (θ 1) > 0, i.e., p > 1 θ. No Invest risk dominates Invest iff p < 1 2, i.e., 1 θ 1 2, i.e., 1 2 θ. Alternatively, if i plays Invest and No Invest with equal probability, then i plays No Invests if and only if 1 2 θ (θ 1) 0, i.e., θ 1 2. Carlsson and van Damme (Econometrica 93): In 2 2 global games that feature dominant regions (θ > 1 and θ < 0), as the noise vanishes the incomplete information game converges (in a specific way) to the complete information game, a unique equilibrum is selected. This unique equilibrium is the risk-dominant equilibrium of the game. More in my lecture on Convergence in Global Games.

5 Strategic Uncertainty in the Investment Game player i player i Invest No Invest Invest θ θ 1 No Invest 0 0 Suppose θ is uncertain: θ R. Players share an improper (Laplacian) prior that θ U(, + ), and they receive conditionally-independent, noisy, private (strategic uncertainty enters here) signals about θ: s i = θ + ν i, with ν i N(0, σ 2 ), where ν i and ν i are independent from each other and from θ. Strategy: for player i is a mapping σ i (s i ) : R {0, 1}, with 0 corresponding to No Invest, and 1 corresponding to Invest. Equilibrium Concept: Bayes Nash. Cutoff/Monotone/Switching Strategy: No Invest if your signal is low, and Invest if your signal is high: σ i (s i ) = 1 if and only if s i > k i for some k i R {± }.

6 Solving for Equilibria player i player i Invest No Invest Invest θ θ 1 No Invest 0 0 i invests if and only if the expected payoff from Invests exceeds that of No Invest. E[Invest σ i, s i ] = Pr(σ i = 1 s i ) E[θ σ i = 1, s i ] +Pr(σ i = 0 s i ) E[θ 1 σ i = 0, s i ] = E[θ s i ] Pr(σ i = 0 s i ) = E[θ s i ] Pr(s i k i s i ). E[No Invest σ i, s i ] = 0. σ i (s i ) = 1 if and only if E[θ s i ] > Pr(s i k i s i ), i.e., s i > Φ( k i s i 2σ ).

7 σ i (s i ) = 1 if and only if s i > Φ( k i s i 2σ ). The LHS is strictly increasing and onto, and the RHS is strictly decreasing. Thus, k i such that s i > Φ( k i s i 2σ ) if s i > k i, s i < Φ( k i s i 2σ ) if s i < k i, k i = Φ( k i k i 2σ ). The best response to a cutoff/monotone strategy is a cutoff/monotone strategy. (Thus, from Athey (2001), there exists a pure strategy monotone equilibrium.) Lets focus on symmetric equilibria: k i = k i = k. Then, k = Φ( k k 2σ ) = Φ(0) = 1 2. Thus, there is a unique symmetric cutoff equilibrium, in which players Invest if and only if their signals is above a cutoff point (threshold) k = 1 2.

8 Public Signals/Proper Priors Now suppose players also receive a public signal p = θ + ɛ, with ɛ N(0, σ 2 p), where ɛ is independent from θ, ν i, and ν i. σ i (s i ) = 1 if and only if E[θ s i, p] > Pr(s i k i s i, p). E[θ s i, p] = σ2 p si +σ 2 p σ 2 p +σ2. Pr(s i k i s i, p) = Φ ( s i s i, p N k i σ2 p si +σ 2 p σ 2 p +σ2 σ 2 σ 2 +2σ 2 σ 2 p σ 2 p +σ2. ) σ 2 p si +σ 2 p σ, σ2 σ 2 +2σ 2 σ 2 p p 2+σ2 σ. Looks complicated? Improper p 2+σ2 prior means σ 2 p, then σ2 σ 2 +2σ 2 σ 2 p σ 2 p +σ2 2σ 2 as we had: 2σ. Still, the LHS is strictly increasing is s i and onto, and the RHS is strictly decreasing in s i, hence there is a unique solution to the indifference equation: E[θ s i, p] = Pr(s i k i s i, p). Moreover, the sign change is from negative to positive as s i traverse the real line, hence, the best response to a monotone strategy is a monotone strategy.

9 Now, focus on symmetric case: σp 2 k + σ 2 p σp 2 + σ 2 = Φ k σ2 p k+σ2 p σ 2 p +σ2 σ 2 σ 2 +2σ 2 σ 2 p σ 2 p +σ2. Recall that any solution k to the above non-linear equation corresponds to a symmetric equilibrium cutoff. How many such solutions exist? Generically, 1 or 3; because the LHS is a straight line, and the RHS is strictly unimodal, i.e., has a unique inflection point, where is switches from being convex to being concave. Now, suppose the noise vanishes σ 2 0. Then LHS k, and RHS 1 2. That is, there is a unique solution. With improper (Laplacian) prior or very diffuse prior (σ 2 p ), there is a unique solution. With banishing noise (σ 2 0), there is a unique solution. Do these observations suggest something about the cause(s) of multiplicity?

10 Continuum of Players Now suppose there are a continuum 1 of players indexed by i [0, 1], who receive conditionally independent private signals. Player i s payoff from Invest is θ + l 1, where l is the proportion of other players who Invest. That is, a player has more incentive to invest the more the others invest (strategic complementarity).i invest if and only if E[θ + l 1 s i ] > 0, i.e., s i + E[l s i ] 1 > 0. E[l s] = Pr(j invests s i ) 1 = Pr(s j > k s i ) = 1 Pr(s j k s i ). Thus, i invests if and only if s i > Pr(s j k s i ). The rest is the same. Assignment: Set up and solve an N-players case.

11 Between Random Variables In many models, this conditional expectation shows up: E[θ x] θ df (θ x) = θ f (θ x)dθ. θ Θ E[θ x] And, we want so know this: x > 0 or < 0?. Alternatively, we want to know what assumptions are needed to make the above derivative, say, positive. The standard general way to do this is to assume f (θ x) has a specific property called monotone likelihood ratio property. f (θ x) exhibits MLRP when f (θ x2) f (θ x 1) is increasing in θ for any x 2 > x 1. That is, Alternatively, for any θ 2 > θ 1 and x 2 > x 1 : θ Θ f (θ x 2 ) > 0. (1) θ f (θ x 1 ) f (θ 2 x 2) f (θ > f (θ1 x2) 2 x 1) f (θ, i.e. 1 x 1) f (θ 2 x 2 ) f (θ 1 x 1 ) > f (θ 2 x 1 ) f (θ 1 x 2 ). (2) Consider a (θ, x) plane, and the joint distribution g(θ, x). g has more weight along the 45 degree line that off that line.

12 Affiliation When two random variables, say, θ and x, exhibit this property, we say that θ and x are affiliated. Affiliation is a notion of positive association along positive correlation, etc. (De Castro s ( 10) Affiliation, etc. ). As you should expect, f (θ x) has MLRP iff f (x θ) has MLRP. Theorem: MLRP FSD (First-order/degree Stochastic Dominance). Proof: (Lehmann 86) Pick an increasing function u(θ). We want to show that if θ and x are affiliated, then u(θ) df (θ x) is increasing θ Θ in x. Pick x 2 > x 1. By affiliation, f (θ x2) f (θ x 1) is increasing, and hence Θ L {θ f (θ x2) f (θ x < 1} is to the left of Θ 1) R {θ f (θ x2) f (θ x 1) > 1}. Define s to be the supremum of u(θ) on Θ L, and i as its infimum on Θ R. Then u(θ) (f (θ x 2 ) f (θ x 1 ))dθ s (f (θ x 2 ) f (θ x 1 ))dθ θ Θ Θ L +i (f (θ x 2 ) f (θ x 1 ))dθ Θ R = (s i) (f (θ x 2 ) f (θ x 1 ))dθ 0. Θ R

13 Putting u(θ) = θ, FSD implies E[θ x 2 ] > E[θ x 1 ] for all x 2 > x 1. Assignment: Read pages of Milgrom and Weber s (Econometrica, 1982) A Theory of Auction and Competitive Bidding, and generalize the 2 by 2 investment (global) game by generalizing the signal structure: There are two players, 1 and 2. Each player receives a private signal s i about θ. s 1, s 2, and θ are affiliated...?

14 Total Positivity definition (Karlin 1968): Let X and Y be subsets of R, and let K : X Y R. K is Totally Positive of order n, TP n, if x 1 < x 2 <... < x n and y 1 < y 2 <... < y n imply that K(x 1, y 1 )... K(x 1, y m ).. 0 K(x m, y 1 )... K(x m, y m ) for each m = 1,..., n. In particular, K(x 1, y 1 ) K(x 2, y 2 ) K(x 2, y 1 ) K(x 1, y 2 ). That is, TP 2 and MLRP/affiliation are the same, at least, in R 2. We talk about higher dimensions in my lecture(s) on supermodular games (See Karlin and Rinott s 1980 Classes of Ordering of Measures and etc. ).

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