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Notes on Global Games

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Notes on Global Games NOTES ON GLOBAL GAMES The material is mainly from Morris and Shin (2003). 1. Introduction Higher-order beliefs play an important role in many situations, especially in situations involving co-ordination. We have seen this in Rubinstein’s e-mail game. The reason players cannot coordinate is not that they have uncertainty regarding the underlying state, but that they are not sure their opponents know they know the opponents know ··· they know the underlying state is favorable. Rubinstein’s e-mail game revisited. Let’s write the underlying game as follows. a b a 0,0 0, θ − 2 b θ − 2, 0 θ, θ Figure 1. E-mail game revisited. The unknown parameter θ = 0 with probability 1 − p, θ = 1 with probability p; Players receive private signals (x1, x2), distributed as follows: If θ = 0, then (x1, x2) = (0, 0) with probability 1; 2n−1 If θ = 1, then for n = 0, 1, ··· ,(x1, x2) = (n, n + 1) with probability (1 − ε) ε, and 2n−2 (x1, x2) = (n, n) with probability (1 − ε) ε. Result: as long as p is not too large or ε sufficiently small, there is a unique equilibrium in which all types play the safe action a; this equilibrium is also the limit as ε → 0. (1) A simple device to model higher-order beliefs: (a) Generate higher-order uncertainty through unknown random parameter and pri- vate noisy signal (2 receives noisy signal about underlying state and 1’s signal, and 1 receives noisy signal about 2’s signal); (b) The private signals are highly correlated; (c) Countably many types for each player; Date: February 6, 2007. 1 2 NOTES (d) Simplification: parameterize payoff uncertainties using a random variable, and al- low players to receive noisy private signals about the parameter, often with a public portion; (2) Robustness of Nash equilibrium: (a) Perturb the complete information game to check how sensitive the Nash equilibrium outcome is to uncertainties in information structure; (b) A rich class of “nearby” games; (c) Strong notion of robustness: in every “nearby” game, there is an equilibrium in that game that yields “close” behavior; (3) Breaking multiple equilibrium: (a) In complete information games, self-fulfilling beliefs result in multiple equilibria; (b) Introducing higher-order uncertainties breaks up the perfect coordination between beliefs and action (equilibrium actions depend on private signals); (c) The unique equilibrium is typically a function of the underlying state; (d) Meaningful comparative statistics and welfare analysis, empirical analysis. 2. Global Games: Definition and Examples Definition 1. Global games are games of incomplete information where the type space is generated by the players each observing a noisy private signal of the underlying state. Note: Global games assume common knowledge of the parameter distribution and noise technology. This is with loss of generality: there is a truly “prior” stage. One could justify the common knowledge of the model in the incomplete information framework a la Hasanyi, but not the CPA. A baseline model. a b a 0,0 0, θ − 1 b θ − 1, 0 θ, θ Figure 2. A Stylized Global Game The unknown parameter θ is uniformly distributed on the real line (improper probability, but we only care about posteriors which is well-defined anyway). 2 Player i observes an private signal xi = θ + εi, where εi ∼ N(0, σ ) and is i.i.d across i, j; Some useful facts: conditional on a signal x, (1) θ|x ∼ N(x, σ2); NOTES 3 2 (2) xj|(xi = x) ∼ N(x, 2σ ); 1 (3) P rob(xj ≤ x|(xi = x)) = 2 . The expected payoff from playing the risky action b is monotonic in the underlying state. This naturally leads to a switching strategy: take the safe action a if and only if the signal ∗ ∗ 1 is below some threshold x , and take the risky action b otherwise. It follows x = 2 is an equilibrium. (Players need to be indifferent between the two actions at the cut-off point.) This strategy is also the unique rationalizable strategy: (1) First round: a and b are strictly dominant strategy for x < 0 and x > 1 respectively; (2) Second round: a is strictly dominant strategy for all x satisfying x < z( √−x ); b is strictly 2σ dominant strategy for all x satisfying x > z( √1−x ). 2σ (To see this, let p = P rob(x < 0|x = x) = z( √−x ), then notice: E (u ) < E [p(θ − j i 2σ i i i 1) + (1 − p)θ|xi = x] < 0; similarly for the other claim.) (3) ··· (4) This process yields two sequences taking the value between 0 and 1. The task is then to show they converge (monotonic real sequences converge), and converge to the same limit (payoffs are strictly increasing in the state). Generalization to a continuum of players where payoffs depend on the degree of coordination. There is a continuum of players with mass 1. Let l denote the fraction of players taking the risky action b. Payoff to playing the risky action b is modified to be θ − 1 + l. Noise technology is the same as before. The same useful observations obtain. In particular, conditional on a signal x, the propor- 1 tion of the opponents who observe a signal no greater than x is exactly 2 . The same argument ∗ 1 leads to x = 2 . A general case that illustrates the point (Atkeson 2000): Consider a coordination game with two actions and a continuum of players. Safe action yields 0, while risky action yields W > 0 if the fraction of players taking the risky action is above some threshold, and it yields L < 0 otherwise. The threshold is a strictly increasing function of the underlying state, denoted by α(θ). Let θ and θ¯ be such that α(θ) = 0, α(θ¯) = 1. (So [θ, θ¯] is the multi-equilibria region.) (The higher the state is, the more demanding the coordination requirement is.) At state θ, each player i receives a noisy private signal xi = θ + εi. The key components are the conditional distribution of θ given a signal x and the conditional distribution of the signal given a state. Assumptions: 4 NOTES (1) The cumulative distribution function for the signal conditional on state P rob(x ≤ x∗|θ): ∗ R → [0, 1] is strictly positive, continuous and decreasing in θ for any x ; (2) The posterior of θ conditional on θ is P rob(θ ≤ θ∗|x), this is assumed to be continuous, decreasing in x for any fixed θ∗. The unique equilibrium, if exists, must be characterized by a pair (x∗, θ∗) that solves the following equation: (1) P rob(x ≤ x∗|θ∗) = α(θ∗), (2) P rob(θ ≤ θ∗|x∗)W + (1 − P rob(θ ≤ θ∗|x∗))L = 0. x∗ is the cut-off signal for equilibrium switching strategy (risky action if below x∗ and safe action otherwise); θ∗ is the threshold state that determines the outcome of the game. Proof via iterated deletion of strictly dominated strategies: (1) First round: risky action is strictly dominating if P rob(θ ≤ θ|x)W + (1 − P rob(θ ≤ θ|x))L ≥ 0, RHS decreases in x, thus it pins down the lower bound signal (a highest x satisfying the inequality) for risky action x0, similarly one can pin down the upper bound signal for safe action x0; (2) Second round: now anyone who receives a signal below x0 takes the dominant risky action. Thus one can find the (highest) threshold state θ0, such that in all states below which the risky action pays off – the new dominant region. Repeat the previous round to find out new threshold signal x1. Similarly for the other end. (3) ··· ∞ n n ∞ (4) We end up with two monotonic sequences {xn, θn}n=0 and {x , θ }n=0. They obviously converge. (5) Look for conditions that ensure them to converge to the same limit. (6) Now relate this to the equations (1)-(2). The limit of the sequences must solve these equations. Thus the question is equivalent to looking for conditions to ensure a unique solution to them. Now check out the conditions Morris and Shin have on pages 19-20 and 27-28. The limiting behavior as the noise goes away. Parameterize the signal technology: xi = θ + σεi and let σ → 0. Under very general assumptions (in 2 × 2 games, for any payoff vector that is drawn from a strictly positive, 8 continuously differentiable, bounded density on R , and any noise term where the components for risky actions are drawn from a continuous density with bounded support, independent of the payoff vector), (1) any sequence of rationalizable strategy profiles converges to a unique limit; (2) limit is independent of the noise distribution; NOTES 5 (3) unique Nash equilibrium in the underlying complete information game if there is one, or the risk dominant Nash equilibrium if there are multiple strict equilibria. Generalization to n × n asymmetric global games: the uniqueness result crucially depends on the supermodularity of the payoffs on action profiles and the underlying state; the noise independent selection result has less structure. 3. Global Games and Robust Equilibrium Motivation: common knowledge of a complete information game is a very demanding assumption. In reality, players are rarely sure about the game itself, and in particular will not be sure about whether the opponents are sure that they themselves are sure ··· what the game is. Thus it makes sense to ask if the outcome in complete information games is robust to small perturbations of players’ information structures. Think of the noise structure as a way to generate rich higher-order belief structures.
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