Coordination Games with Information Aggregation

Transcription

1 1 / 13 Coordination Games with Information Aggregation Mehdi Shadmehr University of Miami Dan Bernhardt University of Illinois MET October, 2012

2 2 / 13 incumbent challenger incumbent h, h w, l challenger l, w θ, θ Lobbying: Two interest groups deciding whether to support an incumbent or an untried challenger. The incubment wins unless both groups support the challenger. If the incumbent wins, he rewards the loyal and punished the disloyal group: l < h < w.

3 2 / 13 incumbent challenger incumbent h, h w, l challenger l, w θ, θ Lobbying: Two interest groups deciding whether to support an incumbent or an untried challenger. The incubment wins unless both groups support the challenger. If the incumbent wins, he rewards the loyal and punished the disloyal group: l < h < w. Investment: Two firms deciding whether to invest in/adopt a new technology that pays off if and only if they both invest. Relationships: Two individuals deciding whether to break up their current relationships to be together. Contentious Politics: Coups, protests, revolutions, wars.

4 3 / 13 Coordination Games with Information Aggregation action 0 action 1 action 0 h, h w, l action 1 l, w θ, θ Citizens receive private signals about θ, then act simultaneously. Uncertainty remains after signals. Miscoordination is costly: l < h.

5 Coordination Games with Information Aggregation action 0 action 1 action 0 h, h w, l action 1 l, w θ, θ Citizens receive private signals about θ, then act simultaneously. Uncertainty remains after signals. Miscoordination is costly: l < h. Economic Forces: Strategic complements: If you take action 1 more, I have more incentive to take action 1 because it is less likley that I get punished. Strategic substitutes: If you take action 1 more, it means you take action 1 after worse signals about θ, implying that E[θ success] is lower. Thus, I have less incentive to take action 1. 3 / 13

6 4 / 13 Coordination Information Aggregation Literature on coordination games, including global games, feature supermodularity. Best responses are monotone.carlsson and va Damme (1993), Morris and Shin (1998), Chassang (2010), Chassang and Padro i Miquel (2010). Many papers do applications. Literature on information aggregation, feature global strategic substitutes. Best responses are monotone. Austen-Smith and Banks (1996), Duggan and Martinelli (2001), Feddersen and Pesendorfer (1996, 1998), McMurray (2012). We analyze the middle ground, where both economic forces exist. Best responses are non-monotone.

7 5 / 13 What We Want to Know Equilibrium Characterization: Number of equilibria. Equilibrium selection. Comparative statics: punishment level (h l) and predatory incentive (w). Should the incumbent punish more harshly or reward more generously? Does an interest group gain if the incumbent reward it more? (Welfare) Does the challenger receive enough support?

8 6 / 13 Best Responses: Single Crossing Property action 0 action 1 action 0 h, h w, l action 1 l, w θ, θ Action 1: Pr(s j > k j s i ) E[θ s j > k j, s i ] + Pr(s j k j s i ) l. Action 0: Pr(s j > k j s i ) w + Pr(s j k j s i )h. µ h l. Player i s expected net payoff from taking action 1 rather than action 0 is (s i ; k j ) = Pr(s j > k j s i ) (E[θ s j > k j, s i ] w + µ) µ. lim (s i ; k j ) < 0 and s i lim (s i ; k j ) > 0. s i

9 7 / 13 Best Responses: Non-monotonicity We want to know ki (k j ) =?. k j ( ) k 1 sign i = sign ( δ(k i ; k j ) ), with k j 2 Sign δ(k i, k j ): δ(k i ; k j ) = E[θ k j, k i (k j )] + h w l. dδ(k i ; k j ) dk j = E[θ kj, k i (k j )] k j + E[θ kj, k i (k j )] f (k j k i ) δ(k j, k i ) k i (k i ;k j ) 1 δ > 0 dδ(ki ;k j ) dk j > 0. 2 δ > 0 if and only if k j is large. k i.

10 7 / 13 Best Responses: Non-monotonicity We want to know ki (k j ) =?. k j ( ) k 1 sign i = sign ( δ(k i ; k j ) ), with k j 2 Sign δ(k i, k j ): δ(k i ; k j ) = E[θ k j, k i (k j )] + h w l. dδ(k i ; k j ) dk j = E[θ kj, k i (k j )] k j + E[θ kj, k i (k j )] f (k j k i ) δ(k j, k i ) k i (k i ;k j ) 1 δ > 0 dδ(ki ;k j ) dk j > 0. 2 δ > 0 if and only if k j is large. k i. There is a critical cutoff k s.t. k i (k j ) exhibits strategic complemts if and only if k j > k. As j takes action 1 more, there is a unique switch from SC to SS.

11 8 / 13 Equilibria action 0 action 1 action 0 h, h w, l action 1 l, w θ, θ If predatory incentives are high, w > w, there are no equilibria in finite cutoff strategies. Otherwise, there are multiple.

12 8 / 13 Equilibria action 0 action 1 action 0 h, h w, l action 1 l, w θ, θ If predatory incentives are high, w > w, there are no equilibria in finite cutoff strategies. Otherwise, there are multiple. With s i = θ + ν i, ν i N, there are two symmetric ones and no asymmetric one.

13 8 / 13 Equilibria action 0 action 1 action 0 h, h w, l action 1 l, w θ, θ If predatory incentives are high, w > w, there are no equilibria in finite cutoff strategies. Otherwise, there are multiple. With s i = θ + ν i, ν i N, there are two symmetric ones and no asymmetric one. Equilibrium Selection: Only the largest is stable, call its associated cutoff k.

14 Equilibria action 0 action 1 action 0 h, h w, l action 1 l, w θ, θ If predatory incentives are high, w > w, there are no equilibria in finite cutoff strategies. Otherwise, there are multiple. With s i = θ + ν i, ν i N, there are two symmetric ones and no asymmetric one. Equilibrium Selection: Only the largest is stable, call its associated cutoff k. Comparative Statics: k w, k µ > 0. 8 / 13

15 9 / 13 Reward or Punishment? Suppose Pr(s j > k s i = k) is decreasing in k. Then, k only if players are sufficiently likley to take action 1. w > k µ > 0 if and To maintain the status quo, harsher punishment is more effective than better rewards if and only if punishment or the reward are already high.

16 10 / 13 Welfare payoff complement/substitutes stategic complement/substitutes. Action are payoff complements if and only if a player s gain from mis-coordination exceeds the other s loss: w > l. Suppose coordination on action 1 is better than coordination on aciton 0. Then, it is socially optimum for palyers to take action 1 more often than they do in equilibrium if and only if w > l.

17 11 / 13 Welfare Effects of Asymmetric Changes incumbent challenger incumbent h, h w i = w + ɛ, l challenger l, w θ, θ 1 Direct Effect: Increases i s welfare. 2 Strategic Effect: i takes action 1 less. If actions are SS, j takes action 1 more. Further increases welfare. If actions are SC, j takes action 1 less. Decreases welfare, and can swamp the positive, direct effect, making i worse off.

18 12 / 13 Welfare Effects of Asymmetric Changes incumbent challenger incumbent h, h w, l challenger l i = l + ɛ, w θ, θ 1 Direct Effect: Increases i s welfare. 2 Strategic Effect: i takes action 1 more. If actions are SS, j takes action 1 less. Decreases welfare, and can swamp the positive, direct effect, making i worse off. If actions are SC, j takes action 1 more. Further increases welfare. SC-SS in Equilibrium: In equilibrium, actions are SS if and only if miscoordination punishment µ is small.

19 13 / 13 Robustness action 0 action 1 action 0 h w action 1 h µ θ no invest invest no invest 0 0 invest α θ k θ α = 1 leads to global strategic complements (supermodular game). We analyze α = 0, where we have both strategic complements and strategic substitutes. With α < 1 strategic substitutes appear again. k = 0 corresponds to infromation aggregation literature.