Job Market Paper - Personal Web Pages - University of Chicago

More documents

Recommendations

Info

Kim: Endogenous Choice of a Mediator g(σ 1,1 (w|w)) ∈ (0, 1) if σ 1,1 (w|w) = 1 2 , and g(σ 1,1(w|w)) = 0 if σ 1,1 (w|w) < 1 2 .27 Then, player 1 (the vetoer) of a weak type would expect to get the payoff of zero if he disagrees, and G µ 5/17,0 is played. But he would also get the payoff of zero if he had voted for the alternative µ 1,0 . Under this belief ¯q 1,1 (w) = 1 and equilibrium strategies, a strong player 2 does have an incentive to not veto, satisfying (T7). For the alternative mediator µ 1,0 , after a veto on µ 1,0 , the players’ beliefs are restricted to a credible vote belief that satisfies (ii) of Definition 2. Lastly, µ 1,0 satisfies (T3) for both types of player 1: U 1 (µ 1,0 |w) = 3.6 ≥ 0.85 · (1 − ψ ′ 1,1(w)) · ( ∑ r∈R σ 1 (r|ww) ∑ d∈D µ 5/17,0 (d|r 2 , ˆr 1 )u 1 (d, ww)), where the right-hand side is always less than or equal to 3.6 for any ψ ′ 1,1 (w) ∈ [0, 1] and for any ˆr 1 ∈ R 1 , and U 1 (µ 1,0 |s) = 0.6 ≥ 0.85 · (1 − ψ ′ 1,1(s)) · ( ∑ r∈R σ 1 (r|t) ∑ d∈D µ 5/17,0 (d|r 2 , ˆr 1 )u 1 (d, sw)), where the right-hand side is always less than or equal to 0 for any ψ ′ 1,1 (s) ∈ [0, 1] and for any ˆr 1 ∈ R 1 because u 1 (d, sw) ≤ 0 for any d. When the status quo is G µ 5/17,0, there is an equilibrium ratification of µ 1,0 because (ii) of Definition 2 is satisfied. Thus, µ 1,0 is a ratifiable mediator against µ 5/17,0 even though it is not individually rational for the weak type with passive updating. That is, a weak type cannot veto the alternative without giving away his identity; and if he gives away his identity, then there is a disadvantage to vetoing. Because the mediator µ 5/17,0 is not immune to the ratification of an alternative, the unique ex ante incentive efficient mediator µ 5/17,0 when ¯p(s) = 0.15 is not threatsecure. The reasoning behind this example is straightforward. A weak type who would receive more of a peace payoff in the status quo finds a switch to the alternative µ 1,0 unprofitable, which puts a higher probability on war, and therefore the weak type gets a lower payoff. Such a player would veto the alternative µ 1,0 in favor of the status quo if the status quo is played with the prior beliefs. But the ratifier can credibly infer that the vetoer is a weak type as a result of the veto. Also, conditional on the weak type vetoing, a strong type has the advantage of not vetoing. So, the vetoer can credibly infer that the ratifier is a strong type. Such inferences from a vote outcome 27 For a given vote belief ¯q·,i, there is more than one equilibrium to G µ 5/17,0 under the ¯q·,i-revised beliefs. However, more than one equilibrium does not present a problem. 42
Kim: Endogenous Choice of a Mediator lead to a ratifier threatening unilateral war in the status quo (more war in the status quo than it would under its original priors), which makes the veto unprofitable because a weak type “fears” war if he does not agree. Hence, with learning, no type wants to veto. If a veto resulted in passive inferences (i.e., no updating), then the alternative µ 1,0 is not individually rational relative to G µ 5/17,0, and thus, not ratifiable. But if the inferences are required to satisfy the credibility conditions, then the alternative is unanimously ratified. Thus, learning in this case has the effect of weakening the participation constraints, by worsening the status quo payoffs for the vetoer, and therefore increases the set of attainable outcomes. Now, assume that two players are involved with the status quo G µ 1,0 that consists of hiring the most ex ante incentive efficient mediator µ 1,0 or going to war. The equilibrium outcome of G µ 1,0 played with the prior beliefs is playing µ 1,0 with honest reporting. Suppose that µ 5/17,0 is an alternative mediator. Without learning, the strong type expects to get U 1 (µ 5/17,0 |s) = 0 from the alternative and U 1 (G µ 1,0 |s) = 0.6 in the status quo; the weak type expects to get U 1 (µ 5/17,0 |w) = 4.14 from the alternative and U 1 (G µ 1,0 |w) = 3.4 in the status quo. Is µ y,0 ratifiable against µ 1,0 when the status quo is G µ 1,0 ? Without loss of generality, suppose that player 2 votes for the alternative. What should both players think if player 1 vetoes µ 5/17,0 , i.e., M = {1}? Suppose that V 1 = {s} and W 2 = {w}. Then, player 2 rationalizes 1’s veto by ¯q 1,1 (s) = 1. Because player 2 did not veto, player 1 updates his belief about player 2’s type to be ¯q 2,1 (w) = 1. With these posteriors, equilibrium of G µ 1,0 σ = ((σ 1,1 (s|s) ∈ [0, 1], σ 1,1 (w|w) = f(σ 2,1 (w|w))), (σ 2,1 (t 2 |t 2 ) ∈ [0, 1], ∀t 2 )) and ψ = ((ψ 1,1 (s) = 1, ψ 1,1 (w) = 0), (ψ 2,1 (t 2 ) ∈ (0, 1), ∀t 2 )), where f(σ 2,1 (w|w)) = 1 if σ 2,1 (w|w) > 1 2 , f(σ 2,1(w|w)) ∈ (0, 1) if σ 2,1 (w|w) = 1 2 , and f(σ 2,1(w|w)) = 0 if σ 2,1 (w|w) < 1 2 . Then, player 1 (the vetoer) of a strong type would expect to get the payoff of zero, is ¯q 2,1 (s)u 1 (d 0 , ss) + ¯q 2,1 (w)u 1 (d 0 , sw) = 0 (6.1) if he disagrees and G µ 1,0 is played. But he would get the payoff of ∑ ¯q 2,1 (t 2 ) ∑ µ 5/17,0 (d|(s, t 2 ))u 1 (d, (s, t 2 )) = ¯q 2,1 (s) · 4 + ¯q 2,1 (w) · (y · 0 + (1 − y) · (−1)) = − 12 /17 t 2 d∈D (6.2) if he had voted for the alternative. Under this belief ¯q 1,1 (s) = 1 and equilibrium strategies (note that ψ 1,1 (s) = 1), a weak player 2 is indifferent between vetoing or not, conditional on player 1 43
Page 1 and 2: Endogenous Choice of a Mediator: In
Page 3 and 4: Kim: Endogenous Choice of a Mediato
Page 41: Kim: Endogenous Choice of a Mediato
Page 67: Kim: Endogenous Choice of a Mediato

Job Market Paper - Personal Web Pages - University of Chicago

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?