Bargaining with uncertain commitment: An agreement theorem Rohan Dutta Work in progress

Inefficiency in bilateral bargaining Asymmetric Information Kennan & Wilson (1990) Irrational Types Abreu & Gul (2000) Behavioral bias Babcock & Loewenstein (1997) Uncertain commitment Schelling (1956) Crawford(1982) Ellingsen & Miettinen(2008)

Acute level of multiplicity Crawford (1982) General Model Players simultaneously demand shares of a pie of size 1 If the demands are compatible payoffs If the demands are incompatible i finds out realization of random variable admits a continuous density with support Then a Bayesian game is played 1-z2+d-k1 , 1-z1+d-k2 z1, 1-z1-k2 0,0 1-z2-k1, z2 Accept Stick Acute level of multiplicity Asymmetric information Uncertain commitment Independent costs

Leading example in Crawford(1982) Cost is drawn from a binomial distribution Prob (ki=2>1) Prob (ki=0) Equilibria with positive probability of disagreement exist Asymmetric information in the second stage not needed for this result. Following incompatible demands the realized costs for both agents becomes common knowledge. 1-z2+d-k1 , 1-z1+d-k2 z1, 1-z1-k2 0,0 1-z2-k1, z2 Accept Stick

1-z2+d-k1 , 1-z1+d-k2 z1, 1-z1-k2 0,0 1-z2-k1, z2 Accept Stick Prob (ki=2>1) (1,1) is an equilibrium demand profile Prob (ki=0) i sticks when cost is high and accepts when ki=0 & k-i=2 If both the costs are 0 then 1 sticks and 2 accepts Payoff to player 1 Payoff to player 2 Deviation payoff to player 1 Deviation payoff to player 2 There are also multiple efficient equilibria No efficient equilibria exists if for both i

Prob (k1=k2=2) Prob (k1=k2=0) Independence of costs? 1-z2+d-2 , 1-z1+d-2 z1, 1-z1-2 0,0 1-z2-2, z2 Accept Stick 1-z2+d , 1-z1+d z1, 1-z1 0,0 1-z2, z2 Accept Stick (1,1) is an equilibrium demand profile If ki = 0 then play (Stick, Accept) If ki = 2 then play (Stick, Stick)

Another Example 1-z2+d-2 , 1-z1+d-2 z1, 1-z1-2 0,0 1-z2-2, z2 Accept Stick 1/3 : k=2 1-z2+d-1/5 , 1-z1+d-1/5 z1, 1-z1-1/5 0,0 1-z2-1/5, z2 Accept Stick 1/3 : k=1/5 1-z2+d , 1-z1+d z1, 1-z1 0,0 1-z2, z2 Accept Stick 1/3 : k=0 (4/5,4/5) is an equilibrium demand profile k=1/5 (Accept, Stick) k=0 (Stick, Accept) (0, 4/5) (4/5, 0) If z1 > 4/5 then (Accept, Stick) when k=0

As Players simultaneously demand shares of a pie of size 1 If the demands are compatible payoffs If the demands are incompatible The cost of backing down k is drawn from a strictly increasing continuous distribution with an interval for its support Player i gets to see k but with some noise Proposition : As the set of equilibrium demand profiles of comprises of only efficient profiles. The noise for i is distributed uniformly over its support and independent of the noise for -i Characterize the set of equilibria of this game as

Risk dominant outcome: (Accept, Stick) 1-z2+d-k , 1-z1+d-k z1, 1-z1-k 0,0 1-z2-k, z2 Accept Stick The highest observation by i where i changes her action from the (Accept, Stick) profile. Risk dominant outcome: (Accept, Stick) Carlsson & van Damme(1993) Pr(A2)=q Pr(A1)=q’

What about the multiplicity of efficient equilibria? With incompatible offers each agent has the incentive to make a lower (but still incompatible) demand than the other agent and force her to always concede in the second stage. No disagreement! What about the multiplicity of efficient equilibria? Small enough Profitable deviation if the probability of high costs is not too high. The (symmetric) set of equilibrium demand profiles that survives depends on the distribution function.

Conclusion: When facing the same cost of backing down, unless the agents believe that intermediate costs are impossible, the bargaining outcome will always be efficient. Claim: Even with independent costs as long as the supports for the random variables are intervals, there will be no inefficient equilibrium. There will, however, be multiple efficient equilibria when the expected cost of backing down is sufficiently high. How does the proof change? 2 dimensional global games argument to get rid of multiplicity in the second stage. For any pair of incompatible demands the state space will be divided into two risk dominance regions. The crucial tradeoff will be between making a higher demand and having a larger risk dominance region where you make the other party concede to your demand.

Sam Spade (The Maltese Falcon): If you kill me, how are you gonna get the bird? And if I know you can't afford to kill me, how are you gonna scare me into giving it to you?