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

2. 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)

3. 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

4. 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

5. 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

6. 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)

7. 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

8. 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

9. 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’

10. 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.

11. 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.

12. 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?