“When two dynamite trucks meet on a road wide enough for one, who backs up?” asks Schelling in his classic essay on bargaining.  There are multiple equilibria.  How can the solution be made determinate?

If one side can make a commitment, Schelling points out a easy solution:

“When one wishes to persuade someone that he would not pay more than $16,000 for a house that is really worth $20,000 to him, what can he do to take advantage of the usually superior credibility of the truth over a false assertion? Answer: make it true…..But suppose the buyer could make an irrevocable and enforceable bet with some third party, duly recorded and certified, according to which he would pay for the house no more than $16,000, or forfeit $5,000.”

But what if both sides can make a commitment?  He says:

“Each must now recognize this possibility of stalemate, and take into account the likelihood that the other already has, or will have, signed his own commitment.”

And it is possible there is incomplete information, further complicating the issue.

In this class, I discuss two player bargaining models whether players can commit to demands.  If a demand is rejected or joint commitments are incompatible, there is a chance of bargaining breakdown or costly delay till an agreement is reached.

First, I begin with complete information.  The classic paper is by Rubinstein and it uses discounting to derive a unique equilibrium.  Since, we want to study commitment, I instead followed Myerson’s analysis in his textbook.  In his model, if a proposer’s demand is accepted, the game ends but if it is rejected, the game ends with probability p. If the game survives till the next period, the responder in the previous round becomes the proposer.  The risk of breakdown acts as a discount factor.  The risk of breakdown is a measure of commitment: the higher is p, the higher is the commitment to the demand.  Myerson shows there is a unique equilibrium which is a function of p.

Second, suppose that with a small probability one player might be an r-insistent type who demands r and rejects any smaller offers.  Then, Myerson shows that even if this player is not the r-insistent type, he can guarantee himself r in any equilibrium.  If he demands r repeatedly, the opponent will give up rather than fight forever as he might be facing the r-insistent type.  The rational player then knows he can get r eventually by pretending to the the r-insistent type.  After a few rounds of haggling, his opponent is forced to give this to him. This solution does not depend on p.  Hence, by adding a small probability that a player might be an r-insistent type, we have changed the equilibrium dramatically from the complete information model.  In some sense, the player with an r-insistent type has a first-mover advantage.  The bound on this player’s payoff varies with r so the equilibrium is not robust in the type that was added to the game.

What happens if both players can commit?  Abreu and Gul study this issue.  They show that essentially all bargaining games devolve into a war of attrition where rational players either pretend to to be r-insistent types at incompatible demands or reveal their rationality and concede, like in the Myerson game.  In equilibrium, the probability that players are r-insistent types must reach one simultaneously.  Otherwise, if say it reaches 1 for player 1 first, the rational type of player 2 must still be dropping out after he knows player 1 will not concede.  But it is better for player 2 to deviate and give in earlier and get surplus rather than waste time.  This idea pins down a property of an endpoint to the war of attrition.  And other arguments can then be used to derive the unique equilibrium.

It is surprising the equilibrium is unique: bargaining games and games with incomplete information typically have multiple equilibria.  The equilibrium is still sensitive to the r-insistent types.  I believe this issue is resolved in later papers by Kambe and Abreu and Pearce.  But I did not get to them.  Here are my slides