Bargaining games Econ 414

General bargaining games A common application of repeated games is to examine situations of two or more parties bargaining over a payment (or division of some total surplus). In each “round”, a player makes an offer, which the other party may accept (in which case the game stops) or reject (in which case the game continues). The game continues for some given number of rounds (which could be infinite), but the surplus decreases in value each round, because of bargaining costs or depreciation.

Example: Alternating offer bargaining The most common bargaining game has two players who alternate offers each round. There is a total surplus S which players are bargaining over, and the value of S drops by a multiplicative discount factor δ each period. The game works as follows: Player 1 proposes some offer x 1 to player 2. Player 2 may then accept (in which case payoffs are S - x 1, x 1 ) or reject the offer (in which case the game continues). Player 2 then proposes an offer y 1 to player 1. Player 1 may accept the offer (payoffs are then δy 1, δ(S – y 1 )) or reject the offer, in which case the game continues. Player 1 then gets to offer again, but over δ 2 S, which if rejected player 2 may make an offer over δ 3 S. This alternating offer process then continues until an offer is accepted.

1 x1x1 2 A R 2 S – x 1, x 1 y1y1 1 A R 1 δy 1, δ(S - y 1 ) x1x1 2

Finite length If the game has a finite number of periods T (ie T total offers), then we have a terminal payoff if the offer in the final round is rejected, which is typically 0,0. Then the game can be solved as a regular dynamic game, finding a subgame perfect nash equilibrium by backward induction. The trick is to note that in each period, the player making the offer will offer just enough such that the player receiving the offer is indifferent between accepting and rejecting the offer (and we will assume that they will accept it). Thus, in the final period, the player making the offer will make an offer of zero, keeping δ T-1 S for themselves. In the preceding period, the player making the offer will offer just enough to make the receiving player accept – ie make an offer so that their payoff is δ T-1 S. This pattern then continues back to the first period. So the result of the game is that the initial offer at the beginning of the game will be accepted, but the offer will depend on how many rounds we have.

Example, T = 4 In round 4, 1 gets 0 if they reject the offer, so they accept any non-negative offer, so 2 offers them y 2 = 0. This gives 2 a round 4 payoff of δ 3 S. In round 3, 1 must make an offer that leaves 2 indifferent between accepting that offer and rejecting the offer (which gives δ 3 S). So 1 will offer x 2 = δS. In round 2, 2 must make an offer that leaves 1 indifferent between accepting that offer and rejecting the offer (which gives δ 2 (S- δS)). So 2 will offer y 1 = δS(1- δ) = δS- δ 2 S. In round 1, 1 must make an offer that leaves 2 indifferent between accepting that offer and rejecting the offer (which gives δ[S – δS(1- δ)]). So 1 will offer x 1 = δ[S – δS(1- δ)] = δS – δ 2 S + δ 3 S. The unique SPNE then is for players to make the offers x i, y i above in the appropriate periods, and to accept any offer of at least x i, y i above in the appropriate periods. Thus, in the unique equilibrium, no bargaining actually occurs; the initial offer is accepted immediately. The pattern in this game continues. If we had T = 6, then x 1 would be δS – δ 2 S + δ 3 S - δ 4 S + δ 5 S

T = ∞ Now, suppose the game is repeated an infinite number of times. We cannot use a normal backward induction process, because there is no last period to solve from. However, there is a recurring pattern in the payoffs and strategies (the game is “stationary”) that we can exploit to find the equilibrium offers. It turns out that the unique equilibrium is one where each player makes the same offer every period (which is the highest that would be accepted in that period). I will not prove this here.

Assume the form of the solution: that player 1 offers x* each period and player 2 offers y* each period. Start at any period N where player 1 makes an offer of x*. Then, if player 2 accepts, they get δ N-1 x*. If they reject, 2 will then offer y*, which will be accepted, so 2 will get δ N (S – y*). Thus, these must be equal, so x* = δ(S – y*). Similarly, stating at a period N+1 where player 2 makes an offer of y*, player 1 gets δ N y*. If player 1 rejects, then 1 will offer x* which will be accepted, giving 1 a payoff of δ N+1 (S – x*). These must be equal, so y* = δ(S – x*). Solve these two bold equations simultaneously, gives x* = y* = δS/(1 + δ).

Convergence The solution to the T-length finite game (for the initial bid) converges to the solution for the infinitely repeated game as T -> ∞. δS – δ 2 S + δ 3 S - δ 4 S + δ 5 S + …. = δS/(1 – δ 2 ) – δ 2 S/(1 – δ 2 ) = δS(1 – δ)/(1 – δ 2 ) = δS/(1 + δ)

Alternate notation There are other common forms of notation for bargaining models which are conceptually identical, but behave slightly differently algebraically. Osborne for eg has players make an offer a pair (x 1, x 2 ) where x 1 is the payoff to player 1 and x 2 is the payoff to player 2. Implicitly, x 2 = S – x 1 in my notation, and S is normally assumed to be 1. Another common notation is again to implicitly assume that S = 1, but to make all offers the payoff to player 1. So offer x 1 if accepted gives x 1, 1 – x 1. y 1 if accepted gives δy 1, δ(1 – y 1 ). X 2 if accepted gives δ 2 x 2, δ 2 (1 – x 2 ). This is functionally the same, but gives us different values for x* and y* (y* = 1 – x*).

Reality? The common feature of nearly all bargaining games is that despite a sometimes complex strategy structure, the game is resolved at the very beginning, when the initial offer is made and accepted. Thus, no bargaining process is ever actually observed in play of the game. But in reality, we often observe protracted bargaining behavior (union negotiations and strikes, competing takeover offers, almost any actual bargaining scenario). So, what is going on? Are people irrational (in the strict game theory sense)? Maybe we have misrepresented preferences or information in the game?