Lecture 1 on Bargaining

Setting the Agenda

This lecture focuses on the well known problem of how to split the gains from trade or, more generally, mutual interaction when the objectives of the bargaining parties diverge.

Resolving conflict

Bargaining is one way of resolving a conflict between two or more parties, chosen when all parties view it more favorably relative to the alternatives (such as courts, theft, warfare).

For example:

1. Unions bargain with their employers about wages and working conditions.

2. Professionals negotiate their employment or work contracts when changing jobs.

3. Builders and their clients bargain over the nature and extent of the work to reach a work contract.

Three dimensions of bargaining

We shall focus on three dimensions of bargaining:

1. How many parties are involved, and what is being traded or shared?

2. What are the bargaining rules and/or how do the parties communicate their messages to each other?

3. How much information do the bargaining parties have about their partners?

Answering these questions helps us to predict the outcome of the negotiations.

Two player ultimatum games

Consider the following three versions of the problem of splitting a dollar between two players. In each case, the rejected proposals yield no gains to either party:

1. The proposer offers anything between 0 and 1, and the responder either accepts or rejects the offer.

2. The proposer makes an offer, and the responder either accepts or rejects the offer, without knowing exactly what the proposer receives.

3. The proposer selects an offer, and the responder simultaneously selects a reservation value. If the reservation value is less than the offer, then the responder receives the offer, but only in that case.

Solution

The solution is the same in all three cases.

The solution is for the proposer to extract (almost) all the surplus, and for the responder to accept the proposal.

Two rounds of bargaining

Suppose that a responder has a richer message space than simply accepting or rejecting the initial proposal.

After an initial proposal is made, we now assume:

1. The responder may accept the proposal, or with probability p, make a counter offer.

2. If the initial offer is rejected, the game ends with probability 1 – p.

3. If a counter offer is made, the original proposer either accepts or rejects it.

4. The game ends when an offer is accepted, but if both offers are rejected, no transaction takes place.

Solution to a 2 round bargaining game

In the final period the second player recognizes that the first will accept any final strictly positive offer, no matter how small.

Therefore the second player reject any offer with a share less than p in the total gains from trade.

The first player anticipates the response of the second player to his initial proposal.

Accordingly the first player offers the second player proportion p, which is accepted.

A finite round bargaining gameThis game can be extended to a finite number of rounds, where two players alternate between making proposals to each other.

Suppose there are T rounds. If the proposal in round t < T is rejected, the bargaining continues for another round with probability p, where 0 < p < 1.

In that case the player who has just rejected the most recent proposal makes a counter offer.

If T proposals are rejected, the bargaining ends.

If no agreement is reached, both players receive nothing. If an agreement is reached, the payoffs reflect the terms of the agreement.

Sub-game perfection

If the game reaches round T - K without reaching an agreement, the player proposing at that time will treat the last K rounds as a K round game in which he leads off with the first proposal.

Therefore the amount a player would initially offer the other in a K round game, is identical to the amount he would offer if there are K rounds to go in T > K round game and it was his turn.

Solution to finite round bargaining game

One can show using the principle of mathematical induction that the value of making the first offer in a T round alternating offer bargaining is:

vT = 1 – p + p2 – . . . + pT

= (1 + pT )/(1 + p)

where T is an odd number.

Observe that as T diverges, vT converges to:

vT = 1 /(1 + p)

Infinite horizonWe now directly investigate the solution of the infinite horizon alternating offer bargaining game.

Let v denote the value of the game to the proposer in an infinite horizon game.

Then the value of the game to the responder is at least pv, since he will be the proposer next period if he rejects the current offer, and there is another offer round.

The proposer can therefore attain a payoff of:

v = 1 – pv => v = 1/(1+p)

which is the limit of the finite horizon game payoff.

Alternatives to taking turns

Bargaining parties do not always take turns. We now explore two alternatives:

1. Only one player is empowered to make offers, and the other can simply respond by accepting or rejecting it.

2. Each period in a finite round game one party is selected at random to make an offer.

When the order is random

Suppose there is a chance of being the proposer in each period. How does the solution differ depend on the chance of being selected?

We first consider a 2 round game, and then an infinite horizon game.

As before p denote the probability of continuing negotiations if no agreement is reach at the end of the first round.

Solution to 2 round random offer game

If the first round proposal is rejected, then the expected payoff to both parties is p/2.

The first round proposer can therefore attain a payoff of:

v = 1 – p/2

Solution to infinite horizonrandom offer game

If the first round proposal is rejected, then the expected payoff to both parties is pv/2.

The first round proposer can therefore attain a payoff of:

v = 1 – pv/2 => 2v = 2 – pv => v = 2/(2 + p)

Note that this is identical to the infinitely repeated game for half the continuation probability.

These examples together demonstrate that the number of offers is not the only determinant of the bargaining outcome.

Multiplayer ultimatum games

We now increase the number of players to N > 2.

Each player is initially allocated a random endowment, which everyone observes.

The proposer proposes a system of taxes and subsidies to everyone.

If at least J < N –1 of the responders accept the proposal, then the tax subsidy system is put in place.

Otherwise the resources are not reallocated, and the players consume their initial endowments.

Solution to multiplayer ultimatum game

Rank the endowments from the poorest responder to the richest one.

Let wn denote the endowment of the nth poorest responder.

The proposer offers the J poorest responders their initial endowment (or very little more) and then expropriate the entire wealth of the N – J remaining responders.

In equilibrium the J poorest responders accept the proposal, the remaining responders reject the proposal, and it is implemented.

Another multiplayer ultimatum game

Now suppose there are 2 proposers and one responder.

The proposers make simultaneous offers to the responder.

Then the responder accepts at most one proposal.

If a proposal is rejected, the proposer receives nothing.

If a proposal is accepted, the proposer and the responder receive the allocation specified in the terms of the proposal.

If both proposals are rejected, nobody receives anything.

The solution to this game

If a proposer makes an offer that does not give the entire surplus to the responder, then the other proposer could make a slightly more attractive offer.

Therefore the solution to this bargaining game is for both proposers to offer the entire gains from trade to the responder, and for the responder to pick either one.

Heterogeneous valuations

As before, there are 2 proposers and one responder, the proposers make simultaneous offers to the responder, the responder accepts at most one proposal.

Also as before if a proposal is rejected, the proposer receives nothing. If a proposal is accepted, the proposer and the responder receive the allocation specified in the terms of the proposal. If both proposals are rejected, nobody receives anything.

But let us now suppose that the proposers have different valuations for the item, say v1 and v2 respectively, where v1 < v2.

Solving heterogeneous valuations game

It is not a best response of either proposer to offer less than the other proposer if the other proposer is offering less than both valuations.

Furthermore offering more than your valuation is weakly dominated by bidding less than your valuation. Consequently the first proposer offers v1 or less.

Therefore the solution of this game is for the second proposer to offer (marginally more than) v1 and for the responder to always accept the offer of the second proposer.

Summary

In today’s session we:

1. began with some general remarks about bargaining and the importance of unions

2. analyzed the (two person) ultimatum game

3. extended the game to treat repeated offers

4. showed what happens as we change the number of bargaining parties

5. broadened the discussion to assignment problems where players match with each other.