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 game
This 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:
v T = 1 – p + p2 – . . . + p T
= (1 + pT )/(1 + p)
where T is an odd number.
Observe that as T diverges, vT converges to:
vT = 1 /(1 + p)
Infinite horizon
We 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 horizon
random 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.