Lecture 5
Bargaining
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.
Bargaining with full information
Two striking features characterize all the
solutions of the bargaining games that we have
played so far:
1. An agreement is always reached.
2. Negotiations end after one round.
This occurs because nothing is learned from
continuing negotiations, yet a cost is sustained
because the opportunity to reach an agreement
is put at risk from delaying it.
Reaching agreement may be costly
Yet there are many situations where conflict is not
instantaneously resolved, and where negotiations
break down:
1. In industrial relations, negotiations can be
drawn out, and sometimes lead to strikes.
2. Plans for construction projects are discussed,
contracts are written up, but left unsigned, so
the projects are cancelled.
3. Weddings are postponed and called off.
The blame game
Consider the following experiment in a multi-round
bargaining game called BLAME. There are two
players, called BBC and a GOVT.
At the beginning of the game BBC makes a
statement, which is a number between zero and
one, denoted N. (Interpret N as a proportion of
blame BBC is prepared to accept.)
The GOVT can agree with the BBC statement N or
refute it. If the GOVT agrees with the statement
then the BBC forfeits £N billion funding, and the
GOVT loses 1 - N proportion of the vote next
election.
Counter proposal
If the GOVT refutes the statement, there is a 20
percent chance that no one at all will be blamed,
because a more newsworthy issue drowns out the
conflict between BBC and GOVT.
If the GOVT refutes the statement, and the issue
remains newsworthy (this happens with probability
0.8), the GOVT issues its own statement P, also a
number between zero and one. (Interpret P as a
proportion of blame the GOVT offers to accept.)
Should the BBC agree with the statement issued by
the GOVT, the GOVT loses P proportion of the vote
in the next election, and the BBC loses £5(1-P)
billion in funds.
Endgame
Otherwise the BBC refutes the statement of the
GOVT, an arbitrator called HUTTON draws a random
variable from a uniform distribution with support
[0,2] denoted H, the BBC is fined £H billion, and the
GOVT loses H/5 proportion of the vote next
election.
What will happen?
The solution can be found using backwards
induction. (See the footnotes or read the press!)
Evolving payoffs and discount factors
Suppose two (or more) parties are jointly liable for a debt
that neither wishes to pay.
The players take turns in announcing how much blame
should be attributed to each player, and the game ends if
a sufficient number of them agree with a tabled proposal.
If a proposal is rejected, the total liability might increase
(since the problem remains unsolved), or decline (if there
is some chance the consequences are less dire than the
players originally thought).
If the players do not reach a verdict after a given number
of rounds, another mechanism, such as an independent
enquiry, ascribes liability to each player.
Summarizing bargaining outcomes
when there is complete information
If the value of the match is constant
throughout the bargaining phase, and is
known by both parties, then the preceding
discussion shows that it will be formed
immediately, or not at all.
The only exception occurs if the current value
of the match changes throughout the
bargaining phase as the players gather new
information together.
Bargaining with incomplete information
If the value of the match is constant throughout
the bargaining phase, and is known by both
parties, then the preceding discussion shows that it
will be formed immediately, or not at all.
In the segment on this topic, we will relax the
assumption that all the bargaining parties are fully
informed.
We now modify the original ultimatum game,
between a proposer and a responder, by changing
the information structure.
Suppose the value to the responder of reaching an
agreement is not known by the proposer.
An experiment
In this game:
1. The proposer demands s from the responder.
2. Then the responder draws a value v from the
probability distribution F(v). For convenience
we normalize v so that v0  v  v1.
3. The responder either accepts or rejects the
demand of s.
4. If the demand is accepted the proposer
receives s and the responder receives v – s,
but if the demand is rejected neither party
receives anything.
The proposer’s objective
The responder accepts the offer if v > s and
rejects the offer otherwise.
Now suppose the proposer maximizes his
expected wealth, which can be expressed as:
Pr{v > s}s = [1 – F(s)]s
Notice the term in the square brackets [1 – F(s)]
is the quantity sold, which declines in price, while
s is the price itself.
Solution to the game
Let so denote the optimal choice of s for the
proposer. Clearly v0  so < v1.
If v0 < so < v1, then so satisfies a first order condition
for this problem:
1 – F’(so) so – F(so) = 0
Otherwise so = v0 and the proposer receives:
[1 – F(v0)]v0 = v0
The revenue generated by solving the first order
condition is compared with v0 to obtain the solution
to the proposer’s problem.
F(s) is a uniform distribution
Suppose:
F(v) = (v – v0)/ (v1 – v0) for all v0  v  v1
which implies
F’(v) = 1 /(v1 – v0) for all v0  v  v1
Thus the first order condition reduces to:
1 – so/(v1 – v0) – (so – v0)/ (v1 – v0) = 0
=>
v1 – v0 – so – so + v0 = 0
=>
2 so = v1
Solving the uniform distribution case
In the interior case so = v1/2. It clearly applies when
v0 = 0, but that is not the only case.
We compare v0 with the expected revenue from the
interior solution v1(v1 – 2v0)/(v1 – v0).
If v0 > 0 define v1 = kv0 for some k > 1.
Then we obtain an interior solution if k > 1 and:
k(k – 2) > k – 1
=>
k2 – 3k + 1 > 0
So an interior solution holds if and only if k exceeds
the larger of the two roots to this equation, that is
k > (3 + 51/2)/2 .
F(s) is [0,1] uniform
More specifically let:
F(v) = v for all 0  v  1
Then the interior solution applies so so = ½,
and F(v) = ½. Thus exchange only occurs half
the time it there are gains from trade.
The trading surplus is:

v1
so
vdF v 
Given our assumption about F(v) it follows that
¼ of the trading surplus is realized, which is ½
of the potential surplus .
Counteroffers
Since there is only one offer, there is no
opportunity for learning to take place during
the bargaining process.
We now extend the bargaining phase by
allowing the player with private information to
make an initial offer. If rejected, the
bargaining continues for one final round.
For convenience we assume throughout this
discussion that F(v) is uniform [0,1].
Solution when there are counteroffers
The textbook analyzes solutions of the following type:
1. There is a threshold valuation v* such that in the
first found every manager with valuation v > v*
offers the same wage w*, and every manager with
valuation v < v* offers lower wages.
2. In the first round the union rejects every offer
below w*, and accepts all other offers.
3. If the bargaining continues to the final round the
union solves the first order condition for the one
round problem using the valuations of the manager
as truncated at v*.
Outcomes of two round bargaining game
Note that if the probability of continuation is
too high, management will not offer anything in
the first round, because it would reveal too
much about its own private value v.
In this case the bargaining process stalls
because management find it strategically
beneficial to withhold information that can be
used against them.
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
6. turned to bargaining games where the players
have incomplete information
7. discussed the role of signaling in such games.