Bargaining, Reputation
and Strategic Investment
45-976
Professor Robert A. Miller
Mini 3, 2005-2006
Course website
The course website is:
Website: http://www.comlabgames.com/45-976/
At the website I shall post:
the course syllabus
power points slide lecture notes
games you can download
handouts related to the projects
Textbook
Robert A. Miller and Vesna Prasnikar,
“Strategic Play” Stanford University Press
(forthcoming)
A draft manuscript is available at:
http://www.comlabgames.com/strategicplay/
We shall focus on upon material found in Chapters 11
through 15 and 1 through 3.
Course objectives
This course will help you to:
1. Recognize opportunities in bargaining and
strategic investment. (Describe the problem
and model it.)
2. Analyze each opportunity to assess its value,
and how that value is attained. (Solve the
model and simulate it.)
3. Persuade your colleagues to follow your
advice. (Analyze the results and report them.)
Lecture 1
Bargaining with complete information
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. In this lecture we avoid the
complications of asymmetrically informed
agents.
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.
Alternative means include:
1. Capitulation
2. Predation and expropriation
3. Warfare and destruction
Bargaining also has these elements in it.
Examples of bargaining situations
Examples of bargaining situations include:
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.
4. Pre nuptial agreements are written by partners
betrothed to be married.
5. No fault divorce law facilitates bargaining over the
division of assets amongst divorcing partners.
Agenda for the lecture
In today’s session we:
1. begin with some general remarks about
bargaining and the importance of unions
2. analyze the (two person) ultimatum game
3. extend the game to treat repeated offers
4. show what happens as we change the number
of bargaining parties
5. broaden the discussion to assignment problems
where players match with each other
6. turn to bargaining games where the players
have incomplete information
7. discuss the role of signaling in such games.
Unions
Unions warrant special mention in discussions of
bargaining and industrial relations.
They are defined as a continuous associations of
wage earners for the purpose of maintaining or
improving their remuneration and the conditions of
their working lives.
In the first half of the 20th century union
membership grew from almost nothing to 35% of
the labor force, only to decline to less that 15% at
the turn of the millennium.
How their composition has
changed
Hidden within these gross trends are three
composition effects worth mentioning :
1. Employment in the government sector increased
from 5% in the early part of the 20th century to
15% in the 1980s, and then stabilized. Union
membership in this sector jumped from about 10%
to about 40% between 1960 and 1975.
2. Employment in agriculture declined from 20% to
3% in the same period. This sector was not
unionized at the turn of the 20th century.
3. Unionization in the nonagricultural private sector
has reflected the aggregate trend, declining to
about 10% of the workforce down from 35%.
Cross sectional characteristics
Within the U.S. membership is highest in the industrial
belt connecting New York with Chicago though
Pittsburgh and Detroit (20 – 30%), lower in upper
New England and the west (10 – 20%), and lowest in
the South and Southwest (10% or less).
Males are 50% more likely to be union members than
females, mainly reflecting their occupational choices.
Union membership differs greatly across countries:
Canada
35%
France
12%
Sweden
85%
United Kingdom 40%
Industrial breakdown and strikes
Strikes are dramatic and newsworthy, but they
are also quite rare:
Less than 5% of union members go on strike
within a typical work year.
Less than 1% of potential working hours of
union members are lost from strikes, before
accounting for compensating overtime.
About 90% of all collective agreements are
renewed without a strike, but the threat of a
strike affects more than 10%.
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.
Ultimatum games
We begin with one of the simplest
bargaining games for 2 or more players.
One player is designated the proposer,
the others are called responders.
The proposer makes a proposal. If
enough responders agree to this
proposal, then it is accepted and
implemented.
Otherwise the proposal is rejected, and a
default plan is implemented instead.
2 player ultimatum games
We consider the problem of splitting a dollar between two
players, and investigate three versions of it:
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 game theoretic solution is the same in all
three cases.
Does the experimental evidence support that
hypothesis?
The solution is for the proposer to extract (almost)
all the surplus, and for the responder to accept
the proposal.
Observe the same outcome would occur if, right at
the outset, the responder had capitulated, or if
the proposer had expropriated the whole surplus.
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 only one player makes offers
In this case, the proposer makes an offer in
the second round, if his first round offer is
rejected.
The solution reverts to the canonical one
period solution.
This simply demonstrates that the rules
about who can make an offer affects the
outcome a lot.
When the order is random
Suppose there is an equal chance of being
the proposer in each period.
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.
Multilateral exchange
In all our previous examples, there is at most one
transaction.
In such games if more than two players were
involved, they competed with each other for the right
to be one of the trading partners.
We now suppose there are opportunities on both
sides of the trading mechanism to form a partnership
with one of a number of different players.
If the prospective partners were identical, then
perhaps a market would form. (But that’s 45-975!)
Examples of assignment problems
We now explore an intermediate case. No two
prospective bargaining partners are alike, but matching
any two partners from either side of the market might
be more productive than not matching them at all.
For example:
1. How are a pool of MBA graduates assigned to
companies as employees?
2. Who gets tenure at at what university?
3. How are partners matched up across different
law firms?
4. How are partners paired for marriage and
parenting?
A multilateral bargaining game
We consider a bargaining game where a fraction of the
players, called publishers, offers royalties to another set
of players, called authors, to publish their manuscripts.
Each author has only one manuscript, so can can
accept at most one offer.
Each publisher can only handle one manuscript, but
can make multiple offers. If more than one offer is
accepted, the publisher may select any one of the
accepted offers
Valuing job matches
Publishers are not a perfect substitutes. Authors
are not identical either.
Each publisher and each author is assigned a
quality index, denoted respectively by pi and aj for
the ith publisher and jth author.
In this lecture we suppose the value of forming a
match between the ith publisher and jth author,
denoted vij is the product of the two index values of
publisher and the author. That is vij = pi aj
The solution
One can show that the best publishers
match up with the best authors. This is
called positive sorting.
It arises because the quality of
manuscripts and the reputation of
publishers are production compliments.
The royalty rate to authors increases with
their index. Likewise the net profits to
publishers is increasing in their index.
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.
Next week we explore the implications of
relaxing these two assumptions.