Week 5
Coordination and Reputation
How are reputations established? In this lecture we
explore three ways, through small changes in the
payoffs (such as limited warranties), affecting the
information set (such as monitoring), and through the
mutual selection of the equilibrium strategy (when
there is more than one). A pure strategy Nash
equilibrium can be interpreted as a self enforcing
agreement. When there is more than one, a natural
question to ask is which pure strategy equilibrium, if
any, will be played.
How many Nash equilibriums are there?
A Nash equilibrium solution to a game
can be found by writing down its strategic
form.
We have already noted that every game
has at least one Nash equilibrium.
However some games have more than
one equilibrium.
The threat of bankruptcy
We consider an industry with weak board of
directors, an organized workforce and an
entrenched management.
Workers and management simultaneously make
demands on the firms resources.
If the sum of their demands is less than or equal
to the total resources of the firm, shareholders
receive the residual.
If the sum exceeds the firm’s total resources, then
the firm is bankrupted by industrial action.
Strategic form of bargaining game
To achieve a bigger share
of the gains from trade,
both sides court
disastrous consequences.
This is sometimes called a
game of chicken, or
attrition.
Extensive form of bargaining game
Best responses illustrated
Multiple pure strategy Nash equilibrium
In this game, there are three pairs of mutual best
responses.
The parties coordinate on an allocation of the pie
without excess demands. Shareholders get nothing.
But any of the three allocations is an equilibrium.
If labor and management do not coordinate on one
of the equilibrium, the firm will bankrupt or
shareholders will receive a dividend.
Quality control
Manufacturers do not consistently produce flawless
products despite legions of consultants who have
advised them against this policy.
Retailers help guard against flawed products by
returning some of the defective items sent, and
lending their brand to the ones they retail.
Consumers cannot judge product quality as well as
retailers and producers, since each one experiences
only a tiny fraction of the end product.
What is an acceptable defect rate, how often should
retailers return defective items, and what are the
implications for consumer demand?
Total quality management
TQM in strategic form
There are two strategies for each player. Having
derived the strategic form of the game, we can easily
locate the pure strategy Nash equilibriums.
There is a unique pure strategy Nash equilibrium, in
which the producer only manufactures flawless
products, the retailer only sells flawless products and
the customer always buys the product.
Why is the pure strategy
Nash equilibrium unconvincing?
But is this Nash equilibrium convincing?
Sure you can’t eliminate any dominated
strategies.
If, however, the producer does manufacture a
defective item, the retailer, but not the
consumer will know, and makes more by
offering the item for sale.
Can the retailer convince consumers that they
really will return defective products?
Is there a mixed strategy equilibrium?
Let q denote the probability that the retailer offers a
defective product item sale.
Let r denote the probability the customer buys the item.
Let p be the probability of producing a flawless item.
Solving for r,
the probability of buying
If 0 < q < 1, then the retailer is indifferent between
offering a defective product and returning it.
In that case:
3r - 2(1 - r) = -1
⇒ 3r – 2 + 2r = -1
⇒ 5r = 1
⇒ r = 0.2
How to solve for p and q
Once we substitute for r = 0.2
in the shopper’s decision, we
are left with the diagram:
1. q is chosen so that the
producer is indifferent
between production methods;
2. p is chosen so that the
shopper is indifferent between
buying and not buying.
Solving q,
the probability of offering the product
The producer will only mix between defective and
flawless items if the benefit from both are equated:
[6r + (1 - r)]q - 3(1 - q) = [3r + (1- r)]
⇒ 2q – 3 + 3q = 1.4
⇒ 5q = 4.4
⇒ q = 0.88
Solving for p, the probability of
producing a flawless product
Investigating the cases above shows that in a mixed
strategy equilibrium r = 0.2 and q = 0.88.
Since the shopper is indifferent between buying the item
versus leaving it on the shelf, there are no expected
benefits of acquiring the item:
9p - 10(1 - p)q = 0 ⇒ (9 +10q)p = 10q
⇒ p = 44/89
Offering a partial refund
We now modify
the game
slightly. If the
customer buys
a defective
product, she
receives partial
compensation.
A different outcome
In this case the manufacturer has a weakly
dominant strategy of specializing in the
production of flawless goods.
Recognizing this, the shopper picks a pure
strategy of buying.
Realizing that the shopper will buy
everything she is offered, the retailer never
returns its merchandise to the
manufacturer (and indeed there is never
any reason too).
Light rail
Alstom, a French company, and Bombardier, a
Canadian company based in Quebec, are the
world’s largest producers of light rail systems.
They frequently compete against each other for
contracts from local governments and airport
authorities.
This industry is characterized by flurries of
contracts interspersed with relatively lean periods.
For this reason we treat each flurry as a known
number of rounds that occur independently of the
last flurry.
Bidding for light rail contracts
The company
charging the lowest
price wins.
If both companies
tender the same
price, they have the
same probability of
winning the contract.
The payoff matrix
illustrates such a
configuration.
The last round in a finite horizon game
Consider the last
round in a
typical flurry.
The dominant
strategy for
each producer is
to cut is price.
This is an
example of the
prisoner’s
dilemma.
The reduced subgame
starting at second last round
Folding back, the strategic
form of the reduced game
starting at the penultimate
round is depicted.
It is obtained by adding (2,2),
the solution payoffs for the
final auction, to each cell.
The dominant strategy of
cutting price is not affected by
this additive transformation.
The reduced game
at the beginning of the first round
Using an induction
argument we can prove
that in the first round,
the expected revenue
each firm will get from
the remaining N –1
tenders is 2(N – 1).
Again the dominance
principle applies, and
both firms cut price in
their first tender.
Solution
The preceding discussion proves the
unique solution is to always cut the price
in this repeated game.
The reason we obtain a tight
characterization of the solution to the
repeated game is that the solution to the
kernel game is unique.
Indeed if a game has a unique solution,
then repeating the game a finite number
of times will simply replicate the solution
to the original kernel game.
Multiple equilibriums
There is no role for coordination and leadership
in situations where the solutions strategies for
each player are uniquely defined.
Thus opportunities for coordination and
leadership arise when there are several
solutions to a game, which we may describe as
self enforcing contracts.
In this case not all the solutions to the overall
game can be found by merely piecing together
the solutions of the kernel games.
Repeated games
Multiplicity is the existence of multiple solutions
within a game (such as a signed contract that still
leaves the bargaining parties discretion about its
implementation)
It sometimes arises when there are ongoing
benefits from continuing a relationship and/or
potential for repeated trade.
If the solutions to all the kernels forming a finite
stage game are unique, then the unique solution to
the stage game is to play those kernel solutions.
In these cases there is no scope for either
leadership or reputation.
An Infinite Horizon Extension
But what if this game did not end at a fixed
point in time?
Consider the following “implicit” agreement
between the two firms:
If neither of us cheat on each other from
now on by cutting price, then we will
continue to hold firm and collect (3,3)
each period.
If either of us ever cheat even once, then
from then on we will always cut price.
This is called a trigger strategy.
When are trigger strategies self enforcing?
The benefit from following this strategy is the
discounted sum of receiving 3 per period.
The discounted sum of breaking the agreement is
receiving 4 in the first period and 2 from the next
period onwards.
The net benefit from breaking the agreement is
therefore the gross of 1 received now, less the cost of
1 unit paid each period from next period onwards.
If the interest rate is r, then the net benefit is
1 – 1/r .
Unless the interest rate exceeds 100 percent the trigger
strategy is self enforcing in this case.
What is a reputation?
In this case not all the solutions to the
overall game can be found by merely piecing
together the solutions of the kernel games.
Dynamic strategies that preserve long term
incentives and cooperation with appropriate
rewards and penalties are, under the right
circumstances, more lucrative than the
outcomes realized from players choosing
say, dominant strategies each period.
Summary
There is no role for coordination in
situations where the solutions strategies
for each player are uniquely defined.
However small changes in the payoffs
induced by product guarantees and
quality verification can bring about large
changes in solution outcomes that are
associated with reputation.
Opportunities for coordination also arise
when there are several solutions to a
game, which we may describe as self
enforcing contracts.