Lecture 4B
Nash Equilibrium
This lecture introduces the
concept of Nash equilibrium.
Incentives in the Workplace
Consider a firm that sells output jointly produced
by a design team and a manufacturing team.
The quality of the output determines the price for
which it can be sold.
To keep matters simple, we assume that for each
extra unit of effort undertaken by either team, up
to 10 units, sales rise by $1.5million per unit.
Any input beyond 10 within either team is wasted
effort, producing no marginal increase in sales.
Cost of Effort
It costs $1million per unit of effort in either team, in
terms of lost sleep, hiring new staff, and buying new
equipment (plant) and materials.
Effort is not observed by the management. It is easy
for each team to disguise sloth and inattention, thus
confusing management about what needs to be
done. The managers realize that.
The managers pitch their plan
To compensate design and manufacturing,
management institutes a profit sharing plan, whereby
production and manufacturing each get one-third of
the sales as compensation.
Furthermore if the firm reaches its sales target of $30
million, it will also distribute a bonus of $100,000
dollars to both teams.
Potential benefits
From the perspective of the firm as a whole,
each unit of effort up to 10 taken by design and
manufacturing costs $20 million and has a
return of $30 million.
If both teams follow the managers directives,
they would earn a net $100,000 each.
This way everyone in the firm benefits,
management and shareholders most of all.
Strategic form representation
The design team chooses its effort
The design team determines its level of effort not
knowing the choice of the production team.
What are its profits if the design team chooses effort
e1 < 10 and manufacturing chooses e2?
Profit1 = Profit share – Cost of effort
Profit1 = (1/3)(1.5e1 + 1.5e2) – e1
Finding the effort level continued
Profit1 = (1/3)(1.5e1 + 1.5e2) – e1
Regardless of e2, Profit1 is decreasing in e1. Therefore
any choice with 10 > e1 > 0 is dominated by e1 = 0.
We only have to check whether both teams
expending effort of 10 is rational. But if one team
works that hard, the other team makes more profit
by spending no effort at all.
Hence design exerts no special effort despite the
profit sharing incentives. The situation for
manufacturing is analogous. Spending no effort is a
dominant strategy.
A large bonus only
Recall that if production and design were efficient,
profits would be $30m and the profit share gave
away 2/3rds of this amount or $20m plus the $200,00
bonus.
Instead, suppose the firm pays each team
compensation of $10 million plus a $50,000 bonus
each if they reach the profit target of $30m.
Otherwise they get nothing.
This is less generous than the profit sharing scheme.
Strategic form representation with
the new bonus
Analysis of the new incentive scheme
Suppose that design expects manufacturing to work
at full capacity to meet the target.
To receive the bonus, design has to work at full
capacity too.
If it doesn’t work at full capacity, the production
goal is not reached, and design incurs losses for
every unit of effort expended. Every effort less than
full capacity is dominated by not working.
Therefore design works at full capacity or not at all.
The teams choose
Manufacturing also faces the same choice.
If they both work at full capacity, profits equal
the bonus less the cost of effort, which nets
design $50,000.
Thus, it is better to work at full capacity than
not at all, providing the other team does too.
Management should stress this point.
We conclude the structure of incentive
schemes (as well as the total amount) can
have a big effect on behavior within groups.
Introduction to Nash equilibrium
The concept of equilibrium is that each
player's behavior can be viewed as the
outcome from him optimizing an
individual objective function that is partly
defined by the solutions of optimization
problems solved by the other players.
Read Chapters 9 and 10 of Strategic Play.
Motivation for best replies
After eliminating those strategies which seem
implausible because they are dominated, or
iteratively dominated, we are sometimes still left with
many possible strategic profiles.
In these cases we must impose more stringent
assumptions on how players behave to reach a
sharper prediction about the outcome of a game.
One approach is to argue that each player forms a
conjecture about the other players’ strategies, and
then maximizes his payoffs subject to this conjecture.
Defining a best reply
In order to develop this concept we first define
the notion of a best reply, which means a pure
or mixed strategy that maximizes the player's
payoff given a strategic profile of choices made
by the other players in the game.
One approach a player might consider to
selecting a strategy is to determine which
strategy maximizes her payoff given the
strategies selected by the other players.
This is called her best reply.
Competition through integration
In this game, a
specialized producer of
components for a durable
good has the option of
integrating all the way
down to forming
dealership, but indirectly
faces competition from a
retailer, which markets
similar final products,
possibly including the
supplier’s.
Supplier
The supplier’s profits are
higher if the retailer only
distributes.
The supplier makes higher
profits by undertaking more
downstream integration if the
retailer integrates upstream, to
avoid being squeezed.
If the retailer confines itself to
distribution, then the best
reply of the supplier is to focus
on its core competency, and
only produce component parts.
Retailer
The profits of the retailer on
this item fall the more
integrated is the supplier.
If the supplier assembles but
does not distribute, then the
retailer should also integrate
upstream, and assemble too.
Otherwise the most profitable
course of action of the retailer
is to focus on distribution.
Nash equilibrium for integration game
The Nash equilibrium is Up (make components) and
Left (distribute only).
To verify this claim note that if the retailer chooses
to distribute only by moving Left, then the best
response of the supplier is to only make
components, by moving Up.
Similarly if the supplier only makes components,
moves to Up, then the best response of the retailer
is to specialize in distribution, moving Left.
By inspecting the other cells one can check there is
no other Nash equilibrium in this game.
A chain of conjectures
The Nash equilibrium is supported by a chain of conjectures
each player might hold about the other. Suppose:
S, the Supplier, thinks that R, the Retailer, plays Left
=> S plays Up;
R thinks that S thinks that R plays Left
=> R thinks that S plays Up
=> R plays L;
S thinks that R thinks that S thinks that R plays Left
=> S thinks that R thinks that S plays Up
=> S thinks that R plays Left
=> S plays Up;
and so on.
Definition of Nash equilibrium
Consider an N player game, where sn is the strategy of
the nth player where 16 n6 N.
Also write s-n for the strategy of every player apart from
player n. That is:
s-n = (s1, . . ., sn-1, sn+1, . . ., sN).
Now ask whether sn is the best response of n to s-n for
all n? If so, nobody has an incentive to unilaterally
deviate from their assigned strategy.
This is called a Nash equilibrium.
Product differentiation
Reduced form of Product
differentiation
The low quality
producer’s strategy of
“reduce price of existing
model” is dominated by
any proper mixture of
the other two strategies.
One we eliminate that
strategy we are left with
a two by two matrix.
Best reply in product differentiation
game – high quality producer
With regards the high quality producer in the
product differentiation game:
1. The best reply to the low quality producer
introducing a new product is to hold a sale.
2. The best reply to the low quality producer
upgrading an existing model is to do the
same thing.
Best reply in product differentiation
game – low quality producer
With regards the low quality producer:
1. The best reply to the high quality producer
introducing a new product is to upgrading the
existing model.
2. The best reply to the high quality producer
upgrading the existing model is to introduce a
new model.
3. The best reply to the high quality producer
holding a sale is to introduce a new model.
Best reply illustrated in the
strategic form
Five Rules
Rule 1: Look ahead and reason back.
Rule 2: If there is a dominant strategy, play it.
Rule 3: Discard dominated strategies.
Rule 4: Iteratively eliminate dominated strategies.
Rule 5: If there is a unique Nash equilibrium, then play
your own Nash equilibrium strategy.