Intermediate Microeconomics Game Theory and Oligopoly

Intermediate Microeconomics

Game Theory and Oligopoly

Game Theory

So far we have only studied situations that were not “strategic”.

The optimal behavior of any given individual or firm did not depend on what other individuals or firms did. E.g.

An individual buys something if its price is less than his willingness to pay.

A firm enters a market if there are positive economic profits to be made at going prices.

Obviously, we might want to expand this. For example, what happens when firms recognize how price will be

affected by their behavior (i.e., not price takers)? Or when one firm or person’s optimal behavior depends on what

another firm or person does.

Game Theory

Game theory helps to model strategic behavior --- or interactions where what is optimal for a given agent depends on what actions are taken by another agent and vice versa.

Applications: The study of oligopolies (industries containing only a few firms) The study of externalities and public goods; e.g. using a common

resource such as a fishery. The study of military strategies. Bargaining. How markets work. Behavior in the courts. Behavior of news media. Crime.

What is a Game?

A game consists of: a set of players

a set of strategies for each player i.e. actions to be performed given any observed state of the world

the payoffs to each player for every possible choice of actions by that player and all the other players.

Simultaneous Move Games

Consider games where players must choose an action without knowing what the other players have chosen. Does a defendant agree to testify against his co-defendants when he

doesn’t know whether or not his co-defendants are going to do the same?

How much should a firm bid for a given item in a silent auction?

Should I act friendly or defensively when I encounter a stranger on an empty street late at night?

How do we model the outcomes in these types of games?


“Prisoners’ Dilemma” Consider a game with 2 players, each player has two options:

Keep quiet (“cooperate” with each other), or talk to Police (“defect”)

Payoffs to Player-1; Cooperate utility of 5 if Player-2 cooperates, utility of 2 if Player-2

defects. Defect utility of 8 if Player-2 cooperates, utility of 3 if Player-2 defects.

Payoffs to Player-2 are analogous.


One way to summarize the payoffs associated with each action is to use a payoff matrix.

Payoff for Player-1 (row player) shown first, followed by payoff for Player-2 (column player)

(5,5) (2,8)

(8,2) (3,3)

Player-2

C D

C

D

Player-1


So how should we think of how to model the outcome of such games?

What is Player-1’s best action to take if Player-2 Cooperates?

What is Player-1’s best action if Player-2 Defects?

How about for Player-2?

So what do think each Player will do?

(5,5) (2,8)

(8,2) (3,3)

Player-2

C D

Player-1C

D


Dominant Strategy - A strategy that gives higher utility than all other strategies given any actions taken by other players.

Does a dominant strategy always exist? Meeting time for dinner?

Wearing a costume to a Halloween party?


“Arms Race” - Consider a game of the following form:

What is Player-1’s best action to take if Player-2 chooses ignore?

What is Player-1’s best action to take if Player-2 chooses attack?

How about for Player-2?

(0,0) (-4,-1)

(-1,-4) (-3,-3)

Player-2

ignore attack

Player-1ignore

attack

Nash Equilibrium

Nash Equilibrium – A set of actions such that each person’s action is (privately) optimal given the actions of others.

Key to a Nash Equilibrium: No person has an incentive to deviate from his Nash equilibrium

action given everyone else behaves according to their Nash equilibrium action.

Nash Equilibrium in “A Beautiful Mind?”


Nash Equilibria of “Prisoner’s Dilemma”?

Both Cooperate?

One Cooperate, other Defect?

Both Defect?

(5,5) (2,8)

(8,2) (3,3)

Player-2

C D

Player-1C

D


Nash Equilibria of Arms Race?

Both Ignore?

One Ignore, other Attack?

Both Attack?

(0,0) (-4,-1)

(-1,-4) (-3,-3)

Player-2

I A

Player-1I

A

Nash Equilibria

Three things to notice: Playing Dominant Strategies are always a Nash Equilibrium

(e.g. Prisoner’s Dilemma).

Nash Equilibria do not have to be Pareto Efficient (e.g. Prisoner’s Dilemma and Arms Race).

There can be multiple equilibria that often that can be Pareto ranked (e.g. Arms Race).

Applications of these types of games?

Game Theory Application: Trade

Suppose Acme Corp. could make a deal with China Corp. to produce widgets abroad. If both stick with the deal (i.e. China Corp. produces quality widgets and Acme

Corp. pays China Corp. the agreed upon fee), Acme’s profits will be $200K while China Corp’s profits will be $50K.

If Acme cheats and pays less than the agreed upon rate after delivery, Acme has profits of $250K and China Corp. ends up losing $50K.

Alternatively, if Acme acts honestly, but China Corp. cheats and produces sub-standard widgets, Acme Corp.’s profits will only be $50K, but China Corp.’s profits will be $90K.

If both act dishonestly, Acme will make only $75K while China Corp. will lose $20K.

If Acme produces widgets domestically, its profits will only be $100K and China Corp. will have profits of $0.

Should trade happen? Will trade happen?

Game Theory Application: Trade

What is the key problem that leads to inefficiency?

How could this problem be overcome?

Continuous Actions

What if players could choose among a continuum of actions?

The standard way to handle such situation is to use Reaction Functions. Reaction function – a function that maps any possible action by Player

b into optimal action for player a.

A Nash Equilibrium will arise at the point where Reaction functions intersect.

Continuous Actions: Cournot Equilibrium

To see the role played by reaction functions we can look at an Oligopoly setting.

So far, we have examined 2 types of market structure.1. Markets where each supplier was small enough that its decision

regarding how much to supply had no effect on price (competition)

2. Markets where there was only one supplier, so its decision regarding how much to supply fully determined price (Monopoly)

What happens with “a few” suppliers who don’t collude (i.e. and oligopoly)?


Cournot Equilibrium Consider a market with two firms, each with cost function equal to

C(q) = 4q

Suppose the market (inverse) demand function is

p(Q) = 84 – 8(Q) where Q = q1 + q2

How do we find optimal quantity supplied by each firm?


So each firm’s reaction function will be:

So what will be Nash Equilibrium?

Will it be efficient?

otherwise

qifqq

0

102/5 221

otherwise

qifqq

0

102/5 112

q2

5

10

q1

5

10

Firm 1’s reaction fn

Firm 2’s reaction fn


What happens as the number of firms increases?

Firm 1 : max p(Q)q1 – C(q1)

FOC:

So what happens when s1 = 1 (i.e. firm 1 is a monopolist)? How about when then number of firms becomes larger, so s1 gets smaller? What about when s1 goes to zero (i.e. firm 1 is an extremely small part of the overall market)?

)()(

)(1)( 1 QMC

Qp

q

Q

QpQp

0)()(

)( 1

QMCq

Q

QpQp

)()(

)(1)( 1 QMC

Q

q

Qp

Q

Q

QpQp

)(1

1)( 1 QMCsQp

divide both terms in brackets by p(Q)

multiply second term in brackets by Q/Q

(s1 = q1/Q or firm 1’s market share)

Continuous Actions: Bertrand Equilibrium

Is Cournot Equilibrium always a realistic model? How else might competition work? What would happen in this case?

Continuous Actions: Bertrand Equilibrium

Bertrand Equilibrium Instead of choosing quantity, firms choose price to sell at. Consider again two firms with C(q) = 4q who face an inverse market

demand function of p(Q) = 84 – 8(Q) or equivalently a market demand curve

Q(p) = 80.5 – p/8

What is the optimal strategy for firm 1 for any given action by firm 2?

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Intermediate Microeconomics Game Theory and Oligopoly