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Intermediate Microeconomics
Game Theory
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.
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Applications:
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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
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a set of strategies for each player
 i.e. actions to be performed given any observed state of the world
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the payoffs to each player for every possible choice of actions by that
player and all the other players.
Simultaneous Move Games
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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?
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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?
Simultaneous Move Games
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“Prisoners’ Dilemma”
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Payoffs to Player-1;
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Consider a game with 2 players, each player has two options:
 Keep quiet (Cooperate with each other)
 Blame partner (Defect against the other)
Cooperate utility of -3 if Player 2 also cooperates, utility of -8 if Player
2 defects.
Defect utility of -5 if Player 2 also defects, utility of -1 if Player-2
cooperates.
Payoffs to Player-2 are analogous.
Players are in different rooms, so they can’t communicate nor know
what their partner does when they make their decision.
Simultaneous Move Games
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One way to summarize the payoffs associated with each
action is to use a payoff matrix.
Player-2
C
C
Player-1
D
(-3,-3)
(-8,-1)
(-1,-8)
(-5,-5)
D
Payoff for Player-1 (row player) shown first, followed by
payoff for Player-2 (column player)
Simultaneous Move Games
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So how should we think of how to model the outcome of such
games?
Player-2
C
C
(-3,-3)
(-8,-1)
D
(-1,-8)
(-5,-5)
Player-1

D
What is Player-1’s best action to take if Player-2 were to keep quiet
(cooperate)?

What is Player-1’s best action if Player-2 were to blame (defect)?

How about for Player-2?

So what do think each Player will do?
Simultaneous Move Games
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Dominant Strategy - A strategy that gives higher utility than all
other strategies given any actions taken by other players.
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Does a dominant strategy always exist?
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Meeting time for dinner?
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Wearing a costume to a Halloween party?
Simultaneous Move Games
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“Arms Race” - Consider a game of the following form:
Player-2
ignore
ignore
Player-1
attack
attack
(0,0)
(-4,-1)
(-1,-4)
(-3,-3)
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What is Player-1’s best action to take if Player-2 chooses ignore?
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What is Player-1’s best action to take if Player-2 chooses attack?
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How about for Player-2?
Simultaneous Move Games
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“Battle of the Sexes” - Consider a game of the following form:
Player-2
Italian
Italian
Player-1
Steak
Steak
(5,2)
(1,1)
(0,0)
(2,5)
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What is Player-1’s best action to take if Player-2 chooses Italian?
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What is Player-1’s best action to take if Player-2 chooses Steak?
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How about for Player-2?
Nash Equilibrium
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Nash Equilibrium – A set of actions such that each person’s action
is (privately) optimal given the actions of others.
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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?”
Simultaneous Move Games
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Nash Equilibria of “Prisoner’s Dilemma”?
Player-2
C
D
C
(-3,-3)
(-8,-1)
D
(-1,-8)
(-5,-5)
Player-1
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Both Cooperate?
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One Cooperate, other Defect?
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Both Defect?
Simultaneous Move Games
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Nash Equilibria of Arms Race?
Player-2
I
Player-1
A
I
A
(0,0)
(-4,-1)
(-1,-4)
(-3,-3)
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Both Ignore?
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One Ignore, other Attack?
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Both Attack?
Simultaneous Move Games
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Nash Equilibria from “Battle of the Sexes” ?
Player-2
Italian
Italian
Player-1
Steak
Steak
(5,2)
(1,1)
(0,0)
(2,5)
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Both Italian?
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Both Steak?
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One Italian, one steak?
Nash Equilibria
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Three things to notice:
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Playing Dominant Strategies are always a Nash Equilibrium
(e.g. Prisoner’s Dilemma).
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Nash Equilibria do not have to be Pareto Efficient (e.g.
Prisoner’s Dilemma and Arms Race).
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There can be multiple equilibria that can be Pareto ranked--everyone is better off in one than the other---(e.g. Arms Race),
or where they cannot be Pareto ranked---players may not agree
on which equilibria is the “best”---(e.g. Battle of Sexes)
Applications of these types of games?
Game Theory Application: Trade

Suppose if Acme produces widgets domestically, its profits will only be
$100K and China Corp. will have profits of $0.
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Suppose Acme Corp. could make a deal with China Corp. to produce
widgets abroad.
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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 substandard 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.
Should trade happen? Will trade happen?
Game Theory Application: Trade
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What is the key problem that leads to inefficiency?
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How could this problem be overcome?
Game Theory: Sequential Games
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One particular way of committing is to act first.
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When there is an ordering to actions we call this a sequential game.
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Equilibria of interest are now what are referred to as Subgame Perfect
Nash Equilibria (SPNE).
 Essentially, each player does not have any incentive to deviate from
his SPNE strategy given other players play SPNE strategy in each
subgame.
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What does this mean? Consider the following examples.
Sequential Move Games
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Consider the Prisoner’s Dilemma game
Player-2
C
Player-1
D
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C
D
(5,5)
(2,8)
(8,2)
(3,3)
But, suppose Player 1 can move first, and
Player 2 can observe what he did.
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“Extensive form” Game Tree?
Sub-game Perfect Nash Equilibria?
 A strategy is needed for each possible subgame.
Sequential Move Games
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Consider the Arms Race game
Player-2
I
Player-1
A
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I
A
(0,0)
(-4,-1)
(-1,-4)
(-3,-3)
But, suppose Player 1 can move first, and
Player 2 can observe what he did.
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“Extensive form” Game Tree?
Sub-game Perfect Nash Equilibria?
Sequential Move Games
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Consider the “Battle of the Sexes” game
Player-2
Italian
Italian
Player-1
Steak
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Steak
(5,2)
(1,1)
(0,0)
(2,5)
But, suppose Player 1 can move first, and
Player 2 can observe what he did.
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“Extensive form” Game Tree?
Sub-game Perfect Nash Equilibria?
Sequential Move Games
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Does Game theory work in the “Real-world”?
http://www.businessinsider.com/golden-balls-game-theory-2012-4