Chapter 5
Game Theory
0 Game theory applied to economics by John Von
Neuman and Oskar Morgenstern
0 Game theory allows us to analyze different social and
economic situations
Games of Strategy Defined
0 Interaction between agents can be represented by a game,
when the rewards to each depends on his actions as well as
those of the other player
0 A game is comprised of
0 Number of players
0 Order of play
0 strategies
0 Chance
0 Information
0 Payoffs
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Example 1:Prisoners’
Dilemma
0 Two people committed a crime and are being interrogated
separately.
0 The are offered the following terms:
0 If both confessed, each spends 8 years in jail.
0 If both remained silent, each spends 1 year in jail.
0 If only one confessed, he will be set free while the other
spends 20 years in jail.
Example 1: Prisoners’
Dilemma
Prisoner 2
Confess
Prisoner 1 Silent
0
0
0
0
confess
silent
8, 8
0, 20
20, 0
1, 1
Numbers represent years in jail
Each has a dominant strategy to confess
Silent is a dominated strategy
Nash equilibrium: Confess Confess
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Example 2: Matching Pennies
Player 2
Player 1
Heads
Tails
Heads
- 1, +1
+1 - 1
Tails
+1 - 1
- 1, +1
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Example 3: Oligopoly Game
General Motors
Ford
High price
Low price
High price
500, 500
100, 700
Low price
700, 100
300, 300
0 Similarly for GM
0 The Nash equilibrium is Price low, Price low
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Game Types
0 Game of perfect information
0 Player – knows prior choices
0 All other players
0 Game of imperfect information
0 Player – doesn’t know prior choices
8
Representing Games
0 The previous examples are of
0 Simultaneous games
0 Games of imperfect information
Games can be represented visually in
0 Bi- matrix form
0 Table
0 Dimensions depend on the number of strategies
0 Game tree
0 Extensive form game
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Matching Pennies
Game of imperfect information
Represented in bi-matrix form
Player 2
Player 1
Heads
Tails
Heads
- 1, +1
+1 - 1
Tails
+1 - 1
- 1, +1
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Extensive form of the game of
matching pennies
Child 1
Heads
Tails
Child 2
Heads
-1
+1
Child 2
Tails
+1
-1
Heads
Tails
+1
-1
-1
+1
Child 2 does not know whether child 1 chose heads or tails.
Therefore, child 2’s information set contains two nodes.
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Strategy
0 A player’s strategy is a plan of action for each of the
other player’s possible actions
Game of perfect information
In extensive form
IBM
1
UNIX
DOS
Toshiba
DOS
600
200
2
3
UNIX
100
100
DOS
100
100
Toshiba
UNIX
200
600
Player 2 (Toshiba) knows whether player 1 (IBM) moved to the left or to the
right. Therefore, player 2 knows at which of two nodes it is located
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Strategies
0 IBM:
0 DOS or UNIX
0 Toshiba
0 DOS if DOS and UNIX if UNIX
0 UNIX if DOS and DOS if UNIX
0 DOS if DOS and DOS if UNIX
0 UNIX if DOS and UNIX if UNIX
Game of perfect information
In normal form
Toshiba
IBM
(DOS | DOS,
DOS | UNIX)
(DOS | DOS,
UNIX | UNIX)
(UNIX | DOS,
UNIX | UNIX)
(UNIX | DOS,
DOS | UNIX)
DOS
600, 200
600, 200
100, 100
100, 100
UNIX
100, 100
200, 600
200, 600
100, 100
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Game of imperfect information
0 Assume instead Toshiba doesn’t know what IBM
chooses
0 The two firms move at the same time
0 Imperfect information
0 Need to modify the game accordingly
Game of imperfect information
In extensive form
IBM
1
UNIX
DOS
Toshiba
DOS
600
200
Information set
2
3
UNIX
100
100
DOS
100
100
Toshiba’s strategies:
• DOS
• UNIX
Toshiba
UNIX
200
600
Toshiba does not know whether IBM moved to the left or to the
right, i.e., whether it is located at node 2 or node 3.
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Game of imperfect information
In normal form
Toshiba
IBM
DOS
UNIX
DOS
600, 200
100, 100
UNIX
100, 100
200, 600
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Equilibrium for Games
Nash Equilibrium
0 Equilibrium
0 state/ outcome
0 Set of strategies
0 Players – don’t want to change behavior
0 Given - behavior of other players
0 Noncooperative games
0 No possibility of communication or binding
commitments
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Nash Equilibria
s*  ( s1* ,..., sn* ) - is a Nash equilibriu m
If i ( s1* ,..., si* ,..., sn* )  i ( s1* ,..., sˆi ,..., sn* )
for all sˆi in Si
s*  ( s1* ,..., sn* ) - array of strategy choices
s*i  strategy choice of player i
*
*
i ( s1 ,..., sn )  payoff
to player i when s * is chosen
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Nash Equilibrium: Toshiba-IBM
imperfect Info game
Toshiba
IBM
DOS
UNIX
DOS
600, 200
100, 100
UNIX
100, 100
200, 600
The strategy pair DOS DOS is a Nash equilibrium. Are there any other equilibria?
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Dominant Strategy Equilibria
0 Strategy A dominates strategy B if
0 A gives a higher payoff than B
0 No matter what opposing players do
0 Dominant strategy
0 Best for a player
0 No matter what opposing players do
0 Dominant-strategy equilibrium
0 All players - dominant strategies
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Oligopoly Game
General Motors
Ford
High price
Low price
High price
500, 500
100, 700
Low price
700, 100
300, 300
0 Ford has a dominant strategy to price low
0 If GM prices high, Ford is better of pricing low
0 If GM prices low, Ford is better of pricing low
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Oligopoly Game
General Motors
Ford
High price
Low price
High price
500, 500
100, 700
Low price
700, 100
300, 300
0 Similarly for GM
0 The Nash equilibrium is Price low, Price low
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Prisoners’ Dilemma
Prisoner 2
Confess
Prisoner 1 Silent
0
0
0
0
confess
silent
8, 8
0, 20
20, 0
1, 1
Numbers represent years in jail
Each has a dominant strategy to confess
Silent is a dominated strategy
Nash equilibrium: Confess Confess
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Prisoners’ Dilemma
0 Each player has a dominant strategy
0 Equilibrium is Pareto dominated
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Elimination of Dominated
Strategies
0 Dominated strategy
0 Strategy dominated by another strategy
0 We can solve games by eliminating dominated
strategies
0 If elimination of dominated strategies results in a
unique outcome, the game is said to be dominance
solvable
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(a) Eliminating dominated strategies
Player 2
Player 1
1
2
3
1
2, 0
2, 4
0, 2
2
0, 6
0, 2
4, 0
(b) One step of elimination
Player 2
Player 1
1
2
1
2, 0
2, 4
2
0, 6
0, 2
(c ) Two steps of elimination
Player 2
Player 1
1
1
2
2, 0
2, 4
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(a) Eliminated dominated strategies
Player 2
Player 1
1
2
3
1
20, 0
10, 1
4, -4
2
20, 2
10, 0
2, -2
(b) Reduced game eliminating column 3 first
Player 2
Player 1
1
2
1
20, 0
10, 1
2
20, 2
10, 0
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Games with Many Equilibria
0 Coordination game
0 Players - common interest: equilibrium
0 For multiple equilibria
0 Preferences - differ
0 At equilibrium: players - no change
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Games with Many Equilibria
Toshiba
IBM
DOS
UNIX
DOS
600, 200
100, 100
UNIX
100, 100
200, 600
The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX
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Normal Form of Matching Numbers:
coordination game with ten Nash equilibria
Player 2
Player 1
1
2
3
4
5
6
7
8
9
10
1
1, 1
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
2
0, 0
2, 2
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
3
0, 0
0, 0
3, 3
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
4
0, 0
0, 0
0, 0
4, 4
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
5
0, 0
0, 0
0, 0
0, 0
5, 5
0, 0
0, 0
0, 0
0, 0
0, 0
6
0, 0
0, 0
0, 0
0, 0
0, 0
6, 6
0, 0
0, 0
0, 0
0, 0
7
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
7, 7
0, 0
0, 0
0, 0
8
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
8, 8
0, 0
0, 0
9
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
9, 9
0, 0
10
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
0, 0
10, 10
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Table 11.12
A game with no equilibria in pure strategies
General 2
Retreat
General 1
Attack
Retreat
5, 8
6, 6
Attack
8, 0
2, 3
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The “I Want to Be Like Mike”
Game
Dave
Wear red
Michael
Wear blue
Wear red
(-1, 2)
(2, -2)
Wear blue
(1, -1)
(-2, 1)
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Credible Threats
0 An equilibrium refinement:
0 Analyzing games in normal form may result in equilibria
that are less satisfactory
0 These equilibria are supported by a non credible threat
0 They can be eliminated by solving the game in extensive
form using backward induction
0 This approach gives us an equilibrium that involve a
credible threat
0 We refer to this equilibrium as a sub-game perfect Nash
equilibrium.
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Non credible threats: IBM-Toshiba
In normal form
Toshiba
IBM
(DOS | DOS,
DOS | UNIX)
(DOS | DOS,
UNIX | UNIX)
(UNIX | DOS,
UNIX | UNIX)
(UNIX | DOS,
DOS | UNIX)
DOS
600, 200
600, 200
100, 100
100, 100
UNIX
100, 100
200, 600
200, 600
100, 100
0 Three Nash equilibria
0 Some involve non credible threats.
0 Example IBM playing UNIX and Toshiba playing UNIX
regardless:
0 Toshiba’s threat is non credible
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Backward induction
IBM
1
UNIX
DOS
Toshiba
DOS
600
200
2
3
UNIX
100
100
DOS
100
100
Toshiba
UNIX
200
600
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Subgame perfect Nash
Equilibrium
0 Subgame perfect Nash equilibrium is
0 IBM: DOS
0 Toshiba: if DOS play DOS and if UNIX play UNIX
0 Toshiba’s threat is credible
0 In the interest of Toshiba to execute its threat
Rotten kid game
0 The kid either goes to Aunt Sophie’s house or refuses
to go
0 If the kid refuses, the parent has to decide whether to
punish him or relent
Player 2 (a parent)
Player 1
(a difficult
child)
Left
(go to Aunt Sophie’s House)
Right
(refuse to go to Aunt Sophie’s
House)
(punish if the
kid refuses)
(relent if the
kid refuses)
1, 1
1, 1
-1, -1
2, 0
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Rotten kid game in extensive
form
Kid
1
Refuse
Go to Aunt
Sophie’s House
2
1
1
Punish if
refuse
-1
-1
Parent
Relent if
refuse
2
0
• The sub game perfect Nash equilibrium is: Refuse and Relent if refuse
• The other Nash equilibrium, Go and Punish if refuse, relies on a non
credible threat by the parent
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