1
Nash-Equilibrium for Two-Person
Games
Chapter 3
2
Zero-sum Games and Constant-sum Games
Definition of zero-sum games Examples: Poker, Battle of the Networks
The arrow diagram for a 2×2 game in normal form
The arrows point towards a Nash equilibrium
Transforming a constant-sum game into a zero-sum game
3
Battle of the Networks, normal form: The payoff matrix
55%, 45%
50%, 50%
52%, 48%
45%, 55%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
4
Battle of the Networks, normal form: Strategy for Network 1
55%, 45%
50%, 50%
52%, 48%
45%, 55%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
Network 1’s Best Responses are underlined.
5
Battle of the Networks, normal form: Strategy for Network 2
55%, 45%
50%, 50%
52%, 48%
45%, 55%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
Network 2’s Best Responses are underlined.
6
Battle of the Networks, normal form: The Equilibrium
55%, 45%
50%, 50%
52%, 48%
45%, 55%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
All Best Responses are underlined.
7
Battle of the Networks, zero-sum form: The payoff matrix
10% , -10%
0, 0
4%, -4%
-10%, 10%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
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Battle of the Networks, zero-sum form: Strategy for Network 1
10%, -10%
0, 0
4%, -4%
-10%, 10%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
Network 1’s Best Responses are underlined.
9
Battle of the Networks, zero-sum form: Strategy for Network 2
10% , -10%
0, 0
4%, -4%
-10%, 10%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
Network 2’s Best Responses are underlined.
10
Battle of the Networks, zero-sum form: The equilibrium
10%, -10%
0, 0
4%, -4%
-10%, 10%
Sitcom Sports
Sitcom
Sports
Network 2
Network 1
All Best Responses are underlined.
11
Why look for Nash equilibrium?
Equilibrium conceptNo player has incentive to change strategy
Analogy to dominance solvabilityPlay of strictly dominant strategies leads to Nash equilibriumWeak dominance solution is also Nash equilibrium
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Nash equilibrium of a game like chess
1
L
R
2
(l, w)L
R (w, l)
(d, d)
a) Extensive Form
(w, l)(l, w)
(d, d) (d, d)
L
L
R
R1
2
b) Normal Form
13
Competitive Advantage
The game Competitive Advantage and the economic realities it reflectsThe solution of Competitive Advantage, showing why firms are driven to adopt new technologies
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Competitive Advantage: The payoff matrix
0, 0
-a, a
a, -a
0 , 0
Stay Put
New Technology
Stay Put
Firm 2
Firm 1New Technology
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Competitive Advantage: Strategy for Firm 1
0, 0
-a, a
a, -a
0 , 0
Stay Put
New Technology
Stay Put
Firm 2
Firm 1New Technology
16
Competitive Advantage: Strategy for Firm 2
0, 0
-a, a
a, -a
0 , 0
Stay Put
New Technology
Stay Put
Firm 2
Firm 1New Technology
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Competitive Advantage: The equilibrium
0, 0
-a, a
a, -a
0 , 0
Stay Put
New Technology
Stay Put
Firm 2
Firm 1New Technology
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1-Card Stud Poker
A model of a more complicated game, made simpler by fewer kinds of cards and smaller handsHow imperfect information is created by the deal of the cardsPrinciples of Poker reflected by the solution of 1-card StudThe concept of dominated strategy
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Stud Poker, the rules
1. Each player places 1 chip into the pot2. Each player is dealt 1 card, face-down
1. Deck has 50% Aces and 50% Kings2. No one looks at their card
3. Each player decides whether to make a further bet of one more chip or not. A player does this by placing 1 chip in hand behind his or her back (or not), out of sight of the other player.
4. Each player simultaneously reveals further bet.5. If only one player has made a further bet, then that player wins
the pot and no cards need be shown. 6. If both players have made a further bet of either 1 or 0 chips,
then there is a showdown. High card wins the pot; in case of a tie, the players split the pot.
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One-Card Stud Poker, extensive form
0 1 2
1/4
1/4
1/4
1/4
(A, A)
(A, K)
(K, A)
(K, K)
Bet
Pass
Bet
Bet
Bet
Pass
Pass
Pass
(0, 0)(a, -a)(-a, a)(0, 0)(a+b, -a-b)
(a, -a)(-a, a)(a, -a)(-a-b, a+b)(a, -a)(-a, a)(-a, a)(0, 0)(a, -a)(-a, a)(0, 0)
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One-Card Stud Poker, normal form: The payoff matrix
0, 0
-a, a
a, -a
0, 0
Pass
Bet
Pass
Bet
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One-Card Stud Poker, normal form: Strategy for player 1
0, 0
-a, a
a, -a
0, 0
Pass
Bet
Pass
Bet
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One-Card Stud Poker, normal form: Strategy for player 2
0, 0
-a, a
a, -a
0, 0
Pass
Bet
Pass
Bet
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One-Card Stud Poker, normal form: The equilibrium
0, 0
-a, a
a, -a
0, 0
Pass
Bet
Pass
Bet
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Nash Equilibria of 2-Player, 0-Sum Games
Two examples of 2-player, 0-sum games with multiple Nash equilibriaEvery equilibrium of a 2-player, 0-sum game has the same payoffs (proved by contradiction for the 2×2 case)
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Two-person, zero-sum game with two solutions: The payoff matrix
1, -1
1, -1
0, 0
0, 0
Right
Left
Right
Player 2
Player 1 Left
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Two-person, zero-sum game with two solutions: Strategy for player 1
1, -1
1, -1
0, 0
0, 0
Right
Left
Right
Player 2
Player 1 Left
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Two-person, zero-sum game with two solutions: Strategy for player 2
1, -1
1, -1
0, 0
0, 0
Right
Left
Right
Player 2
Player 1 Left
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Two-person, zero-sum game with two solutions: Two equilibria
1, -1
1, -1
0, 0
0, 0
Right
Left
Right
Player 2
Player 1 Left
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Two-person, zero-sum game with four solutions: The payoff matrix
1, -1
-1, 1
0, 0
0, 0
Right
Left
Right
Player 2Player 1 Left
0, 0
0, 0
0, 0 1, -1
-1, 1Center
Center
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Two-person, zero-sum game with four solutions: Strategy for Player 1
1, -1
-1, 1
0, 0
0, 0
Right
Left
Right
Player 2Player 1 Left
0, 0
0, 0
0, 0 1, -1
-1, 1Center
Center
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Two-person, zero-sum game with four solutions: Strategy for Player 2
1, -1
-1, 1
0, 0
0, 0
Right
Left
Right
Player 2Player 1 Left
0, 0
0, 0
0, 0 1, -1
-1, 1Center
Center
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Two-person, zero-sum game with four solutions: Four equilibria
1, -1
-1, 1
0, 0
0, 0
Right
Left
Right
Player 2Player 1 Left
0, 0
0, 0
0, 0 1, -1
-1, 1Center
Center
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Von Neumann’s Theorem for2-Player Zero-Sum Games
Every equilibrium of a 2-player zero-sum game has the same payoffsNo benefit to cooperation
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Proof of van Neumann’s theorem: By contradiction, suppose a > b
a, -a
d, -d
c, -c
b, -b
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2-Player Variable Sum Games
The concept of a variable sum gameMost games in business are variable sumA variable sum game may have equilibria with different payoffsNecessary and sufficient conditions for a solutionThe sufficient condition of undominated strategiesGames as parables
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Let’s Make a Deal: The payoff matrix
$15M, $15M
0, 0
0, 0
0, 0
No
Yes
No
Director
Movie Star Yes
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Let’s Make a Deal: Strategy for the Movie Star
$15M, $15M
0, 0
0, 0
0, 0
No
Yes
No
Director
Movie Star Yes
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Let’s Make a Deal: Strategy for the Director
$15M, $15M
0, 0
0, 0
0, 0
No
Yes
No
Director
Movie Star Yes
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Let’s Make a Deal: Two equilibria
$15M, $15M
0, 0
0, 0
0, 0
No
Yes
No
Director
Movie Star Yes
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Video System Coordination: The payoff matrix
1, 1
0, 0
0, 0
1, 1
VHS
Beta
VHS
Firm 2
Firm 1 Beta
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Video System Coordination: Strategy for Firm 1
1, 1
0, 0
0, 0
1, 1
VHS
Beta
VHS
Firm 2
Firm 1 Beta
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Video System Coordination: Strategy for Firm 2
1, 1
0, 0
0, 0
1, 1
VHS
Beta
VHS
Firm 2
Firm 1 Beta
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Video System Coordination: The two equilibria
1, 1
0, 0
0, 0
1, 1
VHS
Beta
VHS
Firm 2
Firm 1 Beta
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-1, -1
2, -2
-2, 2
-1, -1
Confess
Cooperate
Confess
Player 2
Player 1Cooperate
(remain silent)
(remain silent)
All numbers are years of time served
Prisoner’s Dilemma: The payoff matrix
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-1, -1
2, -2
-2, 2
-1, -1
Confess
Cooperate
Confess
Player 2
Player 1Cooperate
(remain silent)
(remain silent)
All numbers are years of time served
Prisoner’s Dilemma: Strategy for Player 1
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-1, -1
2, -2
-2, 2
-1, -1
Confess
Cooperate
Confess
Player 2
Player 1Cooperate
(remain silent)
(remain silent)
All numbers are years of time served
Prisoner’s Dilemma: Strategy for Player 2
48
-1, -1
2, -2
-2, 2
-1, -1
Confess
Cooperate
Confess
Player 2
Player 1Cooperate
(remain silent)
(remain silent)
All numbers are years of time served
Prisoner’s Dilemma: The equilibrium
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Cigarette Advertising on Television
Advertising as a strategic variableHeavy advertising by each firm as a dominant strategy and a game equilibriumHow the ban on cigarette advertising on television in 1971 raised industry profitsThe prisoner’s Dilemma
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Cigarette Television Advertising: The payoff matrix
$60M, $20M
Don’t advertise on television
Company 2
Company 1
$50M, $50M
$27M, $27M
Advertise on television
Advertise on television
Don’t advertise on television $20M, $60M
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Cigarette Television Advertising: Strategy for Company 1
$60M, $20M
Don’t advertise on television
Company 2
Company 1
$50M, $50M
$27M, $27M
Advertise on television
Advertise on television
Don’t advertise on television $20M, $60M
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Cigarette Television Advertising: Strategy for Company 2
$60M, $20M
Don’t advertise on television
Company 2
Company 1
$50M, $50M
$27M, $27M
Advertise on television
Advertise on television
Don’t advertise on television $20M, $60M
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Cigarette Television Advertising: The equilibrium
$60M, $20M
Don’t advertise on television
Company 2
Company 1
$50M, $50M
$27M, $27M
Advertise on television
Advertise on television
Don’t advertise on television $20M, $60M
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2-Player Games with Many Strategies
Using calculus to maximize utility and solve gamesEquilibrium as the solution of a pair of first order conditionsAdvertising budgets as continuous strategic variables
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Game with many strategies,the corners
x2 = 1
Player2
Player 1
1, 1x1 = 1
x1 = 0
x2 = 0
0, 0
0, 00, 0
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Existence of Equilibrium
The best response functionA fixed point theoremThe Existence of Equilibrium
57
Strictly dominance solvable game, best response function
10
0 x1
x2
1
f1(x2)
f2(x1) Nash Equilibrium
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Strictly dominance solvable game, best response mapping
100
x2
1
x1
100000
x2
1000
f(x*)
(500 , 250)x1