n-Person Games in Normal Form - University Of Maryland · Technology Firm 3 - Stay Put Firm 2 Firm...
Transcript of n-Person Games in Normal Form - University Of Maryland · Technology Firm 3 - Stay Put Firm 2 Firm...
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1
n-Person Games
in
Normal Form
Chapter 5
2
Fundamental Differences with 3 Players: the SpoilersCounterexamplesThe theorem for games like Chess does not generalizeThe solution theorem for 0-sum, 2-player games does not generalizeA player playing the spoiler
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Indeterminate three-person game
w, l, l
l, w, l
Top
Bottom
3
4
Multiple solutions, 3-person, zero-sum game
1, -1/2, -1/2 -1/2, 1, -1/2
3-Top 3-Bottom
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Competitive Advantage and Market Niche with 3 PlayersThe row-column matrix representation for 3 playersGames where no player plays the spoilerPure and mixed strategy equilibria for 3-player games
6
Competitive Advantage, three firms
New Technology Stay Put
0, 0, 0
-a, a/2, a/2
New Technology
Stay Put
Firm 3 - New Technology
a/2, -a, a/2
-a/2, -a/2, a
Firm 2
Firm 1New
Technology Stay Put
a/2, a/2, -a
-a/2, a, -a/2
New Technology
Firm 3 - Stay Put
Firm 2
Firm 1
Stay Put
a, -a/2, -a/2
0, 0, 0
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Competitive Advantage, three firms: Strategy for Firm 1
New Technology Stay Put
0, 0, 0
-a, a/2, a/2
New Technology
Stay Put
Firm 3 - New Technology
a/2, -a, a/2
-a/2, -a/2, a
Firm 2
Firm 1New
Technology Stay Put
a/2, a/2, -a
-a/2, a, -a/2
New Technology
Firm 3 - Stay Put
Firm 2
Firm 1
Stay Put
a, -a/2, -a/2
0, 0, 0
8
Competitive Advantage, three firms: Strategy for Firm 2
New Technology Stay Put
0, 0, 0
-a, a/2, a/2
New Technology
Stay Put
Firm 3 - New Technology
a/2, -a, a/2
-a/2, -a/2, a
Firm 2
Firm 1New
Technology Stay Put
a/2, a/2, -a
-a/2, a, -a/2
New Technology
Firm 3 - Stay Put
Firm 2
Firm 1
Stay Put
a, -a/2, -a/2
0, 0, 0
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Competitive Advantage, three firms: Strategy for Firm 3
New Technology Stay Put
0, 0, 0
-a, a/2, a/2
New Technology
Stay Put
Firm 3 - New Technology
a/2, -a, a/2
-a/2, -a/2, a
Firm 2
Firm 1New
Technology Stay Put
a/2, a/2, -a
-a/2, a, -a/2
New Technology
Firm 3 - Stay Put
Firm 2
Firm 1
Stay Put
a, -a/2, -a/2
0, 0, 0
10
Competitive Advantage, three firms:The Nash equilibrium
New Technology Stay Put
0, 0, 0
-a, a/2, a/2
New Technology
Stay Put
Firm 3 - New Technology
a/2, -a, a/2
-a/2, -a/2, a
Firm 2
Firm 1New
Technology Stay Put
a/2, a/2, -a
-a/2, a, -a/2
New Technology
Firm 3 - Stay Put
Firm 2
Firm 1
Stay Put
a, -a/2, -a/2
0, 0, 0
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Market Niche for three firms
Enter Stay Out
-50, -50, -50
0, -50, -50Stay Out
Firm 3 - Enter
-50, 0, -50
Firm 2
Firm 1 Stay Out
-50, -50, 0
0, 100, 0
Firm 3 - Stay Out
Firm 2
Firm 1
Stay Out
100, 0, 0
0, 0, 0
Enter
Enter Enter
0, 0, 100
12
Market Niche, three firms: Strategy for Firm 1
Enter Stay Out
-50, -50, -50
0, -50, -50Stay Out
Firm 3 - Enter
-50, 0, -50
Firm 2
Firm 1 Stay Out
-50, -50, 0
0, 100, 0
Firm 3 - Stay Out
Firm 2
Firm 1
Stay Out
100, 0, 0
0, 0, 0
Enter
Enter Enter
0, 0, 100
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Market Niche, three firms: Strategy for Firm 2
Enter Stay Out
-50, -50, -50
0, -50, -50Stay Out
Firm 3 - Enter
-50, 0, -50
Firm 2
Firm 1 Stay Out
-50, -50, 0
0, 100, 0
Firm 3 - Stay Out
Firm 2
Firm 1
Stay Out
100, 0, 0
0, 0, 0
Enter
Enter Enter
0, 0, 100
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Market Niche, three firms: Strategy for Firm 3
Enter Stay Out
-50, -50, -50
0, -50, -50Stay Out
Firm 3 - Enter
-50, 0, -50
Firm 2
Firm 1 Stay Out
-50, -50, 0
0, 100, 0
Firm 3 - Stay Out
Firm 2
Firm 1
Stay Out
100, 0, 0
0, 0, 0
Enter
Enter Enter
0, 0, 100
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Market Niche, three firms: Three pure strategy equilibria
Enter Stay Out
-50, -50, -50
0, -50, -50Stay Out
Firm 3 - Enter
-50, 0, -50
Firm 2
Firm 1 Stay Out
-50, -50, 0
0, 100, 0
Firm 3 - Stay Out
Firm 2
Firm 1
Stay Out
100, 0, 0
0, 0, 0
Enter
Enter Enter
0, 0, 100
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Mixed Strategy equilibrium in Market Niche with 3 players
From the standpoint of the market, the distribution of number of firms in the market niche, according to mixed strategy equilibria is as follows:
p(3 firms enter) = .08p(2 firms enter) = .31p(1 firm enters) = .42p(No firm enters)= .19
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3-Player Versions of Coordination, Deal-Making, and Advertising
Video System Coordination with 3 firms Let’s Make a Deal with 3 firms Cigarette Advertising on Television with 3 firms
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Video System Coordination, three firms: The payoff matrices
Beta VHS
1, 1, 1
Firm 3 - Beta
Firm 2
Firm 1
Firm 3 - VHS
Firm 2
Firm 1
0, 0, 0
Beta VHS
Beta
VHS
Beta
VHS 1, 1, 1
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
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Video System Coordination, three firms: Strategy for Firm 1
Beta VHS
1, 1, 1
Firm 3 - Beta
Firm 2
Firm 1
Firm 3 - VHS
Firm 2
Firm 1
0, 0, 0
Beta VHS
Beta
VHS
Beta
VHS 1, 1, 1
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
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Video System Coordination, three firms: Strategy for Firm 2
Beta VHS
1, 1, 1
Firm 3 - Beta
Firm 2
Firm 1
Firm 3 - VHS
Firm 2
Firm 1
0, 0, 0
Beta VHS
Beta
VHS
Beta
VHS 1, 1, 1
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
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Video System Coordination, three firms: Strategy for Firm 3
Beta VHS
1, 1, 1
Firm 3 - Beta
Firm 2
Firm 1
Firm 3 - VHS
Firm 2
Firm 1
0, 0, 0
Beta VHS
Beta
VHS
Beta
VHS 1, 1, 1
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
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Video System Coordination, three firms: Two pure strategy equilibria
Beta VHS
1, 1, 1
Firm 3 - Beta
Firm 2
Firm 1
Firm 3 - VHS
Firm 2
Firm 1
0, 0, 0
Beta VHS
Beta
VHS
Beta
VHS 1, 1, 1
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
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Let’s make a deal, three players: Payoffs in millions of dollars
Yes No
5, 5, 5
Player 3 - Yes
Player 2
Player 1
Player 2
Player 1
0, 0, 0
Yes No
Yes
No
Yes
No
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
Player 3 - No
0, 0, 0
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Let’s make a deal, three players: Strategy for player 1
Yes No
5, 5, 5
Player 3 - Yes
Player 2
Player 1
Player 2
Player 1
0, 0, 0
Yes No
Yes
No
Yes
No
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
Player 3 - No
0, 0, 0
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Let’s make a deal, three players: Strategy for player 2
Yes No
5, 5, 5
Player 3 - Yes
Player 2
Player 1
Player 2
Player 1
0, 0, 0
Yes No
Yes
No
Yes
No
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
Player 3 - No
0, 0, 0
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Let’s make a deal, three players: Strategy for player 3
Yes No
5, 5, 5
Player 3 - Yes
Player 2
Player 1
Player 2
Player 1
0, 0, 0
Yes No
Yes
No
Yes
No
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
Player 3 - No
0, 0, 0
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Let’s make a deal, three players: Five pure strategy equilibria
Yes No
5, 5, 5
Player 3 - Yes
Player 2
Player 1
Player 2
Player 1
0, 0, 0
Yes No
Yes
No
Yes
No
0, 0, 0
0, 0, 00, 0, 0 0, 0, 0
0, 0, 0
Player 3 - No
0, 0, 0
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Stonewalling Watergate
Watergate as a 3-person Prisoner’s DilemmaStrictly dominant strategies and uniqueness of equilibriumEquilibria which are bad for the players
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Stonewalling Watergate: D = Dean, E = Ehrlichman, H = Halderman
Stonewall Talk
-3, -3, -3
H - Stonewall
E
D
E
D
-5, -2, -2
Talk
Talk
Stonewall
Talk
-5, -5, -2
-2, -5, -2-2, -5, -5 -2, -2, -5
-5, -2, -2
-4, -4, -4
H - Talk
Stonewall
Stonewall
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Stonewalling Watergate: Strategy for Dean
Stonewall Talk
-3, -3, -3
H - Stonewall
E
D
E
D
-5, -2, -2
Talk
Talk
Stonewall
Talk
-5, -5, -2
-2, -5, -2-2, -5, -5 -2, -2, -5
-5, -2, -5
-4, -4, -4
H - Talk
Stonewall
Stonewall
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Stonewalling Watergate: Strategy for Ehrlichman
Stonewall Talk
-3, -3, -3
H - Stonewall
E
D
E
D
-5, -2, -2
Talk
Talk
Stonewall
Talk
-5, -5, -2
-2, -5, -2-2, -5, -5 -2, -2, -5
-5, -2, -5
-4, -4, -4
H - Talk
Stonewall
Stonewall
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Stonewalling Watergate: Strategy for Halderman
Stonewall Talk
-3, -3, -3
H - Stonewall
E
D
E
D
-5, -2, -2
Talk
Talk
Stonewall
Talk
-5, -5, -2
-2, -5, -2-2, -5, -5 -2, -2, -5
-5, -2, -5
-4, -4, -4
H - Talk
Stonewall
Stonewall
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Stonewalling Watergate: The Nash equilibrium
Stonewall Talk
-3, -3, -3
H - Stonewall
E
D
E
D
-5, -2, -2
Talk
Talk
Stonewall
Talk
-5, -5, -2
-2, -5, -2-2, -5, -5 -2, -2, -5
-5, -2, -5
-4, -4, -4
H - Talk
Stonewall
Stonewall
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Symmetry and Games with Many Players
A compact notation for utility functionsA generalized symmetry sufficient conditionA symmetric game may have asymmetric equilibria
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Solving Symmetric Games with Many Strategies
A test for when a game is symmetricSymmetry makes games easier to solveSolving a game of common interest by exploiting the symmetry of the game
The Nash Demand Game
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The Nash demand game:the payoff matrix
0,0
0, 0
0, 0
0, 0
$0
$2
$0
Player 2Player 1 $2
1, 1
2, 0
0, 2 0, 1
1, 0$1
$1
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The Nash demand game:player 1’s strategy
0,0
0, 0
0, 0
0, 0
$0
$2
$0
Player 2Player 1 $2
1, 1
2, 0
0, 2 0, 1
1, 0$1
$1
38
The Nash demand game:player 2’s strategy
0,0
0, 0
0, 0
0, 0
$0
$2
$0
Player 2Player 1 $2
1, 1
2, 0
0, 2 0, 1
1, 0$1
$1
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The Nash demand game:Nash equilibrium
0,0
0, 0
0, 0
0, 0
$0
$2
$0
Player 2Player 1 $2
1, 1
2, 0
0, 2 0, 1
1, 0$1
$1
40
Stag Hunt
Game requiring cooperation for efficient outcomeAdding third player leads to qualitatively different outcome
additional Nash equilibriaasymmetric outcomespossibility of free riding
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Stag Hunt, two hunters:The payoff matrix
3, 3
1, 0
0, 1
1, 1
hunt big
Hunter 2
Hunter 1 hunt small
hunt big
hunt small
42
Stag Hunt, two hunters:Strategy for hunter 1
3, 3
1, 0
0, 1
1, 1
Hunter 2
Hunter 1 hunt big hunt small
hunt big
hunt small
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Stag Hunt, two hunters:Strategy for hunter 2
3, 3
1, 0
0, 1
1, 1
Hunter 2
Hunter 1 hunt big hunt small
hunt big
hunt small
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Stag Hunt, two hunters:The equilibrium
3, 3
1, 0
0, 1
1, 1
Hunter 2
Hunter 1 hunt big hunt small
hunt big
hunt small
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Stag Hunt, three hunters:The payoff matrix
hunt big hunt big
hunt small hunt small
hunter 1 hunter 1
hunter 2hunt big hunt small hunt big hunt small
hunter 2
hunter 3: hunt big hunter 3: hunt small
3, 3, 3
5, 3, 3 1, 1, 0
3, 5, 3 3, 3, 5 0, 1, 1
1, 0, 1 1, 1, 1
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Stag Hunt, three hunters:Strategy for hunter 1
hunt small
hunt big hunt big
hunt small
hunter 1 hunter 1
hunter 2hunt big hunt small hunt big hunt small
hunter 2
hunter 3: hunt big hunter 3: hunt small
3, 3, 3
5, 3, 3 1, 1, 0
3, 5, 3 3, 3, 5 0, 1, 1
1, 0, 1 1, 1, 1
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Stag Hunt, three hunters:Strategy for hunter 2
hunt small
hunt big hunt big
hunt small
hunter 1 hunter 1
hunter 2hunt big hunt small hunt big hunt small
hunter 2
hunter 3: hunt big hunter 3: hunt small
3, 3, 3
5, 3, 3 1, 1, 0
3, 5, 3 3, 3, 5 0, 1, 1
1, 0, 1 1, 1, 1
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Stag Hunt, three hunters:Strategy for hunter 3
hunt small
hunt big hunt big
hunt small
hunter 1 hunter 1
hunter 2hunt big hunt small hunt big hunt small
hunter 2
hunter 3: hunt big hunter 3: hunt small
3, 3, 3
5, 3, 3 1, 1, 0
3, 5, 3 3, 3, 5 0, 1, 1
1, 0, 1 1, 1, 1
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Stag Hunt, three hunters:Nash equilibria
hunt small
hunt big hunt big
hunt small
hunter 1 hunter 1
hunter 2hunt big hunt small hunt big hunt small
hunter 2
hunter 3: hunt big hunter 3: hunt small
3, 3, 3
5, 3, 3 1, 1, 0
3, 5, 3 3, 3, 5 0, 1, 1
1, 0, 1 1, 1, 1
50
The Tragedy of the Commons
Games played on a commonsThe equilibrium of such a game has a tragic outcomeExternalitiesFirst Welfare TheoremThe case of the Geysers of Northern California
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Tragedy of the Commons:score sheet
Payoff = 5(10 - xi) + xi (23 – 0.25 Σxi)
4
3
2
1
payoffstrategy (xi )
52
Tragedy of the Commons:Commons production function
0 6 12X
F(X)
3.6
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Tragedy of the Commons
0 6 12X
F(X)
5 10
Nash, 5 players
Nash, 10 players
Ultimate Tragic NEF(X)/X = 0.1
Economic Zone Uneconomic Zone
3.6
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Tragedy of the Commons
0 6 12X
F(X)
0.1
average product
marginal product
efficient outcome
tragic outcome
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Appendix. Tragedy of the Commons in the Laboratory
Playing a game in a behavior laboratoryTragic outcomes of a game played on a commons in a laboratoryUnexplained phenomena