game theory
DESCRIPTION
aTRANSCRIPT
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Game Theory and Strategy
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ContentTwo-persons Zero-Sum GamesTwo-Persons Non-Zero-Sum GamesN-Persons Games
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IntroductionAt least 2 playersStrategiesOutcomePayoffs
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Two-persons Zero-Sum GamesPayoffs of each outcome add to zeroPure conflict between 2 players
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Two-persons Zero-Sum Games
Colin
A
B
C
D
Rose
A
7, -7
-1, 1
1, -1
0, 0
B
5, -5
1, -1
6, -6
-9, 9
C
3, -3
2, -2
4, -4
3, -3
D
-8, 8
0, 0
0, 0
8, -8
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Two-persons Zero-Sum Games
Colin
A
B
C
D
Rose
A
7
-1
1
0
B
5
1
6
-9
C
3
2
4
3
D
-8
0
0
8
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Dominance and Dominance Principle Definition: A strategy S dominates a strategy T if every outcome in S is at least as good as the corresponding outcome in T, and at least one outcome in S is strictly better than the corresponding outcome in T.Dominance Principle: A rational player would never play a dominated strategy.
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Saddle Points and Saddle Points PrincipleDefinition: An outcome in a matrix game is called a Saddle Point if the entry at that outcome is both less than or equal to any in its row, and greater than or equal to any entry in its column.Saddle Point Principle: If a matrix game has a saddle point, both players should play a strategy which contains it.
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Value Definition: For a matrix game, if there is a number such that player A has a strategy which guarantees that he will win at least v and player B has a strategy which guarantees player A will win no more than v, then v is called the value of the game.
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Two-persons Zero-Sum Games
Colin
A
B
C
D
Rose
A
7
-1
1
0
B
5
1
6
-9
C
3
2
4
3
D
-8
0
0
8
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Saddle Points
Minimax
Colin
A
B
C
D
Row minimum
Rose
A
4
3
2
5
2
Maximin
B
-10
2
0
1
-10
C
7
5
2
3
2
Maximin
D
0
8
-4
-5
-5
Column Maximum
7
8
2
5
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Saddle Points0 saddle point1 saddle pointmore than 1 saddle points
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Mixed Strategy
Colin
A
B
Rose
A
2
-3
B
0
3
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Mixed StrategyColin plays with probability x for A, (1-x) for BRose A: x(2) + (1-x)(-3) = -3 + 5xRose B: x(0) + (1-x)(3) = 3 - 3xif -3 + 5x = 3 - 3x => x = 0.75Rose A: 0.75(2) + 0.25(-3) = 0.75Rose B: 0.75(0) + 0.25(3) = 0.75
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Mixed StrategyRose plays with probability x for A, (1-x) for BColin A: x(2) + (1-x)(0) = 2xColin B: x(-3) + (1-x)(3) = 3 - 6xif 2x = 3 - 6x => x = 0.375Colin A: 0.375(2) + 0.625(0) = 0.75Colin B: 0.375(-3) + 0.625(3) = 0.75
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Mixed Strategy0.75 as the value of the game0.75A, 0.25B as Colins optimal strategy0.375A. 0.625B as Roses optimal strategy
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Mixed Strategy
Colin
Row difference
Rose oddments
Rose probabilities
A
B
Rose
A
2
-3
2 - (-3) = 5
3
3/8
B
0
3
0 3 = -3
5
5/8
Column difference
2 0 = 2
-3 3 = -6
Colin oddments
6
2
Colin probabilities
6/8
2/8
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Minimax TheoremEvery m x n matrix game has a solution. There is a unique number v, called the value of game, and optimal strategy for the players such thati) player As expected payoff is no less that v, no matter what player B does, andii) player Bs expected payoff is no more that v, no matter what player A doesThe solution can always be found in k x k subgame of the original game
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Minimax Theorem (example)
Colin
A
B
C
D
E
Rose
A
1
12
13
9
10
B
11
2
8
14
5
C
6
7
3
4
15
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Minimax Theorem (example)There is no dominance in the above exampleFrom arrows in the graph, Colin will only choose A, B or C, but not D or E.So the game is reduced into a 3 x 3 subgame
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Example
9-Police
9-0
8-1
7-2
6-3
5-4
7-Guerrillas
7-0
1/2
1/2
1/2
1
1
6-1
1
1/2
1/2
1/2
1
5-2
1
1
1/2
1/2
4-3
1
1
1
1/2
0
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Example
9-Police
7-2
6-3
5-4
7-Guerrillas
7-0
1/2
1
1
6-1
1/2
1/2
1
5-2
1/2
1/2
4-3
1
1/2
0
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Example
9-Police
7-2
6-3
5-4
7-Guerrillas
7-0
1/2
1
1
4-3
1
1/2
0
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Example
9-Police
7-2
5-4
7-Guerrillas
7-0
1/2
1
4-3
1
0
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Mixed Strategy
9-Police
Row difference
Guerrillas oddments
Guerrillas probabilities
7-2
5-4
7-Guerrillas
7-0
1/2
1
-1/2
1
2/3
4-3
1
0
1
1/3
Column difference
1/2
1
Police oddments
1
1/2
Police probabilities
2/3
1/3
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Utility Theory
Colin
A
B
Rose
A
U
V
B
W
X
C
Y
Z
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Utility TheoryRoses order is u, w, x, z, y, vColins order is v, y, z, x, w, u
Colin
A
B
Rose
A
6
1
B
5
4
C
2
3
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Utility Theory
v
w
x
u
i)
0
20
40
60
80
100
ii)
-1
0
1
2
3
4
iii)
17
19
21
23
25
27
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Utility TheoryTransformation can be done using a positive linear function, f(x) = ax + bin this example, f(x) = 0.5(x - 17)
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Colin
A
B
Rose
A
27, -5
17, 0
B
19, -1
23, -3
Colin
A
B
Rose
A
5, -5
0, 0
B
1, -1
3, -3
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Two-Persons Non-Zero-Sum GamesEquilibrium outcomes in non-zero-sum games ~ saddle points in zero-sum games
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Prisoners Dilemma
Colin
Confess
Dont
Rose
Confess
10, 10
0, 20
Dont
20, 0
1, 1
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Nash EquilibriumIf there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium
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Dominant Strategy EquilibriumIf every player in the game has a dominant strategy, and each player plays the dominant strategy, then that combination of strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
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Pareto-optimalIf an outcome cannot be improved upon, ie. no one can be made better off without making somebody else worse off, then the outcome is Pareto-optimal
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Pareto PrincipleTo be acceptable as a solution to a game, an outcome should be Pareto-optimal.
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Prudential Strategy, Security Level and Counter-Prudential StrategyIn a non-zero-sum game, player As optimal strategy in As game is called As prudential strategy.The value of As game is called As security levelAs counter-prudential strategy is As optimal response to his opponents prudential strategy.
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Example
Colin
A
B
Rose
A
2, 4
1, 0
B
3, 1
0, 4
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Exampleconsider only Roses strategysaddle point at AB
Colin
A
B
Rose
A
2
1
B
3
0
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Exampleconsider only Colins strategy
Colin
A
B
Rose
A
4
0
B
1
4
Column difference
4-1=3
0-4=-4
Colin oddments
4
3
Colin probabilities
4/7
3/7
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Example
Rose strategy
Colin strategy
Rose payoff
Colin payoff
prudential
prudential
11/7
16/7
prudential
Counter-prudential
2
4
Counter-prudential
Prudential
12/7
16/7
Counter-prudential
Counter-prudential
3
1
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Example
Rose prudential
A
Colin Prudential
4/7A, 3/7B
Rose Counter-prudential
B
Colin Counter-prudential
A
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Example
BB AA
Equilibrium
BA AB
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Co-operative Solution
Negotiation Set
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Co-operative Solution
Negotiation Set
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Co-operative SolutionConcerns are Trust and Suspicion
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N-Person GamesMore important and common in real lifen is assumed to be at least three
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N-Person Games
Larry A
Colin
A
B
Rose
A
1, 1, -2
-4, 3, 1
B
2, -4, 2
-5, -5, 10
Larry B
Colin
A
B
Rose
A
3, -2, -1
-6, -6, 12
B
2, 2, -4
-2, 3, 1
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N-Person Games
Colin and Larry
AA
BA
AB
BB
Rose
A
1
-4
3
-6
B
2
-5
2
-2
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N-Person Games
Rose and Larry
AA
BA
AB
BB
Colin
A
1
-4
-2
2
B
3
-5
-6
3
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N-Person Games
Rose and Colin
AA
BA
AB
BB
Larry
A
-2
2
1
10
B
-1
-4
12
-1
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N-Person Games
Colin and Larry
BA
BB
Rose optimal
Rose
A
-4
-6
3/5
B
-5
-2
2/5
Colin and Larry optimal
4/5
1/5
Value = -4.4
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N-Person Games
Rose and Larry
BA
AB
Colin optimal
Colin
A
-4
-2
1
B
-5
-6
0
Rose and Larry optimal
1
0
Value = -4
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N-Person Games
Rose and Colin
AA
BA
Larry optimal
Larry
A
-2
2
3/7
B
-1
-4
4/7
Rose and Colin optimal
6/7
1/7
Value = -1.43
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N-Person GamesThe result is
Rose v.s. Colin and Larry
-4.4, -0.64, 5.04
Colin v.s. Rose and Larry
2, -4, 2
Larry v.s. Rose and Colin
2.12, -0.69, -1.43
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SuperaddictiveA characteristic function form game (N, v) is called superadditiveif v(S, T) >= v(S) + v(T) for any two coalitions S and T
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N-Person Prisoners Dilemma
Number of others choosing C
0
1
2
3
4
Player chooses
C
-2
-1
0
1
2
D
-1
0
1
2
3
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N-Person Prisoners DilemmaGeneral form of N-Person Prisoners Dilemmaeach of n players has two strategies, C and Dfor every player, D is a dominant strategyif all players choose D, add will be worse off than if all players had chosen C
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Example
Blues
Red
A
B
B
C
C
D
D
A
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ExampleSincere choice
Blues
Red
1st round
A
B
B
C
2nd round
C
D
D
A
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ExampleOptimal choice
Blues
Red
2nd round
A
B
1st round
B
C
C
D
D
A
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From the bottom up algorithmi) under optimal play, the Reds choice in last round will be the player who is last on the Blues preference list. Mark that player as the Reds last round choice and cross him off both teams listsii) the Blues choice in last round will be the player who is last on the Reds reduced list. Mark the player as Blues and cross him off both teams listsiii) continue like this, finding the choices in the next-to-last round, and on up to the first round
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ExampleSincere choice
Blues
Red
A
B
B
C
2nd round
C
D
D
A
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ExampleSincere choice
Blues
Red
A
B
B
C
2nd round
C
D
D
A
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ExampleSincere choice
Blues
Red
A
B
1st round
B
C
2nd round
C
D
D
A
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ExampleSincere choice
Blues
Red
A
B
1st round
B
C
2nd round
C
D
D
A
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Example of N-Person Prisoners Dilemma
Blues
Red
Green
A
E
C
B
F
F
C
B
E
D
A
D
E
D
A
F
C
B
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Example of N-Person Prisoners DilemmaSincere Choice
Blues
Red
Green
1st round
A
E
C
2nd round
B
F
F
C
B
E
D
A
D
E
D
A
F
C
B
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Example of N-Person Prisoners DilemmaAfter Greens optimal Choice
Blues
Red
Green
1st round
A
E
C
2nd round
B
F
F
C
B
E
D
A
D
E
D
A
F
C
B
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Example of N-Person Prisoners DilemmaAfter Reds optimal Choice
Blues
Red
Green
1st round
A
E
C
B
F
F
C
B
E
2nd round
D
A
D
E
D
A
F
C
B
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Example of N-Person Prisoners DilemmaAfter Blues optimal Choice
Blues
Red
Green
2nd round
A
E
C
B
F
F
1st round
C
B
E
D
A
D
E
D
A
F
C
B
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END