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Game Theory and Strategy

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Game Theory and Strategy Content Two-persons Zero-Sum Games Two-Persons Non-Zero-Sum Games N-Persons Games Introduction At least 2 players Strategies Outcome Payoffs ... – PowerPoint PPT presentation

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Title: Game Theory and Strategy


1
Game Theory and Strategy
2
Content
  • Two-persons Zero-Sum Games
  • Two-Persons Non-Zero-Sum Games
  • N-Persons Games

3
Introduction
  • At least 2 players
  • Strategies
  • Outcome
  • Payoffs

4
Two-persons Zero-Sum Games
  • Payoffs of each outcome add to zero
  • Pure conflict between 2 players

5
Two-persons Zero-Sum Games
6
Two-persons Zero-Sum Games
7
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.

8
Saddle Points and Saddle Points Principle
  • Definition 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.

9
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.

10
Two-persons Zero-Sum Games
11
Saddle Points
  • Minimax

12
Saddle Points
  • 0 saddle point
  • 1 saddle point
  • more than 1 saddle points

13
Mixed Strategy
14
Mixed Strategy
  • Colin plays with probability x for A, (1-x) for B
  • Rose A x(2) (1-x)(-3) -3 5x
  • Rose B x(0) (1-x)(3) 3 - 3x
  • if -3 5x 3 - 3x gt x 0.75
  • Rose A 0.75(2) 0.25(-3) 0.75
  • Rose B 0.75(0) 0.25(3) 0.75

15
Mixed Strategy
  • Rose plays with probability x for A, (1-x) for B
  • Colin A x(2) (1-x)(0) 2x
  • Colin B x(-3) (1-x)(3) 3 - 6x
  • if 2x 3 - 6x gt x 0.375
  • Colin A 0.375(2) 0.625(0) 0.75
  • Colin B 0.375(-3) 0.625(3) 0.75

16
Mixed Strategy
  • 0.75 as the value of the game
  • 0.75A, 0.25B as Colins optimal strategy
  • 0.375A. 0.625B as Roses optimal strategy

17
Mixed Strategy
18
Minimax Theorem
  • Every 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 that
  • i) player As expected payoff is no less that v,
    no matter what player B does, and
  • ii) player Bs expected payoff is no more that v,
    no matter what player A does
  • The solution can always be found in k x k subgame
    of the original game

19
Minimax Theorem (example)
20
Minimax Theorem (example)
  • There is no dominance in the above example
  • From 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

21
Example
22
Example
23
Example
24
Example
25
Mixed Strategy
26
Utility Theory
27
Utility Theory
  • Roses order is u, w, x, z, y, v
  • Colins order is v, y, z, x, w, u

28
Utility Theory
29
Utility Theory
  • Transformation can be done using a positive
    linear function, f(x) ax b
  • in this example, f(x) 0.5(x - 17)

  • --------gt

30
Two-Persons Non-Zero-Sum Games
  • Equilibrium outcomes in non-zero-sum games
    saddle points in zero-sum games

31
Prisoners Dilemma
32
Nash Equilibrium
  • If 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

33
Dominant Strategy Equilibrium
  • If 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.

34
Pareto-optimal
  • If 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

35
Pareto Principle
  • To be acceptable as a solution to a game, an
    outcome should be Pareto-optimal.

36
Prudential Strategy, Security Level and
Counter-Prudential Strategy
  • In 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
    level
  • As counter-prudential strategy is As optimal
    response to his opponents prudential strategy.

37
Example
38
Example
  • consider only Roses strategy
  • saddle point at AB

39
Example
  • consider only Colins strategy

40
Example
41
Example
42
Example
  • BB AA
  • Equilibrium

  • BA
  • AB

43
Co-operative Solution

  • Negotiation Set

44
Co-operative Solution

  • Negotiation Set

45
Co-operative Solution
  • Concerns are Trust and Suspicion

46
N-Person Games
  • More important and common in real life
  • n is assumed to be at least three

47
N-Person Games
48
N-Person Games
49
N-Person Games
50
N-Person Games
51
N-Person Games
52
N-Person Games
53
N-Person Games
54
N-Person Games
  • The result is

55
Superaddictive
  • A characteristic function form game (N, v) is
    called superadditive
  • if v(S, T) gt v(S) v(T) for any two coalitions
    S and T

56
N-Person Prisoners Dilemma
57
N-Person Prisoners Dilemma
  • General form of N-Person Prisoners Dilemma
  • each of n players has two strategies, C and D
  • for every player, D is a dominant strategy
  • if all players choose D, add will be worse off
    than if all players had chosen C

58
Example
59
Example
  • Sincere choice

60
Example
  • Optimal choice

61
From the bottom up algorithm
  • i) 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 lists
  • ii) 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 lists
  • iii) continue like this, finding the choices in
    the next-to-last round, and on up to the first
    round

62
Example
  • Sincere choice

63
Example
  • Sincere choice

64
Example
  • Sincere choice

65
Example
  • Sincere choice

66
Example of N-Person Prisoners Dilemma
67
Example of N-Person Prisoners Dilemma
  • Sincere Choice

68
Example of N-Person Prisoners Dilemma
  • After Greens optimal Choice

69
Example of N-Person Prisoners Dilemma
  • After Reds optimal Choice

70
Example of N-Person Prisoners Dilemma
  • After Blues optimal Choice

71
END
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