Game Theory and Strategy

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Game Theory and Strategy
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Content

• Two-persons Zero-Sum Games
• Two-Persons Non-Zero-Sum Games
• N-Persons Games

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Introduction

• At least 2 players
• Strategies
• Outcome
• Payoffs

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Two-persons Zero-Sum Games

• Payoffs of each outcome add to zero
• Pure conflict between 2 players

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Two-persons Zero-Sum Games
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Two-persons Zero-Sum Games
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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 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.

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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
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Saddle Points

• Minimax

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Saddle Points

• 0 saddle point
• 1 saddle point
• more than 1 saddle points

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Mixed Strategy
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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

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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

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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

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Mixed Strategy
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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

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Minimax Theorem (example)
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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

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Example
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Example
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Example
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Example
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Mixed Strategy
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Utility Theory
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Utility Theory

• Roses order is u, w, x, z, y, v
• Colins order is v, y, z, x, w, u

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Utility Theory
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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

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Two-Persons Non-Zero-Sum Games

• Equilibrium outcomes in non-zero-sum games
  saddle points in zero-sum games

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Prisoners Dilemma
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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

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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.

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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

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Pareto Principle

• To 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 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.

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Example
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Example

• consider only Roses strategy
• saddle point at AB

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Example

• consider only Colins strategy

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Example
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Example
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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 Solution

• Concerns are Trust and Suspicion

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N-Person Games

• More important and common in real life
• n is assumed to be at least three

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N-Person Games
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N-Person Games
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N-Person Games
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N-Person Games
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N-Person Games
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N-Person Games
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N-Person Games
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N-Person Games

• The result is

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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

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N-Person Prisoners Dilemma
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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

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Example
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Example

• Sincere choice

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Example

• Optimal choice

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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

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Example

• Sincere choice

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Example

• Sincere choice

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Example

• Sincere choice

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Example

• Sincere choice

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Example of N-Person Prisoners Dilemma
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Example of N-Person Prisoners Dilemma

• Sincere Choice

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Example of N-Person Prisoners Dilemma

• After Greens optimal Choice

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Example of N-Person Prisoners Dilemma

• After Reds optimal Choice

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Example of N-Person Prisoners Dilemma

• After Blues optimal Choice

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END