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Game Theory and Strategy

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

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Game Theory and Strategy

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  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 => 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 => 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 Colin’s optimal strategy • 0.375A. 0.625B as Rose’s 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 A’s expected payoff is no less that v, no matter what player B does, and • ii) player B’s 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 • Rose’s order is u, w, x, z, y, v • Colin’s 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) -------->

  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 A’s optimal strategy in A’s game is called A’s prudential strategy. • The value of A’s game is called A’s security level • A’s counter-prudential strategy is A’s optimal response to his opponent’s prudential strategy.

  37. Example

  38. Example • consider only Rose’s strategy • saddle point at AB

  39. Example • consider only Colin’s 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

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