Game theory 2011

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Game theory 2011

  1. 1. Game Theory
  2. 2. Characteristics <ul><li>Components </li></ul><ul><ul><li>Players - two or more </li></ul></ul><ul><ul><li>Strategies - finite or infinite </li></ul></ul><ul><ul><li>Payoff table - zero-sum or nonzero-sum </li></ul></ul><ul><li>We consider two-person, zero-sum games with a finite number of strategies </li></ul><ul><li>Assumptions </li></ul><ul><ul><li>Players know about the game structure </li></ul></ul><ul><ul><li>Both players are rational </li></ul></ul>
  3. 3. Pure strategy Row min Column max 1 2 3 4 1 2 3 Player B Player A Maximin = 4 Minimax = 4 Optimal strategy = (2,2)  Saddle point. Value of the game = 4 -3 6 -3 2 6 7 7 4 4 1 8 4 7 6 4 5 8 1 7
  4. 4. Dominance <ul><li>Consider the previous payoff matrix. </li></ul><ul><li>Player B’s 2nd strategy dominates both 1st and 4th strategies. Hence eliminate Player B’s 1st and 4th strategies. </li></ul><ul><li>In the reduced matrix, Player A’s 2nd strategy dominates the 3rd strategy. So, eliminate Player A’s 3rd strategy. </li></ul><ul><li>In the same way, eliminate Player B’s 3rd strategy and then Player A’s 1st strategy. </li></ul>
  5. 5. No dominant strategy Row min Column max 1 2 3 1 2 3 Player A Maximin = 0 Minimax = 0 Optimal strategy = (2,2)  Saddle point. Value of the game = 0  Fair game Player B -4 -4 -2 5 0 -3 6 0 5 2 0 2 6 -2 -3
  6. 6. Mixed strategy Row min Column max 1 2 3 1 2 Player A Maximin = 2 Minimax = 5 Player B Maximin ≠ Minimax  No saddle point, unstable solution Maximin ≤ Value of the game ≤ Minimax 11 5 8 2 1 2 5 8 11 3 1
  7. 7. Graphical solution <ul><li>Let </li></ul><ul><li>x 1 : Probability associated with Player A’s strategy 1 </li></ul><ul><li>x 2 : Probability associated with Player A’s strategy 2 </li></ul><ul><li>y 1 : Probability associated with Player B’s strategy 1 </li></ul><ul><li>y 2 : Probability associated with Player B’s strategy 2 </li></ul><ul><li>y 3 : Probability associated with Player B’s strategy 3 </li></ul><ul><li>such that </li></ul><ul><li>x 1 + x 2 = 1 and y 1 + y 2 + y 3 = 1 </li></ul>
  8. 8. Graphical solution -2x 1 +5 2 9x 1 +2 3 -7x 1 +8 1 A’s expected payoff B’ s pure strategy
  9. 9. Graphical solution 1 3 2 1/4 3/4 1 1/2 1 2 3 4 5 6 7 8 x 1 A’s expected payoff Maximin x 1 = 3/11, x 2 = 8/11, y 1 =0 Value of the game = 49/11
  10. 10. Graphical solution y 2 = 9/11, y 3 = 2/11 3y 2 +2 2 -8y 2 +11 1 A’s expected payoff A’ s pure strategy
  11. 11. Graphical solution 1 2 3 1 2 Player A Player B Solve: 2 6 5 3 6 1
  12. 12. LP solution 1 (y 1 ) 1 (x 1 ) 2 (x 2 ) Player A Player B m (x m )  2 (y 2 ) n (y n )      a mn a 2n a 1n a m2 a m1 a 22 a 21 a 12 a 11
  13. 13. LP solution
  14. 14. LP solution
  15. 15. 1 2 3 1 2 Player A Player B LP solution Example 2 5 8 11 3 1
  16. 16. LP solution

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