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TWO-PERSON ZERO-SUM GAMES

TWO-PERSON ZERO-SUM GAMES. 1.2 2-Person, Zero-Sum Games. Two players Total payoff = 0, thus Payoff to Player I = – Payoff to Player II Information can be summarised nicely in a simple payoff table. Convention. a i = ith action (pure strategy) for Player I

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TWO-PERSON ZERO-SUM GAMES

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  1. TWO-PERSON ZERO-SUM GAMES

  2. 1.2 2-Person, Zero-Sum Games • Two players • Total payoff = 0, thus • Payoff to Player I = – Payoff to Player II • Information can be summarised nicely in a simple payoff table.

  3. Convention

  4. ai = ith action (pure strategy) for Player I • Aj = jth action (pure strategy) for Player II • m = # of actions available to Player I • n = # of actions available to Player II • vij = Payoff to Player I if they select ai and Player II selects action Aj . • Payoff to Player II = – Payoff to Player I

  5. A A A 1 2 3 a1 2 - 4 2 5 a2 6 1 0 - 2 0 Example Player 2 • Player 1 has options: 2 pure strategies a1 or a2. • Player 2 has options: 3 pure strategies A1, A2 or A3. • If player 1 plays pure strategy a1 and Player 2 plays pure strategy A3 then Player 1 wins 25 from Player 2. Player 1

  6. Solution? Independent, unbiased analyst to determine what are the“best” strategies for the two players. Player II Player I

  7. A A A 1 2 3 a1 2 - 4 2 5 a2 6 1 0 - 2 0 1.2.1 Example What are the best strategies for the players?

  8. A A A 1 2 3 a1 2 - 4 2 5 a2 6 1 0 - 2 0 • Conflict: • Player 1 prefers (a1,A3) with payoff = 25 • Player 2 prefers (a2,A3) with payoff = 20 • But what would Player 2 do knowing that Player 1 would go for the 25? What would Player 1 do ........? • What should the players do if they wish to “resolve” the game?

  9. I want the maximum payoff to Player II I want the maximum payoff to Player I Player I Player II

  10. Be sensible, guys! I want the maximum payoff to Player II I want the maximum payoff to Player II I want the maximum payoff to Player I I want the maximum payoff to Player I Player I Player I Player II Player II

  11. Basic Issue Sensible = ?

  12. Bottom Line • We need to agree on some fundamental principles to guide us in deciding what is best for the players. • There are two basic issues to be resolved: • What kind of payoffs should the players expect? • How do you guarantee that the players will accept that solution?

  13. Payoff • Ideal: Maximum possible payoff to each player • Minimum Expected: • Each player gets what she can “secure”, namely the maximum that she can get regardless of what the other player does. • We look at the worst things that can happen and pick the best of these. • Conservative approach. • We seek a strategy such that changing it could achieve a worse outcome.

  14. Philosophy • Conservative approach • We wish to be as secure as possible and not take the risk of larger losses in pursuit of larger gains.

  15. A A A 1 2 3 a1 2 - 4 2 5 a2 6 1 0 - 2 0 Old example Ziegfried • Ideal payoff : Player 1 =25, Player 2 = 20 • Maximum SECURED Payoff: Player 1 = –4 ; Player 2 = –6 • i.e. Player 1 can make sure of doing no worse than losing 4, (regardless of what Player 2 does) by playing a1. James Bond

  16. 1.2.2 Definition • The quantity si := min {vij: j=1,2,...,n} , i=1,2,...,m is called the security level for Player I associated with strategy ai . Similarly, the quantity Sj := max {vij: i=1,2,...,m} , j=1,2,...,n is called the security level for Player II associated with strategy Aj .

  17. Principle I • Player I acts to maximize their security level • Player II acts to minimize their security level

  18. A A A 1 2 3 a 1 1 0 3 1 a - 2 5 1 2 a 1 - 8 1 3 1.2.3 Example • How do we compute the security levels?

  19. A A A s 1 2 3 i a 1 1 0 3 1 1 a - 2 5 1 - 2 2 a 1 - 8 1 - 8 3 S 1 1 0 3 j

  20. A A A s 1 2 3 i a 1 1 0 3 1 1 a - 2 5 1 - 2 2 a 1 - 8 1 - 8 3 S 1 1 0 3 j • Will the players accept this solution? • We apply the same idea to another example: • Maximum security level for Player I = 1. • Minimum security level for Player II = 1. • Suggested solution (a1, A1) .

  21. A A s 1 2 i a 1 5 1 1 a 6 2 2 2 S 6 5 j 1.2.4 Example • The suggested solution (a2, A2) is not stable.

  22. 1.2.2 Definition (Page 10) • A solution (ai, Aj) to a zero-sum two-person game is stable (or in equilibrium) if Player I expecting Player II to Play Aj has nothing to gain by deviating from ai AND Player II expecting Player I to Play ai has nothing to gain by deviating from playing Aj.

  23. Principle II • The players tend to strategy pairs that are in equilibrium, i.e. stable An optimal solution is said to be reached if neither player finds it beneficial to change their strategy.

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