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UNIT II: The Basic Theory

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  1. UNIT II: The Basic Theory • Zero-sum Games • Nonzero-sum Games • Nash Equilibrium: Properties and Problems • Bargaining Games • Bargaining and Negotiation • Review • Midterm 3/24 2/16

  2. Zero-sum Games • The Essentials of a Game • Extensive Game • Matrix Game • Dominant Strategies • Prudent Strategies • Solving the Zero-sum Game • The Minimax Theorem

  3. The Essentials of a Game 1. Players: We require at least 2 players (Players choose actions and receive payoffs.) 2. Actions: Player i chooses from a finite set of actions, S = {s1,s2,…..,sn}. Player j chooses from a finite set of actions T = {t1,t2,……,tm}. 3. Payoffs: We define Pi(s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We require that Pi(s,t) + Pj(s,t) = 0 for all combinations of s and t. 4. Information: What players know (believe) when choosing actions. ZERO-SUM

  4. The Essentials of a Game 4. Information: What players know (believe) when choosing actions. Perfect Information: Players know • their own payoffs • other player(s) payoffs • the history of the game, including other(s) current action* *Actions are sequential(e.g., chess, tic-tac-toe). Common Knowledge

  5. The Essentials of a Game 4. Information: What players know (believe) when choosing actions. Complete Information: Players know • their own payoffs • other player(s) payoffs • the history of the game, excludingother(s) current action* *Actions are simultaneous(e.g., matrix games). Common Knowledge

  6. Extensive Game “Square the Diagonal” (Rapoport: 48-9) Player 1 chooses a = {1, 2 or 3} Player 2 b = {1 or 2} Player 1 c = {1, 2 or 3} Payoffs = a2 + b2 + c2 if /4 leaves remainder of 0 or 1. -(a2 + b2 + c2) if /4 leaves remainder of 2 or 3. Player1’s decision nodes -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 GAME 1. 3 1 2 Player 2’s decision nodes 1 2 3 1 2

  7. Extensive Game • How should the game be played? • Solution: a set of “advisable” strategies, one for each player. • Strategy: a complete plan of action for every possible decision node of the game, including nodes that could only be reached by a mistake at an earlier node. • Player1‘s advisable • Strategy in red • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. Start at the final decision nodes (in red) 3 1 Backwards-induction 2 1 2 3 1 2

  8. Extensive Game • How should the game be played? • Solution: a set of “advisable” strategies, one for each player. • Strategy: a complete plan of action for every possible decision node of the game, including nodes that could only be reached by a mistake at an earlier node. • Player1‘s advisable • Strategy in red • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. Player1’s advisable strategy in red 3 1 2 Player2’s advisable strategy in green 1 2 3 1 2

  9. Extensive Game • How should the game be played? • If both player’s choose their advisable (prudent) strategies, Player1 will start with 2, Player2 will choose 1, then Player1 will choose 2. The outcome will be 9 for Player1 (-9 for Player2). If a player makes a mistake, or deviates, her payoff will be less. • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 3 1 2 1 2 3 1 2

  10. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 1 1 1 2

  11. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 1 1

  12. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 1 1

  13. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 2 1

  14. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 1 1 2

  15. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 1 2

  16. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 1 2

  17. Extensive Game • Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 2 2

  18. Extensive Game • A Clarification: Rapoport (pp. 49-53) claims Player 1 has 27 strategies. However, if we consider inconsistent strategies, the actual number of strategies available to Player 1 is 37 = 2187. • An inconsistent strategy includes actions at decision nodes that would not be reached by correct implementation at earlier nodes, i.e., could only be reached by mistake. • Since we can think of a strategy as a set of instructions (or program) given to an agent or referee (or machine) to implement, a complete strategy must include instructions for what to do after a mistake is made. This greatly expands the number of strategies available, though the essence of Rapoport’s analysis is correct.

  19. Extensive Game • Complete Information: Players know their own payoffs; other player(s) payoffs; history of the game excluding other(s) current action* *Actions aresimultaneous • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 3 1 Information Sets 2 1 2 3 1 2

  20. Extensive Game • In the extensive form game with complete information, Player 2 has only 2 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. T1 (Always choose 1) 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 1 1 1

  21. Extensive Game • In the extensive form game with complete information, Player 2 has only 2 strategies: • -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 • GAME 1. 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 … and T2 (Always choose 2) 2 2 2

  22. Matrix Game T1 T2 S11 S12 S13 S21 S22 S23 S31 S32 S33 Also called “Normal Form” or “Strategic Game” Solution = {(S22, T1)}

  23. Dominant Strategies Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s). T1 T2 T3 T1 T2 T3 -3 0 1 -1 5 2 -2 2 0 -3 0 -10 -1 5 2 -2 -4 0 S1 S2 S3 S1 S2 S3

  24. Dominant Strategies Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s). T1 T2 T3 T1 T2 T3 Sure Thing Principle: If you have a dominant strategy, use it! -3 0 1 -1 5 2 -2 2 0 -3 0 -10 -1 5 2 -2 -4 0 S1 S2 S3 S1 S2 S3

  25. Prudent Strategies Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i. T1 T2 T3 -3 1 -20 -1 5 2 -2 -4 15 S1 S2 S3 Also called, Maximin Strategy

  26. Prudent Strategies Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i. T1 T2 T3 Player 1’s worst payoffs for each strategy are in red. -3 1 -20 -1 5 2 -2 -4 15 S1 S2 S3

  27. Prudent Strategies Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i. T1 T2 T3 Player 2’s worst payoffs for each strategy are in green. -3 1 -20 -15 2 -2 -4 15 S1 S2 S3

  28. Prudent Strategies Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i. Saddlepoint: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff maxmin = minmax. T1 T2 T3 -3 1 -20 -1 5 2 -2 -415 S1 S2 S3 We call the solution {(S2, T1)} a saddlepoint

  29. Prudent Strategies Saddlepoint: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff maxmin = minmax. -3 1 -20 -15 2 -2 -415

  30. Mixed Strategies Player 1 Player 1 hides a button in his Left or Right hand. Player 2 observes Player 1’s choice and then picks either Left or Right. Draw the game in matrix form. Left Right L R L R -2 4 2 -1 Player 2 GAME 2: Button-Button (Rapoport: 65-73.)

  31. Mixed Strategies Player 1 Player 1 has 2 strategies: L or R. Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  32. Mixed Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  33. Mixed Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  34. Mixed Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  35. Mixed Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -242 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  36. Mixed Strategies Player 1 The game can be solve by backwards-induction. Player 2 will … Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  37. Mixed Strategies Player 1 The game can be solve by backwards-induction. … therefore, Player 1 will: Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  38. Mixed Strategies Player 1 What would happen if Player 2 cannot observe Player 1’s choice? Left Right L R L R -2 4 2 -1 Player 2 GAME 2: Button-Button

  39. Mixed Strategies L R L R Player 1 -2 4 2 -1 Left Right L R L R -2 4 2 -1 Player 2 GAME 2: Button-Button

  40. Solving the Zero-sum Game Mixed Strategy: A mixed strategy for player i is a probability distribution over all strategies available to player i. Let (p, 1-p) = prob. Player I chooses L, R. (q, 1-q) = prob. Player 2 chooses L, R. L R GAME 2. -2 4 2 -1 L R

  41. Solving the Zero-sum Game Then Player 1’s expected payoffs are: EP1(L|p) = -2(p) + 2(1-p) = 2 – 4p EP1(R|p) = 4(p) – 1(1-p) = 5p – 1 L R GAME 2. -2 4 2 -1 L R (p) (1-p) 4 -2 EP EP1(R|p) = 5p – 1 2 -1 (q) (1-q) 0 1 p p*=1/3 EP1(L|p) = 2 – 4p

  42. Solving the Zero-sum Game Player 2’s expected payoffs are: EP2(L|q) = 2(q) – 4(1-q) = 6q – 4 EP2(R|q) = -2(q) + 1(1-q) = -3q + 1 EP2(L|q) = EP2(R|q) => q* = 5/9 L R GAME 2. -2 4 2 -1 L R (p) (1-p) (q) (1-q)

  43. Solving the Zero-sum Game Player 1 EP1(L|p) = -2(p) + 2(1-p) = 2 – 4p EP1(R|p) = 4(p) – 1(1-p) = 5p – 1 Player 2 EP2(L|q) = 2(q) – 4(1-q) = 6q – 4 EP2(R|q) = -2(q) + 1(1-q) = -3q + 1 4 -2 -4 2 -EP2 EP1 2 -1 -2 2 2/3 = EP1* = - EP2* =-2/3 This is the Value of the game. 0 p 1 q q*= 5/9 p*=1/3

  44. Solving the Zero-sum Game Then Player 1’s expected payoffs are: EP(T1) = -2(p) + 2(1-p) EP(T2) = 4(p) – 1(1-p) EP(T1) = EP(T2) => p* = 1/3 And Player 2’s expected payoffs are: (V)alue = 2/3 L R GAME 3. (Security) Value: the expected payoff when both (all) players play prudent strategies. Any deviation by an opponent leads to an equal or greater payoff. -2 4 2 -1 L R (p) (1-p) (q) (1-q)

  45. The Minimax Theorem Von Neumann (1928) Every zero sum game has a saddlepoint (in pure or mixed strategies), s.t., there exists a unique value, i.e., an outcome of the game where maxmin = minmax.

  46. Nonzero-sum Games • Examples: • Bargaining • Duopoly • International Trade

  47. Nonzero-sum Games • The Essentials of a Game • Eliminating Dominated Strategies • Best Response • Nash Equilibrium • Duopoly: An Application • Solving the Game • Existence of Nash Equilibrium • Properties and Problems See: Gibbons, Game Theory for Applied Economists (1992): 1-51.

  48. The Essentials of a Game 1. Players: We require at least 2 players (Players choose actions and receive payoffs.) 2. Actions: Player i chooses from a finite set of actions, S = {s1,s2,…..,sn}. Player j chooses from a finite set of actions T = {t1,t2,……,tm}. 3A. Payoffs: We define Pi(s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We allow that Pi(s,t) + Pj(s,t) = 0. 4. Information: What players know (believe) when choosing actions. NONZERO-SUM

  49. Eliminating Dominated Strategies L M R R is strictly dominated by M, so the game can be reduced to Now, B is strictly dominated by T ... 1,0 1,2 0,1 0,3 0,1 2,0 T B T B 1,0 1,2 0,3 0,1 {(T, M)} 1,0 1,2 T

  50. Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff Pi(s’,t) > Pi(s,t) for all s. A dominated strategy will never be played by a rational player. T1 T2 T3 T1 T2 T3 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1 S2 S3