Game Theory Basics - Department of Mathematics

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32 CHAPTER 2. GAMES AS TREES AND IN STRATEGIC FORMcombinations of moves that can arise in figure 2.1 by backward induction are, simplylisting the moves for each player: (XQ,bd), (XQ,cd), and (Y Q,bd). Note that Q andd are always chosen, but that Y can only be chosen in combination with the move b byplayer II.The moves determined by backward induction are therefore, in general, not unique,and possibly interdependent.Backward induction gives a unique recommendation to each player if there is alwaysonly one move that gives maximal payoff. This applies to generic games. A generic gameis a game where the payoff parameters are real numbers that are in no special dependencyof each other (like two payoffs being equal). In particular, it should be allowed to replacethem with values nearby. For example, the payoffs may be given in some practical scenariowhich has some “noise” that effects the precise value of each payoff. Then two suchpayoffs are equal with probability zero, and so this case can be disregarded. In genericgames, the optimal move is always unique, so that backward induction gives a uniqueresult.2.6 Strategies and strategy profilesDefinition 2.1 In a game tree, a strategy of a player specifies a move for every decisionnode of that player. A strategy profile is a tuple of strategies, with one strategy for eachplayer of the game.If the game has only two players, a strategy profile is therefore a pair of strategies,with one strategy for player I and one strategy for player II.In the game tree in figure 2.1, the possible strategies for player I are XP, XQ, Y P, Y Q,ZP, ZQ. The strategies for player II are ad, ae, bd, be, cd, ce. For simplicity of notation,we have thereby specified a strategy simply as a list of moves, one for each decision nodeof the player. When specifying a strategy in that way, this must be done with respect to afixed order of the decision nodes in the tree, in order to identify each move uniquely. Thismatters when a move name appears more than once, for example in figure 2.2(b). In thattree, the strategies of player I are AA, AP, PA, PP, with the understanding that the firstmove refers to the left decision node and the second move to the right decision node ofplayer I.Backward induction defines a move for every decision node of the game tree, andtherefore for every decision node of each player, which in turn gives a strategy for eachplayer. The result of backward induction is therefore a strategy profile.⇒ Exercise 2.1 on page 52 studies a game tree which is not unlike the game in figure2.1. You can already answer part (a) of this exercise, and apply backward induction,which answers (e).
2.7. GAMES IN STRATEGIC FORM 332.7 Games in strategic formAssume a game tree is given. Consider a strategy profile, and assume that players moveaccording to their strategies. If there are no chance moves, their play leads to a unique leaf.If there are chance moves, a strategy profile may lead to a probability distribution on theleaves of the game tree, with resulting expected payoffs. In general, any strategy profiledefines an expected payoff to each player (which also applies to a payoff that is obtaineddeterministically, where the expectation is computed from a probability distribution thatassigns probability one to a single leaf of the game tree).Definition 2.2 The strategic form of a game is defined by specifying for each player theset of strategies, and the payoff to each player for each strategy profile.For two players, the strategic form is conveniently represented by a table. The rowsof the table represent the strategies of player I, and the columns the strategies of player II.A strategy profile is a strategy pair, that is, a row and a column, with a corresponding cellof the table that contains two payoffs, one for player I and the other for player II.If m and n are positive integers, then an m × n game is a two-player game in strategicform with m strategies for player I (the rows of the table) and n strategies for player II (thecolumns of the table).(a)13TI0101B0101IIl r0 20 2(b)❅❅ITBII❅l r3 31 10 20 2Figure 2.3Extensive game (a) and its strategic form (b). In a cell of the strategic form,player I receives the bottom left payoff, and player II the top right payoff.Figure 2.3 shows an extensive game in (a), and its strategic form in (b). In the strategicform, T and B are the strategies of player I given by the top and bottom row of the table,and l and r are the strategies of player II, corresponding to the left and right column ofthe table. The strategic form for the game in figure 2.1 is shown in figure 2.4.A game in strategic form can also be given directly according to definition 2.2, withoutany game tree that it is derived from. Given the strategic form, the game is played asfollows: Each player chooses a strategy, independently from and simultaneously with theother players, and then the players receive their payoffs as given for the resulting strategyprofile.
Page 2: iiThe material in this book has bee
Page 5 and 6: vthe game that is not too complicat
Page 7 and 8: Contents1 Nim and combinatorial gam
Page 9 and 10: CONTENTSix6 Geometric representatio
Page 11 and 12: Chapter 1Nim and combinatorial game
Page 13 and 14: 1.4. WHAT IS COMBINATORIAL GAME THE
Page 15 and 16: 1.6. COMBINATORIAL GAMES, IN PARTIC
Page 17 and 18: 1.8. SUMS OF GAMES 7principle of in
Page 19 and 20: 1.9. EQUIVALENT GAMES 9The games th
Page 21 and 22: 1.9. EQUIVALENT GAMES 11It is an ea
Page 23 and 24: 1.10. SUMS OF NIM HEAPS 13Theorem 1
Page 25 and 26: 1.10. SUMS OF NIM HEAPS 15How does
Page 27 and 28: 1.12. FINDING NIM VALUES 17K of G s
Page 29 and 30: 1.12. FINDING NIM VALUES 19by movin
Page 31 and 32: 1.13. EXERCISES FOR CHAPTER 1 21(b)
Page 33 and 34: 1.13. EXERCISES FOR CHAPTER 1 23end
Page 35 and 36: 1.13. EXERCISES FOR CHAPTER 1 25, ,
Page 37 and 38: Chapter 2Games as trees and in stra
Page 39 and 40: 2.4. DEFINITION OF GAME TREES 29IIa
Page 41: 2.5. BACKWARD INDUCTION 31of the ga
Page 45 and 46: 2.8. SYMMETRIC GAMES 35case letters
Page 47 and 48: 2.9. SYMMETRIES INVOLVING STRATEGIE
Page 49 and 50: 2.10. DOMINANCE AND ELIMINATION OF
Page 51 and 52: 2.11. NASH EQUILIBRIUM 41⇒ Exerci
Page 53 and 54: 2.12. REDUCED STRATEGIES 43⇒ In o
Page 55 and 56: 2.13. SUBGAME PERFECT NASH EQUILIBR
Page 57 and 58: 2.13. SUBGAME PERFECT NASH EQUILIBR
Page 59 and 60: 2.14. COMMITMENT GAMES 49of moves d
Page 61 and 62: 2.14. COMMITMENT GAMES 51This game
Page 63 and 64: 2.15. EXERCISES FOR CHAPTER 2 53010
Page 65 and 66: 2.15. EXERCISES FOR CHAPTER 2 55(a)
Page 67 and 68: 2.15. EXERCISES FOR CHAPTER 2 57(b)
Page 69 and 70: Chapter 3Mixed strategy equilibriaT
Page 71 and 72: 3.4. EXPECTED-UTILITY PAYOFFS 61Sec
Page 73 and 74: 3.5. EXAMPLE: COMPLIANCE INSPECTION
Page 75 and 76: 3.6. BIMATRIX GAMES 65Secondly, mix
Page 77 and 78: 3.7. MATRIX NOTATION FOR EXPECTED P
Page 79 and 80: 3.9. THE BEST RESPONSE CONDITION 69
Page 81 and 82: 3.10. EXISTENCE OF MIXED EQUILIBRIA
Page 83 and 84: 3.11. FINDING MIXED EQUILIBRIA 73is
Page 85 and 86: 3.11. FINDING MIXED EQUILIBRIA 75
Page 87 and 88: 3.12. THE UPPER ENVELOPE METHOD 773
Page 89 and 90: 3.12. THE UPPER ENVELOPE METHOD 79G
Page 91 and 92: 3.13. DEGENERATE GAMES 81Figure 3.9
Page 93 and 94:
3.13. DEGENERATE GAMES 83equilibriu
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3.14. ZERO-SUM GAMES 85Proof. Let x
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3.14. ZERO-SUM GAMES 87numbers. (Ot
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3.14. ZERO-SUM GAMES 89of payoffs t
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3.15. EXERCISES FOR CHAPTER 3 91A f
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3.15. EXERCISES FOR CHAPTER 3 93❅
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Chapter 4Game trees with imperfect
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4.4. INFORMATION SETS 97large compa
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4.4. INFORMATION SETS 99full circle
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4.4. INFORMATION SETS 101We can say
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4.5. EXTENSIVE GAMES 103information
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4.7. REDUCED STRATEGIES 105c 1II❅
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4.8. PERFECT RECALL 107❅❅ITBII
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4.8. PERFECT RECALL 109chance1/21/2
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4.9. PERFECT RECALL AND ORDER OF MO
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4.10. BEHAVIOUR STRATEGIES 113payof
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4.10. BEHAVIOUR STRATEGIES 115The c
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4.11. KUHN’S THEOREM: BEHAVIOUR S
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4.11. KUHN’S THEOREM: BEHAVIOUR S
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4.12. SUBGAMES AND SUBGAME PERFECT
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4.13. EXERCISES FOR CHAPTER 4 123sh
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Chapter 5BargainingThis chapter pre
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5.4. BARGAINING SETS 127and vertica
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5.5. BARGAINING AXIOMS 129game. In
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5.6. THE NASH BARGAINING SOLUTION 1
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5.6. THE NASH BARGAINING SOLUTION 1
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5.7. SPLITTING A UNIT PIE 1351vS( )
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5.8. THE ULTIMATUM GAME 137III0II.0
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5.9. ALTERNATING OFFERS OVER TWO RO
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5.9. ALTERNATING OFFERS OVER TWO RO
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5.10. ALTERNATING OFFERS OVER SEVER
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5.10. ALTERNATING OFFERS OVER SEVER
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5.11. STATIONARY STRATEGIES 147I0x1
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5.11. STATIONARY STRATEGIES 149s <
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5.13. EXERCISES FOR CHAPTER 5 151to
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5.13. EXERCISES FOR CHAPTER 5 153(e
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Chapter 6Geometric representation o
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6.4. BEST RESPONSE REGIONS 157and t
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6.4. BEST RESPONSE REGIONS 159and c
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6.4. BEST RESPONSE REGIONS 161label
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6.6. THE LEMKE-HOWSON ALGORITHM 163
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6.6. THE LEMKE-HOWSON ALGORITHM 165
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6.7. ODD NUMBER OF NASH EQUILIBRIA
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6.8. EXERCISES FOR CHAPTER 6 169(a)
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Chapter 7Linear programming and zer
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7.3. LINEAR PROGRAMS AND DUALITY 17
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7.4. THE MINIMAX THEOREM VIA LP DUA
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7.5. GENERAL LP DUALITY 177x ≥ 0y
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7.5. GENERAL LP DUALITY 179These tw
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7.6. THE LEMMA OF FARKAS AND PROOF
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7.6. THE LEMMA OF FARKAS AND PROOF
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7.7. EXERCISES FOR CHAPTER 7 185but
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Game Theory Basics - Department of Mathematics

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