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© 2015 McGraw-Hill Education. All rights reserved. Introduction Game theory applies mathematics to competitive situations –Places emphasis on the decision-making of the adversaries –Commonly used in business and economics 1994 and 2005 Nobel Prizes for Economic Sciences were given for research in game theory 2

© 2015 McGraw-Hill Education. All rights reserved. 15.1 The Formulation of Two-Person Zero- Sum Games Zero sum games –The net of the winnings is zero –The amount won by one player is lost by the other player Example: Odds and Evens –Game consists of each player simultaneously showing either one finger or two fingers –If the fingers match, the player taking evens wins the match If not, the player taking odds wins the bet 3

© 2015 McGraw-Hill Education. All rights reserved. The Formulation of Two-Person Zero-Sum Games The two strategies of Odds and Evens players –Player shows either one finger or two fingers Payoff table showing payoff to player 1 if player 1 takes evens 4

© 2015 McGraw-Hill Education. All rights reserved. The Formulation of Two-Person Zero-Sum Games For more complicated games involving a series of moves: –Strategy determines the response in each circumstance The primary objective of game theory –To develop rational criteria for strategy selection –Key assumptions Both players are rational, and both choose a strategy to win 5

© 2015 McGraw-Hill Education. All rights reserved. Two politicians running against each other –Both want to campaign in Bigtown and Megalopolis –Both want to travel at night and spend either one full day in each city or two full days in one city –Neither politician will learn the other’s strategy in advance 6 15.2 Solving Simple Games – A Prototype Example

© 2015 McGraw-Hill Education. All rights reserved. Possible strategies for each player –Spend one day in each city –Spend two days in Bigtown –Spend two days in Megalopolis Problem becomes more complicated if: –Player knows what action the other player will take on the first day before finalizing plans for the second day 7 Solving Simple Games – A Prototype Example

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Table15.2 shows form of the payoff table –Entries are in thousands of total net votes 8

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Method of dominated strategies –Rule out a succession of inferior strategies until only one remains Variation 1 of the problem 9

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Variation 1 of the problem (cont’d.) –The table includes no dominated strategies for player 2 –For player 1, strategy 3 is dominated by strategy 1 Strategy 1 has larger payoffs regardless of what player 2 does Eliminate strategy 3 from the payoff table 10

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Variation 1 of the problem (cont’d.) –For reduced payoff table, player 2 has a dominated strategy Strategy 3 is dominated by both 1 and 2 Payoff table is further reduced to: –Strategy 2 becomes dominated by strategy 1 for player 1 at this point Strategy 2 should be eliminated 11

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Variation 2 of the problem –No dominated strategies exist 12

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Variation 2 of the problem –Minimax criterion Player selects moves to minimize maximum losses where resulting choice cannot be exploited by opponent to improve his position –Player 1 should select the strategy whose minimum payoff is the largest –Player 2 should select the strategy whose maximum payoff to player 1 is the smallest 13

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Variation 2 of the problem (cont’d.) –Same entry in the payoff table yields both the maximin and minimax values Saddle point Represents a stable solution (equilibrium solution) 14

© 2015 McGraw-Hill Education. All rights reserved. Solving Simple Games – A Prototype Example Variation 3 of the problem (cont’d.) –Game does not have a stable solution –Whenever one player’s strategy is predictable: Opponent can take advantage to improve his position –Essential feature: neither player should be able to deduce the other player’s strategy Make choices randomly 16

© 2015 McGraw-Hill Education. All rights reserved. 15.3 Games with Mixed Strategies Whenever a game does not possess a saddle point: –Game theory advises each player to assign a probability distribution to each strategy –Plans called mixed strategies Original strategies called pure strategies Performance measure for evaluating mixed strategies –Expected payoff 17

© 2015 McGraw-Hill Education. All rights reserved. Games with Mixed Strategies Expected payoff –Indicates what the average payoff will be if the game is played many times Minimax criterion for games lacking a saddle point –Player should select the mixed strategy that minimizes the maximum expected loss to himself 18

© 2015 McGraw-Hill Education. All rights reserved. Games with Mixed Strategies Minimax theorem –If mixed strategies are allowed, the pair of mixed strategies that is optimal according to the minimax criterion provides a stable solution Neither player can do better by unilaterally changing strategy Best approach: randomly select the pure strategy from the probability distribution 19

© 2015 McGraw-Hill Education. All rights reserved. Graphical Solution Procedure For each pure strategy available to player 2, the expected payoff for player 1 will be: Expected payoff lines are plotted on the graph in Figure 15.1 on next slide 21

© 2015 McGraw-Hill Education. All rights reserved. 15.5 Solving by Linear Programming Solution method for mixed strategy games –Transform the problem to a linear programming problem By applying the minimax theorem and using the definitions of minimax and maximin Expected payoff for player 1 24

© 2015 McGraw-Hill Education. All rights reserved. Solving by Linear Programming Strategy (x 1,x 2,…x m ) is optimal if –For all opposing strategies (y 1,y 2,…y n ) 25

© 2015 McGraw-Hill Education. All rights reserved. Solving by Linear Programming The two formulations are duals of each other –Implies optimal mixed strategies for both players can be found by solving only one linear programming problem Because dual solution is an automatic by-product 28

© 2015 McGraw-Hill Education. All rights reserved. 15.6 Extensions Game theory research extends to more complicated games than are covered here –Two-person, constant-sum game –n-person game –Nonzero-sum game –Noncooperative and cooperative games Defined by amount of pre-game communications –Infinite games Decision variable is continuous 29

© 2015 McGraw-Hill Education. All rights reserved. 15.7 Conclusions General problem addressed by game theory –How to make decisions in a competitive environment Game theory provides a framework for formulating and analyzing simple problems A gap exists between what theory can handle and the complexity of most competitive situations 30