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Two-Person, Zero-Sum Game: Advertising 0 -.6 -.4-1.60.2 -.4.4 -.20 -.6 1.4.60 0 TV N TVN 0 TV N TVN Column Player: Row Player: Matrix of Payoffs to Row Player: Row Minima: Column Maxima:1.4.60 -.6 -.4 -.6 0 MaxiMin MiniMax Game has a saddle point!

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Two-Person, Zero-Sum Game: Mixed Strategies 0 5 10 -2 Column Player: Row Player: Matrix of Payoffs to Row Player: X1R1 X2R2 C1 C2 0 -2 105Column Maxima: Row Minima: MaxiMin MiniMax MaxiMin MiniMax No Saddle Point! Y1Y2

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Graphical Solution 01 X1 VR 10 12/17 50/17 VR < 10(1-X1) VR < -2 +7X1 Optimal Solution: X1=12/17, X2=5/17 VRMAX=50/17

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Graphical Solution 01 X1 VR 10 12/17 50/17 VR < 10(1-X1) VR < -2 +7X1 Optimal Solution: X1=12/17, X2=5/17 VRMAX=50/17 Y1=1 Y1=0 Y1=.75 Y1=.5 Y1=.25

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Two-Person, Zero-Sum Game: Mixed Strategies MODEL: SETS: ROWS/1..2/:X; COLS/1..2/; MATRIX(ROWS,COLS):REW; !REW(I,J) IS THE REWARD MATRIX FOR THE ROW PLAYER; ENDSETS @FOR(COLS(J):@SUM(ROWS(I):REW(I,J)*X(I))>V;); @SUM(ROWS(I):X(I))=1; MAX=V; @FREE(V); DATA: REW=0,5, 10,-2; ENDDATA END

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Two-Person, Zero-Sum Game: Mixed Strategies Optimal solution found at step: 1 Objective value: 2.941176 Variable Value Reduced Cost V 2.941176 0.0000000 X( 1) 0.7058824 0.0000000 X( 2) 0.2941176 0.0000000

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Reward Matrix for Two-Finger Morra 0 2 -30 -200 3 3 00 -4 0 0 -34 (1,1) (1,2) (2,1) (2,2) Column Player:Row Player: Row Minimum: Column Maximum:32433243 -3 -2 -4 -3 (1,1) (1,2) (2,1) (2,2)

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Two-Person, Zero-Sum Game: Mixed Strategies MODEL: !TWO FINGER MORRA GAME; SETS: ROWS/1..4/:X; COLS/1..4/; MATRIX(ROWS,COLS):REW; !REW(I,J) IS THE REWARD MATRIX FOR THE ROW PLAYER; ENDSETS @FOR(COLS(J):@SUM(ROWS(I):REW(I,J)*X(I))>V;); @SUM(ROWS(I):X(I))=1; MAX=V; @FREE(V); DATA: REW=0,2,-3,0, -2,0,0,3, 3,0,0,-4, 0,-3,4,0; ENDDATA END

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Two-Person, Zero-Sum Game: Mixed Strategies Optimal solution found at step: 3 Objective value: 0.0000000E+00 Variable Value Reduced Cost V 0.0000000 0.0000000 X( 1) 0.0000000 0.1428571 X( 2) 0.6000000 0.0000000 X( 3) 0.4000000 0.0000000 X( 4) 0.0000000 0.0000000

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Solving Two-Person Zero-Sum Games 1. Check for a saddle point. If none, go to Step 2. 2. Simplify using iterative dominance. Go to Step 3. 3.If either the number of remaining rows or columns is equal to two the solution can be obtained graphically. Otherwise solve using linear programming methods, e.g. with LINGO.

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Nonzero-Sum Game: Prisoner’s Dilemma (2,2)(0,3) (3,0)(1,1) Harry’s Choices H1 H2 Confess Do not confess S1 Confess Sam’s choices S2 Do not confess Payoffs to (Sam, Harry) (years in prison)

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Nonzero-Sum Game: Prisoner’s Dilemma (2,2)(0,3) (3,0)(1,1) Harry’s Choices H1 H2 Defect Cooperate S1 Defect Sam’s choices S2Cooperate Payoffs to (Sam, Harry) (years in prison)

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Nonzero-Sum Game: Radial Tire Ads on Monday Night Football (0,0)(-2,3) (3,-2) (-1,-1) Sears’s Choices S1 S2 No Yes (Cooperate)(Defect) G1 No (Cooperate) Goodyear’s choices G2 Yes (Defect) Payoffs to (Goodyear,Sears) ($, millions)

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Nonzero-Sum Game: Terminology Dominate Outcome: Dominate Outcome: An outcome that is better for both players than any other Pareto Optimality: Pareto Optimality: Property of an outcome that is not dominated by any other Defect: Defect: To not trust the other player; to consider self-interest only Cooperate: Cooperate: To trust the other player; to seek mutual benefit

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Nonzero-Sum Game: Prisoner’s Dilemma Essential Structure: (Cooperate, Cooperate) (Defect, Cooperate) (Cooperate, Defect) Pareto Optimal (Defect, Defect)Not Pareto Optimal No Dominate Outcome Advantage: Defect

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Battle of the Sexes Prize Fight Ballet Husband Wife 2, 0 0, 2 0, 0 -1, -1 Payoffs to (wife, husband) (pleasure)

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Nonzero-Sum Game: Introduction of New Product (Battle of the Sexes) (0,0)(0,2) (2,0) (-3,-3) Boston C1 C2 No Yes R1No American R2Yes Payoffs to (American,Boston) ($, millions)

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Battle of the Sexes Essential Structure: Two Equilibrium Pairs with different returns to the two players One-time optimal strategy: Deception Repeated-choice optimal strategy: Alternate

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