Presentation on theme: "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."— Presentation transcript:
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!
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
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
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.
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)
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)
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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