Two-Person, Zero-Sum Game: Advertising

Two-Person, Zero-Sum Game: Advertising Matrix of Payoffs to Row Player: Column Player: Row Minima: 0 TV N TVN 0 TV N TVN 0 -.6 -.4 -1 .6 0 .2 -.4 .4 -.2 0 -.6 1 .4 .6 0 -.6 -.4 -.6 0 Row Player: MaxiMin Column Maxima: 1 .4 .6 0 MiniMax Game has a saddle point!

Two-Person, Zero-Sum Game: Mixed Strategies Column Player: Matrix of Payoffs to Row Player: Row Minima: Y1 Y2 C1 C2 MaxiMin X1 R1 X2 R2 0 5 10 -2 0 -2 Row Player: Column Maxima: 10 5 MiniMax MiniMax MaxiMin No Saddle Point!

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

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

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

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

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

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. • 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 Harry’s Choices H1 H2 Confess Do not confess S1 Confess Sam’s choices S2 Do not confess (2,2) (0,3) (3,0) (1,1) Payoffs to (Sam, Harry) (years in prison)

Harry’s Choices H1 H2 Defect Cooperate (2,2) (0,3) (3,0) (1,1) Nonzero-Sum Game: Prisoner’s Dilemma S1 Defect Sam’s choices S2 Cooperate Payoffs to (Sam, Harry) (years in prison)

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

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

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

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

Boston C1 C2 No Yes (0,0) (0,2) (2,0) (-3,-3) Nonzero-Sum Game: Introduction of New Product (Battle of the Sexes) R1 No American R2 Yes Payoffs to (American,Boston) ($, millions)

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

Two-Person, Zero-Sum Game: Advertising