Two-Person, Zero-Sum Game: Advertising

1
Two-Person, Zero-Sum Game Advertising
Matrix of Payoffs to Row Player
Column Player
Row Minima
0 TV N TVN
0 -.6 -.4 -1 .6 0 .2 -.4 .4
-.2 0 -.6 1 .4 .6 0
0 TV N TVN
-.6 -.4 -.6 0
Row Player
MaxiMin
Column Maxima
1 .4 .6 0
MiniMax
Game has a saddle point!
2
Two-Person, Zero-Sum Game Mixed Strategies
Column Player
Matrix of Payoffs to Row Player
Row Minima
Y1 Y2
C1 C2
MaxiMin
0 5 10 -2
X1 R1 X2 R2
0 -2
Row Player
10 5
Column Maxima
MiniMax
MiniMax
MaxiMin
No Saddle Point!
3
Graphical Solution
VR
10
VR lt 10(1-X1)
VR lt -2 7X1
50/17
Optimal Solution X112/17, X25/17 VRMAX50/17
0
1
X1
12/17
4
Graphical Solution
VR
10
VR lt 10(1-X1)
Y11
Y1.75
Y10
VR lt -2 7X1
Y1.5
50/17
Optimal Solution X112/17, X25/17 VRMAX50/17
Y1.25
0
1
X1
12/17
5
Two-Person, Zero-Sum Game Mixed Strategies
MODEL SETS ROWS/1..2/X COLS/1..2/ MATRIX(ROW
S,COLS)REW !REW(I,J) IS THE REWARD MATRIX FOR
THE ROW PLAYER ENDSETS _at_FOR(COLS(J)_at_SUM(ROWS(I)
REW(I,J)X(I))gtV) _at_SUM(ROWS(I)X(I))1 MAXV _at_
FREE(V) DATA REW0,5, 10,-2 ENDDATA END
6
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
7
Reward Matrix for Two-Finger Morra
Column Player
Row 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
8
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 _at_FOR
(COLS(J)_at_SUM(ROWS(I)REW(I,J)X(I))gtV) _at_SUM(ROW
S(I)X(I))1 MAXV _at_FREE(V) DATA REW0,2,-3,0,
-2,0,0,3, 3,0,0,-4, 0,-3,4,0 ENDDATA END
9
Two-Person, Zero-Sum Game Mixed Strategies
Optimal solution found at step 3
Objective value 0.0000000E00
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
10
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.

1. 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.

11
Nonzero-Sum Game Prisoners Dilemma
Harrys Choices H1 H2 Confess
Do not confess
(2,2) (0,3) (3,0) (1,1)
S1 Confess Sams choices S2 Do not
confess
Payoffs to (Sam, Harry) (years in prison)
12
Nonzero-Sum Game Prisoners Dilemma
S1 Defect Sams choices S2 Cooperate
Payoffs to (Sam, Harry) (years in prison)
13
Nonzero-Sum Game Radial Tire Ads on Monday
Night Football
G1 No (Cooperate) Goodyears choices G2
Yes (Defect)
Payoffs to (Goodyear,Sears) (, millions)
14
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
15
Nonzero-Sum Game Prisoners Dilemma
Essential Structure
(Cooperate, Cooperate) (Defect,
Cooperate) (Cooperate, Defect)
Pareto Optimal
Advantage Defect
(Defect, Defect)
Not Pareto Optimal
No Dominate Outcome
16
Battle of the Sexes
Husband
Prize Fight
Ballet
2, 0
0, 0
Ballet
Wife
0, 2
-1, -1
Prize Fight
Payoffs to (wife, husband) (pleasure)
17
Nonzero-Sum Game Introduction of New Product
(Battle of the Sexes)
R1 No American R2 Yes
Payoffs to (American,Boston) (, millions)
18
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