MATH 166 Spring 2016
c
Wen
Liu
G.1
Chapter G Game Theory
G.1 Decision Making
Definition:
• A game is called a two-person constant-sum game if the following are satisfied:
– There are two players (called the row player and the column player).
– The row player must choose one of the m row strategies and simultaneously, the column
player must choose one of the n column strategies.
– If the row player chooses her ith strategy and the column player chooses his jth strategy,
the row player receives aij and the column player receives a given constant less aij .
• The m × n matrix A = (aij ) containing the payoffs when the row player chooses the strategy
in row i and the column player chooses the strategy in column j is called the payoff matrix.
Note: The payoff matrix is constructed from the point of view of the row player.
Example: Consider a game with the following payoff

1 −2 −5
 6 1
2
0 −3 3
matrix

1
3 
−6
where the element in the ith row and jth column represents the amount in dollars that the row player
wins from the column player. (A negative win means the row player loses this amount.) Thus if the
row player chooses row 1 and the column player chooses column 1, then the row player wins $1 from
the column player. If the row player chooses row 1 and the column player chooses column 2, then
the row player loses $2 from the column player. Find the best strategies for each player.
Definition:
• The value $1 in the previous game is called the payoff value. If the payoff value is zero, the
game is called fair. If the value is positive, the row player is favored. If the payoff value is
negative, then the column player is favored.
• Consider a payoff matrix A. If there is an entry, ahk , that is the minimum in its row (row h)
and also the maximum in its column (column k), we call the entry ahk , a saddle point.
• A game with a saddle point is said to be strictly determined.
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MATH 166 Spring 2016
G.1
c
Wen
Liu
Locating a Saddle Point (The m & M method)
• For each row, find the minimum value and place a small “m” next to it.
• For each column, find the maximum value and place a capital “M ” next to it.
• Any entry with both a small “m” and a capital “M ” next to it is a saddle point.
Examples:
1. Determine if the game with the following payoff matrix is strictly determined. If it is, find the
value of the game and the optimal strategies of each of the players.

−3 0 2
A= 1 2 0 
3 1 −1

2. Find all saddle points for the game with the following payoff matrix:


0 1 1 0
B= 1 1 2 1 
1 2 3 1
3. Two players, called Odd and Even, simultaneously put out either 1 or 2 fingers. If the sum of
all the fingers from both players is odd, then Odd wins $1 from Even. If the sum of all the
fingers from both players is even, then Odd loses $1 to Even. Let Odd be the row player and
Even be the column player. Find the payoff matrix and determine if there is a saddle point.
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