9.1 Strictly Determined Games

Game theory is a relatively new branch of mathematics designed to help people who are in conflict situations determine the best course of action out of several possible choices. It has applications in the business world, warfare and political science. The pioneers of game theory are John Von Neumann and Oskar Morgenstern.

Von Neumann

Von Neumann's awareness of results obtained by other mathematicians and the inherent possibilities which they offer is astonishing. Early in his work, a paper by Borel on the minimaxproperty led him to develop ... ideas which culminated later in one of his most original creations, the theory of games.

In game theory von Neumann proved the minimax theorem. He gradually expanded his work in game theory, and with co-author Oskar Morgenstern, he wrote the classic text Theory of Games and Economic Behaviour (1944).

Fundamental principle of Game Theory

1. A matrix game is played repeatedly.

2. Player R tries to maximize winnings.

3. Player C tries to minimize losses.

Two-person zero-sum matrix

1. R chooses (plays) any one of m rows.

2. C chooses (plays) any one of m columns.

An example

Suppose you have $10,000 to invest for a period of 5 years. After some investigation and advice from a financial counselor, you arrive at the following game matrix where you ( R ) are playing against the economy ( C). Each entry in the matrix is the expected payoff after 5 years for an investment of $10,000 in the corresponding row designation with the future state of the economy in the corresponding column section. The economy is regarded as a rational player who can make decisions against the investor – in any case, the investor would like to do the best possible irrespective of what happens to the economy. Find saddle values and optimal strategies for each player.

Finding the saddle point(s) if they exist 1. R strategy: circle the lowest number in each row (worst case scenario)

C’s strategy: Put a square around the greatest value of each column. There are two saddle values located in the first row. So R should choose the first row. C (the economy) can either fall or have no change. In either case the gain of R is the corresponding loss to the economy. The value of the game is 5870 and since this value is not equal to zero, the game is considered to be not fair. .

9.2 Mixed Strategy Games

In this section, we look at non-strictly determined games. For these type of games the payoff matrix has no saddle points.

Non-strictly determined matrix games

A strategy consisting of possible moves and a probability distribution (collection of weights) which corresponds to how frequently each move is to be played. A player would only use a mixed strategy when she is indifferent between several pure strategies, and when keeping the opponent guessing is desirable - that is, when the opponent can benefit from knowing the next move.

Penny matching game

Two players R and C each have a penny , and they simultaneously choose to show the side of the coin of their choice. (H = heads, T = tails) If the pennies match, R wins ( C loses) 1cent. If the pennies do not match, R loses (C wins) 1 cent. In terms of a game matrix , we have

Because there is no saddle point for the penny-matching game, there is no pure strategy for R. We will assign probabilities corresponding to the likelihood that R will choose row 1 or row 2. Similarly, probabilities will be found for C’s likelihood of choosing column one or column 2. The strategy will be to choose the row with a probability that will yield the largest expected value.

Determination of Strategy

Expected Value of a Matrix Game For R

For the matrix game

and strategies

for R and C, respectively, the expected value of the game for R is given by

⎡ ⎤= ⎢ ⎥⎣ ⎦

a bM

c d

[ ]= 1 2P p p⎡ ⎤

= ⎢ ⎥⎣ ⎦

1

2

qQ

q

=( , )E P Q PMQ

Fundamental Theorem of Game Theory (The number v is the value of the game. If v = 0, the game is said to be fair. )

For every m x n matrix game , M , there exists strategies and for R and C , respectively, and a unique number v such that

for every strategy Q of C and

for every strategy P of R.

*P *Q

≥*P MQ v

≤*PMQ v

Solution to a 2 x 2 Non-strictly Determined Matrix Game

For the non-strictly determined game

the optimal strategies and and the value of the game are given by

= =

where

⎡ ⎤= ⎢ ⎥⎣ ⎦

a bM

c d

*P *Q

⎡ ⎤= ⎣ ⎦* * *

1 2P p p ⎡ ⎤= ⎢ ⎥⎣ ⎦

** 1

*2

qQ

q

− −⎡ ⎤⎢ ⎥⎣ ⎦

d c a bD D

−⎡ ⎤⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎣ ⎦

d bD

a cD−

=ad bcv

D

= + − +( ) ( )D a d b c

Original problem :

1. Identify a,b,c,d : a = 1, b = -1, c= -1, d = 1. 2. Find D: (a+d)-(b+c)=4 (not zero)3. The value of the game is v =

v = 0 so it is a fair game 4. Find = = [ 0.5 , 0.5]

5. Find = =

6. Find the expected value of game: E(P,Q)=PMQ=0

−=

ad bcvD

⎡ ⎤= ⎣ ⎦* * *

1 2P p p − −⎡ ⎤⎢ ⎥⎣ ⎦

d c a bD D

⎡ ⎤= ⎢ ⎥⎣ ⎦

** 1

*2

qQ

q

−⎡ ⎤⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎣ ⎦

d bD

a cD

⎡ ⎤⎢ ⎥⎣ ⎦

0.50.5

9.3 Linear programming and 2 x 2 games : A geometric approach

This section will introduce the method of solving a non-strictly determined matrix game without recessive rows or columns. All such games can be converted into linear programming problems. The method applies to a matrix game M that has all positive payoffs.

The method of this section will be illustrated by an example.

For the payoff matrix M

find the optimal strategies for the two players.

−⎡ ⎤=⎢ ⎥−⎣ ⎦

2 41 3

M

Continued ….

Add 5 to each entry of M to make all values positive:

Minimize y subject to the given constraints:

⎡ ⎤=⎢ ⎥

⎣ ⎦

7 14 8

new M = = +

+ ≥

+ ≥

≥

1 2

1 2

1 2

1 2

1

11

, 0

y x xv

ax cxbx dxx x

Continued…

Substitute the values for a, b, c, d into the inequalities…

.= +1 2y x x

+ ≥

+ ≥

≥

1 2

1 2

1 2

7 4 11 8 1

, 0

x xx x

x x

Solve the linear programming problem

Solution is (2/26, 3/26)

Value of the game, v and probability values

The value of the game is given by the equation

The value of the original matrix with negative values is 26/5 – 5 = 1/5

- This is the probability matrix for R:

= =+1 2

1 265

vx x

[ ] ⎡ ⎤= = ⎢ ⎥⎣ ⎦*

1 22 35 5

p vx vx

Solve the second linear programming problem to find the probability matrix for the second player

- -

= +

+ ≤

+ ≤

≥

1 2

1 2

1 2

1 2

7 1 14 8 1

, 0

y z zz zz z

z z

Value of the game and probability matrix for second player

- -

= =+1 2

1 5210

vz z

[ ] ⎡ ⎤= = ⋅ ⋅⎢ ⎥⎣ ⎦⎡ ⎤= ⎢ ⎥⎣ ⎦

*1 2

52 7 52 310 52 10 52

7 310 10

q vz vz

9.4 Linear programming and m x n Games: Simplex Method and the Dual Problem

In this section, the process of solving 2 x 2 matrix games will be generalized to solving m x n matrix games. The procedure will be essentially the same as the process for the 2 x 2 case, but the solution of the linear programming problem will incorporate the simplex method and the dual.

Procedure:

Given the non-strictly determined matrix game M, free of recessive rows and columns,

to find and vproceed as follows:

1. If M is not a positive matrix, add a suitable positive constant k to each element of M to get a new matrix M1

If v1 is the value of game M1 , then the value of the original game M is given by v = v1 – k

⎡ ⎤= ⎢ ⎥⎣ ⎦

1 2 3

1 2 3

r r rM

s s s[ ]=*

1 2P p p⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

1*

2

3

qQ q

q

⎡ ⎤= ⎢ ⎥⎣ ⎦

1 2 31

1 2 3

a a aM

b b b

Procedure continued:

2. Set up the two linear programming problems ( maximization problem is always the dual of the minimization problem) :

A) Minimize

subject to :

B) Maximize:

subject to:

= +

+ ≥+ ≥+ ≥

≥

1 2

1 1 1 2

2 1 2 2

3 1 3 2

1 2

111

, 0

y x xa x b xa x b xa x b xx x

= + +

+ + ≤+ + ≤

≥

1 2 3

1 1 2 2 3 3

1 1 2 2 3 3

1 2 3

11

, , 0

y z z za z a z a bb z b z b z

z z z

Procedure continued:

Step 3. Solve the maximization problem, part (B) , the dual of part (A), using the simplex method as modified in section 5.5. [You will automatically obtain the solution of the minimization problem, Part A, as well, by following this process. ]

Step 4. Use the solutions from the third step to find the value of the game , v1 for game M1 and the optimal strategies and value

V for the original game, M.

= =+ +1

1 2 3

1 1vy z z z

[ ]=*1 1 1 2P v x v x

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

1 1*

2 2

3 3

v zQ v z

v z

= −1v v k

An example

Suppose that an investor wishes to invest $10,000 in long and short term bonds, as well as in gold, and he is concerned about inflation. After some analysis he estimates that the return (in thousands of dollars) at the end of a year will be indicated in the following payoff matrix:

Inflation rateup 3% down 3%

gold

long term bonds

short-term bonds

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

3 33 21 1

Example continued

Assume that fate is a very good player that will attempt to reduce the investor’s return as much as possible. Find the optimal strategies for both the investor and fate. What is the value of the game? 1. We start with the payoff matrix and need to make all entries positive so we choose to add a constant k = 4 to each entry:

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

3 33 21 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

7 12 63 5

Example continued

2. Write the corresponding linear programming problems:

subject to:

Maximize:

subject to:

+ + ≥+ + ≥

≥

1 2 3

1 2 3

1 2 3

7 2 3 11 6 5 1

, , 0

x x xx x x

x x x

= + +1 2 3min y x x x

= +1 2y z z

+ ≤

+ ≤

+ ≤

≥

1 2

1 2

1 2

1 2

7 12 6 13 5 1

, 0

z zz zz z

z z

Example continued:

3. Introduce slack variables and form the simplex tableau and solve the second linear programming problem:

+ + =+ + =+ + =

− − + =

1 2 1

1 2 2

1 2 3

1 2

7 12 6 13 5 1

0

z z xz z xz z xz z y

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥− −⎣ ⎦

1 2 1 2 3

7 1 1 0 0 0 12 6 0 1 0 0 13 5 0 0 1 0 11 1 0 0 0 1 0

z z x x x y

Solution:

After performing the steps, the final solution is displayed below: The value of the game is zero, which means it is a fair game.

[ ]=

⎡ ⎤= ⎢ ⎥⎣ ⎦

=

*

*

0.25 0 0.75

0.50.5

0

P

Q

v