This document provides an overview of game theory and two-person zero-sum games. It defines key concepts such as players, strategies, payoffs, and classifications of games. It also describes the assumptions and solutions for pure strategy and mixed strategy games. Pure strategy games have a saddle point solution found using minimax and maximin rules. Mixed strategy games do not have a saddle point and require determining the optimal probabilities that players select each strategy.

1. Game Theory
OPERES3 Notes

2. Definitions
A game is a generic term, involving conflict
situations of particular sort.
Game Theory is a set of tools and techniques for
decisions under uncertainty involving two or more
intelligent opponents in which each opponent aspires
to optimize his own decision at the expense of the
other opponents. In game theory, an opponent is
referred to as player. Each player has a number of
choices, finite or infinite, called strategies. The
outcomes or payoffs of a game are summarized as
functions of the different strategies for each player.

3. Major Assumptions
1. Players – the number of participants may be
two or more. A player can be a single
individual or a group with the same objective.
2. Timing – the conflicting parties decide
simultaneously.
3. Conflicting Goals – each party is interested
in maximizing his or her goal at the expense
of the other.

4. Major Assumptions
4. Repetition – most instances involve
repetitive solution.
5. Payoff – the payoffs for each combination of
decisions are known by all parties.
6. Information Availability – all parties are
aware of all pertinent information. Each
player knows all possible courses of action
open to the opponent as well as anticipated
payoffs.

5. Classifications of Games
1. Zero-Sum Games – the winner(s)
receive(s) the entire amount of the payoff
which is contributed by the loser (strictly
competitive).
2. Non-Zero Sum Games – the gains of one
player differ from the losses of the other.
Some other parties in the environment may
share in the gain or losses (not strictly
competitive).

6. Two-Person, Zero-Sum Game
– Pure Strategy
Characteristics:
1.There must be two players, each with a finite set
of strategies.
2.Zero-sum implies that the losses of one player is
the exact gain of the other.
3.Pure strategy refers to a prescribed solution in
which one alternative is repeatedly recommended
to each player.
4.Bargaining is not allowed. There could be no
agreement that could be mutually advantageous.

7. Two-Person, Zero-Sum Game
– Pure Strategy
Consider the following game matrix taken
from the point of view of player A.
B1 B2 … Bn
A1 v11 v12 … V1n
A2 v21 v22 … v2n
. . . . .
. . . . .
. . . . .
Am vm1 vm2 … vmn

8. Two-Person, Zero-Sum Game
– Pure Strategy
Example 1:
The labor contract between a company and the union will
terminate in the near future. A new contract must be negotiated.
After a consideration of past experience, the group (Co) agrees
that the feasible strategies for the company to follow are:
C1 = all out attack; hard aggressive bargaining
C2 = a reasoning, logical approach
C3 = a legalistic strategy
C4 = an agreeable conciliatory approach
Assume that the union is considering one of the following set of
approaches:
U1 = all out attack; hard aggressive bargaining
U2 = a reasoning, logical approach
U3 = a legalistic strategy
U4 = an agreeable conciliatory approach

9. Two-Person, Zero-Sum Game
– Pure Strategy
Example 1 (con’t.)
With the aid of an outside mediator, we construct the
following game matrix:
Conditional Gains of Union
Union Strategies Company Strategies
C1 C2 C3 C4
U1 2.0 1.5 1.2 3.5
U2 2.5 1.4 0.8 1.0
U3 4.0 0.2 1.0 0.5
U4 - 0.5 0.4 1.1 0.0

10. Two-Person, Zero-Sum Game
– Pure Strategy
Example 1 (con’t.)
Interpretation of above table or game
matrix
If Co. adopts C1 and Union adopts U1, the final
contract involves a P2.0 increase in wages
(hence, a -P2.0 loss to the company).
From the above table, it is clear that if the
Company decides to adopt C3, Union will adopt
U1. If the Union decides to adopt U3, the
company will adopt C2.

11. Two-Person, Zero-Sum Game
– Pure Strategy
Solution Strategy : Minimax – Maximin
Approach
1. Apply the maximin rule to determine the
optimal strategy for A:
{
max min v ij
j i
[ ]}
2. Aplly the minimax rule to determine the
optimal strategy for B:
{
min max v ij
j i
[ ]}

12. Two-Person, Zero-Sum Game
– Pure Strategy
In the application of the above strategy, the
pure strategy problem results in a saddle
point, i.e., the payoff corresponding to the
maximin rule is identical to the payoff
corresponding to the minimax rule.
Saddle point corresponds to the minimum in
its row and the maximum in its column.

13. Two-Person, Zero-Sum Game
– Pure Strategy
Additional Remarks on Pure Strategy
Problems:
1.Change in Strategy – Since games are
repetitive, both players may change. But in pure
strategy games, there is no incentive to change.
Any player deviating from the prescribed strategy
will usually find a worsening payoff.
2. Multiple Optimal Solutions – Some games
may involve multiple optimal strategies.

14. Two-Person, Zero-Sum Game
– Pure Strategy
Additional Remarks (con’t.)
3. Dominance
Row: The dominating row will have entries which are
larger than and/or equal to (with at least one entry
larger than) to the corresponding entries in the
dominated row.
Column: The dominating column will have entries smaller
than and/or equal to (with at least one entry smaller
than) the dominated column
Dominated rows and columns can be deleted from
the table.

15. Two-Person, Zero-Sum Game
– Pure Strategy
Example 2: Given game matrix showing the
conditional gains of A.
B1 B2 B3 min
A1 7 -1 2 -1
A2 4 4 6 4
A3 6 3 0 0
A4 7 4 5 4
max 7 4 6
Multiple Pure Strategy Solutions: A2, B2 and A4, B2

16. Two-Person, Zero-Sum Game
– Pure Strategy
Example 3: Applying Law of Dominance using game matrix of
Example 1.
U1 dominates U4. U4 can therefore be removed from the
game matrix.
After removing U4, we see that C2 dominates C1. C1 can
likewise be removed from the table.
We are now left with a 3x3 game matrix. This time, we see
that U1 dominates both U2 and U3. U2 and U3 can also be
removed from the table which leaves us with a 1x3 row
vector.
Finally, C3 dominates C2 and C4. This leaves us with a single
value of 1.2 which corresponds to the value under C3 and U1
in the original game matrix.
As we already know, C3 and U1 represents the pure strategy
solution to this game theory problem.

17. Two-Person, Zero-Sum Game
– Pure Strategy
Example 3: Applying Law of Dominance
using game matrix of Example 1.
C1 C2 C3 C4
U1 2.0 1.5 1.2 3.5
U2 2.5 1.4 0.8 1.0
U3 4.0 0.2 1.0 0.5
U4 -0.5 0.4 1.1 0.0

18. Two-Person, Zero-Sum Game
– Mixed Strategy
A mixed strategy problem is one where
players change from alternative to
alternative when the game is repeated.
A mixed strategy problem does not
yield a saddle point.

19. Two-Person, Zero-Sum Game
– Mixed Strategy
Assumptions in Mixed Strategy Problems:
1. The players practice a maximum secrecy with
their plans so that the opponent will not guess
their move.
2. The average payoff is determined by the fraction
of the time that each of the alternatives is played
and there is a certain fraction that is best for each
player.
3. The best strategy for a mixed strategy game is a
random selection of alternatives which conform in
the long run to predetermined proportions.

20. Two-Person, Zero-Sum Game
– Mixed Strategy
Example 4: Using Example 1 but replacing (U3,C3)
value by 1.9.
C1 C2 C3 C4 Min
U1 2.0 1.5 1.2 3.5 1.2
U2 2.4 1.4 0.8 1.0 0.8
U3 4.0 0.2 1.9 0.5 0.2
U4 -0.5 0.4 1.1 0.0 -0.5
max 4.0 1.5 1.9 3.5

21. Two-Person, Zero-Sum Game
– Mixed Strategy
Example 4: (con’t.)
The intersection of these strategies (U1 and C2) is
not an equilibrium or saddle point because 1.5
does not represent both the maximum of its
column and the minimum of its row.
Interpretation: From the above game matrix,
we can see that:
If the Union adopts U1, the Company will adopt C3.
If the Company adopts C3, the Union will adopt U3.
If the Union adopts U3, the Company will adopt C2.
If the Company adopts C2, the Union will adopt U1.
The shift from alternative to alternative becomes a cycle
when the Union goes back to adopt U1.0

22. Two-Person, Zero-Sum Game
– Mixed Strategy
Let
xi = proportion of the time that player A
plays strategy i
yj = proportion of the time that player B
plays strategy j
  m
 x ≥ 0, ∑ x = 1 that will
Player A then selects xi  i i
 i =1
yield
  m m m

max  min  ∑ v i 1 x i , ∑ v i 2 x i , … , ∑ v in x i  
xi
  i =1 i =1 i =1 

23. Two-Person, Zero-Sum Game
– Mixed Strategy
Player B then selects yj that will
 n 
 y j ≥ 0, ∑ y j = 1
 
 j =1 
yield:

  n n n 
min  max  ∑ v1 j y j , ∑ v 2 j y j , … , ∑ v nj y j  
 
yj

  j =1 j =1 j =1 
If xi* and yj* are the optimal solutions
for both players, then the optimal
expected value of the game is:
m n
v =
*
∑∑
i =1 j =1
v ij x i* y *j

24. Two-Person, Zero-Sum Game
– Mixed Strategy
There are several methods for solving
this type of game. It is important to
first use the principle of dominance to
be able to reduce the total number of
alternatives. The above non-linear
optimization model is convertible to a
Linear Programming Model.

25. Two-Person, Zero-Sum Game
– Mixed Strategy
Games Reducible to a 2x2 Matrix
By employing the principle of dominance, it
may be possible to reduce the size of a
game theory problem to a 2x2 matrix.
For player A, the optimal strategy involves
the simultaneous solution of:
x 1 v 11 + x 2 v 21 = x 1 v 12 + x 2 v 22
x1 + x 2 = 1

26. Two-Person, Zero-Sum Game
– Mixed Strategy
Games Reducible to a 2x2 Matrix (con’t)
For player B, the optimal strategy involves
the simultaneous solution of:
y 1 v 11 + y 2 v 12 = y 1 v 21 + y 2 v 22
y1 + y 2 = 1

27. Two-Person, Zero-Sum Game
– Mixed Strategy
Example 5: Using data from Example 4, Reduce
the original game matrix using the principle of
row and column dominance and determine the
mixed strategy solution
C1 C2 C3 C4
U1 2.0 1.5 1.2 3.5
U2 2.4 1.4 0.8 1.0
U3 4.0 0.2 1.9 0.5
U4 -0.5 0.4 1.1 0.0

28. Two-Person, Zero-Sum Game
– Mixed Strategy
Solution of (mxn) Games by Linear
Programming
As given previously, the following optimization
model solves for the optimal strategy of Player
A:
 m m m

maxmin ∑vi1 xi , ∑vi 2 xi ,…, ∑vin xi 
xi
  i=1 i =1 i =1 
m
s .t . ∑x
i =1
i =1
xi ≥ 0 ∀i

29. Two-Person, Zero-Sum Game
– Mixed Strategy
Solution of (mxn) Games by Linear
Programming (con’t.)
This model can be converted to linear
programming using the following:
Let
 m m m

v = min ∑ vi1 xi , ∑ vi 2 xi ,…, ∑ vin xi 
 i =1 i =1 i =1 

30. Two-Person, Zero-Sum Game
– Mixed Strategy
Solution of (mxn) Games by Linear
Programming (con’t.)
Then, the LP model is given by:
Max Z = v
m
s .t . ∑v
i =1
ij xi ≥ v ∀j
m
∑i =1
xi = 1
xi ≥ 0 ∀i

31. Two-Person, Zero-Sum Game
– Mixed Strategy
Solution of (mxn) Games by Linear
Programming (con’t.)
Assuming that v>0, we divide all
constraints by v and let Xi=xi/v. Since
1
max v ≡ min , the model for Player A becomes:
v
m
min z = ∑
i=1
X i
m
s .t . ∑
i=1
v ij X i ≥ 1 ∀ j
X i ≥ 0 ∀ i

32. Two-Person, Zero-Sum Game
– Mixed Strategy
Solution of (mxn) Games by Linear
Programming (con’t.)
Using the same principle, player B’s
optimization problem is given by:
n
max w = ∑j =1
Y j
n
s .t . ∑
j =1
v ij Y j ≤ 1 ∀i
Y j ≥ 0 ∀j

33. Two-Person, Zero-Sum Game
– Mixed Strategy
Solution of (mxn) Games by Linear
Programming (con’t.)
Note: In cases where the payoff matrix
contains negative payoffs, we scale up all
entries by adding a fixed number T which
will render all values non-negative. Scaling
does not affect the optimal solution except
to increase its value by T.

34. Two-Person, Zero-Sum Game
– Mixed Strategy
Example 6: Using the game matrix below,
find the mixed strategy solution.
y1 y2 y3
x1 6 -4 -14
x2 -9 6 -4
x3 1 -9 1