GAME THEORY IN OPERATIONS RESEARCH

1. Game Theory
Aman Jindal

2. Definition of game theory
• The branch of mathematics concerned with
the analysis of strategies for dealing with
competitive situations where the outcome of
a participant’s choice of action depends
critically on the actions of other participants.
Game theory has been applied to contexts in
war, business, and biology.

3. Terms Used in Game theory
• Number of Players
• How many players are there?
• If a game involves only two players (competitors), then it
is called a two-person game. However, if the number of
players is more, the game is referred to as n-person game.
• Strategies The strategy for a player is the list of all possible
actions (moves or courses of action) that he will take for every
payoff (outcome) that might arise. It is assumed that the
rules governing the choices are known in advance to the
players.

4. • Payoffs are the consequences for each player
for every possible profile of strategy choices
for all players.
• Zero-sum (or constant-sum) game : one player's
winnings are the others' losses, so the net gain is
zero across all players
• Optimal Strategy The particular strategy by which
a player optimises his gains or losses without
knowing the competitor's strategies.
• Value of game The expected outcome per play
when players follow their optimal strategy.

5. Assumptions of the Game theory
• Each player has available to him a finite
number of possible strategies (courses of
action). The list may not be same for each
player.
• Player A attempts to maximize gains and
player B minimise losses.
• The decision of both players are made
individually prior to the play with no
communication between them.

6. • The decisions are made simultaneously and
also announced simultaneously so that
neither player has an advantage resulting from
direct knowledge of the other player’s
decision.
• Both the players know not only possible
payoffs to themselves but also of each other.

7. Note
• By convention, the payoff table for the player
whose strategies are represented by rows (say
player A) is constructed.

8. Types of Strategies
• Pure Strategy It is the decision rule which is
always used by the player to select the
particular strategy. Thus, each player knows in
advance of all strategies out of which he
always selects only one particular strategy
regardless of the other player’s strategy, and
the objective of the player is to maximize
profit or minimize losses.

9. • Mixed strategy Courses of action that are to
be selected on a particular occasion with
some fixed probability are called mixed
strategies.

10. Pure Strategy
• Maximin – minimax principle
• Maximin Criterion: The player who is
maximizing his outcome or payoff finds out his
minimum gains from each strategy (course of
action) and selects the maximum value out of
these minimum gains.
• Minimax Criterion: In this criterion the
minimizing player determines the maximum
loss from each strategy and then selects with
minimum loss out of the maximum loss list.

11. Example 1
Player
A
Player B
B1 B2 B3
A1
A2
-1 2 -2
6 4 -6
For the game with payoff matrix:
Determine the best strategies for
players A and B. Also determine the
value of game. Is this game (i) fair?
(ii) strictly determinable?

12. Example 1
Player A Player B
B1 B2 B3
Row Minimum
A1
A2
-1 2 -2
6 4 -6
-2
-6
Column Maximum 6 4 -2
Maximin
Minimax
Player A adopts A1 strategy.
Player B adopts B3 strategy.
Value of game V = -2
Not fair but strictly determinable.

13. Saddle Point or Equilibrium Point
• In a payoff matrix the value, which is the
smallest in its row and the largest in the
column, is called the saddle point.

14. Example 2
• A company management and the labour union
are negotiating a new three year settlement.
Each of these has 4 strategies:
(i) Hard and aggressive bargaining
(ii) Reasoning and negotiating approach
(iii)Legalistic strategy
(iv)Conciliatory approach

15. • The cost to the company are given for every
pair of strategy choice
Union Strategies
Company Strategies
I II III IV
I
II
III
IV
20 15 12 35
25 14 8 10
40 2 10 5
-5 4 11 0
What strategy will the two
sides adopt? Also determine
the value of the game.

16. Union Strategies
Company Strategies
I II III IV
I
II
III
IV
20 15 12 35
25 14 8 10
40 2 10 5
-5 4 11 0
Union Strategies
Company Strategies
I II III IV
I
II
III
IV
20 15 12 35
25 14 8 10
40 2 10 5
-5 4 11 0
The company will adopt strategy III
And union will always adopt strategy I.
Value of game V = 12

17. Mixed Strategies
• A method of playing a matrix game in which
the player attaches a probability weight to
each of the possible options, the probability
weights being nonnegative numbers whose
sum is unity, and then operates a chance
device that chooses among the options with
probabilities equal to the corresponding
weights.

18. 1. Odds Method (2X2 matrix)
• If payoff matrix for player A is given by
The following formulae are used to find the
value of game and optimal strategies:
Player A
Player B
B1 B2
A1
A2
a11 a12
a21 a22

19. Player A
Player B
B1 B2
Odds
A1
A2
a1 a2
b1 b2
b1- b2
a1 – a2
Odds a2- b2 a1 – b1
Value V ={a1( b1 – b2 ) + b1( a1 – a2 )}/{( b1-b2 )+( a1 – a2 )}
Prob. of A1 = ( b1 – b2 )/ {( b1-b2 )+( a1 – a2 )}
Prob. of A2 = ( a1 – a2 ) / {( b1-b2 )+( a1 – a2 )}
Prob. of B1 = ( a2 – b2 ) / {( a2 – b2 )+( a1 – b1 )}
Prob. of B2 = ( a1 – b1 ) / {( a2 – b2 )+( a1 – b1 )}

20. Example 2
• Two players A and B are involved in a game of
matching coins. When there are both heads,
player A wins 100 points and wins 0 when
there are two tails. When there is one head
and one tail, B wins 50 points. Determine the
payoff matrix, the best strategy for both
players A and B. Find the value of game to A.

21. Player A
Player B
H T
Odds
H
T
100 -50
-50 0
50
150
Odds 50 150
Value V = {100 (50) + (-50)(150)} / {(-50) + (150)}
= -12.5
Prob. of A selecting strategy H = 50/200 = 1/4
Prob. of A selecting strategy T = 150/200 = 3/4
Prob. of B selecting strategy H = 50/200 = 1/4
Prob. of B selecting strategy T = 150/200 = 3/4

22. Dominance Method
• Rule 1. If all the elements in a row (say ith
row) of a payoff matrix are less than or equal
to the corresponding elements of the other
row (say jth row) then the player A will never
choose the ith strategy or in other words the
ith strategy is dominated by the jth strategy.

23. • Rule 2. If all the elements in a column (say rth
column) of a payoff matrix are greater than or
equal to the corresponding elements of the
other column (say sth column) then the player
B will never choose the rth strategy or in other
words the rth strategy is dominated by the sth
strategy.
• Rule 3. A pure strategy may be dominated if it
is inferior to average of two or more other
pure stategies.

24. Example 3
• Reduce the following game by dominance
method and find the game value:
Player B
Player A
I II III IV
I 3 2 4 0
II 3 4 2 4
III 4 2 4 0
IV 0 4 0 8

25. Example 4
• Using the dominance probability, obtain the
optimal strategies for both the players and
determine the value of the game. The payoff
matrix for player A is given
Player B
Player A
I II III IV V
I 2 4 3 8 4
II 5 6 3 7 8
III 6 7 9 8 7
IV 4 2 8 4 3

26. Graphic Method (mX2 or 2Xn)
Example 5. Solve the game with payoff matrix
Player A
Player B
B1 B2 B3
A1
A2
1 2 0
0 -2 2

27. Lower Envelope
Maximin

28. Example 6
Solve the following game graphically.
A
B
1 2 3 4 5
1
2
-5 5 0 -1 8
8 -4 -1 6 -5

29. Example 7
• Solve the game graphically where payoff
matrix for player A has been prepared:
Player B
Player A
A1 A2 A3 A4 A5
B1
B2
1 5 -7 4 2
2 4 9 -3 1