Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
CHAPTER
THEORY OF GAMES
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
15
Learning Objectives
• Assumptions, Definitions and Classification of
Games.
• Two-person Zero Sum Games
• Saddle Point and Pure Strategies
• Dominance
• Mixed Strategies
• Graphical Solution of Games
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
Page 3
Theory of Games
• It deals with decision making under
conditions of competition. It deals with
situations where different parties have
opposing interests and both take
decisions so that there interests are
promoted.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Assumptions
• Conflict must be focussed on a central issue. Players
(parties to the conflict) must have opposing interests or the
same objective.
• Players act rationally and intelligently.
• Each player has a finite set of alternatives to choose from.
Players must make simultaneous and concurrent decisions
as to which of the set of alternatives to employ. These
decisions, once made, cannot be reversed during a
particular round of conflict.
• Each player must measure the worth of all possible
outcomes on the same scale.
• Each player must exert some influence to bear on the
situation but must not be able to exert complete control over
the situation by himself.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Definitions
• Person. The term refers to one of the opposing
interests or players. It could be an individual, a group
of individuals or an organisation.
• Game. The set of rules that defines what can and
cannot be done; the payoffs in terms of monetary
values or utility values and pay off methods. Rules
must be complete and no change is permitted during
the game.
• Zero Sum Game. A game in which the total winnings
equal the total losses.
• Play of Game. The choosing of a course of action by
a player along with an exchange of resulting payoffs.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Definitions
• Strategy. A players course of action that is
complete and ready before commencement
of the game.
• Optimal Strategy. A strategy which
guarantees a player the best he can expect
regardless of what the opponent player does.
• Value of the Game. Expected payoff when
each player is using his/her optimal strategy.
• Solution. An optimal strategy for each player
and the value of the game.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Classification of Games
• If a game has two players we refer to it as a
two person game. When the players are three
or more, the game is referred to as an nperson game. Solution of n-person games is
still being developed
• If the sum of total gains and losses is zero,
the game is referred to as a two person zero
sum game.
• If the sum of total gains and losses is not
zero, the game is referred to as a two person
non-zero sum game.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Two-Person Zero Sum Games
• Consider the game:
Strategy X
Player B
Strategy M
Strategy N
B wins 2 points
B wins 1 point
Strategy Y
A wins 3 points
Player A
A wins 4 points
• As can be seen, the game is heavily in favour of A.
Given a choice A will play only Strategy Y, as he will
always win. B will try to minimise his loss by always
playing Strategy M. This is a two-person zero sum
game.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
Page 9
Two-Person Zero Sum Game
• The game is written as a matrix with payoffs always
written from the row player’s point of view.
Player A
Strategy M
Player B
Strategy N
Strategy X
B wins 2 points
B wins 1 point
Strategy Y
A wins 3 points
A wins 4 points
B
 2 1
A

3
4


Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Saddle Point and Pure Strategies
Example
•
•
•
•
Blue and Red Scooters may both indulge in no advertising,
moderate advertising and heavy advertising.
If Blue does no advertising – it will get 50% market share if Red
also does no advertising; 40% market share if Red advertise
moderately and 28% market share if Red advertises heavily.
Blue use moderate advertising – it will get 70% of the market
share with no advertising by Red, 50% of the market share if
Red also resorts to moderate advertising, and 45% of the market
share if Red does heavy advertising.
Blue uses heavy advertising – in such an event it gets 75% of
the market share if Red does not advertise, 55% if Red
advertises moderately and 50% if Red advertises heavily.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Saddle Point and Pure Strategies
Example
•
The problem may be written as:
Red
Blue
None
Moderate
Heavy
None
50
40
28
Moderate
70
50
45
Heavy
75
55
50
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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• Both players will attempt to maximise their gains or minimise their
losses. Blue will choose a criterion of maximin – best of the worst.
• Red will also attempt to minimise his losses by using a similar strategy.
Since the payoffs are written from Blue’s point of view, Red will follow the
criterion of minimax – minimum of the maximums as the matrix
represents loss or regret for him.
Red
Blue
None
Moderate
Heavy
Maximin
None
50
40
28
28
Moderate
70
50
45
45
Heavy
75
55
50
50
Minimax
75
55
50
• Blue should advertise heavily and Red should also advertise heavily.
• This course will always yield a payoff of 50.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Red
Blue
None
Moderate
Heavy
Maximin
None
50
40
28
28
Moderate
70
50
45
45
Heavy
75
55
50
50
Minimax
75
55
50
• There is a single strategy for both players which would
maximise their gains (or minimise their losses). Both players
will play the same strategy every time the game is played.
Each player has a pure strategy, one that he will play all the
time.
• The payoff obtained when each player plays his pure strategy
is called a saddle point, or the saddle point is the value of the
game when each player has a pure strategy.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Mixed Strategies
• When a pure strategy is not available for the players, they will
have to use a mix of strategies.
• It is imperative that the strategies are mixed in such a manner
that the payoffs are optimised.
• When using mixed strategies, the players have to maintain
security of their game plan. It must be ensured that the opponent
does not come to know of the strategy that will be employed. If
he can get prior information, he will choose an appropriate
strategy to counter the one used by the other player.
• The strategies will have to played randomly in the desirable
proportions so that no patterns of play can be established.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
Page 15
Mixed Strategies - Example
Player B
Player
A
b1
B2
a1
1
0
a2
-4
3
• Let p be the proportion of time A plays row a1 , then
(1 – p) is the proportion of time that he plays a2.
• He would expect the same payoff irrespective of
whether B plays b1 or b2.
• Expected Value if B plays b1, p  1  (1  p )  ( 4)  5 p  4
• Expected Value if B plays b2, p  0  (1  p )  3  3  3 p
5 p  4  3  3 p; p 
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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8
, (1  p ) 
1
8
Page 16
Mixed Strategies - Example
Player B
Player
A
b1
B2
a1
1
0
a2
-4
3
• Let q be the proportion of time B plays row b1 , then
(1 – q) is the proportion of time that he plays b2.
• He would expect the same payoff irrespective of
whether A plays a1 or a2.
q  1  (1  q )  0  q
• Expected Value if A plays a1,
• Expected Value if A plays a2, q  4  (1  q )  3  3  7q
q  3  7q; q 
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
3
8
, (1  q ) 
5
8
Page 17
Mixed Strategies - Example
Player B
Player
A
b1
B2
a1
1
0
a2
-4
3
• A should play strategy a1 for 7/8 of the time and a2 for
1/8 of the time.
• B should play strategy b1 for 3/8 of the time and b2 for
5/8 of the time.
• Value of the game is 3/8
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Mixed Strategies – Short Cut
Method
Player B
Player A
b1
b2
a1
1
0
7/8
17
a2
-4
3
1/8
71
3/8
5
3
5/8
53
• Subtract the smaller payoff in each row from the
larger payoff, and the smaller payoff in each column
from the larger payoff.
• Interchange each of these pairs of subtracted numbers
• Divide each pair of numbers by the sum of the pair
• Strategy for A (7/8, 1/8) and for B (3/8, 5/8)
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
Page 19
Dominance
• In a game, it is often possible to find an entire row (or column)
which one player will avoid when there is another row (or
column) which is always better for him to play. In that case, we
say that the row (or column) is dominated by another row (or
column).
• If all values of elements of Row A are greater than corresponding
values of Row B, we can eliminate row B as Row A dominates
Row B. The Row player will never play strategy B, if he has a
choice between strategy A and B as A always gives him more
payoff.
• Similarly, if the value of elements in Column A are all smaller
than corresponding values of Column B, then Column A
dominates Column B and Column B can be eliminated.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
Page 20
Dominance - Example
• Two breakfast food manufacturing firms A and B are
competing for an increased market share. To improve
their market share both the firms have the following
options:
– Give coupons – a1, b1
– Decrease price – a2, b2
– Maintain present strategy – a3, b3
– Increase advertising – a4, b4
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Firm B
Firm A
b1
b2
b3
b4
a1
35
65
25
5
a2
30
20
15
0
a3
40
50
0
10
a4
55
60
10
15
15
10
10/25 15/25
20
5/25
5
20/25
• All payoffs in the row a1 are higher than the payoffs in the row
a2. a1 dominates a2. Eliminate a2
• All payoffs in the row a4 are higher than the payoffs in the row
a3. a4 dominates a3. Eliminate a3
• All payoffs in the column b4 are lower than the payoffs in the
column b1. b4 dominates b1. Eliminate b1
• b4 also dominates column b2.. Eliminate b2
• Solve by short cut method. A’s Strategy (1/5,0,0,4/5) B’s
Strategy (0,0,2/5,3/5). Value of game 13.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
Page 22
Graphical Solution of Games
• Used for solving two person zero sum 2 X n or m X 2 games
( 2 rows and n columns or m rows and 2 columns).
B
A
a1
a2
b1
7
-6
b2
-7
4
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
b3
4
2
b4
-8
6
• Assume that Player A plays
strategy a1 for p proportion of
time and strategy a2 for (1 p) part of the time, then if B
plays strategy b1, A’s payoff
will be: 13p – 6
• Work out for other strategies
of B and plot on a graph.
Page 23
8
6
b1
b4
b3
4
2
P
O0
-2
0
0.2
0.4
P’
-6
-8
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
0.8
b2
-4
-10
0.6
p
1
• Player A’s aim is to
maximise his
minimum payoff.
Consider the lower
shaded envelope
and find the highest
point, which in this
case is P.
• OP gives the value
of p = 5/12
• PP’ is value of game
= -7/12
• P lies on the
intersection of
strategies b1 and b2.
Solve as a 2 X 2
game
Page 24
B
8
6
b1
b4
A
b3
4
P
O0
0
0.2
0.4
P’
-6
-8
-10
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
0.6
0.8
b2
-4
p
-7
10/24
-6
4
14/24
11/24 13/24
2
-2
7
1
• The game reduces
to the one shown
above.
• Solving by short cut
method, strategy for
A is (5/12,7/12) and
for B
(11/24,13/24,0,0)
• Value of the game is
-7/12.
Page 25
Graphical Solution of Games
• If the game is of the form m X 2, the procedure remains the
same, except that the upper envelope is considered, as the
column player is minimising his maximum loss.
Player B
Player A
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
b1
b2
a1
7
-5
a2
2
6
a3
-6
3
a4
8
-1
Page 26
10
8
6
a2
Q
4
a3
2
0O
0
-2
-4
a4
0.2
0.4
a1
Q’
0.6
0.8
1
• OQ’ gives the value of
q
= 7/13
• QQ’ gives the value of the
game = 50/13
• The game can be reduced
to a 2 X 2 game with the
strategies being a2, a4,b1
and b2.
• Strategies for A (0, 9/13,0,
4/13) and for B (7/13, 6/13)
-6
-8
q
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
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Summary
• The game theory has too many assumptions
to be of any real practical value. It can be
used only in a limited context.
• n person games are more complex as some
players may collaborate against others. There
may be negotiations and so on. Solutions to
such games are currently not available.
Quantitative Techniques for Decision Making
M.P. Gupta & R.B. Khanna
© Prentice Hall India
Page 28