Game theory project...

Game Theory…

-Prof. In charge – Prof. Chitnis

Group No. 1
• SagarAgarwal 01
• Raj Bazaz 06
• ZubinChawda 10
• ChiragDihrawani(GL) 14
• AkshayGadkari 17
• NeenadKadam 24
• Harshit Patel 40
• TusharPatil 42
• Tejas Shah 55

Applications Of Game Theory…
• Example 1 (Principle of Dominance):
In an election campaign, the strategies adopted by the ruling and opposition party along
with pay-offs (ruling party's % share in votes polled) are given below:

Opposition Party's Strategies

Ruling Party's Campaign Campaign Spend two days
Strategies one day in two days in in large.
Campaign one 55 40 35
day in each city
Campaign two 70 70 55
days in large
towns
Spend two days 75 55 65
in large rural
sectors

Assume a zero sum game. Find optimum strategies for both parties and expected payoff
to ruling party.

Let A1, A2 and A3 be the strategies of the ruling party and B1, B2 and B3 be those of the
opposition. Then

Player B
B1 B2 B3
A1 55 40 35
Player A
A2 70 70 55
A3 75 55 65

Here, one party knows his strategy as well as other party's strategy and one person's gain is
another person's loss.

Now, with the given matrix:

Player B
B1 B2 B3 Row
minimax
Player A A1 55 40 35 35
A2 70 70 55 55
A3 75 55 65 65
Column 75 70 65
maxima

As maximin = 55 and minimax= 65 , there is no saddle point.

Row 1 is dominated by row 2 and column 1 is dominated by column 2 giving the
reduced 2 x 2 matrix as :

B2 B3
A2 70 55
A3 55 65

P= d–c
(a + c) – (b + c)
= 65 – 55
(70 + 65)- (55 + 55)
= 2
5
1-P = 1– 2
5
= 3
5

q = d–b
(a + d) – (b + c)
= 65 – 55
(70 + 65) – (55 + 55)
= 2
5
1–q = 3
5

v = ad – bc
(a + b) – (b + c)
= 70 * 65 – 55 * 55
25
= 4550 – 3025
25
= 61

By equating expected losses of opposition party, regardless of what ruling party would
choose, we get

15 q1 + 55 = -l0ql + 65 so that q1 = 2/5 and (1 – q1) = 3/5

Hence opposition party would choose strategy B2 and B3 with a probability of

0.4 and 0.6 respectively.

The value of the game is determined by substituting the value of p1 and q1 in any of the
expected values and is determined as 61, i. e.,

Expected gain to ruling party:

(i) 15 x 0.4 + 55 = 61
(ii)-10 x 0.4 + 65 = 61

Expected loss to opposition party:

(i) 15 x 0.4 + 55 = 61
(ii) -10 x 0.4 + 65 = 61

EXAMPLE 2.

Even though there are several manufacturers of scooters, two
firms with branch names Janta and Praja, control their market in Western India. If both
manufacturers make model changes of the same type for this market segment in the
same year, their respective market shares remain constant. Likewise, if neither makes
model changes, then also their market shares remain constant. The pay-off matrix in
terms of increased/decreased percentage market share under different possible
conditions is given below:

JANTA PARAJ
NO CHANGE MINOR MAJOR
CHANGE CHANGE

NO CHANGE 0 -4 -10
MINORCHAN 3 0 5
GE
MAJOR 8 1 0
CHANGE

Solution.
(i) Clearly, the game has no saddle point. Making use of dominance principle, since the
first row is dominated by the third row, we delete the first row. Similarly, first
column is dominated by the second column and hence we delete the first column.
The reduced pay-off matrix will be as follows:

PRAJA
JANTA MINOR CHANGE MAJOR CHANGE
MINOR CHANGE 0 5
MAJOR CHANGE 1 0

As the reduced pay-off matrix does not possess any saddle point, the players will use
mixed strategies. The optimum mixed strategy for player A is determined by

P = d– c
(a + d) – (b + c)
= 0-1
(0+0)-(5+1)
= 1
6

1-p = 1-1
6
= 5
6

q = d-b
(a + d) - (b + c)
= 0–5
(0 – 0) – (5 - 1)
= 5
6

1–q = 1–5
6
= 1
6

V = ad-bc
(a + d) - (b + c)

= (0 x 0) – (5 x 1)
(0 + 0) – (5+1)
= 5
6

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