Question

(b) Two companies, A and B, sell two brands of flu medicine and they are both considering three possibilities: advertise in radio, television or newspapers. Each company will select just one advertising method and can capture a portion of the market from each other. The following matrix summarises the percentage of the market captured by company A for each combination of alternatives A and Bj. B1 (Radio) B2 (TV) B3 (Papers) 5 8 7 A1 (Radio) A2 (TV) 6 5 8 A3 (Papers) 5 7 6 Each company must make its selection before learning the decision of the other company. Determine how each company will select its advertising method. [13 Marks]

Answer 1

Given any of the three strategies taken by 'B', the payoffs of A in A₃ is dominated by those in A₁. So, remove the A₃ row from the matrix.

A / B	B1	B2	B3
A1	5	8	7
A2	6	5	8

Similarly, for any of the two strategies taken by 'A', the payoffs of B in B₃ is dominated by those in B₁. So, remove the B₃ column from the matrix.

A / B	B1	B2
A1	5	8
A2	6	5

Now write the payoffs of B as well knowing that this is a zero-sum game. Underline the best payoffs of A (and B) for each strategy taken by B (and A).

A / B	B1	B2
A1	5, -5	8, -8
A2	6, -6	5, -5

As it is found, there is no strategy in the pure form where both the payoffs are underlined. So, there is no Nash equilibrium in pure form.

So, find the Nash equilibrium in the mixed form.

Let 'p' and (1 - p) be the probabilities of A playing A₁ and A₂ respectively. Similarly, let 'q' and (1 - q) be the probabilities of B playing B₁ and B₂ respectively.

So,

-5*p - 6*(1 - p) = -8*p - 5*(1 - p)
or, p - 6 = -3*p - 5
or, 4*p = 1
or, p = 1/4 and (1 - p) = 3/4

Also,

5*q + 8*(1 - q) = 6*q + 5*(1 - q)
or, 8 - 3*q = q + 5
or, 4*q = 3
or, q = 3/4 and (1 - q) = 1/4

So, A will take A₁ to 1/4^th of its full value and A₂ to 3/4^th of its full value. Similarly, B will take B₁ to 3/4^th of its full value and B₂ to 1/4^th of its full value.

The value of the game = 5*3/4 + 8*1/4 = 5.75