(a) Construct a flowchart to show the logic solution of a zero-sum game.

b) Two manufacturers compete in a market for a specialized calculator. Company A controls 75% of the market while company B controls 25% of the market. Company A is considering a vigorous annual marketing campaign which will cost Sh.35,000,000. The total market for the specialize calculator is 100,000 units per year. The profit contribution per unit is Sh.3,000.

Company B is debating how much money to invest in research and development every year. It is considering three alternatives: Sh.25,000,000, Sh.50,000,000 and Sh.80,000,000. It is estimated that if company A runs a vigorous annual marketing campaign, its share of the market after one yea will be either 79% or 73%, depending on company B‟s investment in research and development (Sh.25,000,000, 50,000,000 and Sh.80,000,000 respectively).

On the other hand, if company A does not run the marketing campaign, company B‟s share of the market will decrease by 1% of the total market if it invests Sh.25,000,000 in research and development, increase by 1% if it invests Sh.50,000,000 in research and

development and increase by 3% if Sh.80,000,000 is invested.

Required:

i Using the share of the market percentages only, convert the above into a zero sum game, and hence solve for the optimal strategies for both companies.

ii Obtain a pay off table consisting of contribution to profit in monetary terms, and hence solve the game.

(i)Let A1 be company A undertakes a vigorous market campaign.

A2 be Company A does not run the market campaign

B1 = Company B invests Sh.25 m in Research and Development (R & D) B2 = Company B invests Sh.50 m in R & D

B3 = Company B invests Sh.80 m in R & D

The game has a saddle point occurring at strategy A3 . These are the optimal strategies with a game value of 73% of the market share of A implying that B will get a market share of 27%.

**A‟s Reasonings**

– If B plays strategy B1 , then A should play strategy A2 to maximize his winnings.

– If B plays strategy B2 , then A should play strategy A2.. In all cases A plays strategy A2B‟s reasonings.

– If A plays strategy A1 then B should play strategy B1 to maximize his profits.

– If A plays strategy A2 then B should play strategy B1 .

The saddle point occurs when A plays strategy A2 and B plays strategy B1 . The profit contribution will be:

Company A: Sh.228 million

Company B: Sh. 47 million