Two manufacturers, A and B, are competing for sales in the same area. Both make and sell Product 1 and Product 2. The other producers of the same products are minor and can be neglected. If for a manufacturer its share of sales of Product 1 is a1 and that of Product 2 is a2 then it controls (a1 + a2)/2 of the market. At the moment Manufacturer B has sales volume three times as large as Manufacturer A for each of the two products. Because of the recent technological breakthrough both companies will be making major improvements in both products. Each of them can develop both products improvements simultaneously and will have them ready for sale in 12 months. Alternatively, each of them can have a ‘crash programme' for one of the products followed by a similar programme for the other product. If one product is developed first, it can be marketed ahead of the competitor. The crash programme for the first product (either Product 1 or Product 2) will take 9 months for Manufacturer B and 10 months for Manufacturer A. The subsequent crash programme for the other product will require 9 months for each of the manufacturers. If both manufacturers market their improved models simultaneously then Manufacturer A will increase its share for that product by 2%. Similarly, if Manufacturer A is ahead of its competitor by 2, 6 or 8 months, then its share for the corresponding product will increase by 20, 30 and 40 percent, respectively. If Manufacturer B is ahead of its competitor by 1, 3, 7 or 10 months, then Manufacturer A will lose 6, 10, 12 and 14 percent of its share for the corresponding product, respectively.
(i) Verify that the situation in which each manufacturer is aimed at maximising its market share can be formulated as a matrix games with the matrix.
B1 B2 B3
A1 27 35 35
A2 29 19 38
A3 29 38 19
Clearly state the players, the strategies of each player and describe how the entries of the pay-off matrix are derived.
(ii) Assume that the board of Manufacturer B due to technical reasons has decided not to develop both products simultaneously, which is equivalent to the removal of strategy B1 from the table above. Solve the obtained game geometrically. Give interpretation and explanation of the found solution.