Question

Construct and analyze a 2x2 one-shot game between two manufacturers (A and B) of competing for game consoles in which each manufacturer must choose between two different software technologies(x and y) which are not compatible with each other. What payoffs are necessary to generate two Nash equilibrium in which each manufacturer chooses the same technology? Would making your game a sequential-move encounter (in which A chooses its software platform before B)make it easier to predict the outcome of this game

Answer 1

	C	D
A	10 , 16	14 , 24
B	15 , 20	06 , 12

Neither player has a dominant strategy. For example, if plays A and plays D then A's payoff is 14. But if A plays B and D plays C, then A's payoff is 15. A similar argument shows that D also does not have a dominant strategy.

The two pure NE are:

• (B, C): If A plays B, then D’ best response is C with a payoff of 20 rather than 12 if D played D; if D plays C, then A’s best response is B with a payoff of 15 rather than 10 if A plays A.

• (A, D): Similar reasoning shows that (A, D) is a pure NE. The worst NE is (B, C) with SW of 35 while the optimum outcome is (A, D) with SW of 38'

Let D play C with probability p and hence play D with probability (1 − p). By the principle of indifference, it must be that the expected payoff for A is the same whether she plays A or B. Thus: p(10) + (1 − p)(14) = p(15) + (1 − p)(6)

so that p = 8 13 .

Let A play A with probability q and hence plays B with probability (1 − q).

In order to insure that D payoff’s are the same for strategies C and D, we have that:

q(16) + (1 − q)(20) = q(24) + (1 − q)(12) so that q = 1 2 .

To verify that p = 8 13 , q = 1 2 is a mixed NE, we calculate •

A’s payoff for A is 8 13 (10) + 5 13 (14) = 150 13 •

B’s payoff for B is 8 13 (15) + 5 13 (6) = 150 13 •

C’ payoff for C is 1 2 (16) + 1 2 (20) = 18 •

D’ payoff for D is 1 2 (24) + 1 2 (12) = 18

By the principle of indifference, it does not matter which strategy D plays.