Athletes are preparing for the coming Olympics. Let us now assume the intrinsic superiority of Adidas is not as great, and that network effects are stronger for Nike. These modifications are reflected in different payoffs. Now, the payoff from having Nike is 60n and from having Adidas is 15+5a; x number of athletes are simultaneously deciding between Nike and Adidas.

a. Find all Nash Equilibria.

b. With these new payoffs, let us now suppose that a third option exists, which is to buy Puma; it has a payoff of 1,000. Athletes simultaneously decide among Nike, Adidas, and Puma. Find all Nash equilibria

 

Nash equilibrium is defined as the best outcome that players can achieve where no player has an incentive to deviate from the derived outcome as it maximizes the payoff.

Step-by-step explanation

Nash equilibrium is defined as the best outcome that players can achieve where no player has an incentive to deviate from the derived outcome as it maximizes the payoff.

a) Athletes will choose Nike, if the payoff from Nike is higher than Adidas.

60n > 15 + 5a

or

a < 12n - 3

Athletes will choose Adidas, if the payoff from Adidas is higher than Nike.

15 + 5a > 60n

or

a > 12n - 3

b) when Puma is introduced with a payoff of 1000, then

Athletes will choose Puma, if the payoff from Puma is higher than Adidas and Nike.

1000 > 60n > 15 + 15a

or

1000 > 15 + 15a > 60n

Athletes will choose Nike, if the payoff from Nike is higher than Adidas and Puma.

60n > 15 + 15a > 1000

or

60n > 1000 > 15 + 15a

Athletes will choose Adidas, if the payoff from Adidas is higher than Nike and Puma.

15 + 15a > 1000 > 60n

or

15 + 15a > 60n > 1000