Online Gaming:

Consider a two-stage game between two firms that produce online games. In the first stage, they play a Cournot competition game (each chooses a quantity qi) with demand function p = 100 - q and zero marginal production costs (c_i(q_i) = 0 for i = 1, 2). In the second stage, after observing the pair (q₁, q₂) and after profits have been distributed, the players play a simultaneous-move "access" game in which each can open its platform to allow players on the other platform to play online with players on its platform (O for player 1, o for player 2), or choose to keep its platform non compatible (N for player 1, n for player 2), in which case each platform's players can play only with others on the same platform.

If they choose (N, n) then secondstage payoffs are (0, 0). If only one firm chooses to open its platform, it bears a cost of -10 with no benefit, since the other firm did not allow open access. Finally if both firms choose (O, o) then each firm gets many more eyeballs for advertising, and payoffs for each are 2500. Both players use the same discount factor δ to discount future payoffs.

a. Find the unique Nash equilibrium in the first-stage Cournot game and all of the pure-strategy Nash equilibria of the second-stage access game. Find all the pure-strategy subgame-perfect equilibria with extreme discounting (δ = 0). Be precise in defining history-contingent strategies for both players.

b. Now let δ = 1. Find a subgame-perfect equilibrium for the twostage game in which the players choose the monopoly (total profit-maximizing) quantities and split them equally (a symmetric equilibrium).

c. What is the lowest value of δ for which the subgame-perfect equilibrium you found in (b) survives?

d. Now let δ = 0.4. Can you support a subgame-perfect equilibrium for the two-stage game in which the players choose the monopoly quantities and split them equally? If not, what are the highest profits that the firms can make in a symmetric equilibrium?