TITLE: Finding mixed equilibria in a two-player infinite stage game
QUESTION [1 upvotes]: Find the mixed and pure nash equilibria in the following game.
Two companies spend a dollar each to enter a market on the first day. Because of the market's limitation, each day that they both operate, they lose 2 dollars each. However, if one operates alone, it earns 3 dollars every day. They can leave the market from the second day, but they can't come back once they leave the market. The payoff for each company is the average payoff in its operation days.
It's easy to see that there are only two pure equilibria, (stay forever, leave on the second day) with the payoff $(3,-1)$ and vice versa. But I don't know how we can look at the mixed equilibria and if there are any?

REPLY [0 votes]: Let’s just consider the situation when on some day players $1$ and $2$ have to decide whether to play the next day. Then payoff for the first player is $p_1(3-5p_2)$ where $p_i$ is the probability that the player $i$ plays the next day. The best response for the player $1$ is to play $p_1 = 1$ when $p_2 = 0$, $0$ in case the opponent plays $1$ and any choice if $p_2 = 0.6$. Due to the symmetry the only equilibrium seems to be when both choose $0.6$  Perhaps you can similarly treat the original case.
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