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| | h | l |
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| H | (4, 3) | (0, 2) |
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| L | (5, 0) | (3, 1) |
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Suppose that stage game is infinitely repeated in the following way.
Player1 is infinitely lived with discount factor $\delta\in (0,1)$, and in each period a new player2 arrives, plays the game once and then leaves. (Thus, in each period, player2 is concerned only with his payoff in that period and not future payoffs.)
In each period, both player 1 and 2 observe the actions of all previous periods.
Describe a subgame perfect equilibrium strategy profile in which player 1 chooses H in every period on the equilibrium path, and derive the lower bound on $\delta$ needed.
The static game has two Nash equilibria (in pure strategies): (H,h) and (L,l). The analog of a grim trigger strategy may be used to sustain cooperation: long-run 1 and short-run 2 play (H,h) in the first round and keep playing (H,h) in period t; if someone deviates, they both go to (L,l) forever after.
To check subgame perfection, by the one-deviation principle, it suffices to check that the long-run payoff for 1 $4/(1- \delta)$ is greater than the sum of short-term gain 5 and the long-term continuation value $3\delta/(1-\delta)$. This gives $$\frac{4}{1 - \delta} \ge 5 + \frac{3\delta}{1 - \delta}$$ or $\delta \ge 1/2$.