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### One shot deviation principle in games with many players

It's a bit more complicated. Treating open-loop strategies as closed-loop strategies- that happen to play a given action in each period irrespective of the history- will not give you a subgame-perfect ...
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The $r$, or rather $\frac{1}{1/r}$, is the "average" part of the average discounted payoff. As you note, $\mathrm e^{-rs}$ is the discount factor. Observe that \int_0^\infty ...