# One shot deviation principle in games with many players

This question is motivated by an example in Section 4.7.3. Open-Loop and Closed-Loop Equilibria in Games with many players in Game Theory by Fudenberg and Tirole.

The definitions of closed-loop and open-loop in the book are given below.

Closed-loop and Open-loop strategies(Section 4.7.1 in FT): Our definition of a multi-stage game with observed actions corresponds to the closed-loop information structure, where players condition their play at time $$t$$ on the history of play until that date. The corresponding strategies are called closed-loop strategies, while open-loop strategies are functions of calendar time alone.

The following example is given to illustrate that an open-loop equilibrium can also be subgame-perfect equilibrium.

Section 4.7.3 in FT: Consider a game has a continuum of nonatomic individuals of each player type--a continuum of player 1s, a continuum of player 2s, and so on. (Let the set of individuals be copies of the unit interval endowed with Lebesgue measure for concreteness). Suppose further that each player $$i$$'s payoff is independent of the actions of any subset of opponents with measure $$0$$. Then if one individual player $$j$$ deviates, and all players $$k\not=i,j$$ ignore $$j$$'s deviation, it is clearly optimal for player $$i$$ to ignore the deviation as well. Thus the outcome of an open-loop equilibrium is subgame perfect.

My question is that, in the argument above, why do we need the part that it's optimal for player $$i$$ to ignore $$j$$'s deviation given others doing so?

To my understanding, to prove an open-loop equilibrium to be subgame perfect, we need to apply the one-shot deviation princinple. Therefore suppose player $$j$$ deviates, then because of the assumption and the fact that the measure of player $$j$$ is $$0$$, this will not affect the payoff of any player. Hence if a strategy profile constitutes an open-loop equilibrium, it's also subgame perfect.

However, the argument in the book seems to indicate that we not only need to check whether player $$j$$'s deviation is profitable, we also need to check whether there is a profitable deviation for player $$i$$ in the subgame in which $$j$$ already deviates. I am a little confused about how to apply the one-shot deviation principle.

To make my confusion more specific, consider an infinitely repeated version of the two-person prisoners' dilemma, and consider the following strategy profile: Both players will choose $$C$$, and if one player chooses $$D$$ at some $$t$$, both players will play $$D$$ for all $$\tau>t$$.

Suppose we want to check whether the above strategy profile constitutes a subgame perfect equilibrium, we apply the one-shot deviation principle by the following steps.

Step 1:We assume player $$i$$ deviates to $$D$$ at some time $$t$$, and no deviation has occurred until $$t$$, we check whether this deviation is profitable.

Step 2:Next we assume player $$i$$ deviates to $$C$$ in some subgame in which a deviation has already occurred and check whether such deviation is profitable.

Step 3:If the answer is no in both steps, we say that the above strategy profile constitutes a subgame-perfect equilibrium.

In this particular setting, my question is that, in step 1, do we also need to check whether there is a profitable deviation for player $$-i$$ after the deviation of $$i$$ from $$C$$ to $$D$$? How to distinguish between deviations in all subgames and one-shot deviation?