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?
Thanks in advance for your help!
