I’ve written previously on game theory, about the generality of Pure Strategy Nash Equilibria (PSNE), and the drawbacks of Sub-Game Perfect Nash Equilibria (SGPE). In this post I have another limitation for SGPE.

First, some definitions:
PSNE: “No player can change one of their strategies and improve their payoff, given the strategies of all other players.”
Subgame: “A subset of any extensive-form game that includes an initial node (which doesn’t share an information set with other nodes) and all its successor nodes.”
Subgame Equilibrium (SGE): “The PSNE of the Subgame”
SGPE: “The set of PSNE that are also SGE”

Clearly, there is nothing inconsistent about the above definitions. The reason that SGPE emerged was because some PSNE assert that a player would be willing to choose strategies that do not maximize conditional payoffs in subgames that are off of the equilibrium path. So, people often characterize the SGPE as a player ‘being rational each step of the way in each subgame’.

But, there is a problem. “Each step of the way” and “in each subgame” are not the same thing. Each step of the way implies that a player is rational at each decision – ie, at each information set. But, not every information set is a subgame! So, a SGPE can include rationality at each SGE while also permitting some irrationality at individual information sets. Since economists like to identify the bounds of their claims, let me emphasize the word can. In order to be correct, I need only identify one case in which the claim is true.

Here is that case:

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