Economist Writing Every Day

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:

Finding the PSNE for the entire game is easy. It’s just a matter of converting the game to matrix form and underlining each of the optimal conditional strategies. Below is that matrix with the underlining. We can see that the set of 2 PSNE are: {(C,EF);(C,FF)}.

Next, in subgame2, SGE2 is F. Therefore, any PSNE in the entire game that refers to the strategy in information set2 for player 2 must be F if it will also be a SGPE. So, among the set of PSNE, both strategies associated with information set 2 are F. Therefore, there are two SGPE.

What’s the big deal? Strictly speaking, nothing. We followed the definition and found the SGPE. But notice, player 2 choosing E in information set 1 would never happen if player 2 found himself acting rationally at all information sets. Strategy E is strictly dominated by F. But because there is no subgame that includes merely player 2’s info set 1, E has been permitted in the set of SGPE.

In other words, SGPE does NOT assume rationality “each step of the way” because it permits strictly dominated strategies. We must be careful to note that SGPE is the set of equilibria that satisfies rationality in each subgame. That’s not the same as assuming rationality at each information set – ie, for each decision.

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*Further, we can say that EE, EF, FE, & FF are four stand-alone strategies such that *EF* is not strictly dominated. Again, that’s consistent with rationality at each subgame, and not each information set or decision node