The Imperfection of Subgame Perfection

I’ve written previously about Pure Strategy Nash Equilibria (PSNE). They are the set of strategies that players can adopt in equilibrium – with no incentive to change their strategy. Students have an intuition that PSNE aren’t great because some outcomes that they identify depend on players making silly decisions in the past. In jargon, we can say that some PSNE depend on players choosing irrationally in a subgame while still reaching a PSNE.

See the extensive form game (below right). There are two players, each with two strategies per information set, and player two has two information sets. All PSNE will include a strategy for each information set. We can present the same game in normal form in order to make it easier to identify the PSNE (below left).

Player 1 (P1) can choose the row (B or C) and Player 2 (P2) can choose the column. Importantly, whether P1 might want to change his mind depends on P2’s strategy at the decision node in the alternative information set. Therefore, P2 must have two strategies, one per information set.

The four PSNE strategies and payoffs are underlined in the above table and they are noted in red on the below extensive form games. Again, the logic of PSNE states that no player can improve their payoff by changing only their own strategy, given the opposing player’s strategy. After all, a player can control their own strategy, but not that of their opponent. For example, note PSNE II. In the left subgame, P2 chooses M. His payoff would be unchanged if he changed his strategy, given the strategy of P1.

This is where students get excited. They say “But why?! P2 would never choose M in the left subgame!” To which I say “Correct!”. I tell my students that PSNE don’t address the why. They address what is possible in equilibrium. At this point, students briefly get incredulous and denigrate the usefulness of game theory.

Introducing Subgame Perfect Nash Equilibrium

“Look”, I say, “you now know how to identify PSNE. That tool is like a hammer. But not every problem is a nail. You are right that P2 would never want to choose M in the left subgame *if* P1 had already chosen B.”  Let’s add a new requirement to our PSNE. Let’s say that players must also choose what is best for themselves at each subgame. Then, in the left subgame, P2 would never choose M. Though, in the right subgame, P2 can still choose either M or N.  Introducing this additional constraint narrows our set of four possible equilibria down to two more seemingly reasonable equilibria (III & IV). The name of these two remaining equilibria are the Subgame Perfect Nash Equilibria (SGPNE). They are the PSNE that also include rational decisions in each subgame.

We are much closer to satisfying the question of ‘why’ players do what they do. It doesn’t take long, however, for students to take their ‘why’ to the next level. They are still not satisfied with the SGPNE. Students recognize that PSNE III is better for P2 than is PSNE IV. So, they reason, P2 should just always play NM and never play NN. Then P2 would be guaranteed a payoff of 5 rather than 4 because P1 would always choose B rather than C.

Students really want P2 to be able achieve his highest payoff. Unfortunately, in the right subgame, P2 is indifferent between M and N because they both provide the same payoff. Once P1 has chosen C, P2 has no incentive to choose either M or N. This is also unfortunate for P1 because he runs the risk of doing either better or worse than if he had chosen B.*

Herein lies the imperfection of SGPNE. Once P1 chooses C, he has credibly committed to not choosing B. There is no going back. Due to P2’s indifference and the fact that he chooses second, a possible and rational outcome is (C, NM) even though that is not a PSNE for the entire game. Once P1 has chosen his strategy, the left subgame strategy ceases to be relevant and the right subgame equilibria drive the outcome of the entire game. Therefore, not only has the set of SGPNE failed us in our attempt to predict outcomes – so has the set of PSNE!

The only equilibrium concept that wasn’t violated was that of the subgame equilibria (SGE) among the proper subgames. Here, I use the word ‘proper’ in a set theory sense to denote that I am not referring to the subgame that includes the entire game. Therefore, our game theory does appear quite limited – in this game anyway. Depending on the game, the SGE of the final decision maker can exclude anywhere between nearly zero and all but one possible outcome. Which, isn’t exactly satisfying…

Indeed, game theory is a set of tools. And no set of tools will always be appropriate for all problems. Part of teaching includes helping students to have the wisdom and prudence to use the right model for the job.

*P2 could choose randomly between M and N and be just as happy either way. I won’t discuss Mixed Strategy Nash Equilibria (MSNE), though they exist.

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