Subgame Perfection

In order to deepen our understanding of extensive-form games, we need an example with more interesting structure than the PD offers.

Consider the game described by this tree:

Figure 9

This game is not intended to fit any preconceived situation; it is simply a mathematical object in search of an application. (L and R here just denote ‘left’ and ‘right’ respectively.)

Now consider the strategic form of this game:

Figure 10

If you are confused by this, remember that a strategy must tell a player what to do at every information set where that player has an action. Since each player chooses between two actions at each of two information sets here, each player has four strategies in total. The first letter in each strategy designation tells each player what to do if he or she reaches their first information set, the second what to do if their second information set is reached. I.e., LR for Player II tells II to play L if information set 5 is reached and R if information set 6 is reached.

If you examine the matrix in Figure 10, you will discover that (LL, RL) is among the NE. This is a bit puzzling, since if Player I reaches her second information set (7) in the extensive-form game, she would hardly wish to play L there; she earns a higher payoff by playing R at node 7. Mere NE analysis doesn't notice this because NE is insensitive to what happens off the path of play. Player I, in choosing L at node 4, ensures that node 7 will not be reached; this is what is meant by saying that it is ‘off the path of play’. In analyzing extensive-form games, however, we should care what happens off the path of play, because consideration of this is crucial to what happens on the path. For example, it is the fact that Player I would play R if node 7 were reached that would cause Player II to play L if node 6 were reached, and this is why Player I won't choose R at node 4. We are throwing away information relevant to game solutions if we ignore off-path outcomes, as mere NE analysis does. Notice that this reason for doubting that NE is a wholly satisfactory equilibrium concept in itself has nothing to do with intuitions about rationality, as in the case of the refinement concepts discussed in Section 2.5.

Now apply Zermelo's algorithm to the extensive form of our current example. Begin, again, with the last subgame, that descending from node 7. This is Player I's move, and she would choose R because she prefers her payoff of 5 to the payoff of 4 she gets by playing L. Therefore, we assign the payoff (5, −1) to node 7. Thus at node 6 II faces a choice between (−1, 0) and (5, −1). He chooses L. At node 5 II chooses R. At node 4 I is thus choosing between (0, 5) and (−1, 0), and so plays L. Note that, as in the PD, an outcome appears at a terminal node—(4, 5) from node 7—that is Pareto superior to the NE. Again, however, the dynamics of the game prevent it from being reached.

The fact that Zermelo's algorithm picks out the strategy vector (LR, RL) as the unique solution to the game shows that it's yielding something other than just an NE. In fact, it is generating the game's subgame perfect equilibrium (SPE). It gives an outcome that yields a NE not just in the whole game but in every subgame as well. This is a persuasive solution concept because, again unlike the refinements of Section 2.5, it does not demand ‘extra’ rationality of agents in the sense of expecting them to have and use philosophical intuitions about ‘what makes sense’. It does, however, assume that players not only know everything strategically relevant to their situation but also use all of that information. In arguments about the foundations of economics, this is often referred to as an aspect of rationality (as in the phrase ‘rational expectations’. It is helpful to be careful not to confuse rationality in general with computational power and the possession of budgets, in time and energy, to make the most of it.

An agent playing a subgame perfect strategy simply chooses, at every node she reaches, the path that brings her the highest payoff in the subgame emanating from that node. SPE predicts a game's outcome just in case, in solving the game, the players foresee that they will all do that.

A main value of analyzing extensive-form games for SPE is that this can help us to locate structural barriers to social optimization. In our current example, Player I would be better off, and Player II no worse off, at the left-hand node emanating from node 7 than at the SPE outcome. But Player I's basic economic rationality, and Player II's awareness of this, blocks the socially efficient outcome. If our players wish to bring about the more socially efficient outcome (4,5) here, they must do so by redesigning their institutions so as to change the structure of the game. The enterprise of changing institutional and informational structures so as to make efficient outcomes more likely in the games that agents (that is, people, corporations, governments, etc.) actually play is known as mechanism design, and is one of the leading areas of application of game theory. The main techniques are reviewed in Hurwicz and Reiter (2006), the first author of which was awarded the Nobel Prize for his pioneering work in the area.

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