Recall, rationality and political economy
It has been observed that much literature in game theory relies on simplifying assumptions that can frustrate the application of the theory, particularly to public policy. The objective of this chapter is to give arguments why several other assumptions are problematic and to sketch some possible alternatives. We will begin with a common (often tacit) assumption of noncooperative game theory and then proceed to explore two further issues of cooperative game theory and an ambiguity in the concept of rationality.
“BEHAVIOR STRATEGIES SUFFICE”
We now have the technical apparatus to reconsider the role of contingent and behavior strategies in game theory, and the idea that, thanks to Kuhn’s demonstration, “behavior strategies suffice.” As we recall from Chapter 3, Kuhn had demonstrated that an important family of games in extensive form can be analyzed by using behavior strategies only, choosing local (generally randomized) best responses. It was noted, however, that this analysis is not applicable to games of imperfect recall (CGT, 1997, pp. 146-68), nor to any cooperative game (Selten, 1964), nor does it recover all Nash equilibria (Selten, 1975). It was also stated in Chapter 3 that Kuhn’s reasoning does not apply to noncooperative equilibrium concepts other than the Nash equilibrium. This will now be discussed.
In particular, correlated strategies cannot be derived from the local determination of behavior strategies as best responses at each information set. Consider Game 9.1, shown in extensive form by Figure 9.1. No “story” or application will be given for this game, which is offered strictly to illustrate the relation between contingent and behavior strategies. The agents are a and b and the game proceeds in just two stages. First, a chooses between behavior strategies u, c, and d, and then (depending on a’s play at the first stage) b chooses between t1 and b1 (at information set B1) or t2 and b2 (at information set B2).
As usual, agent a’s contingent strategies need not be distinguished from
Figure 9.1 Game 9.1 in extensive form his behavior strategies, since he makes the first play. Agent b has four contingent strategies:
Notice that this list is highly redundant, as always when behavior strategies are translated to contingent strategies. The reason for this redundancy is that contingent strategies 1 and 3 differ only with respect to play at information set b2, which is never reached in Nash-equilibrial play, and similarly strategies 2 and 4. Notice also that any cooperative solution to this game will correspond to a strategy of d by player 1, followed by any behavior strategy of agent b and an offsetting side payment. But this can never be realized if behavior strategies are chosen as local best responses.
Game 9.1 in strategic normal form is given by Table 9.1. This game has a number of Nash equilibria, due to the redundancy that has been mentioned, but they fall into two categories: pure strategy equilibria yielding 3,3 and randomized strategies yielding expected values of 1.5, 1.5. The pure strategy equilibria provide relatively good outcomes but, as usual, they raise questions since they seem to require consultation or information that the agents are assumed (in Figure 9.1) not to have. However,
Table 9.1 Game 9.1 in strategic normal form
this game has a simple correlated equilibrium solution. Let the two agents flip a coin and play according to the rules:
For agent a: “If H then u else c.”
For agent b: “If H then 1 else 2,” or “if H and (u or c) then t1 else t2 else if T and (u or c) then b1 else t2.”
What a correlated strategy does, in effect, is to imbed the game in a larger game in which the first step is the signal, in this case flipping the coin. Neither agent has any reason to deviate from play by these rules: in a’s case, a deviation to d or to play the “wrong” strategy from u and c will leave him with nothing, and in b’s case, a deviation to play b1 on H or t1 on T, a “wrong” behavior strategy, will similarly leave him with nothing. However, whether the original game is expressed in contingent or behavior strategies, the strategies of play in the larger game must themselves be contingent strategies. To be specific, unless agent b knows that agent a will play according to the contingent strategy “if H then u else c,” agent b has no information that would allow him to choose a behavior strategy that would produce a pure strategy equilibrium. This is the logical issue in coordination games, and local choice of behavior strategy offers no escape from it. Local choice of best-response behavior strategies cannot produce a correlated equilibrium in this example.