Paper Review: Subjective Probability and the Theory of Games: Comments on Kadane and Larkey’s Paper

Last week I reviewed Kadane and Larkey’s paper. In short, the main claim of their paper was that since the solution concepts used in game theory do not depend on the beliefs of the players they are irrelevant to game theory.

Haranyi wrote a response to this paper, which it what I will review today.

***The original paper can be found here.***

Harsanyi starts off by noting that in some textbooks there are two versions of Bayesian decision described, both of which are really “empty caricatures”. The first of these is the subjectivist version, which allows an agent to use any prior she wishes. The other is the necessitarian, which entirely specifies the probabilities an agent in that context must use.

Harsanyi thinks something like a middle ground is the more correct view. In real life cases there are some priors that are more reasonable than others, and the quantity of different priors and the extent to which they differ will depend on the context. As an example he mentions thermodynamics, in which “one can derive the classical prior probability distributions in a uniform and systematic way by using the maximum entropy principle in conjunction with certain physical invariance principles” (p. 121).

So the idea is that we use information about the situation in order to construct a prior for that situation. Harsanyi claims that this is no different in game theory, and that the solution concepts used in game theory encode precisely the kind of information that is useful for a player to construct her prior.

This is where Harsanyi disagrees with Kadane and Larkey’s approach to game theory, which takes attention away from solution concepts. He writes

Kadane and Larkey oppose any use of normative solution concepts and oppose imposing any rationality criteria on the players’ choice of subjective probabilities. They do not seem to realize that their approach would amount to throwing away essential information, viz., the assumption (even in cases where this is a realistic assumption) that the players will act rationally and will also expect each other to act rationally. Indeed, their approach would trivialize game theory by depriving it of its most interesting problem, that of how to translate the intuitive assumption of mutually expected rationality into mathematically precise behavioral terms (solution concepts).

p. 121

Haranyi is saying that when rational players find themselves in the strategic context with common knowledge of rationality, then this is relevant information for constructing the appropriate prior. Just like in the thermodynamics case, we should use all available information to constrain our priors. The fun of game theory then is figuring out how to translate this information into mathematically precise constraints.

The thing I think I disagree about here is how much Kadane and Larkey would disagree. From their paper, it seems to me that they would agree with this idea. If the agent believes herself and her opponent to be rational, and believes that they have common knowledge of rationality, then this would be reflected in both her actions and her prior. This seems to be reflected in their idea that “solution concepts are a basis for particular prior probability distributions” (Kadane and Larkey, p. 166).

I might be missing a deeper lesson of the debate here, but so far it seems to me that the real disagreement is one of emphasis here. Kadane and Larkey seem to want to emphasize the role that the subjective probability distribution plays in decision making; for them the effect of the strategic context on the agents’ decision is entirely mediated its effect on their priors. Thus more attention should be paid to the priors.

For Harsanyi the emphasis is better placed on the solution concepts. On his view, all that Kadane and Larkey’s view amounts to is

the highly uninformative statement that in a two-person zero-sum game, just as in any other game, each player should try to maximize his expected payoff in terms of his subjective probabilities.

p. 122

This would indeed be very uninformative. To be fair to Kadane and Larkey, though, based on the specific cases they discuss (for example, repeated games) it seems like their approach isn’t that empty. Rather, as I said before, I think it de-emphasizes the role that the solution concepts from game theory should play in prior formation, and want to emphasize other considerations, like that happens when one player doesn’t believe the other is rational.

Following that, if there is a more substantial disagreement between them, it seems more at the object level about quite how useful the recommendations from normative game theory that use solution concepts are. For example, Harsanyi writes:

This is not to imply that normative game theory cannot provide very valuable heuristic help in developing a psychological theory of actual—often quite error-prone—human behavior in game situations. To the contrary, normative theories of rational behavior can often suggest very fruitful hypotheses to empirical psychology because irrational actions can often best be interpreted as psychologically understandable deviations from the normative standards of rationality.

p. 122

If I am in a strategic context, and I am trying to form beliefs about what my opponents will do, then if Harsanyi is right in this passage, normative game theory can be a useful heuristic for constructing my expectations about how the other person will behave, even if they don’t do it perfectly. Kadane and Larkey, on the other hand, don’t mention this as explicitly. But I don’t see any reason that their reason would commit them to being opposed to it. Furthermore, if the status of how useful these normative notions are for predicting actual behavior is the locus of disagreement, this seems to be a largely empirical question, not to be settled by these kind of theoretical debates.

So, to conclude, my best model of this debate is that it is more one of emphasis, or at most one about the empirical success, of the solution concepts from game theory. If it is the former, then it isn’t much of a substantial disagreement over context as one about how we should discuss content. If it is the latter, its resolution will belong to empirical scientists.

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