In yesterday’s post, I talked about how people don’t actually need to be rational in order for rational choice theory to be a good way of describing and predicting their behavior. I would put forth that the reason for this is that although there is asymmetry in a superficial dimension (the actual reasons that people make the choices they do), there is a deeper symmetry in the relevant dimension (the choices they make). If the goal of developing models is to predict behavior, then you never need to resolve the question as to whether or not people are rational, you only need to define the operator which maps the specific kind of irrational behavior to your model of rational behavior.

So far, the only way I can see that could throw a major monkey-wrench into the task and make everything very complicated is if preferences don’t commute – that is to say that if you prefer A to B and B to C you necessarily prefer A to C. If preferences commute, then you are working entirely with an Abelian group, but if they don’t commute, you are working with a non-Abelian group and their properties are very different. If preferences are commutative, you can easily convert preferences directly into money and back into preferences (would you rather have A or $x, B or $y, C or $z), which means that you can define a simple measure of value. However, if they do not commute, you have to define a much more complicated function that defines the relative value associated with preferences. Note that you still can define a measure of value and map this type of irrationality onto a map of preferences, it’s just not as simple.

I think one thing that is often overlooked is that most of these economic models that “assume rationality” are formulated in an extremely general way, which makes it much easier to define a map from irrationality to preferences. For example, in game theory a game is defined by its payoff matrix, not its outcome matrix. For example, in the prisoner’s dilemma, the outcome matrix is as set forth in Fig. 1, but the “game” of the prisoner’s dilemma is anything with a payoff matrix of the form Fig 2, where a_1/2 < b_1/2 and a_1/2 < d_1/2. Just because we expect rational people only considering the length of their sentences to have a payoff matrix of that form when the outcome matrix looks like Fig 1. doesn’t mean that any time you are presented with that outcome matrix you can assume you are in a prisoner’s dilemma. A rational person might simply prefer not to rat out their compatriot – the stronger the preference for keeping tight-lipped, the lower that person’s payoff for defecting. Even if you found that no people, when put in a literal prisoner’s dilemma ended up in the Nash equilibrium, that doesn’t necessarily mean that the Nash equilibrium analysis doesn’t work, it could very well mean that the players are playing a different game (with a different payout matrix).

Overall, I think the debate seems like it’s being framed very poorly. It’s very easy to model the behavior of individuals if you assume that you know their utility functions at all points in the decision-space and that they are rational always. However, if either of those assumptions breaks down, you don’t have much to go on. Additionally, deviations from either of those assumptions can be mapped onto a model of deviations from the other, so it is sufficient to choose whichever one you like more and assign all prediction errors to a breakdown of the other assumption. Effectively, what this means to me is that there’s no reason to debate whether or not people are rational – you can assume that they are rational and measure peoples’ utility functions (you are going to have to measure them anyway, the only difference here is that you have to account for the function having a more complicated form).

Updated: Oops, accidentally switched the “cooperate” and “defect” labels in the figures. Thanks to Iron Man for pointing out the error. The situation has now been remedied.

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Economics, Game Theory, Rationality, Symmetry by Paul Ganssle