$\renewcommand\Pr{\mathbb{P}}$ $\newcommand\E{\mathbb{E}}$

Tuesday, July 22, 2014

The unfalsifiability of rationality

Spurred by a recent discussion on "rational" versus "adaptive" models and economics, I recalled something I worked on briefly in 2011. How can we actually define rationality in the first place in order to obtain a falsiable hypothesis.

For simplicity, let us restrict ourselves to the following setting. A set of $n$ subjects participate in a psychological experiment. In this experiment, the $i$-th subject obtains some sequence of observations $x_{i,t} \in X$, with After each observation, it performs an action $a_{i,t} \in A$. We assume that some observations and action sequences are preferred to others. In particular, the observations may entail the dispersal of monetary rewards or presumably pleasurable stimuli. How can we tell whether subjects are rational?

Let us construct a family of belief and utility models with parameters $\theta \in \Theta$. That is, for a subject with parameters $\theta_i$, we define the following probabilities for the next observation \[ \Pr_{\theta_i} (x_{i,t+1} \mid x_{i,1:t}, a_{i,1:t}) \] conditioned on the observation-action history. We also assume that each subject has a utility function $U_i : X^* \times A^* \to \mathbb{R}$ and that it tries to maximise its expected utility \[ \E^{\pi_i}_{\theta_i} U_i \] with some policy $\pi_i : X^* \times A^* \to A$ for selecting actions.

The falsiability problem occurs because, even if we assume a priori a particular utility function, any subject's actions will always be consistent with some belief model, if $\Theta$ is large enough. Consequently, we must forget thinking about formal rationality in the first place, and start discussing reasonable models instead.

What are reasonable models? One standard answer comes from Occam's razor. Given a nice family of distributions (or programs), we could simply bet that most subjects would put most of their belief on the simplest parts of the family.

However, for any hypothesis test, we need an alternative hypothesis. Unfortunately, no obvious way of building "unreasonable" models exists. However, one could always think about the $\epsilon$-contamination class with "oracle" beliefs: these assign maximum probability to the observed data. I then performed an analysis which seemed to support the null hypothesis that the beliefs are "reasonable", when the subjects (in this case algorithms) are indeed reasonable, even in the absence of a known alternative hypothesis.

I am not sure if there is interest in that sort of thing. If yes, then perhaps a bit of refinement of the alternative class model may be in order.

No comments: