Learnability and Models of Decision Making under Uncertainty

Abstract

We study whether some of the most important models of decision-making under uncertainty are uniformly learnable, in the sense of PAC (probably approximately correct) learnability. Many studies in economics rely on Savage's model of (subjective) expected utility. The expected utility model is known to predict behavior that runs counter to how many agents actually make decisions (the contradiction usually takes the form of agents' choices in the Ellsberg paradox). As a consequence, economists have developed models of choice under uncertainty that seek to generalize the basic expected utility model. The resulting models are more general and therefore more flexible, and more prone to overfitting. The purpose of our paper is to understand this added flexibility better. We focus on the classical expected utility (EU) model, and its two most important generalizations: Choquet expected utility (CEU) and Max-min Expected Utility (MEU). Our setting involves an analyst whose task is to estimate or learn an agent's preference based on data available on the agent's choices. A model of preferences is PAC learnable if the analyst can construct a learning rule to precisely learn the agent's preference with enough data. When a model is not learnable we interpret it as the model being susceptible to overfitting. PAC learnability is known to be characterized by the model's VC dimension: thus our paper takes the form of a study of the VC dimension of economic models of choice under uncertainty. We show that EU and CEU have finite VC dimension, and are consequently learnable. Morever, the sample complexity of the former is linear, and of the latter is exponential, in the number of states of uncertainty. The MEU model is learnable when there are two states but is not learnable when there are at least three states, in which case the VC dimension is infinite. Our results also exhibit a close relationship between learnability and the underlying axioms which characterise the model.

Learning Time Dependent Choice

“…The data points are drawn according to some unknown distribution, and the analyst has no control over the data he is presented with. Our main result here is a structural criterion on preference models that allows for a drastic improvement over the PAC learning complexity bounds achieved in [2]. We stipulate that the agent weights time-delayed payoffs according to polynomials, which allows for considerable freedom in how payoffs are weighted.…”

“…Problems of learning economic parameters have received recent attention from computer scientists; see, e.g., [1,2,3,16,22]. Inspired by a general theme of demanding computational robustness from economic models (Echenique, Golovin, and Wierman provide a nice discussion of this topic in [11]), the tools of learning theory provide relevant and exciting perspectives from which to view economic models that have been around for several decades.…”

“…Our paper contributes to the line of work that specifically studies models of choice using the perspectives of learning theory. This confluence of decision theory and learning theory was initiated by Basu and Echenique [2], who consider the learning problem for models of choice under uncertainty. Our investigation in this paper is motivated by models of how agents make choices over time.…”

Recovering Preferences From Finite Data

“…Below in Sections 4 and 5, we apply Theorem 3 in different environments. Basu and Echenique (2018) compute the VC dimension of some common models of choice, they show, in particular, that the class of expected utility, Choquet expected utility, and two-state max-min preferences have finite VC dimension.…”

“…More closely related to our paper, Matzkin (2003) and Blundell, Kristensen, and Matzkin (2010) consider identification in an econometric model of stochastic demand data (see Matzkin, 2007, for a general discussion). Recently, Basu and Echenique (2018) investigate the learnability of four standard models of choice under uncertainty using the notion of Probably Approximately Correct (PAC) learning from computational learning theory. Basu (2019) applies several other measures of model complexity to a study of stochastic choice.…”