Multinomial logit processes and preference discovery: inside and outside the black box

Abstract

We provide two characterizations, one axiomatic and the other neuro-computational, of the dependence of choice probabilities on deadlines, within the widely used softmax representation \[ p_{t}\left( a,A\right) =\dfrac{e^{\frac{u\left( a\right) }{λ\left( t\right) }+α\left( a\right) }}{\sum_{b\in A}e^{\frac{u\left( b\right) }{λ\left( t\right) }+α\left( b\right) }}% \] where $p_t\left(a,A\right)$ is the probability that alternative $a$ is selected from the set $A$ of feasible alternatives if $t$ is the time available to decide, $\lambda$ is a time dependent noise parameter measuring the unit cost of information, $u$ is a time independent utility function, and $\alpha$ is an alternative-specific bias that determines the initial choice probabilities reflecting prior information and memory anchoring. Our axiomatic analysis provides a behavioral foundation of softmax (also known as Multinomial Logit Model when $\alpha$ is constant). Our neuro-computational derivation provides a biologically inspired algorithm that may explain the emergence of softmax in choice behavior. Jointly, the two approaches provide a thorough understanding of soft-maximization in terms of internal causes (neurophysiological mechanisms) and external effects (testable implications).