Multinomial Logit Processes and Preference Discovery: Inside and Outside the Black Box

Abstract

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 $$\begin{align*} p_{t}\left( a,A\right) =\dfrac{e^{\frac{u\left( a\right) }{\lambda\left( t\right) }+\alpha\left( a\right) }}{\sum_{b\in A}e^{\frac{u\left( b\right) }{\lambda\left( t\right) }+\alpha\left( b\right) }}, \end{align*}$$ 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 behavioural 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 behaviour. Jointly, the two approaches provide a thorough understanding of softmaximization in terms of internal causes (neuro-physiological mechanisms) and external effects (testable implications).