# Luce's choice axiom

In probability theory, Luce's choice axiom, formulated by R. Duncan Luce (1959), states that the probability of selecting one item over another from a pool of many items is not affected by the presence or absence of other items in the pool. Selection of this kind is said to have "independence from irrelevant alternatives" (IIA).

## Overview

Mathematically, the axiom states that the probability of selecting item i from a pool of j items is given by:

${\displaystyle P(i)={\frac {w_{i}}{\sum _{j}{w_{j}}}}}$

where w indicates the weight (a measure of some typically salient property) of a particular item.

This function used elsewhere in mathematics and science, where it is known as the normalized exponential function or softmax, and dates to the Boltzmann distribution in statistical mechanics; see Softmax function § History.

## Applications

The axiom is often encountered in economics, where it can be used to model a consumer's tendency to choose one brand of product over another. It is also found in psychology, particularly in cognitive science where it is used to model approximately rational decision processes.