R. Duncan Luce's Choice Axiom, introduced in his 1959 book Individual Choice Behavior, is a foundational principle in mathematical psychology and behavioral economics. It formalizes the idea that each alternative has an intrinsic "strength" or "value," and that choice probabilities are proportional to these values.
Key property: P(a|{a,b}) / P(b|{a,b}) = v(a)/v(b)
This ratio is independent of other alternatives
Independence from Irrelevant Alternatives
The defining property of Luce's Choice Axiom is the Independence from Irrelevant Alternatives (IIA): the relative odds of choosing a over b are unaffected by the presence or absence of other alternatives in the choice set. This property, while mathematically elegant, is empirically violated in many contexts — the similarity effect, attraction effect, and compromise effect all demonstrate that adding a new option can change the relative preference between existing options.
Legacy
Despite its empirical limitations, Luce's Choice Axiom has been enormously influential. It provides the mathematical foundation for multinomial logit models in economics and marketing, Bradley-Terry models in sports ranking, and softmax functions in machine learning and neural network models of decision making.