The relationship between Luce's Choice Axiom, Thurstone's Theory of Comparative Judgment, and the double exponential distribution

By J. Yellott. 1977

Abstract Holman and Marley have shown that Thurstone's Case V model becomes equivalent to the Choice Axiom if its discriminal processes are assumed to be independent double exponential random variables instead of normal ones. It is shown here that for pair comparisons, this representation is not unique; other discriminal process distributions (specifiable only in terms of their characteristic functions) also yield a model equivalent to the Choice Axiom. However, none of these models is equivalent to the Choice Axiom for triple comparisons: There the double exponential representation is unique. It is also shown that within the framework of Thurstone's theory, the double exponential distribution, and hence the Choice Axiom, is implied by a weaker assumption, called “invariance under uniform expansions of the choice set.”

Subfields: Statistics and Probability, Game & Decision Theory

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Citations in the corpus

A* Sampling (2014, background): “Moreover, the max and argmax are independent random variables, see Appendix for proofs.”
The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables (2016, methodology): “In contrast, for the general n-ary case the Gumbel is a crucial 1 and the Gumbel-Max trick cannot be generalized for other additive noise distributions [45].”
Offline A/B Testing for Recommender Systems (2018, methodology): “Good examples of distributions are the BradleyTerry Luce (BTL) model [3], the Placket-Luce model [4, 7, 20] or more generally Thurstonian models [26, 28].”
Reparameterizable Subset Sampling via Continuous Relaxations (2019, methodology): “The Gumbel-max trick [Yellott, 1977] generates random keys r̂i = log(wi) + gi by perturbing logits with Gumbel noise gi ∼ Gumbel(0, 1), then taking xi∗ such that i∗ = arg maxi r̂i as a sample.”
Potential demand for alternative fuel vehicles (2002, background): “Yellott (1977) has proved that for IIA to hold the random variables { must be type III extreme value distributed, i.e., Vj( ) = + ε }ε j j ε j j S, ∈ (2.1) ( ) ( )P x ej xε ≤ = − −exp .”
Stochastic Choice and Consideration Sets (2012, background): “A Random Utility Maximization (RUM) rule [3] is defined by a probability distribution π on the possible rankings R of the alternatives and the assumption that the agent picks the top element of the R extracted according to π. Block and Marschak [3], McFadden [26] and Yellot [45] have shown that the ”
Counterfactual Off-Policy Evaluation with Gumbel-Max Structural Causal Models (2019, methodology): “We recall the following fact, known as the Gumbel-Max trick (Luce, 1959; Yellott, 1977; Yuille.”
Stochastic Optimization of Sorting Networks via Continuous Relaxations (2019, methodology): “We give the proof sketch below and refer the reader to Yellott Jr (1977) for more details.”