In a seminal publication more than four decades ago, Strauss (1979) presented groundbreaking findings. One of his key contributions was the expansion of the multivariate distribution of utilities, moving beyond independent double-exponentials (or Gumbels) to generate choice probabilities that are consistent with the multinomial logit model. He introduced a significant generalization, referred to in this paper as the Strauss model, which incorporates correlated double-exponentials and extends to a broader class of distributions. This paper will examine and define this generalization in detail.

Although the Strauss model introduces a specific type of correlation among the alternatives and perturbs the original double-exponential margi-nals, the choice probabilities remain invariant and align with the logit formula, which Strauss refers to as the “choice axiom”, remaining faithful to the terminology of the foundational work by Luce (1959).

At a more detailed level of choice, the ranking probabilities — i.e., the probabilities associated with all possible rankings or permutations of the available alternatives — follow the exploded logit model, which Strauss designates as “complete decomposition”. It is crucial to note that if the ranking probabilities are consistent with the exploded logit model, then the choice probabilities must necessarily follow the multinomial logit model. However, whether the converse is true — are there random utility models for which choice probabilities are consistent with the multinomial logit model, but ranking probabilities do not follow the exploded logit model? — remains an unresolved question.1

In this paper, we revisit the Strauss model by presenting minimal assumptions regarding the support of the marginal distributions, followed by a discussion of the necessary and sufficient conditions for the model’s validity, drawing upon contemporary copula theory (for an authoritative reference, see Nelsen, 2006). Although Strauss did not employ copulas in his work, we prove that his model aligns with archimedean copulas that are applied to specific marginals. We draw upon the existing literature on archimedean copulas to establish the conditions for the model’s validity. This methodology capitalizes on the advanced theoretical constructs offered by archimedean copulas, thus providing rigor and precision to our findings by relying on well-established results. Consequently, we circumvent the redundancy of complex proofs, focusing on elucidating the essential conditions while ensuring a robust and precise understanding of the model.

It is noteworthy that copulas, have been studied within the discrete choice models framework. Tony Marley, in his paper (Marley, 1982), was a pioneer, as he often was for many other issues, by applying copulas to the random utility model. He applied Sklar’s theorem in the specific context of ARUMs with independent and identically distributed random terms (independent Thurstonian model). A key theorem in his paper proves that, in this context, any choice probability of a given alternative within a choice set can be represented by a copula applied to all the binary choice probabilities involving the chosen alternative and the remaining alternatives in that set.2

More recently, Schwiebert (2016) proves that modeling the multivariate distribution of utilities with archimedean copulas aids the computation of integrals necessary for determining choice probabilities. This approach yields closed-form expressions for the partial derivatives of the cumulative distribution functions, which act as integrands within these computations. Consequently, the integration of probabilities is simplified through the use of techniques including Gauss–Hermite quadrature or Monte Carlo methods. It is not our objective to furnish an exhaustive survey of the literature on the application of copulas to random utility models; rather, our emphasis lies exclusively on utilizing copulas to define the boundaries of the family of distributions encompassed by the Strauss model.

It should be noted that Beggs et al. (1981) provided an elegant proof, although two years after (Strauss, 1979), establishing that the multinomial logit model, derived from a random utility model with independent double-exponentials, results in the exploded logit model. However, Strauss showed that the exploded logit model can be achieved within a much broader framework that includes a larger family of distributions. By revisiting the Strauss model, we highlight the broader scope of his results, which include many models that conform to the exploded logit model beyond the specific case of independent double-exponentials.

The subsequent sections of this paper are structured as follows. Section 2 elaborates on the formulation of the Strauss model alongside the underlying assumptions. Section 3 provides a representation of the Strauss model via copulas, elucidating that this representation utilizes archimedean copulas. Section 4 details examples of distributions that conform to the framework of the Strauss model. In Section 5, we explicate the conformity of the ranking probabilities within the Strauss model to the exploded logit model, providing a new proof of this claim. Lastly, Section 6 summarizes the findings of the paper and suggests avenues for future research.