Probabilistic Choice Behavior Theory: Axioms as Constraints in Optimization

In choice behavior theory, models specifying the probability of a response R _a1a2…am often take the form https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0076_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>where a ₁ a ₂, …, a_m are independent variables, h ₁, h ₂, …, h_m are real valued functions, H is a real valued function in m variables, and F is a distribution function. Occasionally, F and H are completely specified. [For example, in Thurstone-Case V (1927), we have https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0077_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>as a special case of the above equation.]. It is remarked that the estimation of the parameters, h_i (a_t ), can be profitably made via an equivalent model https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0078_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>where the θs are solutions of the functional equation https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0079_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>(e.g., in Thurstone-Case V: θ _ab = h (a) - h (b);i.e., θ _ac = θ _ab + θ _bc ). This idea is explored in three different models. One example involving the representation https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0080_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>is developed in detail. Maximum likelihood estimators are provided, and a likelihood ratio statistic is derived.

94In a variety of choice behavior situations, data are collected to estimate the probability of some response R _a1a2…am , characterized by an m-tuple (a ₁, a ₂, …, a _m) ∈ A ₁ X A ₂ X … X A _m of aspects. For instance, a ₁ ∈ A ₁ might identify the subject, a ₂ ∈ A ₂ might be some feature of the experimental situation, and (a ₃ ∈ A ₃ to a_m ∈ A_m might describe the stimuli. (If one wishes, a ₁, a ₂, …, a_m may be taken as independent variables.) A typical model constraining these probabilities is symbolized by the equation https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0081_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>in which h_i (1 ≤ i ≤ m) is a function mapping A_i into the reals, H is a real function in m real arguments (e.g., a polynomial), and F is some (cumulative) distribution function (d.f.)- Frequently, F and H are completely specified. For example, F might be the d.f. of a standard normal or logistic random variable (r.v.). A classic example is offered by the so-called Thurstone-Case V (1927), in which the probability that a will be judged “greater than” b is given by https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0082_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>specializing Eq. (1) (Ф is the d.f. of a standard, normal r.v.).

A problem is that of estimating the h_i (a_i ) in (1), considered as parameters of a statistical model. The main point of this chapter is that, in solving this problem, it may be helpful to analyze the function H from the viewpoint of the solutions Ө of the functional equation https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0083_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

In other terms, this involves finding axioms on the parameters θ _a1a2…am ensuring the existence of h_i (1 ≤ i ≤ m) and H such that (2) holds. This approach provides a clarifying analysis of the constraints set on the probabilities P{R _a1a2…am }by the function H, and also simplifies the practical computation of estimators. To illustrate, consider again Thurstone-Case V. Rather than estimating directly the parameters h (a), h (b), …, we could estimate some parameters θ _ab , satisfying both https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0084_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>It is not difficult to show that the two approaches yield equivalent results. In 95particular, in the framework of appropriate side conditions, the last equation implies that θ can be written https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315802473/ad96c89e-6b28-4aee-be3e-e2d6e41fb1bc/content/fig0085_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>for some function h (Aczél, 1966). Thurstone-Case V is not discussed in detail here but turns out to be a subcase of two of the models investigated in this chapter. Let us consider three slightly more complex cases of Eq. (1).