Dual choice axiom and probabilistic choice

Abstract

A decision maker chooses in a probabilistic manner if she does not necessarily prefer the same choice alternative when repeatedly presented with the same choice set. Probabilistic choice may occur for a variety of reasons such as unobserved attributes of choice alternatives, imprecision of preferences, or random errors/noise in decisions. The Luce choice model (also known as strict utility or multinomial logit) is derived from the choice axiom (also known as the independence from irrelevant alternatives). This axiom postulates that the relative likelihood of choosing one choice alternative A over another choice alternative B is not affected by the presence or absence of other choice alternatives in the choice set. This paper presents a dual choice axiom: the relative probability of NOT choosing A over the probability of NOT choosing B is independent from irrelevant alternatives. A new model of probabilistic choice is derived from this dual axiom. This model coincides with Luce’s choice model only in the case of a binary choice. The new model has similar properties as the Luce choice model: the higher is the utility of a choice alternative, the higher is the probability that a decision maker chooses this alternative and the lower is the probability that he or she chooses any other alternative. The new model differs from the Luce choice model in two aspects: utility of choice alternatives is bounded (from above and below) and choice probabilities are more sensitive to differences in utility of choice alternatives.

Appendix

1.1 Proof of Proposition 1

For any two alternatives A,B∈S, dual choice axiom implies Eq. (3). Similarly, for any two alternatives B,C∈S, dual choice axiom implies Eq. (14).

$$ \frac{1-P\left(B|S\right)}{1-P\left(C|S\right)}=\frac{P\left(C,B\right)}{P\left(B,C\right)} $$

(14)

Multiplying Eq. (3) by Eq. (14) yields Eq. (15).

$$ \frac{1-P\left(A|S\right)}{1-P\left(C|S\right)}=\frac{P\left(B,A\right)}{P\left(A,B\right)}\frac{P\left(C,B\right)}{P\left(B,C\right)} $$

(15)

According to dual choice axiom the left-hand side of Eq. (15) is equal to P(C, A)/P(A, C). Using this result we can rewrite Eq. (15) as Eq. (16).

$$ P\left(A,B\right)P\left(B,C\right)P\left(C,A\right)=P\left(A,C\right)P\left(C,B\right)P\left(B,A\right) $$

(16)

Equation (16) is known as the product rule (e.g., Estes 1960, p. 272; Luce and Suppes 1965, definition 25, p. 341). According to Theorem 48 in Luce and Suppes (1965, p. 350), a binary choice probability function P : S × S → ℝ satisfies the product rule (16) if and only if it is a binary strict utility (4). Q.E.D.

1.2 Proof of Proposition 2

Using the result of proposition 1, for any two alternatives A,B∈S, we can rewrite dual choice axiom (3) as Eq. (17).

$$ \frac{1-P\left(A|S\right)}{1-P\left(B|S\right)}=\frac{U(B)}{U(A)} $$

(17)

Equation (17) can be rearranged as Eq. (18).

$$ P\left(B|S\right)=1-\left[1-P\left(A|S\right)\right]\frac{U(A)}{U(B)} $$

(18)

Since Eq. (18) holds for any alternative B∈S, we can sum it over all B∈S to obtain Eq. (19).

$$ {\sum}_{B\in S}P\left(B|S\right)=n-\left[1-P\left(A|S\right)\right]{\sum}_{B\in S}\frac{U(A)}{U(B)} $$

(19)

The left-hand side of Eq. (19) is equal to one due to (1). Thus, we can rewrite Eq. (19) as Eq. (20).

$$ \left[1-P\left(A|S\right)\right]{\sum}_{B\in S}\frac{U(A)}{U(B)}=n-1 $$

(20)

A simple rearranging of Eq. (20) then yields model of probabilistic choice (5). Q.E.D.