Probabilistic Choice Systems

A probabilistic choice system (PCS) is defined formally by a vector (I, Z, 38, S, P), where I is a set indexing alternatives, Z is the universe of vectors of measured attributes of alternatives, § :I Z is a mapping specifying the observed attributes of alternatives, ^ is a family of finite, nonempty choice (or budget) sets from I, S is the universe of vectors of measured characteristics of individuals, and P :l x 08 x S -»[0, 1] is a choice probability.

The index set I is imposed by the analyst and is assumed to be external to the actual choice process. Any natural or intrinsic indexing of alternatives which may affect choice is included in the vector of measured attributes zeZ. The universe of measured attributes Z will be treated here as an abstract set; in later applications it will usually be assumed to be a rectangle, or else a countable dense set, in finite-dimensional Euclidean space. The choice probability P{i | B, s) specifies the probability of choosing ie I, given that a selection must be made from the choice set Be J and that the decision-maker has characteristics seS. We use the notation P(C | B, s) = 2, gCP(i | B, s). Choice probabilities are assumed to satisfy the following two conditions:

pcs 5.1: Choice probabilities are nonnegative and sum to one, with P(B|B, s) =1.

PCS 5.2: Choice probabilities depend only on the measured attributes of alternatives and individual characteristics; if B = {/x, ..., /'„}

and B' = {«i /;} have zk = f(4) = ((¿¿) for k = 1, . . . , n, then

It should be noted that a PCS is analogous to a conventional econometric specification of a demand system, with the functional specification of the demand structure and the distribution of errors combined to specify the distribution of demand.

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