# A representation theorem for finite best-worst random utility models

Choosing an element from an offered set of alternatives is arguably the most basic paradigm of preference behavior. Typically, if the same set is offered several times, the choice will not always be the same. This is often attributed to the participantâ€™s preference fluctuating over time due to the effect of various alternatives to be compared, or to the difficulty of distinguishing between similar alternatives.Theories of best-choice behavior try to account for the probability of choosing an alternative y from an offered set Y, a subset of base set X. This intrinsic randomness leads naturally to postulating the existence of a random variable U(x) , for each alternative x in Y, representing the momentary strength of preference for alternative x. Alternative y choosen from Y if the momentary (sampled) value of U(y) exceeds that of any other alternative, aka random utility model (RUM). Falmagne (1978) showed that nonnegativity of certain linear combinations of choice probabilities (Block-Marschak polynomials) is necessary and sufficient for the existence of a RUM representation of best-choice probabilities. Marley & Louviere (2005) proposed an alternative task, where a participant is asked to select both the best and the worst option in the available subset of options Y. Let B(b,w,Y) be the probability that a participant chooses b as best and w as worst alternative in the set Y. Here I show that non-negativity of best-worst Block-Marschak polynomials, appropriately defined, is necessary and sufficient for the existence of a RUM representation of best-worst choice probabilities. The theorem is obtained by extending proof techniques for the corresponding result on best choices (Falmagne, 1978).

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