Combinatoric and Geometric Aspects of Some Probabilistic Choice Models — A Review

Two recent developments in random utility theory are reviewed, with special attention devoted to their combinatoric and geometric underpinnings. One concerns a new class of stochastic models describing the evolution of preferences, and the other some probabilistic models for approval voting. After recalling various commonly used preference relations, we discuss the fundamental property of ‘wellgradedness’ which is satisfied by certain important families of relations, such as the semiorder and the biorder families. The wellgradedness property plays a crucial role in the design of recent stochastic models of preference. Social choice, and approval voting in particular, provide natural arenas for the application of probabilistic models. We examine some partial results regarding the so-called ‘approval voting polytope’ which can be used for the characterization of a particular model of subset choices. We review several families of facets of this polytope and list some unsolved problems. An example illustrates how these geometric results help understand competing models of subset choice.

This paper reviews recent developments in two areas of random utility theory. One concerns a new class of stochastic models describing the evolution of preferences, and the other some probabilistic models for approval voting (cf. Doignon and Falmagne, 1994, Doignon and Falmagne, 1997, Doignon and Regenwetter, 1997, Doi-gnon and Regenwetter, in preparation, Falmagne, 1997, Falmagne and Doignon, 1997, Falmagne and Regenwetter, 1996, Falmagne, Regenwetter and Grofman, 1997, Regenwetter, 1996, Regenwetter, 1997, Regenwetter and Doignon, 1998, Regenwetter, Falmagne and Grofman, 1998, Regenwetter and Grofman, 1998a, Regenwetter and Grofman, 1998b, Regenwetter, Marley and Joe, 1998). Sections 1, 2 and 3 are devoted to the stochastic models and their combinatoric structure, and Sections 4 to 7 review some results on the geometric underpinnings of the approval voting model. Section 8 reviews related recent geometric structures, and the last section provides a conclusion and outlook.