Aggregation Theorems and the Combination of Probabilistic Rank Orders

There are many situations where we wish to combine multiple rank orders or other preference information on a fixed set of options to obtain a combined rank order. Two of the most common applications are determining a social rank order on a set of options from a set of individual rank orders on those options, and predicting (or prescribing) an individual’s overall rank order on a set of options from the rank orders on a set of component dimensions of the options. In this paper, I develop solutions to this class of problems when the rank orders can occur probabilistically. I develop aggregation theorems that are motivated by recent theoretical work on the combination of expert opinions and I discuss various models that have the property that the representations are ‘of the same form’ for both the component and overall rank order probabilities. I also briefly discuss difficulties in actually using such probabilistic ranking models in the social choice situation.

[1]  Eugene Galanter,et al.  Handbook of mathematical psychology: I. , 1963 .

[2]  P. Fishburn.  Probabilistic Social Choice Based on Simple Voting Comparisons , 1984 .

[3]  A. A. J. Marley,et al.  Context dependent probabilistic choice models based on measures of binary advantage , 1991 .

[4]  Peter C. Fishburn,et al.  A Probabilistic Model of Social Choice: Comment' , 1975 .

[5]  D. Critchlow.  Metric Methods for Analyzing Partially Ranked Data , 1986 .

[6]  Salvador Barberà,et al.  Preference aggregation with randomized social orderings , 1978 .

[7]  Peter C. Fishburn,et al.  Expected utility and approval voting , 1981 .

[8]  Christian Genest,et al.  Combining Probability Distributions: A Critique and an Annotated Bibliography , 1986 .

[9]  C. Genest.  Pooling operators with the marginalization property , 1984 .

[10]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[11]  Bezalel Peleg,et al.  Distribution of Power under Stochastic Social Choice Rules , 1986 .

[12]  R. Duncan Luce,et al.  Individual Choice Behavior , 1959 .

[13]  Peter C. Fishburn,et al.  Towards a Theory of Elections with Probabilistic Preferences , 1977 .

[14]  Christian Genest,et al.  Allocating the weights in the linear opinion pool , 1990 .

[15]  Peter C. Fishburn,et al.  Axioms for approval voting: Direct proof , 1978 .

[16]  Jean-Claude Falmagne,et al.  On a recurrent misuse of a classical functional equation result , 1981 .

[17]  J. Kacprzyk,et al.  Multiperson decision making models using fuzzy sets and possibility theory , 1990 .

[18]  A. A. J. Marley,et al.  Some Probabilistic Models of Simple Choice and Ranking , 1968 .

[19]  P. J. Huber.  Pairwise Comparison and Ranking: Optimum Properties of the Row Sum Procedure , 1963 .

[20]  A. Gibbard.  Manipulation of Schemes That Mix Voting with Chance , 1977 .

[21]  Michael D. Intriligator,et al.  A Probabilistic Model of Social Choice , 1973 .

[22]  J. Aczél,et al.  Aggregation theorems for allocation problems , 1984 .

[23]  A. A. J. Marley,et al.  Aggregation theorems and multidimensional stochastic choice models , 1991 .

[24]  Stephen A. Clark.  The representative agent model of probabilistic social choice , 1992 .

[25]  Peter J. Huber,et al.  Pairwise Comparison and Ranking in Tournaments , 1963 .

[26]  P. Fishburn.  Multiperson Decision Making: A Selective Review , 1990 .

[27]  Joseph S. Verducci,et al.  Probability models on rankings. , 1991 .

[28]  C. Robertson,et al.  A characterization theorem for random utility variables , 1981 .

[29]  K. McConway.  Marginalization and Linear Opinion Pools , 1981 .