Signed orders, choice probabilities, and linear polytopes
The number of subsets of a finite set can be huge and it is unreasonable to suppose that an individual can order these subsets. On the other hand, one can consider what the author calls signed orders. Given a set \(S=\{1,\dots,b\}\), there is a set \(S^*=\{1^*,\dots,n^*\}\) and the order is defined over \(S\cup S^*\). The elements of \(S^*\) signify exclusion from the subset. For instance, a linear signed order for \(n=3\) given by \(1\succ 2^*\succ 3\succ 3^*\succ 2\succ 1^*\) would mean that the first priority is to have 1 in some committee, the second priority is to have 2 excluded from the committee etc. In this paper, the author studies the relationship between binary choice probabilities \(p(x,y)\) for distinct \(x,y\in S\cup S^*\) and probability distributions on the set of all linear signed orders on \(S\cup S^*\). In particular, he searches for necessary and sufficient conditions for the existence of a probability distribution \(P\) on the set \(L_n\) of linear signed orders such that \(p(x,y)=\Sigma\{P(l): l\in L_n\) and \(xly\}\). The cases where \(n=2\) and \(n=3\) are dealt with.
scientific article; zbMATH DE number 1749373
scientific article