Majority voting on orders.

*(English)*Zbl 0994.91011The authors consider the situation where voters have individual preferences over a finite set of alternatives, \(X\). However, voters must choose among specified ranked lists of these alternatives, called orders. One can think of orders as specific political platforms. A method of making such a choice is called a preference extension rule.

More specifically, the authors suppose that the voters construct an order, \(P\), through majority voting on pairwise comparisons of the alternatives (no ties or cycles). A preference extension rule is then called tournament consistent if the order \(P\) is chosen by majority voting on the orders on \(X\).

The authors add two other axioms. A neutrality axiom says that name is irrelevant and an independence axiom says that if one order is preferred to another over a subset \(B\) of \(X\), then the addition of an element \(a\) which is either preferred to all alternatives in \(B\) or for which every alternative in \(B\) is preferred to \(a\) does not affect the preference of the orders.

The main result is that a preference extension rule satisfies these three axioms if and only if it is either a top lexicographic order or a bottom lexicographic order, where a top lexicographic order compares orders in strict lexicographical form from the top of the ranking and a bottom lexicographic order compares them from the bottom of the ranking.

The authors also consider the consequences of relaxing the independence axiom, showing that replacing it with weaker conditions allows for additional extension rules.

More specifically, the authors suppose that the voters construct an order, \(P\), through majority voting on pairwise comparisons of the alternatives (no ties or cycles). A preference extension rule is then called tournament consistent if the order \(P\) is chosen by majority voting on the orders on \(X\).

The authors add two other axioms. A neutrality axiom says that name is irrelevant and an independence axiom says that if one order is preferred to another over a subset \(B\) of \(X\), then the addition of an element \(a\) which is either preferred to all alternatives in \(B\) or for which every alternative in \(B\) is preferred to \(a\) does not affect the preference of the orders.

The main result is that a preference extension rule satisfies these three axioms if and only if it is either a top lexicographic order or a bottom lexicographic order, where a top lexicographic order compares orders in strict lexicographical form from the top of the ranking and a bottom lexicographic order compares them from the bottom of the ranking.

The authors also consider the consequences of relaxing the independence axiom, showing that replacing it with weaker conditions allows for additional extension rules.

Reviewer: Duncan Melville (Canton)

##### MSC:

91B12 | Voting theory |