Which of 20 Voting Procedures Satisfy or Violate the Subset Choice Condition (SCC) in a Restricted Domain?

Abstract

The relative desirability of a voting procedure is assessed, inter alia, by verifying which axioms, or postulates, it satisfies or violates. One of these axioms is the subset choice condition (SCC). This axiom requires that if a candidate, x, is elected under a given voting procedure, f, in a profile consisting of n voters and k competing candidates (n, k > 1), then x ought to be elected by f also in such profiles over any proper subset of candidates that contain x and that preserve the pairwise preference relations of the original profile. Most known voting procedures violate, in general, this axiom. However, we were interested to find out which voting procedures satisfy or violate this axiom under a restricted domain assumption where a Condorcet winner exists and is elected in the initial profile by the investigated voting procedure. It turns out that, obviously, all conceivable Condorcet-consistent voting procedures satisfy SCC under this restricted domain assumption. However, most known non-Condorcet-consistent procedures continue to violate SCC even under the restricted domain assumption.

Appendices

Exercises for Chapter 6

• Problem 6.1

• Construct a profile showing that the Plurality Voting procedure does not satisfy SCC when there is no Condorcet winner.

• Problem 6.2

• Construct a profile with the following properties:

• (i) there is a Condorcet winner, say z;

• (ii) z is not elected under the Plurality with Runoff procedure;

• (iii) z is elected by the Plurality with Runoff procedure in all proper subsets of candidates containing z.

• Problem 6.3

• Construct a similar example for the Alternative Vote procedure.

• Problem 6.4

• Consider the following procedure called Borda with (single) Runoff: given any profile over a set of candidates, the Borda score of each candidate is computed, whereupon those two candidates with the largest score are selected for runoff. In this binary comparison, the candidate ranked higher than the other by more voters is elected. Is this procedure Condorcet-consistent?

• Problem 6.5

• Does the preceding Borda with (single) Runoff procedure satisfy SCC?

Answers to Exercises for Chapter 6

• Problem 6.1

3 voters::

x > y > z

2 voters::

y > z > x

2 voters::

z > x > y

There is no Condorcet winner in this profile, x is the Plurality Voting winner, but z defeats x in the subset {x, z}.

• Problem 6.2

4 voters::

x > z > y

3 voters::

y > z > x

2 voters::

z > y > x

Here z is the Condorcet winner, but is not elected in the Plurality Runoff contest. Yet, z is elected in both proper subsets it belongs to.

• Problem 6.3

• Since the Alternative Vote and Plurality with Runoff procedures are equivalent in three-candidate contests, the preceding example applies here too.

• Problem 6.4

• No, it is not. See the following profile:

3 voters::

x > y > z > w

2 voters::

x > z > y > w

2 voters::

y > z > w > x

2 voters::

z > y > w > x

Here x is the (Absolute) Condorcet winner, but y and z are the runoff contestants (in which y wins). Thus x is not elected by the Borda with (single) Runoff procedure.

• Problem 6.5

• No, it does not. The preceding example demonstrates this. There y wins, but would not win in the {x, y} subset.