Measuring Violations of Positive Involvement in Voting

TL;DR

This paper examines how often and under what conditions voting methods violate positive involvement, revealing that some standard methods undermine voter influence despite seeming fair.

Contribution

It introduces new measures for quantifying violations of positive involvement and analyzes these violations through computer simulations under various models.

Findings

Certain voting methods frequently violate positive involvement.

Violations are more significant when considering voter potency and conditional probabilities.

Some methods diminish voter influence even when violations are rare.

Abstract

In the context of computational social choice, we study voting methods that assign a set of winners to each profile of voter preferences. A voting method satisfies the property of positive involvement (PI) if for any election in which a candidate x would be among the winners, adding another voter to the election who ranks x first does not cause x to lose. Surprisingly, a number of standard voting methods violate this natural property. In this paper, we investigate different ways of measuring the extent to which a voting method violates PI, using computer simulations. We consider the probability (under different probability models for preferences) of PI violations in randomly drawn profiles vs. profile-coalition pairs (involving coalitions of different sizes). We argue that in order to choose between a voting method that satisfies PI and one that does not, we should consider the probability of PI violation conditional on the voting methods choosing different winners. We should also relativize the probability of PI violation to what we call voter potency, the probability that a voter causes a candidate to lose. Although absolute frequencies of PI violations may be low, after this conditioning and relativization, we see that under certain voting methods that violate PI, much of a voter's potency is turned against them - in particular, against their desire to see their favorite candidate elected.