Plurality with Runoff (COMSOC)

comsoc

Definition

Plurality with Runoff

In COMSOC, Plurality with Runoff is a two-round voting rule that first selects the two alternatives with the highest plurality scores, and then runs plurality again between only those two finalists.

Let $C$ be a set of alternatives and let

$$R:=({\succ }_1,\dots ,{\succ }_n)$$

be a preference profile. In the first round, compute the plurality score

$${\mathrm{plu}}_R(x)$$

of each alternative $x\in C$. Keep two alternatives $p,q\in C$ with highest plurality scores. In the second round, each voter votes for whichever of $p$ and $q$ they rank higher. The alternative with more second-round votes is a Plurality-with-Runoff winner.

If a tie occurs when selecting the finalists or in the runoff, a tie-breaking convention is needed unless the rule is treated as an irresolute rule returning all tied winners.

Mechanism

Two rounds

Plurality with Runoff uses the following procedure:

1. Compute the first-place plurality scores of all alternatives.
2. Delete all alternatives except two alternatives with the highest plurality scores.
3. Compare the two remaining alternatives by plurality restricted to them.
4. Return the alternative preferred by more voters among the two finalists.

Properties

Uses more information than plurality

Ordinary Plurality only counts first places. Plurality with Runoff also uses how voters compare the two finalists, so lower-ranked alternatives can matter after the first round.

No-show paradox

The lecture notes state that Plurality with Runoff can suffer from the no-show paradox: in some profiles, a voter may obtain a better outcome by not voting.

Example

Course profile

In the lecture profile, the first-round plurality scores are

$${\mathrm{plu}}_R(A)=30,{\mathrm{plu}}_R(B)=18,{\mathrm{plu}}_R(C)=28,{\mathrm{plu}}_R(D)=24.$$

The two finalists are therefore $A$ and $C$.

In the runoff between $A$ and $C$:

voters	ranking	runoff vote
$30$	$A\succ B\succ C\succ D$	$A$
$18$	$B\succ D\succ C\succ A$	$C$
$28$	$C\succ B\succ A\succ D$	$C$
$24$	$D\succ C\succ B\succ A$	$C$

Hence $A$ receives $30$ runoff votes and $C$ receives $70$ runoff votes. Thus $C$ is the unique Plurality-with-Runoff winner.