Plurality with Runoff (COMSOC)

Definition

Plurality with Runoff

In COMSOC, plurality with runoff is a two-round voting rule for a profile $R$ over at least two alternatives:

1. Select finalists: retain two alternatives $p,q$ with the highest plurality scores.
2. Run off: delete all other alternatives from each ranking and apply plurality again. Each voter supports whichever finalist they prefer:

$$\begin{array}{rl}s(p)& =|\{i:p{\succ }_{i}q\}|,s(q)=|\{i:q{\succ }_{i}p\}|,\\ {\lambda }_{\mathrm{Runoff}}(R)& =\arg \underset{x\in \{p,q\}}{\max }s(x).\end{array}$$

A tie-breaking convention must specify which two alternatives advance if first-round scores tie at the cutoff. A tied runoff returns both finalists unless a further tie-break selects one.

Lower Preferences Can Decide the Runoff

Ordinary plurality ignores every position below first place. In the runoff, lower positions matter whenever a voter’s first choice has been eliminated. The reported ranking stays unchanged; removing other alternatives reveals which finalist the voter prefers.

Issues

Condorcet-inconsistency

The 100-voter example below elects the Condorcet winner $C$, but agreement on one profile does not establish Condorcet consistency. Consider instead:

$$\begin{array}{rl}35\text{ voters}:& A\succ B\succ C,\\ 33\text{ voters}:& C\succ B\succ A,\\ 32\text{ voters}:& B\succ C\succ A.\end{array}$$

$B$ is the Condorcet winner, since

$$\begin{array}{rl}{N}_R(B,A)& =33+32=65>n/2=100/2=50,\\ N_R(B,C)& =35+32=67>50.\end{array}$$

But first-place scores are $(A,B,C)=(35,32,33)$. Only $A,C$ advance, excluding $B$. In their runoff, $C$ wins by $33+32=65$ votes to $A$‘s 35.

A Condorcet winner would defeat any opponent if it reached the runoff, but the first-round filter need not admit it. Hence plurality with runoff is not Condorcet consistent. No tie-breaking is involved here.

No-show Paradox

Definition

No-show Paradox ( COMSOC)

A voting rule exhibits the no-show paradox if a voter can obtain a strictly preferred outcome by abstaining rather than reporting their true preferences, with all other votes unchanged. For a single-winner rule $f$ defined for varying numbers of voters, this means that some profile $R=({\succ }_i,R_{-i})$ satisfies

$$\underset{\text{winner without voter }i}{\underbrace{f(R_{-i})}}{\succ }_i\underset{\text{winner with voter }i}{\underbrace{f({\succ }_i,R_{-i})}}.$$

Here $R_{-i}$ removes voter $i$‘s ballot, and ${\succ }_i$ is their true preference ranking. Participating hurts this voter according to their own preferences; they benefit by not showing up.

Unlike manipulation by a false report, the voter submits no ballot. For a set-valued rule, comparing outcomes additionally requires a preference over winner sets or a fixed tie-breaking convention.

Link to original

Plurality with runoff can exhibit this paradox: participating can change which two alternatives reach the runoff, resulting in a less-preferred winner.

The diagram shows the group version: two voters with true preferences $C\succ B\succ A$ are better off both abstaining than both participating. No tie-breaking affects either winner; when $A$ and $C$ tie in the first round, both advance.

This demonstrates a coordinated abstention benefit, not a single voter’s ability to change the outcome alone. The separate example below illustrates the ordinary two-round procedure.

Example

The same 100 voters, a different winner

Each row groups voters with the same complete ranking. The plurality diagram shows the first-round counts; its winner $A$ is the plurality leader, not yet the runoff winner.

The two highest scores are $30$ for $A$ and $28$ for $C$, so retain $A,C$ and delete $B,D$. The rankings restricted to the finalists are

$$\begin{array}{rl}30\text{ voters}:& A\succ C,\\ 18\text{ voters}:& C\succ A,\\ 28\text{ voters}:& C\succ A,\\ 24\text{ voters}:& C\succ A.\end{array}$$

In particular, the second group’s $B\succ D\succ C\succ A$ becomes $C\succ A$: they now support $C$ without changing their report. Hence

$$\begin{array}{rl}s(A)& =30,\\ s(C)& =18+28+24=70,\\ {\lambda }_{\mathrm{Runoff}}(R)& =\{C\}.\end{array}$$

Plurality elects $A$; plurality with runoff elects $C$.