Plurality (COMSOC)

Definition

Plurality

In COMSOC, plurality is the voting rule that selects the alternatives with the most first-place votes. For a profile $R=({\succ }_1,\dots ,{\succ }_n)$ over $C$,

$$\begin{array}{rl}{\mathrm{plu}}_R(x)& :=\underset{\text{one point per first-place vote}}{\underbrace{|\{i:x\text{ is first in }{\succ }_i\}|}},\\ & \\ {\lambda }_{\mathrm{Plurality}}(R)& :=\underset{\text{all alternatives with the highest score}}{\underbrace{\arg \underset{x\in C}{\max }{\mathrm{plu}}_R(x)}}.\end{array}$$

Each voter contributes exactly one point; lower-ranked alternatives receive none. Tied leaders are all co-winners, so plurality is not generally resolute without tie-breaking.

Issues

Condorcet-inconsistency

In the 100-voter example below, $C$ is the Condorcet winner:

$$\begin{array}{rl}{N}_R(C,A)& =18+28+24=70,\\ N_R(C,B)& =28+24=52,\\ N_R(C,D)& =30+28=58.\end{array}$$

All three counts exceed $n/2=100/2=50$.

Nevertheless, plurality selects $A$ with 30 first-place votes rather than $C$ with 28. In their head-to-head contest, 70 voters prefer $C$ to $A$. Counting only first places misses this majority, so plurality is not Condorcet consistent.

Ignoring Lower Preferences

Plurality is simple: count only first-place votes. All lower preferences are discarded, so rearranging positions $2,\dots ,m$ cannot change the winner. The rule cannot distinguish an alternative ranked second by everyone else from one ranked last by everyone else, provided their first-place counts agree.

Prone to Manipulation

The rule sees only the rankings voters report, not what they genuinely prefer. Manipulation means reporting a different ranking to obtain a winner they prefer to the winner under truthful voting.

Consider the 18 voters in the second group below. Keep all other voters’ reports fixed.

1. What they want: their true preference is $B\succ D\succ C\succ A$. They want $B$ most and $A$ least.
2. What honesty gives them: voting for $B$ gives it only 18 points. It loses to $A$, which has 30. Supporting their favourite therefore leaves them with their least favourite winner.
3. What they can change: $D$ already has 24 points. If all 18 report $D\succ B\succ C\succ A$, each now gives their single point to $D$, because plurality counts the reported first place:

Nothing is automatically passed from a losing alternative to the next one. We are comparing two possible sets of ballots for the same election: one where these voters put $B$ first, and one where they put $D$ first. Recounting the changed ballots gives

$$\begin{array}{rl}B:& 18-18=0,\\ D:& 24+18=42>\underset{A}{\underbrace{30}}>\underset{C}{\underbrace{28}}.\end{array}$$

Now $D$ wins. They have not made $B$ win, but they have replaced their worst outcome by their second-best:

$$\underset{\text{winner when truthful}}{\underbrace{A}}\prec \underset{\text{winner after the false report}}{\underbrace{D}}\text{according to their unchanged true preferences}.$$

That improvement is why the false report benefits them. This example involves a coordinated group; it does not show that one voter alone could change this outcome.

Example

A profile of 100 voters

Each row groups voters who reported the same complete ranking. For example, “30 voters: $A\succ B\succ C\succ D$” means that 30 distinct voters submitted exactly this order—not merely that they put $A$ first. The four rows represent

$$30+18+28+24=100$$

voters. Plurality counts only their first-place choices.

Alternative $A$ wins with $30$ first-place votes, despite being ranked last by $18+24=42$ voters. Lower positions do not affect plurality scores.