Plurality (COMSOC)

comsoc

Definition

Plurality

In COMSOC, Plurality is a voting rule where every voter gives one point to their most preferred alternative and no points to all other alternatives.

Let

$$C:=\{c_1,\dots ,c_m\}$$

be the set of alternatives, and let

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

be a preference profile. The plurality score of an alternative $x\in C$ is

$${\mathrm{plu}}_R(x):=\left|\{i\in \{1,\dots ,n\}:x\text{ is ranked first in }{\succ }_i\}\right|.$$

Plurality returns the alternatives with maximum plurality score:

$${\lambda }_{\mathrm{Plurality}}(R):=\arg \underset{x\in C}{\max }{\mathrm{plu}}_R(x).$$

Mechanism

First-place counting

Plurality only reads the first position of each ranking. If a voter ranks

$$A\succ B\succ C\succ D,$$

then $A$ receives one plurality point, while $B$, $C$, and $D$ receive none from that voter.

Example

Course profile

In the lecture profile, the voters rank the four alternatives as follows:

voters	ranking	first-ranked alternative
$30$	$A\succ B\succ C\succ D$	$A$
$18$	$B\succ D\succ C\succ A$	$B$
$28$	$C\succ B\succ A\succ D$	$C$
$24$	$D\succ C\succ B\succ A$	$D$

Hence

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

Since $A$ has the highest plurality score, $A$ is the unique plurality winner.

Properties

Uses little preference information

Plurality is simple, but it ignores all information below the first position. This makes it easy to compute, but it can be sensitive to strategic behaviour because lower-ranked comparisons do not affect the score.

Polynomial-time winner determination

To determine whether a given alternative $c^{\ast }\in C$ is a plurality winner, compute the plurality score of every alternative and compare $c^{\ast }$ with the maximum score.

Thus plurality winner determination is solvable in polynomial time.