Applications to voting and social choice

Applications to Voting and Social Choice

Linear algebra provides mathematical tools for analyzing voting systems, preference aggregation, and collective decision-making.

Preference Vectors

In ranked voting, each voter expresses a preference ordering over candidates. These preferences can be represented as vectors or matrices:

• Pairwise comparison matrix: Entry a_ij counts how many voters prefer candidate i over candidate j
• Score vectors: Each candidate receives a numerical score based on ranking position

Eigenvector-Based Ranking Methods

Perron-Frobenius Theorem and eigenvector analysis underlie several ranking methods:

Kendall Method: Uses the principal eigenvector of a pairwise comparison matrix to produce a collective ranking.

PageRank Adaptation: Similar to Google algorithm, eigenvector centrality ranks candidates based on the strength of preferences pointing to them.

Voting as Linear Transformations

Different voting methods (plurality, Borda count, ranked pairs) can be viewed as linear transformations of preference data into social rankings.

Borda Count Example

For n candidates, the Borda count assigns points (n-1, n-2, ..., 1, 0) to ranking positions. The total scores form a vector:

s = B^T v

where B encodes the ballot matrix and v is the vote distribution vector.

Fair Division

Linear programming and matrix methods solve fair division problems: dividing resources among parties with different valuations to achieve equitable outcomes.