# Applications to voting and social choice

Put content here## Applications to Voting and Social Choice
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Linear algebra provides mathematical tools for analyzing voting systems, preference aggregation, and collective decision-making.
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### Preference Vectors
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In ranked voting, each voter expresses a preference ordering over candidates. These preferences can be represented as vectors or matrices:
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- **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
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### Eigenvector-Based Ranking Methods
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**Perron-Frobenius Theorem** and eigenvector analysis underlie several ranking methods:
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**Kendall Method**: Uses the principal eigenvector of a pairwise comparison matrix to produce a collective ranking.
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**PageRank Adaptation**: Similar to Google algorithm, eigenvector centrality ranks candidates based on the strength of preferences pointing to them.
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### Voting as Linear Transformations
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Different voting methods (plurality, Borda count, ranked pairs) can be viewed as linear transformations of preference data into social rankings.
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### Borda Count Example
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For n candidates, the Borda count assigns points (n-1, n-2, ..., 1, 0) to ranking positions. The total scores form a vector:
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**s = B^T v**
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where B encodes the ballot matrix and v is the vote distribution vector.
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### Fair Division
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Linear programming and matrix methods solve fair division problems: dividing resources among parties with different valuations to achieve equitable outcomes.

# Parents

* Applications