Misrepresentation in district voting

Voting systems in which voters are partitioned to districts encourage accountability by providing voters an easily identifiable district representative, but can result in a selection of representatives not representative of the electorate's preferences. In some cases, a party may have a majority of the popular vote, but lose the elections due to districting effects. We define the Misrepresentation Ratio which quantifies the deviation from proportional representation in a district-based election, and provide bounds for this ratio under various voting rules. We also examine probabilistic models for election outcomes, and provide an algorithm for approximating the expected Misrepresentation Ratio under a given probabilistic election model. Finally, we provide simulation results for several such probabilistic election models, showing the effects of the number of voters and candidates on the misrepresentation ratio.