Avoiding Paradoxes in Positional Voting Systems

It is well-known that no voting system can be entirely fair: this is Arrow’s Theorem, which states that under a certain natural definition of fairness, the only fair voting system is a dictatorship. Thus, in democratic political systems, one expects voting paradoxes in which different voting systems give different results. We analyze the mathematical basis of paradoxes amongst a certain class of voting systems, the positional voting systems, and prove several conditions under which all positional voting systems give identical results. Our framework for this work uses basic representation theory to encapsulate the symmetry inherent in voting. We use the same framework to re-derive a sufficient condition for agreement between the Borda Count and Condorcet methods. Summary It can be mathematically proven that no voting system is entirely fair: a famous result called Arrow’s Theorem proves that there is no fair, deterministic voting system, under a precise but natural definition of fair. This leads to a proliferation of many different kinds of voting systems, which often give contradictory results with the same votes; this is the notion of a “voting paradox.” In this paper, we analyze the mathematical causes of paradoxes within a specific popular class of voting systems, the positional voting systems, which includes the commonly used plurality voting system. We also re-derive a condition for when there are no paradoxes between one positional system and another type of voting system based on head-to-head comparisons.