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An Overview of Voting System Terminology

There are many systems for determining the winner for an election: plurality, approval, the various Condorcet methods, and the infamous Borda count. These methods all have their supporters, and the arguments for various methods are often based on desirable criteria possessed by the systems. To that end, I present an overview of these criteria, their dependencies, and their results.

The list of implications among these chosen criteria: Schwartz implies Smith; LIIAC implies Smith; Smith implies Condorcet, mutual majority, and Condorcet loser; Condorcet loser implies majority loser; mutual majority implies majority loser and majority; Condorcet implies majority; majority implies unanimity; strong Pareto implies weak Pareto implies unanimity; unanimity implies non-imposition.

Majoritarian Criteria

Unanimity Critria

Richness Criteria

Other Criteria

Terminology

Voting Models

These terms are easy to gloss over, but they are important, since results do not carry easily from one to the others. Also note that the terminology changes meaning based on the scope: for a method producing only a winner, the Pareto principle means less. Suppose all voters preferred A to B and all voters preferred B to C. In a system producing a winner, this means neither B nor C can win, while in a system returning a linear order B must rank strictly between A and C. A voting system producing a ranking could be Pareto-efficient as an SDF but not as an SWF.

Table of voting models, above. The voters' choice is specified on the rows, while the result is specified on the columns. Note that the obvious extension to the table (adding elements corresponding to nonempty set of winners and unique winner to voters choices) would allow for restricted voting methods similar to approval voting and pluraliy voting, respectively.

linear orderingweak orderingnonempty setunique winner
linear orderingVCGVS
weak orderingresolute SWFSWFSDFMVS

Voting Results

References

This is a list of papers, books, and articles noted above. The references simply indicate use; the paper in question need not have invented the concept or term. Strong references, however, are papers generally considered as the origin of the concept (although their terminology may not match mine or current academic practice).

Other works referenced, which I have not yet properly reviewed: