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.

Majoritarian Criteria

• Schwartz criterion: The winner is chosen from the Schwartz set.
• Smith criterion: The winner is chosen from the Smith set. [Smith 1973]: Condorcet criterion.
• Condorcet criterion: The winner is the Condorcet winner, if one exists.
• Condorcet loser criterion: The winner is not the Condorcet loser. [SY 1998]
• mutual majority criterion: If a set of candidates are preferred by a majority to all members outside that set, then the winner will be chosen from that set.
• majority criterion: If a candidate is the first choice of a majority, that candidate is the winner.
• majority loser criterion: If a candidate is the last choice of a majority then that candidate is not the winner. [SY 1998]: strong no imposition power (SNIP)

Unanimity Critria

• strong Pareto principle: If some voters prefer one candidate to another, and no voters prefer the latter, the former is ranked strictly higher than the latter. [HSSX 2005]; [Smith 1973]: strict unanimity
• weak Pareto principle: If all voters prefer one candidate to another, the former is ranked strictly higher than the latter. [HSSX 2005]; [Arrow1963]: Condition P; [Tanaka 2003]: weak Pareto optimality; [Smith 2005]: unanimity and pair-comparison unanimity; [Smith 1973]: unanimity.
• near-unanimity: If a condidate is the first choice of all voters except possibly one, that candidate is ranked first. [Smith 2005]; [Eckert 2004]: weak no-veto principle.
• unanimity: If a candidate is the first choice of all voters, then that candidate is the winner. [MS 2004], [DJL 2001]

Richness Criteria

• strong non-imposition: Any ranking of candidates is possible.
• strong pairwise non-imposition: For any two candidates, either can be ranked strictly above the other. [Tanaka 2003]: strict non-imposition; [Arrow 1950]: citizens' sovereignty
• weak pairwise non-imposition: For any two candidates, either can be ranked above the other unless they are ranked indifferently. [Tanaka 2003]: non-imposition
• non-imposition: Any candidate can be a winner.
• weak non-imposition: At least two outcomes are possible. [Eckert 2004]: non-imposition

Other Criteria

• independence of irrelevant alternatives (IIA): Adding a candidate does not change the winner unless the new candidate is the winner. [Arrow 1950]; [Arrow 1951]: axiom of independence of irrelevant alternatives (AIIA); [Cool 2001]: axiom of pairwise decision making (APDM)
• independence of clones: No candidates outside a clone set will rank strictly between members of that set. [Tideman 1987]
• local independence from irrelevant alternatives (LIIA): Adding a candidate outside the Smith set does not change the winner. [LY 1978]
• 'independence from totally irrelevant alternatives': Adding a candidate who is ranked last by all voters does not change the winner.
• separability: The outcome for disjoint subsets of voters is the same as that of their union. [Smith 1973]; [Young 1975]: consistency

Terminology

• Condorcet winner: a candidate that pairwise defeats every other candidate. [CK 2003]: strong Condorcet winner.
• Schwartz set: the union of all smallest sets unbeatable by members outside the set. [Schwartz 1986]: generalized optimal choice axiom (GOCHA); union of minimal undominated sets
• Smith set: the smallest nonempty set such that every member of the set pairwise defeats every member outside the set. [Schwartz 1986]: generalized top choice axiom (GETCHA); [KL 2004]: minimal dominant set
• Condorcet loser: a candidate that is pairwise defeated by every other candidate.

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.

• collective choice rule (CCR): a function mapping weak orderings to social preferences (which need not be transitive or complete). [Tanaka 2003]; [Eckert 2004]: social aggregation rule (SAR)
• social decision function (SDF): a function mapping weak orderings to a nonempty set of winners. [Cool 2001]: also constitution; [Taylor 2005]: voting rule; [Smith 2005]: Gibbardian voting system with ties (GVS with ties)
• resolute voting rule: a function mapping weak orderings to a winner. [Taylor 2005]; [Smith 2005]: most peoples' voting system (MVS)
• social welfare function (SWF): a function mapping weak orderings to a weak ordering. [Taylor 2005]; [Tanaka 2003]: Arrovian social welfare function; [Smith 2005]: Arrovian voting system (AVS); [Cool 2001]: social welfare function generating mechanisms (SWF-GM).
• voting correspondance (VC): a function mapping linear orderings to a nonempty set of winners. [Pérez 2001]; [SY 1998]: social choice rule (scr)
• resolute social welfare function: a function mapping weak orderings to a linear ordering. [Taylor 2005]
• Gibbardian voting system (GVS): a function mapping linear orderings to a winner. [Smith 2005]; [MS 2004]: social choice function (SCF); [DJL 2001]: voting procedure

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.

Voting Results

• linear ordering: a preference ordering not allowing ties (a complete transitive antisymmetric binary relation). (Sloane's A000142) [Taylor 2005]; [Smith 2005]: Gibbardian preference ordering
• weak ordering: a preference ordering allowing ties (a complete transitive binary relation). (Sloane's A000670) [Smith 1973], [Taylor 2005], [Tanaka 2003]; [Smith 2005]: Arrovian preference ordering
• nonempty set of winners: one or more winners. (Sloane's A000225)
• unique winner: a single winner.

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).

• [Arrow 1950] Kenneth J. Arrow, A difficulty in the concept of social welfare, The Journal of Political Economy, Vol. 58, No. 4 (1950), pp. 328–346.
• [Arrow 1963] Kenneth J. Arrow, Social Choice and Individual Values, 2nd edition, 1963.
• [CK 2003] Donald E. Campbell and Jerry S. Kelly, A strategy-proofness characterization of majority rule, Economic Theory, vol. 22, No. 3 (2003), pp. 557–568.
• [CK 2002] Donald E. Campbell and Jerry S. Kelly, Non-monotonicity does not imply the no-show paradox, Social Choice and Welfare, vol. 19, No. 3 (2002), pp. 513–515.
• [Cool 2001] Thomas Cool, Voting theory for democracy, Thomas Cool Consultancy & Econometrics (2001), ISBN 90-804774-3-5.
• [DJL 2001] Bhaskar Dutta, Matthew O. Jackson, and Michel Le Breton, Strategic candidacy and voting procedures, Econometrica, Vol. 69, No. 4 (2001), pp. 1013–1037.
• [Eckert 2004] Daniel Eckert, Proximity preservation in an anonymous framework, Economics Bulletin, Vol. 4, No. 6 (2004), pp. 1–6.
• [HSSX 2005] Chiaki Hara, Tomoichi Shinotsuka, Kotaro Suzumura, and Yongsheng Xu, On the possibility of continuous, Paretian and egalitarian evaluation of infinite utility streams, 2005.
• [Igersheim 2005] Herrade Igersheim, Extending Xu's results to Arrow's Impossibility Theorem, Economics Bulletin, Vol. 4, No. 13 (2005), pp. 1–6.
• [KL 2004] László Á. Kóczy and Luc Lauwers, The minimal dominant set is a non-empty core-extension, Institute of Economics Hungarian Academy of Sciences, Budapest, 2004.
• [LY 1978] A. Levenglick and H. Young, A consistent extension of Condorcet's election principle (abstract), SIAM Journal of Applied Mathematics, Vol. 35, No. 2 (1978), pp. 285–300.
• [MS 2004] Dipjyoti Majumdar and Arunava Sen, Ordinally Bayesian incentive compatible voting rules, Econometrica, Vol. 72, No. 2 (2004), pp. 523–540.
• [Pérez 2001] Joaquín Pérez, The strong no-show paradoxes are a common flaw in Condorcet voting correspondences (early version), Social Choice and Welfare, Vol. 18 (2001), pp. 601–616.
• [SY 1998] Murat R. Sertel and Bilge Yılmaz, The Majoritarian Compromise is Majoritarian-Optimal and Subgame-Perfect Implementable, 1998.
• [Smith 1973] John H. Smith, Aggregation of preferences with variable electorate, Econometrica, vol. 41 (1973), pp. 1027–1041.
• [Smith 2005] Warren D. Smith, The voting impossibilities of Arrow, Gibbard & Satterthwaite, and Young, 2005.
• [Tanaka 2003] Yasuhito Tanaka, A necessary and sufficient condition for Wilson's impossibility theorem with strict non-imposition, Economics Bulletin, Vol. 4, No. 17 (2003), pp. 1–8.
• [Taylor 2005] Alan D. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press, 2005.
• [Tideman 1987] T. Nicolaus Tideman, Independence of clones as a criterion for voting rules (abstract), Social Choice and Welfare, Vol. 4, No. 3 (1987), pp. 185–206.
• [Young 1975] H. P. Young, Social choice scoring functions (abstract), SIAM Journal of Applied Mathematics, Vol. 28, No. 4 (1975), pp. 824–838.

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

• [Arrow 1951] Kenneth J. Arrow, Social Choice and Individual Values, 1951.
• [Schwartz 1986] Thomas Schwartz, The Logic of Collective Choice, Columbia University Press, 1986.