Negative responsiveness paradox

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A diagram showing who would win an IRV election for different electorates. The win region for each candidate is erratic, with random pixels dotting the image and jagged, star-shaped (convex) regions occupying much of the image. Moving the electorate to the left can cause a right-wing candidate to win, and vice versa.
A 4-candidate Yee diagram under IRV. The diagram shows who would win an IRV election if the electorate is centered at a particular point. Moving the electorate to the left can cause a right-wing candidate to win, and vice versa. Black lines show the optimal solution (achieved by Condorcet or score voting).

In social choice, the negative responsiveness,[1][2] perversity,[3] or additional support paradox[4] is a pathological behavior of some voting rules, where a candidate loses as a result of having "too much support" from some voters, or wins because they had "too much opposition". In other words, increasing (decreasing) a candidate's ranking or rating causes that candidate to lose (win).[4] Electoral systems that do not exhibit perversity are said to satisfy the positive response or monotonicity criterion.[5]

Perversity is often described by social choice theorists as an exceptionally severe kind of electoral pathology.[6] Systems that allow for perverse response can create situations where a voter's ballot has a reversed effect on the election, thus treating the well-being of some voters as "less than worthless".[7] Similar arguments have led to constitutional prohibitions on such systems as violating the right to equal and direct suffrage.[8][9] Negative response is often cited as an example of a perverse incentive, as voting rules with perverse response incentivize politicians to take unpopular or extreme positions in an attempt to shed excess votes.

Most ranked methods (including Borda and all common round-robin rules) satisfy positive response,[5] as do all common rated voting methods (including approval, highest medians, and score).[note 1]

Perversity occurs in instant-runoff voting (IRV),[10] the single transferable vote,[3] and quota-based apportionment methods.[2] According to statistical culture models of elections, the paradox is especially common in RCV/IRV and the two-round system.[citation needed] The randomized Condorcet method can violate monotonicity in the case of cyclic ties.

The participation criterion is a closely-related, but different, concept. While positive responsiveness deals with a voter changing their opinion (or vote), participation deals with situations where a voter choosing to cast a ballot has a reversed effect on the election.[11]

  1. ^ May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN 0012-9682. JSTOR 1907651.
  2. ^ a b Pukelsheim, Friedrich (2014). Proportional representation: apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.
  3. ^ a b Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN 0092-5853. JSTOR 2110496.
  4. ^ a b Felsenthal, Dan S. (April 2010). "Review of paradoxes afflicting various voting procedures where one out of m candidates (m ≥ 2) must be elected". GBR. pp. 1–52.
  5. ^ a b D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996
  6. ^ Felsenthal, Dan S.; Tideman, Nicolaus (2014-01-01). "Interacting double monotonicity failure with direction of impact under five voting methods". Mathematical Social Sciences. 67: 57–66. doi:10.1016/j.mathsocsci.2013.08.001. ISSN 0165-4896. it is generally agreed among social choice theorists that a voting method that is susceptible to any type of monotonicity failure suffers from a particularly serious defect.
  7. ^ Arrow, Kenneth J. (2017-12-13). Social Choice and Individual Values. p. 25. doi:10.12987/9780300186987. ISBN 978-0-300-18698-7. Since we are trying to describe social welfare and not some sort of illfare, we must assume that the social welfare function is such that the social ordering responds positively to alterations in individual values, or at least not negatively. Hence, if one alternative social state rises or remains still in the ordering of every individual without any other change in those orderings, we expect that it rises, or at least does not fall, in the social ordering.
  8. ^ Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.
  9. ^ dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.
  10. ^ Ornstein, Joseph T.; Norman, Robert Z. (2014-10-01). "Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections". Public Choice. 161 (1–2): 1–9. doi:10.1007/s11127-013-0118-2. ISSN 0048-5829. S2CID 30833409.
  11. ^ Dančišin, Vladimír (2017-01-01). "No-show paradox in Slovak party-list proportional system". Human Affairs. 27 (1): 15–21. doi:10.1515/humaff-2017-0002. ISSN 1337-401X.


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