Monotonicity criterion

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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). IRV Yee.svg
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).

The positive response/association, [1] [2] monotonicity, or nonperversitycriterion [3] is a principle of social choice theory that says that increasing a candidate's ranking or rating should not cause them to lose. [4] Positive response rules out cases where a candidate loses an election as a result of receiving too much support from voters (i.e. being "too popular to win"). Rules that violate positive response are also called perverse [5] and are said to suffer the more-is-less paradox. Such paradoxes are especially common in ranked-choice voting (RCV-IRV), [6] a behavior which can lead to the elimination of moderate candidates and the election of extremists. [7]

Contents

Systems that violate positive response can create situations where a voter's ballot has a reversed effect on the election, making it "less than worthless"; as such, perversity is generally considered to be an exceptionally severe pathology, [8] and German courts have previously struck down nonmonotonic systems for violating the right to equal and direct suffrage. [9] [10]

Most ranked methods (including Borda and all common tournament solutions) satisfy positive response, [4] as do all commonly-used rated voting methods (including approval, highest medians, and score). [note 1]

However, the criterion is violated by instant-runoff voting, [11] the single transferable vote, [12] and Hamilton's apportionment method. [2]

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.

By method

Runoff voting

Runoff-based voting systems, such as ranked choice voting (instant-runoff) fail the positive response criterion. A notable example is the 2009 Burlington mayoral election, the United States' second instant-runoff election in the modern era, where Bob Kiss won the election as a result of 750 ballots ranking him in last place. [13]

An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party initially loses. However, they are elected after running an unsuccessful campaign and adopting an unpopular platform, which pushes their supporters away from the party and into the Top party.

Popular BottomUnpopular Bottom
Round 1Round 2Round 1Round 2
Top25%X mark.svg+6%Top31%46%
Center30%55%Yes check.svg↗Center30% X mark.svg
Bottom45%45%-6%Bottom39%54% Yes check.svg

This election is an example of a center-squeeze, a class of elections where instant-runoff and plurality have difficulties electing the majority-preferred candidate, because the first-round vote is split between an extremist and a moderate. Here, the loss of support for Bottom policies makes the Top party more popular, allowing it to defeat the Center party in the first round.

A famous example of a less-is-more paradox can be seen in the 2022 Alaska at-large special election.

Quota rules

Proportional representation systems using largest remainders for apportionment do not pass the positive response criterion. This happened in the 2005 German federal election, when CDU voters in Dresden were instructed to vote for the FDP, a strategy that allowed the party an additional seat. [2] As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of equal and direct suffrage. [14]

Frequency of violations

For electoral methods failing positive value, the frequency of less-is-more paradoxes will depend on the electoral method, the candidates, and the distribution of outcomes. Negative voting weights tend to be most common with instant-runoff, with what some researchers have described as an "unacceptably high" frequency. [6]

Theoretical models

Results using the impartial culture model estimate about 15% of elections with 3 candidates; [15] [16] however, the true probability may be much higher, especially when restricting observation to close elections. [17] For moderate numbers of candidates, the probability of a less-is-more paradoxes quickly approaches 100%.[ citation needed ]

A 2013 study using a two-dimensional spatial model of voting estimated at least 15% of IRV elections would be nonmonotonic in the best-case scenario (with only three equally-competitive candidates). The researchers concluded that "three-way competitive races will exhibit unacceptably frequent monotonicity failures" and "In light of these results, those seeking to implement a fairer multi-candidate election system should be wary of adopting IRV." [6]

Real-world situations

Alaska 2022

Alaska's first-ever instant-runoff election resulted in negative vote weights for many Republican supporters of Sarah Palin, who could have defeated Mary Peltola by placing her first on their ballots. [18]

Burlington, Vermont

In Burlington's second IRV election, incumbent Bob Kiss was re-elected, despite losing in a head-to-head matchup with Democrat Andy Montroll (the Condorcet winner). However, if Kiss had gained more support from Wright voters, Kiss would have lost. [13]

Survey of nonmonotonic elections

A survey of 185 American instant-runoff elections where no candidate was ranked first by a majority of voters found five additional elections containing monotonicity failures. [13]

2005 German Election in Dresden

A negative voting weight event famously resulted in the abolition of Hamilton's method for apportionment in Germany after the 2005 federal election. CDU voters in Dresden were instructed to strategically vote for the FDP, a strategy that allowed the party to earn an additional seat, causing substantial controversy. As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of equal and direct suffrage. [2]

See also

Notes

  1. ↑ Apart from majority judgment, these systems satisfy an even stronger form of positive responsiveness: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.

Related Research Articles

Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting, but also lets voters give partial (in-between) approval ratings to candidates.

In social choice theory and politics, the spoiler effect or Arrow's paradox refers to a situation where a losing candidate affects the results of an election. A voting system that is not affected by spoilers satisfies independence of irrelevant alternatives or independence of spoilers.

<span class="mw-page-title-main">Condorcet method</span> Pairwise-comparison electoral system

A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.

Arrow's impossibility theorem is a key result in social choice showing that no rank-order method for collective decision-making can satisfy the requirements of rational choice. Specifically, any such rule violates independence of irrelevant alternatives, the principle that a choice between and should not depend on the quality of a third, unrelated option .

Independence of irrelevant alternatives (IIA) is a major axiom of decision theory which codifies the intuition that a choice between and should not depend on the quality of a third, unrelated outcome . There are several different variations of this axiom, which are generally equivalent under mild conditions. As a result of its importance, the axiom has been independently rediscovered in various forms across a wide variety of fields, including economics, cognitive science, social choice, fair division, rational choice, artificial intelligence, probability, and game theory. It is closely tied to many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski-Young theorem, and the money pump arguments.

<span class="mw-page-title-main">Pathological (mathematics)</span> Mathematical phenomena whose properties are counterintuitive

In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or nice. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved.

In an election, a candidate is called a majority winner or majority-preferred candidate if more than half of all voters would support them in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the majority-rule principle, because they extend the principle of majority rule to elections with multiple candidates.

The participation criterion, sometimes called votermonotonicity, is a voting system criterion that says candidates should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob.

In political science and social choice theory, Black'smedian voter theorem states that if voters and candidates are distributed along a one-dimensional spectrum and voters have single peaked preferences, any voting method satisfying the Condorcet criterion will elect the candidate preferred by the median voter.

A voting system satisfies join-consistency if combining two sets of votes, both electing A over B, always results in a combined electorate that ranks A over B. It is a stronger form of the participation criterion. Systems that fail the consistency criterion with high frequency are susceptible to the multiple-district paradox: in other words, it is possible to draw boundaries in such a way that a candidate who wins the overall election still fails to carry even a single electoral district.

The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method. Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.

Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause a higher-ranked candidate to lose.

Schulze STV is a draft single transferable vote (STV) ranked voting system designed to achieve proportional representation. It was invented by Markus Schulze, who developed the Schulze method for resolving ties using a Condorcet method. Schulze STV is similar to CPO-STV in that it compares possible winning candidate pairs and selects the Condorcet winner. It is not used in parliamentary elections.

Instant-runoff voting (IRV), also known as ranked-choice voting or the alternative vote (AV), combines ranked voting together with a system for choosing winners from these rankings by repeatedly eliminating the candidate with the fewest first-place votes and reassigning their votes until only one candidate is left. It can be seen as a modified form of a runoff election or exhaustive ballot in which, after eliminating some candidates, the choice among the rest is made from already-given voter rankings rather than from a separate election. Many sources conflate this system of choosing winners with ranked-choice voting more generally, for which several other systems of choosing winners have also been used.

<span class="mw-page-title-main">2009 Burlington mayoral election</span> American municipal election in Vermont

The 2009 Burlington mayoral election was the second mayoral election since the city's 2005 change to instant-runoff voting (IRV), after the 2006 mayoral election. In the 2009 election, incumbent Burlington mayor won reelection as a member of the Vermont Progressive Party, defeating Kurt Wright in the final round with 48% of the vote.

A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.

<span class="mw-page-title-main">STAR voting</span> Single-winner electoral system

STAR voting is an electoral system for single-seat elections. The name stands for "Score then Automatic Runoff", referring to the fact that this system is a combination of score voting, to pick two finalists with the highest total scores, followed by an "automatic runoff" in which the finalist who is preferred on more ballots wins. It is a type of cardinal voting electoral system.

The sincere favorite or no favorite-betrayal criterion is a property of some voting systems, that says voters should have no incentive to vote for someone else over their favorite. It protects voters from having to engage in a kind of strategy called lesser evil voting or decapitation.

Entitlement-ratio, weight-ratio, vote-ratio, or population-ratio monotonicity is a property of apportionment methods. It says that if the entitlement for A increases proportionally to that of B, then A should not lose any seats to B. Apportionments violating this rule are called population paradoxes; a particularly severe variant, where voting for a party causes it to lose seats, is called a no-show paradox.

<span class="mw-page-title-main">Center squeeze</span> Bias of some electoral systems that favors extremists

A center squeeze is a type of spoiler effect where a majority-preferred and socially-optimal candidate is eliminated in favor of more extreme candidates in plurality-runoff methods, like the two-round and ranked-choice runoff (RCV) rules. In a center squeeze, the presence of more-extreme candidates "squeezes" a candidate trapped between them, starving them of the first-preference votes they need to survive in earlier rounds.

References

  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. 1 2 3 4 Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
  3. ↑ 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.
  4. 1 2 D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters , Issue 6, 1996
  5. ↑ 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.
  6. 1 2 3 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.
  7. ↑ McGann, Anthony J.; Koetzle, William; Grofman, Bernard (2002). "How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections". American Journal of Political Science. 46 (1): 134–147. doi:10.2307/3088418. ISSN   0092-5853. JSTOR   3088418.
  8. ↑ 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.
  9. ↑ Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
  10. ↑ dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN   0044-2070 . Retrieved 2024-05-02.
  11. ↑ 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.
  12. ↑ 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.
  13. 1 2 3 Graham-Squire, Adam T.; McCune, David (2023-06-12). "An Examination of Ranked-Choice Voting in the United States, 2004–2022". Representation: 1–19. arXiv: 2301.12075 . doi:10.1080/00344893.2023.2221689.
  14. ↑ dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN   0044-2070 . Retrieved 2024-05-02.
  15. ↑ Miller, Nicholas R. (2016). "Monotonicity Failure in IRV Elections with Three Candidates: Closeness Matters" (PDF). University of Maryland Baltimore County (2nd ed.). Table 2. Retrieved 2020-07-26. Impartial Culture Profiles: All, TMF: 15.1%
  16. ↑ Miller, Nicholas R. (2012). MONOTONICITY FAILURE IN IRV ELECTIONS WITH THREE ANDIDATES (PowerPoint). p. 23. Impartial Culture Profiles: All, Total MF: 15.0%
  17. ↑ Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN   0219-4937.
  18. ↑ Graham-Squire, Adam; McCune, David (2024-01-02). "Ranked Choice Wackiness in Alaska". Math Horizons. 31 (1): 24–27. doi:10.1080/10724117.2023.2224675. ISSN   1072-4117.
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