# Tactical voting

In all non-dictatorial electoral systems, some form of tactical voting (or strategic voting) occurs when a voter misrepresents their sincere preferences in order to gain a more favorable outcome. Any minimally useful voting system has some form of tactical voting, as shown by the Arrow's theorem, Gibbard's theorem, and the Gibbard-Satterthwaite theorem. However, the type of tactical voting and the extent to which it affects the timbre of the campaign and the results of the election vary dramatically from one voting system to another.

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## Types of tactical voting

There are different types of tactical voting. Many of these can be summarized as involving "order-reversal" (you indicate you prefer Y over X though you prefer X to Y). Note that most ranked methods can incentivize order-reversal (though to varying degrees), while score voting does so only very rarely. Avoiding order-reversal is rather weak, as a voter indicating they prefer all candidates equally would not be order-reversing; yet the fact that no ordinal voting method can meet this test is seen as a huge argument against them by cardinal advocates.

### Compromising

Compromising (sometimes favorite-burying or useful vote) is a type of tactical voting in which a voter insincerely ranks or rates an alternative higher (more generally, increases their support for that alternative) in the hope of getting it elected. For example, in the first-past-the-post election, a voter may vote for an option they perceive as having a greater chance of winning over an option they prefer (e.g., a left-wing voter voting for a popular moderate candidate over an unpopular leftist candidate). Duverger's law suggests that, for this reason, first-past-the-post election systems will lead to two party systems in most cases.

Compromising-compression is a compromising strategy that involves insincerely giving two candidates an equal ranking (or equal rating). Compromising-reversal is a compromising strategy that involves insincerely reversing the order of two candidates on the ballot.

A simple example with approval voting using approval thresholds:

30 A| >B>C

20 B| >A>C

31 C| >A=B

C has the most approvals (31), but if A-top voters decide to also approve B (vote A>B| >C), then they can make B win instead with 50 approvals, a result that they prefer.

1 A>B>C

1 B>C>A

1 C>A>B

This is an example of a Condorcet cycle where each candidate pairwise beats another. If any voter here decides to swap their 1st choice and 2nd choice, then they can make their 2nd choice win in any Condorcet method i.e. if the A-top voter instead votes B>A>C, then B becomes a majority's 1st choice.

### Burying

Burying is a type of tactical voting in which a voter insincerely ranks (or rates) an alternative lower in the hopes of defeating it. For example, in the Borda count, a voter may insincerely rank a perceived strong alternative last in order to help their preferred alternative beat it. A real-world analogy would be voters of one party crossing over to vote in the other party's primary against the candidate they think might beat the candidate of their party.

Burying-compression is a burying strategy that involves insincerely giving two candidates an equal ranking or rating (or truncating, which generally amounts to the same thing).

Burying-reversal is a burying strategy that involves insincerely reversing the order of two candidates on the ballot.

30: A>B
25: B>A
40: C

A is the Condorcet winner here (pairwise beats B 30 to 25 and C 55 to 40). But if A-top voters vote A>C instead, then they can make A win in several Condorcet methods, such as Schulze, Minimax, etc. This is because they start a Condorcet cycle where A has the weakest pairwise defeat of the three (A loses 30 to 40 to C, while B loses 25 to 70 to C and C loses 40 to 55 to A). This is an example of the chicken dilemma.

Burying is often discussed in the context of Condorcet methods, where it can be used to create strategic Condorcet cycles. Also see later-no-help for some examples of burying.

A method that passes both later-no-harm and later-no-help is impervious to burying strategy. This because, if a voter prefers candidate C to W, then whether the voter expresses a later preference for W neither increases (later-no-help) nor decreases (later-no-harm) C's chance of winning. Instant-runoff voting and Plurality are examples of such methods.

Neither later-no-harm nor later-no-help on its own provides complete resistance to burying, however. If a method only passes later-no-harm, it's possible that one particular way of filling in a ballot that expresses a preference for C will help C, thus making C win, while another won't; and it's possible that the former is one where the current winner is buried where the latter is the honest ballot. Similarly, if the method only passes later-no-help, the honest ballot may still harm C whereas the burial ballot does not. If the method passes later-no-help, then truncation works at least as well as burial, but burial may still work.

### Pushover

There are two types of strategies referred to as pushover:

• A narrow type, which involves encouraging voters to rank (or score) a candidate (called "B" in this example) lower than another candidate (called "A" in this example) in hopes that "B" is elected. This strategy won't work in systems that pass the mono-raise criterion.
• A broader type (also known as turkey-raising or the pied-piper strategy) which can happen in two-round systems.[1][2] This broader type requires three candidates to explain: "A", "B" and "X". Let's say that voters are asked to choose (in the first round of an election) between "B" and "X". Voters who prefer "A" in the second round of the election may hope to have other voters vote for "the turkey" (candidate "X") who cannot beat "A", rather than see candidate "B" advance to the second round of the election, and may vote for "X" over "B" if they are allowed.[3]

### Free riding

Free riding is a form of tactical voting which affects any Multi-Member System that has a mechanisms to increase the level of Proportional representation. The strategy is to lower your endorsement for candidates which you expect to be elected without your support. This allows more of your vote power to go into electing other candidates, because the voting method takes less of your voting power.

2-winner example:

30: A1>A2
14: B1
5: C1

In any PR method that spends a Droop quota or more of ballots when a candidate is elected, A1 is likely to win first, and then at least 16.333 A voters' ballots will be spent, leaving them with 13.666 ballots to support A2. This allows B1 to win the second seat in most methods. However, if the A1 voters had split into two groups of 15 voters each, with one bullet voting (only voting for) A1 and the other only for A2, then they guarantee that both A1 and A2 win in most methods, because the two candidates have more votes each (15) than the other candidates (B1 with 14 and C1 with only 5), and when the votes in favor of one are spent, only the 15 voters who chose that candidate lose their ballot weight.

### Other types of strategic voting

One-sided strategy is when only the side/faction that benefits from the strategy (i.e. those who prefer the candidate the strategic voting is intended to benefit to the candidates it is intended to hurt) votes strategically, while the side(s) that would be hurt don't.

#### Coordinated strategy

It's important to differentiate between coordinated strategy, and uncoordinated strategy, as well as informed strategy vs. uninformed strategy.

For example, approval voting and score voting guarantee that at least half of the voters can force their preferred candidates to tie or win, and force their dispreferred candidates to tie or lose (meaning they pass a weak form of mutual majority). However, this crucially hinges on these half of the voters of voters knowing a) that they all prefer those candidates, and b) that they all plan to use the strategy. Otherwise, those who attempt the strategy may either fail to support all of the candidates supported by the group of voters, resulting in the strategy not always working, or they may do it while not everyone else in the group does, which potentially weakens their own vote's ability to influence who wins among the candidates not maximally preferred by that half of the voters. So strategy comes in difficulty levels of execution.

## Strategy-free voting methods

It has been shown by the Gibbard-Satterthwaite theorem that it is impossible for a voting method to be both strategy-free and deterministic (that is, select the same outcome every time it is applied to the same set of ballots). The Random Ballot voting method, which selects the ballot of a random voter and uses this to determine the outcome, is strategy-free, but may result in different choices being selected if applied multiple times to the same set of ballots.

However, the extent to which tactical voting affects the timbre and results of the campaign varies dramatically from system to system: see below.

## Strategy-resistant voting methods

While no deterministic voting method may be strategy-free, the degree that they reward strategy differ greatly. Plurality voting and the Borda count often reward tactical voting, while Condorcet-IRV hybrid methods are considerably more robust.[4]

François Durand found that for voting methods passing a weak form of the majority criterion, modifying the method to elect the Condorcet winner whenever one exists can never increase the susceptibility to strategy. He also found that, given an independence assumption, asking for more information than ranks can't unlock higher levels of strategy resistance. Durand thus argues that a search for the most strategy-resistant voting method can be restricted to ranked methods that pass the Condorcet criterion.[5]

A variety of criteria have been devised to indicate forms of strategy resistance. See, for instance, dominant mutual third burial resistance.

## Examples in real elections

In United Kingdom elections, there are three main parties represented in the Parliament: the Labour party, the Conservative party and the Liberal Democrats. Of these three, Labour and the Liberal Democrats are most similar. Many people who prefer the Liberal Democrats vote for the Labour candidate where Labour is stronger and vice-versa where the Liberal Democrats are stronger, in order to prevent the Conservative candidate from winning.

In 2010, Liberal and Conservative governments shared the vote of the UK voters creating a hung government, it was decided that Conservatives and Liberal Democrats will perform as a power-sharing government. However this was not the first time the country has been run in a similar fashion as Liberal and Conservative governments alternated in power until World War I and Labour formed two short-lived minority governments in 1923-24 and 1929-31.

In the 1997 UK General Election, the Democratic Left organised GROT - Get Rid Of Them - a tactical voter campaign. In 2001, the Democratic Left's successor organisation the New Politics Network organised a similar campaign tacticalvoter.net. Since then tactical voting has become a real consideration in British politics as is reflected in by-elections and by the growth in sites such as www.tacticalvoting.com who encourage tactical voting as a way of defusing the two party system and empowering the individual voter.

## Rational voter model

Academic analysis of tactical voting is based on the rational voter model, derived from w:rational choice theory. In this model, voters are short-term instrumentally rational. That is, voters are only voting in order to make an impact on one election at a time (not, say, to build the political party for next election); voters have a set of sincere preferences, or utility rankings, by which to rate candidates; voters have some knowledge of each other's preferences; and voters understand how best to use tactical voting to their advantage. The extent to which this model resembles real-life elections is the subject of considerable academic debate.

### Predisposition to sincerity

Some experiments have found that voters tend to behave sincerely more often than the instrumentally rational model indicates. In an experiment designed to have a low barrier to sophisticated voting, Herzberg and Wilson found that only 20%-40% of the voters made use of the opportunity; the rest voted sincerely.[6]

Blais and Nadeau use a two-step analysis procedure to argue that 30% of the voters who would have benefited from strategic voting in the 1988 Canadian election actually did vote strategically.[7] They furthermore reason that tactical voting is more prevalent if the voters have only a weak intensity of preference for their first choice over their second, or if the election is a close race between their second and third choice.

However, the dominance of the two major parties in the United States (typically pulling well over 90% of the vote) suggest this predisposition can be overwhelmed when the incentives for strategy become too large.

### Myerson-Weber strategy

One type of general rational voter strategy is given by Myerson and Weber.[8] It consists of each voter estimating how likely it is that pairs of candidates are going to be tied, and then voting to optimally break the most likely ties.

The model assumes that the voter's utility depends only on who wins, not (for instance) whether a losing candidate the voter supports is seen to have put up a good fight.

For a point-summing system the strategy can be formally described as follows. Let there be k candidates and define

vi = the number of points to be voted for candidate i
ui = the voter's gain in utility if candidate i wins the election
pij = the (voter's perceived) pivot probability that candidates i and j will be tied for the most total points to win the election.

Then the voter's prospective rating for a candidate i is defined as:

${\displaystyle R_{i}=\sum _{j\neq i}\;p_{ij}\cdot (u_{i}-u_{j})\,}$

The gain in expected utility for a given vote is given by:

${\displaystyle G(p,v,u)=\sum _{i=1}^{k}\;v_{i}\cdot R_{i}\,}$

Formal strategies like the Myerson-Weber can be incorporated into voting methods to produce a declared strategy voting method. For instance, Range voting can be thus augmented to produce SARVO-Range.

## Pre-election influence

Because tactical voting relies heavily on voters' perception of how other voters intend to vote, campaigns in electoral systems that promote compromise frequently focus on affecting voters' perception of campaign viability. Most campaigns craft refined media strategies to shape the way voters see their candidacy. During this phase, there can be an analogous effect where campaign donors and activists may decide whether or not to support candidates tactically with their money and labor.

In rolling elections, or runoff votes, where some voters have information about previous voters' preferences (e.g. presidential primaries in the United States, French presidential elections), candidates put disproportionate resources into competing strongly in the first few stages, because those stages affect the reaction of latter stages.

## Views on tactical voting

Some people view tactical voting as providing misleading information. In this view, a ballot paper is asking the question "which of these candidates is the best?". This means that if one votes for a candidate who one does not believe is the best, then one is lying. Labour Party politician Anne Begg considers tactical voting dangerous: [1]

Tactical voting is fine in theory and as an intellectual discussion in the drawing room or living rooms around the country, but when you actually get to polling day and you have to vote against your principles, then it is much harder to do.

While most agree that tactical voting is generally a problem, there are some cases when a strictly limited amount of it may bring about an more democratic result. Since the Gibbard-Satterthwaite theorem shows that all systems are vulnerable to tactical voting, the question becomes what kinds of tactical voting are encouraged by each system, and how often, more than whether a method is immune to strategy.

For ranked voting systems, Arrow's impossibility theorem proves that any voting system is arguably undemocratic in at least some case. However, tactical voting may be used to mitigate the issues of such systems. For instance, under purely honest voting, Condorcet method-like systems tend to settle on consensus candidates, while Instant-Runoff Voting favors those candidates which have a stronger polarizing faction - who may often be more fringe in beliefs. An electorate using one of these two systems but which (in the general or the specific case) preferred the characteristics of the other system could consciously use strategy to achieve a result more characteristic of the other system. Under Condorcet, they may be able to win by "burying" the consensus candidate (although this risks throwing the election to the opposing faction); while under IRV, they could always compromise and vote for the consensus above their true favorite.

The problem is that such tactical voting would tend to overshoot and give undesired results. This greatly complicates the comparative analysis of voting systems. If tactical voting were to become significant, the perceived "advantages" of a given voting system could turn into disadvantages - and, more surprisingly, vice versa.

Finally, any voting system that relies on a particular strategy to produce good results can be replaced by another voting system that executes that strategy on behalf of the voters - a so-called declared strategy voting method. This is a consequence of the w:revelation principle. It is thus not possible to get around impossibility results by relying on tactical voting.

One might try to salvage immunity to strategy by engineering a method where voting tactically is always dangerous and prone to overshooting. Such a method could make strategy too risky to engage in. However, when the Gibbard-Satterthwaite theorem applies, it is impossible to completely eliminate the incentive for strategy in this manner: there will always exist some situation where strategic voting is safe.[9]

## Definitions

Frontrunner/viable candidate: A candidate expected to have a significant chance of winning.

Truncation: When a voter doesn't show support for some of their less-preferred candidates (i.e. an A>B>C voter truncates and only votes A>B or A).

Bullet voting: When a voter only supports one candidate (usually defined as also maximally supporting them in rated methods). It is a special case of truncation.

Min-maxing: When a voter gives maximal support to some candidates (usually defined here as ranking or rating them all equally) and no support to all other candidates.

## Tactical voting in particular systems

Steven Brams and Dudley R. Herschbach argued in a paper in Science magazine in 2000 that approval voting was the system least amenable to tactical perturbations. This may be related to the fact that approval voting does not permit preferences ('likes' or 'dislikes') to be stated at all, permitting only a statement of tolerances, that is, "which candidate could you stand to see win", as opposed to "which candidate would you like to see win".

Due to the especially deep impact of tactical voting in first past the post electoral systems, some argue that systems with three or more strong or persistent parties become in effect forms of disapproval voting, where the expression of disapproval, to keep an opponent out of office overwhelms the expression of approval, to approve a desirable candidate. Ralph Nader refers to this as the "least worst" choice, and argues that the similarity of parties and the candidates grows stronger due to the need to avoid this disapproval.

Sirin Botan et al. showed that every Condorcet method of a particular type sometimes incentivizes the creation of Condorcet cycles when there's a sincere Condorcet winner. The types covered are Condorcet methods that only use pairwise defeat information and don't always tie when there's no Condorcet winner. [10] This category includes, among others, ranked pairs and Copeland's method, but not Smith//IRV or Condorcet-cardinal hybrid methods.

There are arguments about the best voting strategy to take in different systems, but the general consensus is:

• Score voting (including approval): Give the highest score to all candidates better than the expected value of the winner (or better than the frontrunner, if you don't know the expected values). Give the lowest score to all the other candidates. This is known as the threshold strategy or min-max-ing.
• Methods failing No Favorite Betrayal: Rank your favorite frontrunner first and your least-favorite frontrunner last.

## Notes

### Voting for the lesser of two evils

Much voting strategy revolves around a voter deciding whether to back one of the frontrunners or not; this often reduces further to deciding which of 2 frontrunners to back, which results in essentially a head-to-head matchup between the two. This is often referred to as deciding whether to "vote for the lesser of two evils or waste your vote". One of the goals of voting reform is to allow voters to be able to be as sincere as possible in expressing their preference for nonviable candidates.

### Information in strategic voting

An important thing to consider with strategic voting is how difficult it is for voters to figure out how to strategically vote. Distinctions are made between zero-info strategy (strategy that can be applied to get a better result without any information of other voters' preferences) and strategies that revolve around having various amounts of (accurate) polling information. In addition, the likelihood of a strategy working, and the risk/amount of harm (see utility) coming from it backfiring is also studied. Another common measure of a voting method's resistance to strategic voting is manipulability, which measures how often a voter or group of voters can vote strategically to improve the election results from their point of view.

### Multi-winner methods

The Duggan-Schwartz theorem extends Gibbard-Satterthwaite to multi-winner voting methods. It states that either some candidates can never win, or some voters are treated differently than others, or the outcome consists of some group of voters' first preferences, or the method is manipulable. It therefore isn't possible to escape tactical voting by making the method elect multiple winners (unless it elects so many that everybody's first preference is elected).