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# Tactical voting

In voting systems, tactical voting (or strategic voting or sophisticated voting) occurs when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.

It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is completely strategy-free must be either dictatorial or nondeterministic (that is, might not select the same outcome every time it is applied to the same set of voter preferences). For instance, the random ballot voting method, which randomly selects the ballot of a single 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 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.

## Types of tactical voting

### Compromising

Compromising (sometimes "useful vote") is a type of tactical voting in which a voter insincerely ranks an alternative higher in the hope of getting it elected. For example, in the first-past-the-post election, voters 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 in order to help defeat a strong right-wing candidate.) Duverger's law suggests that, for this reason, first-past-the-post election systems will lead to two-party systems in most cases. In those proportional representation systems that include a minimum percentage of votes that a party must achieve to receive any seats, people might vote tactically for a minor party to prevent it from dropping below that percentage, which would make the votes it does receive useless for the larger political camp that party belongs to.

### Burying

Burying is a type of tactical voting in which a voter insincerely ranks an alternative lower in the hopes of defeating it. For example, in the Borda count or in a Condorcet method, a voter may insincerely rank a perceived strong alternative last in order to help their preferred alternative beat it.

### Push-over

Push-over is a type of tactical voting in which a voter ranks a perceived weak alternative higher, but not in the hopes of getting it elected. This primarily occurs in runoff voting when a voter already believes that her favored candidate will make it to the next round – the voter then ranks an unpreferred, but easily beatable, candidate higher so that her preferred candidate can win later. In the United States, for instance, voters of one party sometimes vote in the other party's primary to nominate a candidate who will be easy for their favorite to beat, especially after that favorite has secured his party's own nomination.

### Bullet voting

Bullet voting is a type of tactical voting used in elections where voters select more than one representative from a pool of candidates. In this instance a voter can cast as many votes as there are representatives to be elected. The winners are the candidates that receive the highest vote totals. By not casting the maximum number of votes, a voter helps his or her preferred candidate by not supplying votes to potential rivals.

It is also a vital strategy within cumulative voting.

## Examples in real elections

As the United States has, by and large, a two-party system, there is almost no opportunity for tactical voting to occur in general elections. However, one high-profile example of tactical voting was the situation that led to the 2003 California recall. During the primaries, Republicans Richard Riordan (former mayor of Los Angeles) and Bill Simon (a self-financed businessman) were vying for a chance to compete against the unpopular Governor of California, Gray Davis. As California holds open primaries in which anyone can vote for any candidate he or she pleases, Davis supporters were rumored to have voted for Simon because Riordan was perceived as a greater threat to Davis; this combined with a negative advertising campaign by Davis describing Riordan as a "big-city liberal", and Simon ultimately won the primary despite a last-minute business scandal. However, he lost the election against Davis; discontent soon led to the recall.

In the 1997 United Kingdom general election in Winchester, Mark Oaten (Liberal Democrat) beat the incumbent Conservative MP Gerry Malone with a majority of 2 votes. Malone successfully challenged the election in the High Court, which declared it void. A by-election was held which returned Mark Oaten as MP, this time with a larger majority of 21,556. This was due to the majority of Labour voters voting Liberal in the by-election.

The candidacy of Richard Huggett in both 1997 elections as a "Literal Democrat" candidate led in part to the creation of the Registration of Political Parties Act 1998.

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 considered most similar by many voters. 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 the 1997 UK general election, Democratic Left helped Bruce Kent set up GROT - Get Rid Of Them - a tactical voter campaign whose sole aim was to help prevent the Conservative Party from gaining a 5th term in office. This coalition was drawn from individuals in all the main opposition parties and many who were not aligned with any party. While it would be hard to prove that GROT swung the election itself, it did attract significant media attention and brought tactical voting into the mainstream for the first time in UK politics. 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 tacticalvoting.com who encourage tactical voting as a way of defusing the two party system and empowering the individual voter. In the 2005 UK General Election individuals set up tacticalvoting.net to balance the tactical voting debate.

In the Ontario general election, 1999, strategic voting was widely encouraged by opponents of the Progressive Conservative government of Mike Harris. This failed to unseat Harris, and succeeded only in suppressing the New Democratic Party vote to a historic low.

In the Canadian general election, 2004 and to a lesser extent in the Canadian general election, 2006, strategic voting was a concern for the federal New Democratic Party. In the 2004 election, the governing Liberal Party was able to convince many New Democratic voters to vote Liberal in order to avoid a Conservative government. In the 2006 elections, the Liberal Party attempted the same strategy, with Prime Minister Paul Martin asking New Democrats and Greens to vote for the Liberal Party in order to prevent a Conservative win. The New Democratic Party leader Jack Layton would respond by asking voters to "lend" their votes to his party, suggesting that the Liberal Party would be bound to lose the election regardless of strategic voting.

In the 2006 local elections in London, tactical voting is being promoted by sites such as London Strategic Voter in a response to national and international issues. The question of whether this approach acts to undermine local democracy is receiving much debate.

In Northern Ireland, it is widely thought that (predominantly Protestant) Unionist voters in Nationalist strongholds have voted for the Social Democratic and Labour Party (SDLP) to prevent Sinn Féin from capturing such seats.

## Rational voter model

Academic analysis of tactical voting is based on the rational voter model, derived from 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.

### Myerson-Weber strategy

An example of a rational voter strategy is described by Myerson and Weber . The strategy is broadly applicable to a number of single-winner voting systems that are additive point systems, such as Plurality, Borda, Approval, and Range. The strategy is optimal in the sense that the strategy will maximize the voter's expected utility when the number of voters is sufficiently large.

This rational voter model assumes that the voter's utility of the election result is dependent only on which candidate wins and not on any other aspect of the election, for example showing support for a losing candidate in the vote tallies. The model also assumes the voter chooses how to vote individually and not in collaboration with other voters.

Given a set of k candidates and a voter let:

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:

$R_i = Sigma_\left\{j neq i\right\} ; p_\left\{ij\right\} * \left(u_i - u_j\right),$

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

$G\left(p,v,u\right) = Sigma_\left\{i=1\right\}^k ; v_i * R_i,$

The gain in expected utility can be maximized by choosing a vote with suitable values of vi, depending on the voting system and the voter's prospective ratings for each candidate. For specific voting systems, the gain can be maximized using the following rules:

• Plurality: Vote for the candidate with the highest prospective rating. This is to be distinguished from choosing the best of the frontrunners, which is a common but imprecise plurality tactic. The highest prospective rating can in fact belong to a weak candidate, even the weakest.
• Borda: Rank the candidates in decreasing order of prospective rating.
• Approval: Vote for all candidates that have a positive prospective rating; do not vote for any candidates that have a negative prospective rating.
• Range: Vote the maximum points for all candidates that have a positive prospective rating; vote the minimum allowed value for all candidates that have a negative prospective rating; vote any number of points for a candidate with a prospective rating of zero.

An important special case occurs when the voter has no information about how other voters will vote. This is sometimes referred to as the zero information strategy. In this special case, the pij pivot probabilities are all equal and the rules for the specific voting systems become:

• Plurality: Vote for the most preferred (highest utility) candidate. This is the sincere plurality vote.
• Borda: Rank the candidates in decreasing order preference (decreasing order of utility). This is the sincere ranking of the candidates.
• Approval: Calculate the average utility of all candidates. Vote for all candidates that have a higher-than-average utility; do not vote for any candidates that have a lower-than-average utility.
• Range: Calculate the average utility of all candidates. Vote the maximum points for all candidates that have a higher-than-average utility; vote the minimum points for all candidates that have a lower-than-average utility; vote any value for a candidate with a utility equal to the average.

Myerson and Weber also describe voting equilibria that require all voters use the optimal strategy and all voters share a common set of pij pivot probabilities. Because of these additional requirements, such equilibria may in practice be less widely applicable than the strategies.

## 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 voter's 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. British Labour Party politician Anne Begg considers tactical voting dangerous:

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.

Tactical voting is commonly regarded as a problem, since it makes the actual ballot into a nontrivial game, where voters react and counter-react to what they expect other voters' strategies to be. British mathematician Charles Dodgson (a.k.a. Lewis Carroll) was, apparently, the first to acknowledge this fact. A game such as this might even result in a worse alternative being chosen, because most of the voters used it as a strategic tool.

Though Arrow's impossibility theorem and Gibbard-Satterthwaite theorem prove that any useful voting system is prone to some kind of manipulation, some use game theory to search for some kind of "minimally manipulatable" voting schemes.

Game theory can also be used to analyze the pros and cons of different methods. For instance, under purely honest voting, Condorcet method-like systems tend to settle on compromise candidates, while Instant-Runoff Voting favors those candidates which have strong core support - who may often be more extremist. 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 compromise candidate (although this risks throwing the election to the opposing extreme); while under IRV, they could always "compromise". It could be argued that in this case the option to vote tactically or not actually helps the electorate express its will, not only on which candidate is better, but on whether compromise is desirable. (This never applies to "sneakier" tactics such as push-over.)

Tactical voting greatly complicates the comparative analysis of voting systems. If tactical voting were to become significant, the perceived "advantages" of a given voting system (that is, tending towards compromise or favoring core support) could turn into disadvantages - and, more surprisingly, vice versa.

## Tactical voting in particular systems

### First past the post / plurality voting

Tactical voting by compromising is exceedingly common in plurality elections.

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 in order to keep an opponent out of office overwhelms the expression of approval to elect a desirable candidate. Ralph Nader refers to this as the "least worst" choice, and argues that the similarity of parties and the candidates in first past the post systems grows stronger due to the need to avoid this disapproval.

One often-overlooked flaw in the first past the post system is that, in single member districts, voters can invariably select only one candidate, whilst in multi-member districts they can never select more candidates than the number of seats. Approval voting, by contrast, allows voters to cast a vote for as many candidates as they wish. This in turn allows for "voting against" a certain despised candidate without having to guess at who the strongest contender is to defeat that candidate. Such a system would also serve to reduce the spoiler effect.

### Approval voting

Steven Brams and Dudley R. Herschbach argued in a paper in Science magazine in 2001 that approval voting was the system least amenable to tactical perturbations. Authors of a 2004 paper in Public Choice disagreed, saying, "AV is one of the most susceptible systems to manipulation by small groups of voters (for example, small, maverick groups could determine the AV outcome.)

### Instant runoff voting

Instant runoff voting has a reduced incentive for the compromising strategy, plus a minor vulnerability to the push-over strategy.

### Condorcet

Condorcet methods have a further-reduced incentive for the compromising strategy, but they have some vulnerability to the burying strategy. The extent of this vulnerability depends on the particular Condorcet method. Some Condorcet methods arguably reduce the vulnerability to burying to the point where it is no longer a significant problem.

### Borda

The Borda count has both a strong compromising incentive and a large vulnerability to burying. Here is a hypothetical example of both factors at the same time: If there are two candidates whom a voter considers to be the most likely to win, the voter can maximize their impact on the contest between these candidates by ranking the candidate whom they like more in first place, and ranking the candidate whom they like less in last place. If neither candidate is their sincere first or last choice, the voter is employing both the compromising and burying strategies at once.

### Single Transferable Vote

The compromising incentive exists in the Single Transferable Vote. If one's top-choice candidate is elected, only a fraction of one's vote will be transferred to one's next-favoured candidate. If one feels the favoured candidate is certain to be elected in any case, insincerely ranking the second candidate first guarantees them a full vote if needed. However, the greater the certainty of the first candidate being elected, the bigger their likely surplus, the higher the fraction of the vote that would be transferred to the next candidate, and hence the lower the proportionate benefit of tactical voting.

More sophisticated tactics may be practicable where the number of candidates, voters and/or seats to be filled is relatively small.

Some forms of STV allow tactical voters to gain an advantage by listing a candidate who is very likely to lose in first place, as a form of pushover. Meek's method essentially eliminates this strategy.

## Tactical unwind

The term "tactical unwind" is used by some political scientists and commentators to refer to the phenomenon when tactical voting takes place in one general election but in subsequent elections voters revert to their normal patterns.