Strategic Resistance Analysis

When choosing a voting method for use in a competitive political elections, it is important that the method be resistant to strategic voting that alters the outcome. If voters were impartial judges who cast ballots dispassionately, strategic resistance would not be a high priority, but in political contests, voters are necessarily invested in the result. In elections for public office, voters have a conflict of interest by design.

According to the Gibbard-Satterthwaite theorem, all voting methods are vulnerable to some form of strategic voting. However, voting methods are not equal in their resistance to such strategies. From our analysis below, we conclude that Ranked Choice Voting is the method most resistant to strategic voting.

Our analysis considers 4 potential voting strategies: bullet-voting, burying, compromise, and push-over:

  • Bullet voting: to insincerely express a preference for only a single candidate to increase the probability that candidate wins. Everything this analysis says of bullet voting applies equally to any degree of insincere preference truncation, e.g. expressing a preference for 2 candidates when one's sincere preference includes 3.
  • Burying: to insincerely express a lower preference for a candidate to decrease the probability that candidates is elected. A voter that is burying normally intends to defeat the strongest opponent of their sincere favorite.
  • Compromising: to insincerely express a higher preference for a candidate to increase the probability that candidate is elected. A voter that is compromising normally intends to elect their "compromise candidate" (the beneficiary of the insincere higher preference) when they deem their sincere favorite unlikely to win.
  • Push-over: to insincerely express a higher preference for a candidate to increase the probability that a different candidate wins. A voter that is using push-over normally intends for the "push-over candidate" (the beneficiary of the insincere higher preference) to defeat the strongest opponent to their sincere favorite, before their sincere favorite then defeats the push-over candidate.

A two-part analysis

Our analysis is similar in several respects to a 2012 analysis by Professor Jack Nagel. For each voting method, we answer two questions:

  • How likely are voters to use a strategy to change the outcome, independent of the particular voting method?
  • How often is a the voting method in a state where use of the strategy would change the outcome?

If we view each of these answers as probabilities, then the probability an election can by manipulated by a strategy is the product of the two. Unfortunately, it's not possible to assign precise numerical values to the probability voters will use a strategy, because that depends on social factors and often on the availability of accurate polling in advance of the election, as well. We instead identify several properties that increase the likelihood each strategy will be used. As for the probability that a method can be manipulated by strategy, prior research has approximated numerical values for some method-strategy combinations, subject of course to the assumptions made by those statistical models, and we cite those results where available.

Evaluating Voting Strategies

We begin with the question of how likely voters are to use a strategy to change the outcome, independent of the particular voting method. We compare each of the 4 strategies with respect to 4 properties: sincere order, campaign incentive, zero knowledge, and counterstrategy. Each of these properties are "negative" qualities, in that possessing one increases the likelihood the strategy will be used in practice:

  • Sincere order: the strategy does not require an inversion of the voter's sincere preference order. If a voter sincerely prefers X to Y, a strategy with "sincere order" never requires the voter to express a strict preference of Y to X. It only allows for "exaggerated" preferences, such as ones that show X preferred to Y by a stronger degree, or ones that show indifference between X and Y. Although the word "sincere" makes it sound benign, strategies with with this property are particularly insidious for two reasons. (1) They are more intuitive, because it is easier for voters to deduce that exaggerated preferences may benefit them than inverted ones. (2) They raise fewer moral qualms, because it is less egregious to "bend" the truth with an exaggeration than the clear dishonesty of an inversion.
  • Campaign incentive: it is in interest of political campaigns to encourage their supporters to use the strategy. By "supporters," we mean voters for whom that candidate is the favorite, i.e. who would rank that candidate first. If a strategy possesses "campaign incentive," the campaign or campaign surrogates are likely to encourage their supporters to engage in the strategy in a coordinated fashion.
  • Zero knowledge: the strategy can be carried out without advanced knowledge of how others will vote. If a strategy can be carried out with no or little advanced knowledge, then it can be executed even when accurate polling information is unavailable.
  • Counterstrategy: the strategy motivates others to employ the strategy themselves in opposition. If a strategy possesses counterstrategy, use of it can trigger a strategic "arms race," further increasing its prevalence.

Bullet voting: Very likely

Bullet voting is the only strategy to possess all four properties. It has "sincere order," because voting for only a single candidate (or generally truncating one's ballot) does not require inverting one's sincere preferences, only exaggerating them. Since bullet voting benefits the voter's favorite candidate, it is in the interest of campaigns to encourage their supporters to engage in it. Bullet voting doesn't require any advanced polling data to use correctly, so it suffers from "zero knowledge," as well. Lastly, if some voters bullet vote, thereby leaving their opponents off the ballot altogether, the opponents are motivated to retaliate by bullet voting themselves. The combination of all four properties makes voters very likely to bullet vote when it is advantageous.

Burying: Likely

Burying involves the inversion of preferences, so it does not have sincere order. However like bullet voting, it does benefit a voter's favorite candidate, and thus suffers from campaign incentive. Burying requires more than zero knowledge, but not too much more: just an accurate prediction of the front-runners. It also has counterstrategy, because if supporters of candidate A bury candidate B, supporters of B may retaliate by burying A.

Compromising: Somewhat likely

Compromising is the only strategy to lack campaign incentive. The point of the compromising strategy is to elect someone other than one's favorite, so its not in a campaign's interest to encourage. Like burying, compromising also lacks sincere order and requires a correct prediction of just the front-runners to carry out. There is no counterstrategy to compromising.

Push-over: Not likely

Push-over is an implausible strategy in real political elections. It requires detailed advanced knowledge of how everyone else will vote, very far from "zero knowledge." It also lacks sincere order. It does technically possess some degree of campaign incentive, because the strategy benefits a voter's favorite candidate, but it carries too high-level a risk for campaigns to be realistic. A campaign would need to convince a surgical subset of their supporters — not too few, not too many — to invert their preferences while everyone else votes honestly. In their study of the 2007 French legislative elections, which used two-round runoff, Dolez and Laurent found exactly zero voters who attempted it. Pushover has no counterstrategy.

Evaluating Voting Methods

Factoring in prior evaluation of the strategies themselves, we now evaluate the overall strategic resistance of the following voting methods:

Ranked Choice Voting: Very High Resistance

Ranked choice voting is immune to the strategies with the highest likelihood of use: bullet-voting and burying. RCV is immune to bullet-voting because it satisfies a criterion known as later-no-harm, which means that ranking an additional choice on the ballot doesn't hurt the chances an earlier choice is elected. While RCV is vulnerable to compromising, the situations in which it is vulnerable are rare, measured to be "low" by James Green-Armytage's statistical analysis; plus we evaluated compromising to be only a "somewhat likely" strategy to begin with. It is technically vulnerable to the push-over strategy, but that strategy is too risky and difficult to pull off in a political election.

Two-round Runoff: High Resistance

Like Ranked Choice Voting, Two-Round Runoff is immune to the bullet-voting and burying strategies. It also vulnerable to push-over, but the implausibility of that strategy, as confirmed by the above-mentioned Dolez and Laurent study, confirm that it is a non-issue in political elections. It is, however, more vulnerable than RCV to compromising. In crowded fields, there is a motivation to insincerely vote for a candidate that is presumed to be one of the top-two vote-getters. Dolez and Laurent found this to be about 1.6% of voters in the 2007 French legislative elections — quite low, but present nonetheless.

Plurality Voting: Moderate Resistance

Plurality is trivially immune to bullet voting, burying, and push-over, since the voter cannot vote for more than one choice. It is vulnerable to compromising, which we evaluated to be only a "somewhat likely" strategy for voters to use when advantageous. That said, plurality is very frequently vulnerable to compromising, more often than any other method studied by Green-Armytage's statistical analysis.

Condorcet Methods: Moderate Resistance

Condorcet methods are vulnerable to all four voting strategies, to varying degrees. All Condorcet methods violate later-no-harm and so are vulnerable to bullet-voting. However, Condorcet methods also violate the later-no-help, meaning that a later choice can help an earlier choice, which may deter a voter who is considering a bullet vote. While Condorcet methods are technical vulnerable to compromising, it is only advantageous in the rare case that no Condorcet winner exists. There are also situations in which Condorcet methods are vulnerable to push-over, but as is the case for other methods, push-over is too risky and complicated to be plausible in a political elections. The most serious strategic problem for Condorcet methods is their frequency of burying vulnerabilities, exacerbated by the fact that burying possesses campaign incentive, near-zero knowledge, and counter-strategy.

Despite being theoretical vulnerability to many strategies, Condorcet methods make it difficult for a voter to know when and how to take advantage of the vulnerability when it exists. However, voters may bullet vote or bury instinctively, even when it isn't necessarily advantageous, because these strategies intuitively "should" work, and voter education could not truthfully teach that they never work. Of particular concern is what would happen under Condorcet when burying an opponent motivates counter-burying in retaliation. If two front-runners encourage their respective supporters to bury the other front-runner, it could lead to the election of neither: ironically it could elect the Condorcet loser at equilibrium, including in situations where even plurality voting would elect the Condorcet winner.

Approval and Range Voting: Very Low Resistance

Approval and Range Voting have very low strategic resistance. The biggest strategic concern with these methods is their vulnerability to bullet voting, a strategy that suffers from all of the motivating factors: sincere order, campaign incentive, zero knowledge, and counterstrategy. Also, unlike Condorcet methods, Approval and Range satisfy later-no-help, so bullet voting can never backfire under these methods. Approval and Range are also frequently vulnerable to both burying and compromising, rated a "high" vulnerability by the Green-Armytage statistical analysis. They are immune to push-over.

Range voting is "spreading" strategy, in which all the candidates are given either the maximum or minimum scores. That strategy can be seen as a combination of compromising and burying simultaneously.

A particular strategic concern with Approval and Range voting is the combination of bullet voting and counter-bullet-voting. Two candidates that comprise a mutual majority may face a "chicken dilemma," also known as the "Burr dilemma," wherein their supporters have to choose between approving of both, to increase the odds that at least of them wins, or bullet vote for their favorite at the risk that neither wins. If the supporters of two front-runners bullet vote and counter-bullet vote, then the outcome will degrade to the plurality result.