1 Introduction

ARROW (1951, 1963) and later Gibbard (1973) and Satterthwaite (1975) show that all voting systems violate seemingly reasonable expectations under some circumstances. The question is, how often do these anomalies occur, and how seriously? Further, what is the likelihood that a voting rule will identify the correct winner? What is the definition of the correct winner?

These questions lead to the exploration of models of voter behavior and analyses of various voting rules under these models. Tideman and Plassmann (2010) showed that a spatial model of voter behavior fits the observed data so well that it is hard to imagine a model that fits them better. They further conclude that “any conclusions about the probability of voting phenomena in actual elections are suspect if they are reached on the basis of models other than the spatial model”. This conclusion is of profound importance.

Non-spatial models of voter behavior do not allow for the possibility that candidates could alter outcomes by altering their positions, simply because there is no “position” and no “space” in which to alter them. Conversely, spatial models imply that candidates must consider the strategic implications of their ideological position or risk defeat by another candidate who does. Consider an election held under Condorcet-minimax, with three candidates drawn from the same distribution as the voters. A fourth candidate could choose to enter the race at or near the median voter and win almost every time. It is implausible that candidates would not consider this possibility in actual elections. So, while candidate strategy cannot play a role under simulations using the Impartial Culture or Impartial Anonymous Culture models of the distribution of voters, candidate strategy must be considered under a spatial model, to reflect what we should reasonably expect to happen.

The concept of candidate strategy in a spatial model of voter preference also has implications for measuring and comparing the impact of voting anomalies. If the positions of candidates are consistent across voting rules, it is sufficient to state that a voting rule did or did not select the optimal candidate. The potential impact of strategic voting has been measured as the frequency of the possibility of changing the outcome (Green-Armytage, et al., 2016), rather than by a measure of its impact on the “correctness” of the winning candidate. When candidates behave strategically, the impact of a voting rule anomaly can vary because the positions of the candidates are variable.

More importantly, the strategy favored by the voting rule can cause deleterious decisions by the candidates before the race even begins. The voting rule may appear to avoid undesirable outcomes, but only because good candidates were eliminated or altered their positions for strategic advantage before the election was held.

Tideman and Plassmann (2008) suggest that “The goal of such an analysis is to find a voting method that is on average more likely than other methods to identify the correct outcome.” This is correct when all voting rules are using the same set of candidates. When the candidates are forced to make different strategic decisions by different voting rules, this needs to be amended to “finding the voting method that on average produces the best outcome.”

Arrow, and later Gibbard and Satterthwaite proved that all voting rules for ranked ballots violate some seemingly reasonable expectations some of the time. Since then, a metric of the frequency with which a rule violates those reasonable expectations has developed. Given a spatial voting model and strategic candidates, this goal should be refined to include a measure of the social utility of the resulting representation, which for spatial models can reasonably be specified as the negative of the mean distance between a voter and the winning candidate.

2 The voting model

Voters use a preference model to choose their preferred candidate as follows:

$$S = - \left\| {I_{C} - I_{V} } \right\| + Qc + U_{VC}$$

where: S is the score for the candidate, IC and IV are the ideology vectors of the candidate and voter respectively, Q is the quality of the candidate; Q is held at zero in these simulations, Uvc is the uncertainty in voter V’s perception of candidate C’s location; it is a unit Gaussian * 0.2σ.

For each voter, a score is computed for each candidate; the voter ranks the candidates in the order of the scores, highest first. An example of the result is shown in Fig. 1.

Fig. 1
figure 1

A spatial model without uncertainty. Without uncertainty, A receives all the votes to the left, B receives a tiny slice of gray in the center and C gets all the votes to the right. A more realistic outcome would be that all three candidates share voters everywhere with A having an advantage to the right and C to the left

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U, Uncertainty is not included in some simulation models. In many models, voters are considered to have perfect information about both the candidate’s positions and their own. The consequence is that simulations for three candidates who are very close to each other are unrealistic. Figure 1 shows an example of 3 candidates without uncertainty, where the candidates are close together. In reality, voters have different and imperfect information. Voters to the right will be slightly more likely to favor C over B over A, but the difference will be slight. A more realistic outcome is for all three candidates to have approximately equal vote share with any difference being due to candidate quality rather than ideology. Accounting for uncertainty is essential in elections with more than two candidates.

3 Strategy simulation methodology

How do candidates “choose” their ideology? How does the model account for candidates altering their ideology simultaneously? I have chosen solutions to these problems that are a step in the right direction, but are by no means perfect. Future investigators will doubtless refine each element of this process.

Candidates have ideologies that are drawn from a normal distribution with σ = 0.5, candidates with ideology > 1.5 σ are rejected, and groups of candidates that do not span the range of − 0.25 σ to 0.25 σ are rejected.

Each candidate attempts to maximize their expected utility. The utility function rewards them for winning the race, but that reward decreases as they move away from their preferred initial ideology.

$$U = Max(0,1 - \left\| {I_{x} - I_{o} } \right\|/f) \cdot P(W\left| {I_{x} ,I_{cc} } \right.)$$

where Ix is the proposed new ideology, Io is the starting ideology, f is the flexibility, set to 0.7 σ, Icc is the ideology of the other candidates in the race, \(P(W\left| {I_{x} ,I_{cc} } \right.)\) is the probability of winning given Ix and Icc.

For computational reasons the ideology space is limited to a single dimension and divided into 21 discrete bins spanning a range from − 1.5 σ to 1.5 σ. Each bin is ~ 0.14 σ wide.

After initial seeding, each candidate tests their utility at each bin within 3 of their starting bin at an ideology uniformly chosen within that bin. The candidate adopts the ideology that maximizes their utility. For example: if a candidate has a starting ideology that maps to bin 4, they are allowed to test their utility in bins 1 through 7. Three rounds of adjustment are performed, with an annealing process where the search range for an optimum utility is set to 3, 2, and then to 1 bin. The binning process helps model the uncertainty in each candidate’s estimate of the voters’ opinion of the ideology of their competitors.

The objective is to simulate an election where candidates adjust their policy positions based on the positions and policies of other candidates. Candidates can make larger changes early in the race but lose credibility for making large changes later in the race.

4 W, win probability

The as yet unexplained element of the utility function is the win probability. W could be approximated by simulating a large number of elections for each ideology under consideration. This was deemed intractable; the annealing requires enough simulations to establish a probability for each of 75 different potential configurations for each election with strategic candidates to be simulated.

A relatively simple neural net was chosen to predict the probability of victory for a candidate, given the ideologies of the other candidates in the race. The architecture is simple, consisting of 21 input bins, and 3 fully connected layers of 512 neurons with dropout of 0.3 applied to each layer. The output layer again has 21 output neurons with a softmax cross-entropy error producing a probability of win for each neuron.

For training, 100,000 elections are simulated, producing 1e6 training samples. Each simulated election generates 10 samples, by eliminating one candidate at a time from the input space and then flipping the input space due to symmetry. The total number of elections possible is 21^5 (4e6) for a five-candidate race.

Error is masked for all output neurons other than the one corresponding to the eliminated candidate, because no information was gathered about the chance of victory for any bin other than the one the candidate held when the election was simulated.

The network was implemented with Python 3.9 and Tensorflow 2.5. Training was performed on a 16 core I9 Mac Pro, without GPU acceleration. Training time was under 2 h.

5 Results

Figure 2 shows the frequency of the ideology of winning candidates without strategy under IRV.

Fig. 2
figure 2

IRV winners with random candidates

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First and foremost, the distribution of outcomes under IRV is radically different when candidates use strategy. Under IRV, a normally distributed electorate produces a polarized distribution of outcomes. Candidates with the highest social utility are unlikely to win under IRV when candidates behave strategically. Colors indicate starting ideologies (Fig. 3).

Fig. 3
figure 3

IRV with strategic candidates

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Figures 4 and 5 show the distributions of outcomes under Condorcet-minimax, when candidates maintain their randomly chosen positions and when they use strategy. Condorcet-minimax shows a concentration of the distribution on winners with high social utility.

Fig. 4
figure 4

Condorcet-minimax with random candidates

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Fig. 5
figure 5

Condorcet minimax winners with strategic candidates

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Figure 6 shows the frequency of the location choices of all candidates, including those who did not win under IRV.

Fig. 6
figure 6

IRV candidate choices

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Candidates do not win when near the center under IRV and avoid the center for their preferred ideology. This gives IRV the appearance of not eliminating the Condorcet winner, however the Condorcet winner is eliminated before the election even begins, because being a candidate with high social utility is so disadvantageous to one’s probability of winning.

Figure 7 shows that under Condorcet-Minimax, candidates choose to run at ideologies with high social utility in nearly all cases.

Fig. 7
figure 7

Condorcet-minimax candidate choices

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Merrill (1984) defines Social Utility Efficiency by:

$$Efficiency = \mathop {\lim }\limits_{n \to \infty } \left( {\frac{{E_{w} - E_{avg} }}{{E_{max} - E_{avg} }}} \right)$$

where n is the number of voters, Eavg is the average utility of a candidate, Emax is the maximum utility of the candidates in a race, Ew is the utility of the winning candidate.

This seems straightforward, but how should one take account of candidate strategy? The original metric measures improvement over random selection by a voting rule that ignores strategy. When candidates use strategy, their positions and utility change. Eavg and Emax are the average and maximum of a candidate’s utility at the beginning of the process, before they adjust their positions due to strategy. If, instead, they measured utility at the time of the election, a rule which induced candidates to move to positions of less social utility would score well even though the winning candidate was of inferior social utility. In this way, Merrill’s Social Utility Efficiency measures the combination of strategic changes induced by the voting rule and the rule’s ability to select well among the final group of candidates.

Figure 8 shows the impact of strategy as a function of flexibility. Flexibility of zero is “strategy-free” as candidates cannot adjust their initial position. Flexibility of 1σ shows [the consequences of assuming that the flexibility factor in the candidate utility function is increased from 0.7 to 1.0.]

Fig. 8
figure 8

Social utility efficiency by flexibility for IRV and Condorcet-minimax

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The efficiency score for IRV is significantly lower than others have reported, because the initial distribution of candidates used here has higher average social utility than drawing randomly from the population. While drawing randomly from the population is traditional, it produces many unrealistic scenarios, causing all voting rules to appear better and closer together than they are with better candidates (higher average social utility). The candidate generation process used here eliminates many unrealistic candidate scenarios from Eavg. The higher baseline utility of the distribution of candidates results in a smaller degree of improvement for IRV.

6 Likelihood of a Condorcet-tie with strategy

Candidates in an election run with a Condorcet-consistent voting rule might move closer to the median voter and consequently be more likely to create a circumstance where no candidate defeats every other candidate. This possibility is plausible (Fig. 9a) if candidate quality, Qc, is perfectly equal across candidates. Figure 9b shows that adding even a small amount of variance to candidate quality reduces the percentage of ties to a negligible level.

Fig. 9
figure 9

Chance of a Condorcet-tie with and without variance in candidate quality

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These ties, should they occur, are the byproduct of all candidates moving close to the point of maximum social utility. They can only occur if at least 3 candidates are so close to the maximum social utility that no candidate is clearly preferred to the other two. Such a circumstance seems both unrealistic and something to be celebrated in the unlikely event that it should occur.

7 The likelihood of electing the Condorcet candidate under IRV

With randomly chosen candidates and an ideology space of higher dimensionality, IRV often selects the Condorcet winner. When candidates make strategic position decisions to maximize their expected utility, a different question is relevant: What is the likelihood that the winning candidate under IRV would have a social utility score greater than or equal to the social utility score of the Condorcet-Minimax winner? One thousand elections were simulated in which, in each simulation, a set of candidate starting positions and a sample of 1000 voters was given to both IRV and Condorcet-Minimax. Candidates then changed their positions to improve their chances of winning. The result was that the winner under IRV had a social utility score greater than or equal to that of the Condorcet-Minimax winner in 2 outcomes out of 1000. In other words, in 99.8% of the trials, Condorcet-Minimax produced a socially better result than IRV.

8 Understanding the results

Strategy under Condorcet-Minimax is apparent; the candidate nearest the median voter (and with the highest social utility) has an advantage in any two-candidate race. Another candidate can defeat a candidate at the median but they must have a quality advantage and be close enough to the median voter for that quality advantage to be decisive. The results show that when candidates make strategic decisions, they move towards the point of maximal social utility. The result is that, if 1 is the social utility score for the best candidate in the absence of strategy, it is possible, through strategy, to score better than the perfect choice without strategy and achieve an efficiency greater than 1.

Under IRV, it has been established that two partisan candidates on opposite sides can bracket a more representative candidate and starve them of first-place votes. The more representative candidate is eliminated due to a lack of first-place votes; the race is then settled between the less representative candidates (Munger, 2021). These bracketing candidates can be at the 17th and 83rd percentiles of the ideological distribution of voters and still force the more representative candidate from the race, when three candidates are competing under IRV (Fig. 10).

Fig. 10
figure 10

Bracketing. Candidate B would easily defeat either A or C but is bracketed by them. Starting with two candidates with one (labeled B) candidate at or near the median voter (50th percentile) and another candidate (labeled A) somewhere between the 20th percentile and the candidate at the median voter. In this figure, A receives 33.6%, B 32.2%, and C 34.4%. B is eliminated and C wins after B’s votes are redistributed

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Bracketing is an effective strategy regardless of the dimensionality of the ideological space and requires strategic behavior by only one candidate. To extend the example above to an arbitrary number of dimensions, consider B at the origin and A at any location such that A would receive more than 1/3 of the voters in a two-candidate race. Rotate the coordinate system such that the X axis lies along the line connecting A to B. C may now enter the race on the X axis opposite to A. If C is closer to B than A then C will win (Fig. 11).

Fig. 11
figure 11

A fourth candidate

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If a fourth candidate, D, enters the race, it is possible to force out C. In this example, D enters the race outside of C. This takes first-place votes from C and forces C from the race first, and B wins the election. So, while two candidates can bracket a candidate near the median voter, a fourth candidate can upset that strategy.

It should be noted, however, that D does not and cannot win the race. A more successful strategy for D would be to enter at the same location as C and hope that C is eliminated in the first round. If C is eliminated, D would then go on to assume C’s winning position. In the simulations, very few candidates near the median voter won, and candidates near a candidate like C chose to replicate C’s position under IRV.

Munger (2021) suggested that these strategies would be used under IRV, but it was unclear how often they would occur when candidates acted in their self-interest. The simulations show that the answer is nearly always. Further, these strategies are effective in a political space with any number of dimensions.

9 Conclusions

The first and most important conclusion is a general one, not about any particular voting rule. When candidates make strategic decisions, the results are significantly different than when they do not. Like previous refinements, this calls into question prior analyses that do not consider strategic behavior. The advantage obtained by strategic behavior implies that public figures can be expected to make strategic decisions. Thus, survey data and ballots have limited utility for analyzing voting rules other than the ones under which they are gathered. They are not without value, but one must carefully consider the implications of strategic behavior when using them. For instance, it would be unwise to declare that the results under two election systems would be equivalent based on ballot analysis, when the optimal candidate strategy differs between the two methods.

The second implication is that the optimal candidate strategy under Condorcet-minimax is clear and simple: be as close to the median voter as possible. This is not controversial and will not surprise any observer familiar with Condorcet-consistent voting rules.

Also unsurprising is that, as candidates move closer to the median voter, there is a higher likelihood of a Condorcet-tie. This likelihood decreases if there is variance in the quality of the candidates, as would be expected in actual elections. Fortunately, the policy implications of these ties are limited. The tie is more probable because several candidates are close to the median voter; consequently, the policy implications of choosing any one of them are minimal. These ties happen precisely when at least three candidates have moved near the point of maximum social utility. In the current environment, this seems both improbable and desirable. Any voting rule that could induce such an outcome seems ideal.

Candidate strategy under IRV is somewhat of a surprise. It was known (Munger, 2021) that it was possible for two candidates to bracket and eliminate a more representative candidate. It was not clear that candidates would fall into that cooperative strategy as they optimize their individual utility. These results show that this pattern is very consistent, happening nearly every time. When candidates optimize their utility, outcomes are significantly different than when candidates do not.

IRV places candidates with the highest social utility at a disadvantage. Not only do candidates with high social utility fail to win many elections, but the disadvantage is so extreme that candidates with high social utility move to inferior positions. It also calls into question the claim that IRV often elects the Condorcet candidate. While technically true, strategic behavior under IRV nearly always eliminates the Condorcet candidate before the race even begins.

IRV creates a polarized distribution of outcomes from a normal distribution of voters without the need for primaries, political parties, social media or any of the other things often blamed for creating or promoting polarization. This analysis does not consider the impact of political parties, but it seems unlikely that political parties would reduce the polarization inherent in IRV.