### Utilitarian vs. Majoritarian Election Methods

by Clay Shentrup

I've been asked to describe why a utilitarian election system (e.g. Score Voting or Approval Voting) is preferable to a majoritarian one. I want to start by looking at some different electoral scenarios to demonstrate the complexity associated with the term "majority winner", which is typically thought of as being a straightforward concept.

## Scenario one: when no candidate is favored by a majority of voters

When there are more than two candidates, the notion of a "majority winner" can get a bit murky. Take the following set of ranked preferences for a hypothetical four-candidate election.

 % of voters their ranking 35% W > Z > X > Y 32% X > Z > Y > W 17% Y > X > Z > W 16% Z > Y > X > W

This means that the top row represents the 35% of voters who prefer candidate W over Z over X over Y. No candidate is the favorite of a majority. Let's consider what would happen here with a few different voting systems.

#### Plurality Voting (aka First-past-the-post)

If we went only on first-place votes, W would be the winner, with 35%. But that would seem to be undemocratic, since a whopping 65% majority of the voters would take any other candidate over W!

#### Top-Two Runoff (aka TTR)

This system attempts to find a majority winner by pitting the top two finishers in a second head-to-head matchup, if no one gets a majority of votes in the first round. In this case, W and X would go to the second round. Based on these preferences, X would obliterate W, 65% to 35%. But let's not forget that people can change their minds in the interim between the initial election and the runoff. And different people may show up to vote in the second round. So we can't say for certain which of these candidates will win. But making the reasonable assumption that X is the winner, is this a majoritarian outcome? Consider that a 51% majority of the voters prefer Z over X.

#### Instant Runoff Voting (aka IRV)

IRV attempts to find a majority winner by removing the remaining candidate with the fewest first-place rankings. In this case, it would start by eliminating Z, with only 16% of the first-place votes. Those votes would then transfer to Y, the second choice of those voters. Y would then have 33% of the vote (16% + 17%). Thus X, with only 32% of the vote, would be eliminated next, transferring 32% of the votes to Y. Y would be the winner, with 65% of the votes.

Again, is this a majoritarian outcome? Consider that a large 67% majority of voters prefer X to Y. And X gets almost twice as many first-place votes as Y — 32% compared to 17%. And Z is preferred to Y by a gigantic 83% of the voters, and gets just a bit fewer first-place votes — 16% compared to 17%.

#### Condorcet voting

Condorcet (French, pronounced condor-SAY) systems elect the candidate who would beat all rivals in a head-to-head matchup. They differ in how they resolve cyclic "ties", such as when a majority prefers X to Y, and Y to Z, and Z to X.

In this particular election, there is a beats-all winner, so any Condorcet method would produce the same result. Z wins. Z is preferred to W by a 65% majority. Z is preferred to Y by an 83% majority. Z is preferred to X by a 51% majority. Thus one could argue that Z is the undisputed majority winner.

### Conclusion

Three different systems each find a different "majority winner". This demonstrates the arbitrariness that is inherent to the very goal of electing a majority winner. It is also worth noting that both TTR and IRV only guarantee that the winner is preferred by a majority to at least one other candidate. For instance, IRV elects Y. Y is preferred by a majority to W, but would lose by a majority vs. any other candidate.

We cannot say how Score Voting would behave in this scenario, since lots of different scorings could produce these rankings. But we can say that a lose-to-all candidate (aka "Condorcet loser") like W would be unlikely to win with Score Voting. For instance, computerized election simulations have shown that Score Voting has at least a 77% chance of electing a beats-all winner when one exists (see table 2 here). And there is a theorem that, given plausible expectations of voter tactics, Score Voting is more likely to elect beats-all winners than actual beats-all election systems.

When no candidate has an outright majority, I believe it's reasonable to elect the candidate who was the most preferred by the most voters.

## Scenario two: when a candidate is the favorite of a majority of voters

This rarely happens with multiple candidates, which is often the case in large consequential elections. But when there are just two candidates, there is always a clear majority winner. Score Voting (including it's simplified form of Approval Voting) are utilitarian voting methods, and have faced some criticism for potentially thwarting the will of a majority. Here's a simplified example with three voters and two candidates.

 Democrat Republican Voter 1 9 6 Voter 2 7 3 Voter 3 1 9

The Republican wins with 18 points, compared to the Democrat's 17 points. Whereas a majoritarian system would of course elect the Democrat. Before we delve into the controversial philosophical discussion about whether this is the right thing to do, I want to focus on the practical reality here. This scenario is extremely unlikely. Especially considering that voters with the preferences I just described would tend to vote more like this.

 Democrat Republican Voter 1 10 0 Voter 2 10 0 Voter 3 0 10

That is, they would tend to "normalize" their scores to the full range of allowed values — in this case 0 to 10. You could call this "tactical" voting, but it's still relatively sincere, in that voters don't have any incentive to misorder the candidates. It's just using the full power of their ballot.

Bear in mind that voters can also do this in multi-candidate elections. Any time a majority of voters prefers a certain candidate, they can force that candidate to be elected by giving that candidate a maximum score, and giving all other candidates a minimum score. They may choose not to do that for one of the following reasons:
• They enjoy the expressive act of voicing their opinions on multiple candidates.
• They are trying to maximize their expected value. E.g. a voter who feels X=10, Y=9, Z=0, in a tight race between X and Z where Y is most people's second choice, may want to "hedge his bets" by also approving Y. He knows that if that causes Y to win instead of X, it only costs him one point of satisfaction, whereas if it causes Y to win instead of Z, it saves him nine points. The cost-benefit analysis there is highly tilted in favor of supporting Y. Therefore this voter should not feel disenfranchised if it turns out that X was the favorite of a majority of voters, but lost to Y. The voter knew this going in, and decided that the payoff was bigger than the risk.
• They may want to be benevolent, knowing that elections produce more broadly pleasing outcomes when there is more sincere voting.
The bottom line is that utilitarian voting methods cannot deprive a majority of the ability to get their favored candidate.

### Conclusion: utilitarian voting systems will rarely conflict with the majoritarian principle

To summarize:
• Most large significant elections have more than two candidates.
• In those elections, there will rarely be a candidate preferred by the majority of voters.
• Given that no candidate has an outright majority, a "majoritarian" can be satisfied with the election of any candidate who is not the lose-to-all candidate (aka "Condorcet loser").
• Score Voting is extremely unlikely to elect a lose-to-all candidate.
• Score Voting may be, in practice, more likely to elect a beats-all candidate than even beats-all election systems, due to tactical voter behavior.
• When there is a majority-preferred candidate in a 3-or-more-candidate race, Score Voting will almost always elect that candidate.
• In a two-candidate election, Score Voting will almost always elect the majority-preferred candidate. And the more decisive the majority, the less probable it will be that Score Voting will violate majority rule.
• A majority can always get their way with Score Voting, if they choose to, by simply giving their favorite option the maximum score, and a minimum score to all other options.
In short, it will be extremely uncommon for Score Voting to clearly contradict the majoritarian principle, and even when it does, that will only be because the majority voluntarily gave up their right to have their way. The tiny chance of a "non-majoritarian" outcome is more than outweighed by the numerous practical benefits. So you can reasonably support Score Voting and/or Approval Voting even if you support majority rule.

## Why utilitarianism is actually preferable to majoritarianism

The preceding argumentation was meant to be pragmatic. I realize that I'm not going to talk most people out of supporting majority rule, so instead I'm simply pointing out that Score Voting and Approval Voting really aren't all that much in conflict with the majoritarian perspective.

But now I want to take a look at the theory of social choice, to argue that actually utilitarianism is preferable to majoritarianism.

### Selfishness

If you're being rational, the best outcome for you as an individual is the one that maximizes your expected utility. For instance, say you believe that the utility of three options is X=4, Y=2, Z=0. Then a guarantee of getting Y is equally preferable to you as a 50/50 chance of getting X or Z. Either way, your expected utility is 2. But a 75/25 chance of getting X or Z would be better than the guarantee of getting Y, because it would have an expected value of 3.

The economist John Harsanyi observed that the choice with the greatest utility from the group is also the one that provides each individual within the group the highest expected value, to the extent that they are unsure of their identity.

For example, say a three-member organization to which you belong decides to have a dinner party at a restaurant. Let's say your utilities for the two restaurants are as follows.

 Restaurant 1 Restaurant 2 Member 1 9 6 Member 2 7 3 Member 3 1 9

If you don't know which of these members represents you, then you'd be better off with restaurant 2. Your expected utility is slightly higher. If you were to find out that you were actually one of the members in the majority, then of course you'd be better off with restaurant 1.

The point is, if you're choosing a voting method to be used for a series of future elections, whose outcome you cannot predict, then you'll be better off using the most utilitarian voting system. Or, more precisely, the voting system with the lowest Bayesian Regret.

### It's mathematically proven that the group can prefer X even if a majority of its members prefer Y

That may seem counter-intuitive and even impossible, but it's mathematically proven. Most people fail to appreciate that this was actually the core of Arrow's Theorem.

Further, it's mathematically proven that (given certain seemingly irrefutable premises) utilitarianism is the only "social choice aggregation function" that can even possibly be correct. I.e. all others lead to self-contradictions about which candidate is actually preferred by the group.

This latter point means that utilitarianism is not only better in the selfish sense, but also in terms of what's best for the group as a whole.

### Pizza

Consider this simplistic scenario.

A table tennis coach is buying his three students dinner after an out-of-town tournament. They head to a small pizza shop which offers only cheese pizza or pepperoni. Two of the students would slightly prefer pepperoni. But the third is devoutly Jewish, and does not eat pork. He would therefore strongly prefer cheese to pepperoni.

majoritarian view would say that pepperoni is the better choice, since it's preferred by a majority (two out of three) of the students. Whereas a utilitarian view would say that cheese was the better choice, because it would only make the pepperoni fans a little less satisfied, while making the Jewish student vastly more satisfied. Assuming you were that coach, trying to do what was best for those three students, do you really think it would be better to respect the majority instead of the overall good?