# Utilitarian vs Majoritarian in single winner

There is still a lot of confusion about utilitarian vs majoritarian in single winner systems so I figure I will clear it up. An illustrative example for 100 voters with candidates A, B, C ,D is

20 = A:5, B:2, C:2, D:0
20 = A:2, B:5, C:2, D:0
20 = A:2, B:2, C:5, D:0
40 = A:0, B:0, C:0, D:5

D is the score winner with 200 and all others have 180. I think D is the correct answer because it yields the most Utility. It is worth explaining why this is. First lets look at the argument for majoritarianism. If only A and D were running we would get.

60 = A:5, D:0
40 = A:0, D:5

And A would win so it looks as if B and C are spoilers for A. IRV would yield either A, B or C as the winner because it is Majoritarian. It is not the case that B and C are spoilers for A. Score relies on there being enough candidates to properly get utilities. In the absence of a representative sample of candidates score would yield the majoritarian winner A. This does not mean that score gives the correct answer in the case of less candidates but that it does not work as intended because the voters do not have enough choice. It is important to then note that partisanship is fundamentally incompatible with score. If there is no party to put up candidates for a significant portion of the voters then score will not yield accurate utilities. In party based systems there are never parties for all groups. This is why I also advocate for other reforms which reduce partisanship. IRV is more compatible with partisanship which is why collectivists have always favoured majoritarian solutions and individualists favour utilitarian solutions.

Of course one could strategically vote under score and give

20 = A:5, B:5, C:5, D:0
20 = A:5, B:5, C:5, D:0
20 = A:5, B:5, C:5, D:0
40 = A:0, B:0, C:0, D:5

But this would not really happen if you look at it from a game theory perspective. The voters who favour A,B and C are in competition and are actually not so ideologically aligned. In the end some might give a little more and this might be enough to win in this scenario. For this reason some advocate that Approval Voting is the appropriate system because it forces voters into the Nash Equilibrium. I would rather let the voter decide and get a better view on utility in less contrived situations than this one. The proper argument for Approval is simplicity.

Another game theory perspective to consider is that elections do not exist in a vacuum. Utilitarianism leads to an equilibrium where more candidates run who are centrist to try to find the right balance to please the most. Majoritarianism leads to tyranny of the majority. Polls just do not show that there are two clumps of voters. Ideologically people are Gaussian distributed around the center. To view the problem as warring factions is factually incorrect and will undermine democracy and representation. In this scenario there would be another candidate who would get a nonzero score from both D voters and one of the other groups. This means that this scenario is a bit of a strawman.

It is worth pointing out that STAR is not a ‘solution’. STAR would not give D as the winner but adding a clone of D as E would give

20 = A:5, B:2, C:2, D:0, E:0
20 = A:2, B:5, C:2, D:0, E:0
20 = A:2, B:2, C:5, D:0, E:0
40 = A:0, B:0, C:0, D:5, E:5

Which would then give either E or D as the winner. So STAR does not really “solve” the issue of having to choose between majoritarian and utilitarian. It is a half step towards majoritarianism. I am not saying that half step is not worth it. Pure theory answers are not always the best in the real world so maybe STAR compensates for the inevitable partisanship in all systems better than score but we do not really know.

I hope this clears things up. Score would lead to a long term better situation even if it is hard to see how that would look from out current partisan polarized situation. If you are still unconvinced, consider that majoritarianism evolved out of the inability to do utilitarianism properly. Giving everybody 1 indivisible unit of utility is where the justification arouse for majoritarianism in the first place.

The main points I’d make regarding majoritarianism are 1) the Smith Set candidates are a better representation of majoritarianisn than the IRV winner, and 2) in Condorcet methods, while you can’t indicate weak preferences, you can indicate no preference between candidates you weakly prefer, so generally a situation like

51 A5 B4
49 B5

strikes me as an incorrect or at least rather unconvincing critique of majoritarianism, as there is nothing stopping a A5 B4 voter from indicating A=B. The other thing is, why would a voter in the majority wish to force their preference if they suspect there is a consensus candidate? I’d tend to think voters tend to honestly prefer candidates who are well-liked by all, since that translates into a better mandate and more progress for that voter’s priorities than the voter’s favorite but more polarizing candidate.

This is the problem with analyzing majoritarianism using IRV: IRV doesn’t provide for the idea of “overlapping majorities” the same way Condorcet does. Simple example:

26 A>B
25 B>A
49 C>B

There’s a mutual majority for A and B, so both IRV and Smith-efficient methods guarantee one of them win. IRV looks at as if only the opinions of the voters in the majority matter essentially, so it eliminates B and lets a majority of the majority decide the winner (A). Smith-efficient methods see it as "the mutual majority disqualifies C, so eliminating C, we get:

26 A>B
74 B

and B must win as the majority’s 1st choice among the uneliminated candidates. Two perspectives that both can be thought of via elimination, but with very different results.

Edit: Probably the main predictor of whether someone prefers utilitarianism or majority rule for a particular use case is whether they believe there is likely to be a consensus candidate that can represent most of the electorate well, since when there are such candidates, it usually doesn’t matter too much which one you pick, so even a points-based ambiguity in deciding who should win is acceptable, whereas if one expects there to be significant two-sidedness among voters, then majority rule may be necessary to ensure the “right” side wins. This is another point in favor of (Smith-efficient, in particular, though even something like Condorcet//IRV passes mutual majority) Condorcet methods, since they can do the former to a significant extent and guarantee the latter.