# Utility

Though often used in voting theory to refer to cardinal utility i.e. rated method utility, it can also be used for discussing ordinal utility i.e. ranked-preference utility. (The Borda count uses ranked ballots to find what this article would describe as rated utilities).

An example would be (using a preference-approval):

51: A>B|

49: B|>A

A majority prefer A over B, but are willing to support either of the two, whereas a minority both prefer and only support B. Therefore, ordinal utility says A is best, while rated utility says B is best.

## Two-candidate case

In the two-candidate case, the two approaches differ; cardinal/rated utility says that the candidate who makes voters net-happier should win (if everyone measured their happiness on a scale), whereas ranked/ordinal utility requires majority rule, which can be thought of as at least satisfying the majority criterion.

Self-referential Smith-efficient Condorcet methods that always elect the utilitarian (rated utility) winner in the two-candidate case will be Approval voting or Score voting. For majority rule, the equivalent is Smith-efficient Condorcet methods.

Note that in the two-candidate case, voters using rated utilities can exaggerate the difference in utility between the candidates to derive majority rule, while voters using ranked utilities can (in the limit) approximate rated utility by using a probability proportional to their personal difference in utility between the two candidates to decide whether to vote for their preferred candidate of the two, or not vote. Example:

100,001: A:1 B:0.8

100,000: A:0 B:1

The points are roughly A 100,000, B 180,000, so there is a rated utility margin of 80,000 points in favor of B.

If, for cardinal utility, the A>B voters give B a 0, they can make A have slightly more points, i.e. majority rule. And if, in majority rule, the A>B voters use a 20% probability of voting A>B and 80% for voting A=B (i.e. a 20% probability of picking A and a 80% probability of not voting for either candidate), then in the limit, A will have ~20,000 votes and B ~100,000, which is an 80,000 vote margin in favor of B, thus effectively simulating the rated utility margin.
Another consideration is whether there should be a "satisfaction threshold" at which point increasing someone's utility matters less. For example, between a candidate who gives 100% utility to 60% of the voters and a candidate who gives 51% utility to all voters, some would consider the latter candidate better, despite them giving less cardinal utility., because all voters get significant utility from them, while 49% of voters get nothing from the first candidate. See [1] for an example.

## Notes

One notable contrast between ordinal and cardinal utility is that with ordinal, one voter can shift their preference to make a good majority-preferred candidate become a bad minority-preferred candidate, whereas with cardinal utility, there is a degree of damage i.e. it is not too bad a thing to elect a candidate with slightly less utility than the utilitarian winner.

Many forms of utility are mostly additive i.e. it's not the voters' individual preferences that are of utmost importance, but rather the values produced by adding them up. For example, a candidate given 5 points when voters had rated ballots on a scale of 0 to 5 could've been given 1 point by 5 voters or 5 points by 1 voter.

Likewise, with pairwise preferences, if candidate A has 5 votes against B's 4 in the A vs B matchup, this could be equivalent to 5 voters ranking A 1st and B 2nd, or 5 voters ranking A 2nd-to-last and B last.

## Utility vs utility

Note that utility in the democratic context refers to "giving voters what they want", or making them happy with the outcome of the election, but does not imply that what they want is actually good for them, or that it will positively affect their well-being in the future (which "utilitarianism" would imply in other contexts). Voting methods can only work with the preference data they are given.

1. "r/EndFPTP - Mock ballot: who do you think should win in this election, and why?". reddit. Retrieved 2020-05-11.