# Social utility efficiency

Efficiency of several voting systems with an impartial culture model and 25 voters[1]

Social utility efficiency (SUE) is a measurement of the utilitarian performance of voting methods—how likely they are to elect the candidate who best represents the voters' preferences.[2]

It is also known as utilitarian efficiency,[3][4] voter satisfaction index (VSI)[5][6] or voter satisfaction efficiency (VSE).[7]

## Definition

Social utility efficiency is defined as the ratio between the social utility of the candidate who is actually elected by a given voting method and that of the candidate who would maximize social utility, where ${\displaystyle E[]}$is the expected value over many iterations of the sum of all voter utilities for a given candidate:[8]

${\displaystyle \operatorname {SUE} ={\frac {E[{\text{selected candidate}}]-E[{\text{random candidate}}]}{E[{\text{maximizing candidate}}]-E[{\text{random candidate}}]}}}$

A voting method with 100% efficiency would always pick the candidate that maximizes voter utility. A method that chooses a winner randomly would have efficiency of 0%, and a (pathological) method that did worse than a random pick would have less than 0% efficiency.

SUE is not only affected by the voting method, but is a function of the number of voters, number of candidates, and of any strategies used by the voters.[1]

## History

The concept was originally introduced as a system's "effectiveness" by Robert J. Weber in 1977, defined as:[2]

${\displaystyle \operatorname {Effectiveness} =\lim _{n\to \infty }{\frac {E_{\text{elected}}(m,n)-E_{\text{random}}(m)}{E_{\text{maximal}}(m,n)-E_{\text{random}}(m)}}}$

Where ${\displaystyle E}$ is the expected social utility of the given candidate, ${\displaystyle n}$ is the number of voters, and ${\displaystyle m}$ is the number of candidates. He used a random society (impartial culture) model to analytically calculate the effectiveness of FPTP, two Approval variants, and Borda, as the number of voters approaches infinity.

It was given the name "social utility efficiency" and extended to the more realistic spatial model of voting by Samuel Merrill III[1] in the 1980s, calculated statistically from random samples, with 25–201 voters and 2–10 candidates.[9] This analysis included FPTP, Runoff, IRV, Coombs, Approval, Black, and Borda (in increasing order of efficiency). (Merrill's model normalizes individual voter utility before finding the utility winner, while Weber's does not, so that Merrill considers all 2-candidate voting systems to have an SUE of 100%, decreasing with more candidates, while Weber considers them to have an effectiveness of ${\displaystyle {\sqrt {2/3}}}$ = 81.6%, with some systems increasing with more candidates.)

In 2017, Jameson Quinn studied SUE under the name "voter satisfaction efficiency",[10] using more complex and arguably more realistic parameters, examining a wider variety of scenarios and using a hierarchical cluster model of voter behavior. He analyzed a number of methods that had not been included in previous simulations, and his unpublished results found the best performers to be STAR voting, his own method 3-2-1 voting, Ranked pairs, or Score voting, depending on the scenario tested.[10][11]

A similar metric, referred to as "Bayesian regret",[12] measures the same property, but inverted and non-normalized.[5][13][14][15][16] They are related by the formula:[7]

${\displaystyle \operatorname {SUE} ({\text{method}})=1-{\frac {\operatorname {BR} ({\text{method}})}{\operatorname {BR} ({\text{random winner}})}}}$

where "random winner" refers to the hypothetical election method of choosing a candidate at random regardless of the opinions of the electorate (not the random ballot voting method, which is weighted towards candidates who receive more votes).

## References

1. ^ a b c Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. ISSN 0092-5853. JSTOR 2110786.
2. ^ a b Weber, Robert J. (September 1978). "Comparison of Public Choice Systems". Cowles Foundation Discussion Papers. Cowles Foundation for Research in Economics: 16, 38, 62. No. 498.
3. ^ Mueller, Dennis C. (2003). Public choice III. Cambridge: Cambridge University Press. ISBN 0-511-06504-3. OCLC 191952945.
4. ^ Duddy, Conal (2017). "Geometry of run-off elections". Public Choice. 173 (3–4): 267–288. doi:10.1007/s11127-017-0476-2. ISSN 0048-5829.
5. ^ a b Shentrup, Clay (2007-07-07). "Voter Satisfaction Index". Center for Range Voting. Retrieved 2019-07-24. Voter satisfaction index, or "VSI" for short (also called "social utility efficiency" ... a lower number is actually better, and this can confuse people who are new to the concept. ... the utility units have an arbitrary magnitude, making it difficult to compare Bayesian regret figures
6. ^ Huang, John (January 11, 2020). "Alternative Voting Methods — How well do they perform in the best case?". Americans for Representation. Retrieved 2021-01-31. For this blog post, I’m going to stick with something called “Voter Satisfaction Index”.
7. ^ a b Quinn, Jameson (2017-02-10). "Voter Satisfaction Efficiency FAQ". GitHub Pages. Retrieved 2019-07-24.
8. ^ Merrill, Samuel (2014-07-14). Making Multicandidate Elections More Democratic. Princeton University Press. ISBN 9781400859504. If the ratings are interpreted as Von Neumann-Morgenstern utilities … I define the social utility of a candidate as the sum of all voter utilities for that candidate.
9. ^ Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. ISSN 0092-5853. JSTOR 2110786.
10. ^ a b "Voter Satisfaction Efficiency (VSE) FAQ". Jameson Quinn. Retrieved 2021-02-03.
11. ^ Frohnmayer, Mark. "The Election Science Behind the Reform Movement". Equal Vote. Retrieved 28 December 2020.
12. ^ Smith, Warren D. (2006). "Bayesian Regret for dummies". RangeVoting.org. Retrieved 2021-01-31.
13. ^ Quinn, Jameson. "Answer to 'How should you judge the quality of a voting system?'". Quora. Retrieved 2019-07-24. The simulation method, originally called "Bayesian Regret" by Warren Smith in his systematic exploration of this question, and more recently re-christened as "Voter Satisfaction Efficiency" (VSE)
14. ^ Quinn, Jameson (2018-04-12). "A voting theory primer for rationalists". LessWrong. Retrieved 2019-07-24. Nowadays the name … is "Voter Satisfaction Efficiency" (VSE).
15. ^ Hansen, Jeremy A (2014). "Comparing Approval At-Large to Plurality At-Large in Multi-Member Districts". Conference: Fifth International Workshop on Computational Social Choice. Social-utility efficiency … Smith referred to a similar formulation as Bayesian regret
16. ^ "Range voting with mixtures of honest and strategic voters". RangeVoting.org. Retrieved 2019-07-26. SociallyBest 0 ... RandomWinner 1 ... SociallyWorst 2.0024