Social utility efficiency explained

Social utility efficiency (SUE) or voter satisfaction efficiency (VSE)^[1] is a metric for comparing voting methods which compares them based on the average well-being of voters.^{[2] [3]}

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

is the expected value over many iterations of the sum of all voter utilities for a given candidate:^[4]

$$\backslash operatorname=\; \backslash frac$$

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.^[5]

History

The concept was originally introduced as a system's "effectiveness" by Robert J. Weber in 1977, defined as:^[6]

$$\backslash operatorname=\backslash lim\_\; \backslash frac$$

Where

is the expected social utility of the given candidate, is the number of voters, and 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 in the 1980s, calculated statistically from random samples, with 25–201 voters and 2–10 candidates.^[7] 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

= 81.6%, with some systems increasing with more candidates.)

In 2017, Jameson Quinn studied SUE under the name "voter satisfaction efficiency",^[8] using more complex and arguably more realistic parameters, examining a wider variety of scenarios and using a hierarchical cluster model of voter behavior. He found the best performers to be ranked pairs, STAR voting, and score voting, depending on the scenario tested.^[9]

A similar metric, referred to as "Bayesian regret",^[10] measures the same property, but inverted.^{[11] [12] [13]} They are related by the formula:^[14]

$$\backslash operatorname(\backslash text)\; =\; 1\; -\; \backslash frac$$

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).

See also

• Condorcet efficiency
• Comparison of electoral systems

References

1. Wolk . Sara . Quinn . Jameson . Ogren . Marcus . 2023-03-20 . STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform . Constitutional Political Economy . 34 . 3 . 310–334 . en . 10.1007/s10602-022-09389-3 . 257653868 . 1043-4062. free .
2. Book: Mueller, Dennis C.. Public choice III. 2003. Cambridge University Press. 0-511-06504-3. Cambridge. 191952945.
3. Duddy. Conal. 2017. Geometry of run-off elections. Public Choice. en. 173. 3–4. 267–288. 10.1007/s11127-017-0476-2. 254935333 . 0048-5829.
4. Book: Merrill, Samuel. Making Multicandidate Elections More Democratic. 2014-07-14. Princeton University Press. 9781400859504. en. 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.. Samuel Merrill III.
5. Merrill. Samuel. Samuel Merrill III. 1984. A Comparison of Efficiency of Multicandidate Electoral Systems. American Journal of Political Science. 28. 1. 23–48. 10.2307/2110786. 0092-5853. 2110786.
6. 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.
7. Merrill. Samuel. Samuel Merrill III. 1984. A Comparison of Efficiency of Multicandidate Electoral Systems. American Journal of Political Science. 28. 1. 23–48. 10.2307/2110786. 0092-5853. 2110786.
8. Web site: Voter Satisfaction Efficiency (VSE) FAQ. 2021-02-03. Jameson Quinn. en-US.
9. Web site: Frohnmayer. Mark. The Election Science Behind the Reform Movement. 28 December 2020. Equal Vote.
10. Web site: Smith. Warren D.. 2006. Bayesian Regret for dummies. 2021-01-31. RangeVoting.org.
11. 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.
12. Web site: Range voting with mixtures of honest and strategic voters. 2019-07-26. RangeVoting.org. SociallyBest 0 ... RandomWinner 1 ... SociallyWorst 2.0024.
13. Web site: Smith . Warren D. . The case for score voting . Constitutional Political Economy . Springer Nature . 30 November 2024 . 297–309 . en . 10.1007/s10602-023-09403-2 . 1 September 2023.
14. Web site: Voter Satisfaction Efficiency FAQ. Quinn. Jameson. 2017-02-10. GitHub Pages. en-US. 2019-07-24.