Social utility efficiency

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 $E[]$ is the expected value over many iterations of the sum of all voter utilities for a given candidate:{{Cite book|last=Merrill|first=Samuel|url=https://books.google.com/books?id=5gMABAAAQBAJ&pg=PA32|title=Making Multicandidate Elections More Democratic|date=2014-07-14|publisher=Princeton University Press|isbn=9781400859504|location=|pages=|language=en|quote=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.|author-link=Samuel Merrill III}}

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

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.{{Cite journal|last=Merrill|first=Samuel|author-link=Samuel Merrill III|date=1984|title=A Comparison of Efficiency of Multicandidate Electoral Systems|journal=American Journal of Political Science|volume=28|issue=1|pages=23–48|doi=10.2307/2110786|issn=0092-5853|jstor=2110786}}

The concept was originally introduced as a system's "effectiveness" by Robert J. Weber in 1977, defined as:{{Cite journal|last=Weber|first=Robert J.|date=September 1978|title=Comparison of Public Choice Systems|url=https://cowles.yale.edu/publications/cfdp/cfdp-498|journal=Cowles Foundation Discussion Papers|publisher=Cowles Foundation for Research in Economics|volume=|pages=16, 38, 62|id=No. 498|via=}}

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

Where $E$ is the expected social utility of the given candidate, $n$ is the number of voters, and $m$ 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 in the 1980s, calculated statistically from random samples, with 25–201 voters and 2–10 candidates.{{Cite journal|last=Merrill|first=Samuel|author-link=Samuel Merrill III|date=1984|title=A Comparison of Efficiency of Multicandidate Electoral Systems|journal=American Journal of Political Science|volume=28|issue=1|pages=23–48|doi=10.2307/2110786|issn=0092-5853|jstor=2110786}} 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 $\backslash sqrt\{2/3\}$ = 81.6%, with some systems increasing with more candidates.)

In 2017, Jameson Quinn studied SUE under the name "voter satisfaction efficiency",{{Cite web|title=Voter Satisfaction Efficiency (VSE) FAQ|url=http://electionscience.github.io/vse-sim/VSE/|access-date=2021-02-03|website=Jameson Quinn|language=en-US}} 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.{{cite web|last1=Frohnmayer|first1=Mark|title=The Election Science Behind the Reform Movement|url=https://www.equal.vote/science|access-date=28 December 2020|website=Equal Vote}}

A similar metric, referred to as "Bayesian regret",{{Cite web|last=Smith|first=Warren D.|date=2006|title=Bayesian Regret for dummies|url=https://www.rangevoting.org/BayRegDum.html|archive-url=|archive-date=|access-date=2021-01-31|website=RangeVoting.org}} measures the same property, but inverted.{{Cite journal|last=Hansen|first=Jeremy A|date=2014|title=Comparing Approval At-Large to Plurality At-Large in Multi-Member Districts|url=https://www.researchgate.net/publication/275274570|journal=Conference: Fifth International Workshop on Computational Social Choice|volume=|pages=|quote=Social-utility efficiency … Smith referred to a similar formulation as Bayesian regret|via=}}{{Cite web|title=Range voting with mixtures of honest and strategic voters|url=https://rangevoting.org/StratHonMix.html|access-date=2019-07-26|website=RangeVoting.org|quote=SociallyBest 0 ... RandomWinner 1 ... SociallyWorst 2.0024}}{{cite journal |last1=Smith |first1=Warren D. |title=The case for score voting |url=https://link.springer.com/article/10.1007/s10602-023-09403-2 |journal=Constitutional Political Economy |publisher=Springer Nature |access-date=30 November 2024 |pages=297–309 |language=en |doi=10.1007/s10602-023-09403-2 |date=1 September 2023|volume=34 |issue=3 |doi-access=free }} They are related by the formula:{{Cite web|url=http://electionscience.github.io/vse-sim/VSE/|title=Voter Satisfaction Efficiency FAQ|last=Quinn|first=Jameson|date=2017-02-10|website=GitHub Pages|language=en-US|access-date=2019-07-24}}

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

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).

• Condorcet efficiency
• Comparison of electoral systems

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Category:Voting theory