Satisfaction approval voting (SAV), also known as equal and even cumulative voting, is an electoral system that is a form of multiwinner approval voting as well as a form of cumulative voting. In the academic literature, the rule was studied by Steven Brams and Marc Kilgour in 2010.[1] In this system, voters may approve a number of candidates, and each approved candidate receives an equal fraction of the vote. For example, if a voter approves 4 candidates, then each candidate receives a 0.25 fractional vote. The election winners are those candidates that receive the highest fractional vote count.

This election system has been used for the election of the city council in Peoria, Illinois, since 1991.[2][3]

Formal description

Satisfaction approval voting aims to maximise the electorate's satisfaction, rather like proportional approval voting (PAV), however SAV calculates a voter's satisfaction differently to the way used in PAV. The satisfaction gained by a voter when a candidate they approve of is elected is equal to 1/n where n is the number of candidates that they voted for.[4] This has the effect of giving everyone a single vote that they split between the n candidates that they vote for. This makes calculating the winners much easier than for PAV,[5] as a voter's satisfaction gained for each elected candidate under this method is independent of how many of their choices have been elected, making satisfaction additive.[1]

Example

There are 10 voters, 4 candidates (Alice, Bob, Carol and Dan) and 2 seats. The votes are:

  • 4 voters vote for both Alice and Bob (each of these two candidates with get half a vote from each of these four voters).
  • 3 voters vote only for Carol (this candidate gets a whole vote from each of these three voters).
  • 3 voters vote only for Dan (this candidate gets a whole vote from each of these three voters).

Using the methodology used in PAV:

Potential winners of the two seats
Alice and BobAlice and CarolAlice and DanBob and CarolBob and DanCarol and Dan
Satisfaction of voters for

Alice and Bob

 422220
Satisfaction of voters for Carol030303
Satisfaction of voters for Dan003033
total satisfaction455556

Therefore, C and D win.

Alternatively, making use of the system's additive satisfaction property:

AliceBobCarolDan
Alice and Bob voters – total vote2200
Carol voters – total vote0030
Dan voters – total vote0003
overall vote2233

Comparison to approval voting

SAV tends to elect committees that better represent the diversity of voters than AV. SAV is also less susceptible to manipulation by candidate-cloning: if a winning candidate is cloned (such that the same voters who vote for the original candidate also vote for the clone), then in AV, both the original candidate and the clone will have the same score as the original candidate had before the cloning, so the clone will win and displace a winner with a lower score (if any). In contrast, in SAV, both the original and the clone will have a lower score (due to the division of each voter's vote by the number of approved candidates), so it is less likely to displace another winner.

In party-approval voting

Party-approval voting is a special case of approval voting in which each voter can approve one or more parties, rather than directly approving candidates. SAV can be applied to this setting as follows.

  • For each party, compute its upper quota - the fraction of votes it received, times the total number of seats, rounded up.
  • For each voter, define the satisfaction score as the number of seats allocated to his approved party divided by the upper-quota of this party.
  • Allocate to each party either its upper-quota or its lower-quota; choose the set of parties receiving their upper-quota in a way that maximizes the voter satisfaction.

When every voter votes for a single party, the result that maximizes voter satisfaction coincides with the Quota method of Balinsky and Young - a quota-capped variant of D'Hondt method of apportionment; hence, it satisfies both upper and lower quota.[1]: Prop.9 

References

  1. 1 2 3 Brams, Steven J.; Kilgour, D. Marc (2010). "Satisfaction Approval Voting" (PDF). Paper presented at the Annual National Conference of the Midwest Political Science Association, Chicago, Illinois, in April 2010.
  2. ↑ "FairVote - Spotlight on Reform: Peoria, IL". archive.fairvote.org. Retrieved 2023-12-20.
  3. ↑ Adams, Pam (2011-11-01). "Cumulative voting worked, surviving plaintiffs says". Peoria Journal Star. Retrieved 2023-12-20.
  4. ↑ Brams, Steven J.; D. Marc Kilgour (2014). "Satisfaction Approval Voting" (PDF). In Rudolf Fara; Dennis Leech; Maurice Salles (eds.). Voting Power and Procedures: Essays in Honour of Dan Felsenthal and Moshe Machover. Springer. pp. 322–346. doi:10.1007/978-3-319-05158-1_18. ISBN 978-3-319-05158-1.
  5. ↑ Aziz, Haris; Serge Gaspers, Joachim Gudmundsson, Simon Mackenzie, Nicholas Mattei, Toby Walsh (2014). "Computational Aspects of Multi-Winner Approval Voting". Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems. pp. 107–115. arXiv:1407.3247v1. ISBN 978-1-4503-3413-6.{{cite book}}: CS1 maint: multiple names: authors list (link)
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