Clay Shentrup & Warren D. Smith, Aug 2015
Suppose that somewhere in the universe, there exist two possible events X and Y, and three voters Amy, Bob, and Cal, such that each of the voters has the following (see table) personal valuations for X and Y; and they each will simply value the combined event X&Y as the sum of their X and Y values.
The four possible combination-outcomes are ∅ (i.e. neither X nor Y happen), just X, just Y, and "X&Y both happen," For simplicity we're assuming each of our voters values X and Y independently; i.e. the value of X (in the view of each voter) is unaffected by whether Y happens, and vice versa. One also could consider voters for whom X and Y are dependent events, e.g. the value of X to such a voter would depend on whether Y happens. Then our examples would still work provided the voters value X and Y nearly-enough to independently, e.g. the effects of Y on the value of X, and vice versa, are small enough.
Example election #1:
| Voter | X value | Y value | combination values ∅, X, Y, X&Y |
|---|---|---|---|
| Amy | 3 | –6 | 0, 3, –6, –3 |
| Bob | –6 | 3 | 0, –6, 3, –3 |
| Cal | 4 | 5 | 0, 4, 5, 9 |
What would happen if we handled this election with simple-majority 2-choice votes?
In election #1 using simple majority votes:
These three simple majority votes contradict each other.
This makes us realize that the naive philosophy that "the majority should win" in any simple-majority two-choice vote, is logically untenable.
Example election #2 (same as election #1, except Cal replaced by Deb):
| Voter | X value | Y value | combination values ∅, X, Y, X&Y |
|---|---|---|---|
| Amy | 3 | –6 | 0, 3, –6, –3 |
| Bob | –6 | 3 | 0, –6, 3, –3 |
| Deb | 2 | 1 | 0, 2, 1, 3 |
The exact same three simple majority votes would happen in election #2 (except replace "Cal" by "Deb").
Given that naive majoritarianism is untenable, what should replace it? We believe the best available philosophy is "utilitarianism." I.e, if these two example elections had been decided by summing everybody's honest valuations of all 4 options, then the summed-scores would have been
| Election #1 | ∅=0 | X=1 | Y=2 | X&Y=3 |
|---|---|---|---|---|
| Election #2 | ∅=0 | X=–1 | Y=–2 | X&Y=–3 |
We devised the two example elections so that their sums-of-honest-scores yield exactly opposite conclusions. But neither election features any self-contradiction, and their respective winners would have been the best (reckoned by summed honest values) for their respective societies.
The philosophy of utilitarianism would be instantiated by "honest score voting." I.e, if these two example elections had been held using score voting on all 4 options, their summed-score winners with honest scores would indeed have been X&Y and ∅ respectively.
However, in 2-choice elections score voting degenerates to simple-majority voting if all voters "exaggerate" their scores to the max and min ends of the allowed-score range. So with all-exaggerating voters, the same contradictions would happen if the same three 2-choice elections were held. With a full 4-choice election, score voting with honest votes would enact X&Y in election #1 but neither (∅) in election #2; but with "normalized" scores (i.e. honest scores except each voter scales them to employ the full allowed-score interval), both elections with 4-choice score voting would elect X&Y – which in election #1 would be the best, but in election #2 the worst, possible outcome. With honest-voter plain plurality voting both elections would yield 3-way exact ties between X, Y, and X&Y. With instant runoff IRV election #1 would be a 2-way tie between Y and X&Y; with Condorcet systems it would be a a cycle "∅ > X&Y > Y > X > ∅."
Or you just don't like numerical valuation at all? No problemo... Example election #3:
| #Voters | Their preferences |
|---|---|
| 35% | X > ∅ > X&Y > Y |
| 33% | Y > ∅ > X&Y > X |
| 32% | X&Y > X > Y > ∅ |
In this version, the voters have only preference orderings among the 4 possible outcomes. If we handled this election with simple-majority 2-choice votes:
Again, these three simple majority votes contradict each other.
Plain plurality, Borda, and IRV all would choose "X" in this situation with honest preference orderings as votes, but Condorcet systems would regard it as a cycle "∅ > X&Y > X > Y > ∅." Approval voting (if all voters approved top two) would choose ∅.