by Abd ul-Rahman Lomax & Warren D. Smith

We consider "3-slot range" voting where you can give each candidate a score in the set {0,1,2}; highest sum-score wins.

The voter (you) has these utilities for the three candidates ABC:

We contrast three voting strategies in a 3-candidate election:

- honest "Borda-style" 3-slot range voting: A=2, B=1, C=0.
- exaggerated "AntiPlurality-style" voting: A=2, B=2, C=0.
- exaggerated "Plurality-style" voting: A=2, B=0, C=0.

Which strategy works better for you?

Here we shall determine "utility" in cases where your vote could be "relevant" – i.e. it could raise average utility above not voting at all. This eliminates from consideration all the vote patterns in which your vote could not possibly change the winner. In other words, we only examine 2-way near-ties or 3-way near-ties.

We assume all such patterns of vote totals
are equally likely, except that *three*-way near-ties are regarded as
*less* likely (say T times the probability, for some T with 0<T≤1;
although actually we could also permit these to be *more* likely, i.e. T>1,
the math below will not mind).
Finally, we also assume ties are broken by unbiased random chance.

Then there are
exactly 15 vote-total patterns for which the other voters are relevant and
somebody is out of contention ("–∞"), and exactly 25
vote-total patterns for which the other voters are relevant and
*everybody* has a chance to win or tie.
(For each, an offset is subtracted such that one
of the candidates – it doesn't matter which –
has a net vote of 2. The rivals' vote-totals thus vary from 0 to 4.
If the other candidate is far behind, he has a score of "–∞".)

Totals from Result if Result if you Result if you other voters you honestly exaggerate exaggerate ------------ vote 210 utility "220" utility "200" utility A B C -------- ------- ------- ------- ------- ------- -∞ 0 2 C wins 0 BC tie 5+3e C wins 0 -∞ 1 2 BC tie 5+3e B wins 10+6e C wins 0 -∞ 2 2 B wins 10+6e B wins 10+6e BC tie 5+3e -∞ 3 2 B wins 10+6e B wins 10+6e B wins 10+6e -∞ 4 2 B wins 10+6e B wins 10+6e B wins 10+6e 0 -∞ 2 AC tie 10 AC tie 10 AC tie 10 1 -∞ 2 A wins 20 A wins 20 A wins 20 2 -∞ 2 A wins 20 A wins 20 A wins 20 3 -∞ 2 A wins 20 A wins 20 A wins 20 4 -∞ 2 A wins 20 A wins 20 A wins 20 0 2 -∞ B wins 10+6e B wins 10+6e AB tie 15+3e 1 2 -∞ AB tie 15+3e B wins 10+6e A wins 20 2 2 -∞ A wins 20 AB tie 15+3e A wins 20 3 2 -∞ A wins 20 A wins 20 A wins 20 4 2 -∞ A wins 20 A wins 20 A wins 20 The below 25 configurations each have (smaller?) relative likelihood T: ----------------------------------------------------------------------- 0 0 2 AC tie 10 ABC tie 10+2e AC tie 10 0 1 2 ABC tie 10+2e B wins 10+6e AC tie 10 0 2 2 B wins 10+6e B wins 10+6e ABC tie 10+2e 0 3 2 B wins 10+6e B wins 10+6e B wins 10+6e 0 4 2 B wins 10+6e B wins 10+6e B wins 10+6e 1 0 2 A wins 20 A wins 20 A wins 20 1 1 2 A wins 20 AB tie 15+3e A wins 20 1 2 2 AB tie 15+3e B wins 10+6e A wins 20 1 3 2 B wins 10+6e B wins 10+6e AB tie 15+3e 1 4 2 B wins 10+6e B wins 10+6e B wins 10+6e 2 0 2 A wins 20 A wins 20 A wins 20 2 1 2 A wins 20 A wins 20 A wins 20 2 2 2 A wins 20 AB tie 15+3e A wins 20 2 3 2 AB tie 15+3e B wins 10+6e A wins 20 2 4 2 B wins 10+6e B wins 10+6e AB tie 15+3e 3 0 2 A wins 20 A wins 20 A wins 20 3 1 2 A wins 20 A wins 20 A wins 20 3 2 2 A wins 20 A wins 20 A wins 20 3 3 2 A wins 20 AB tie 15+3e A wins 20 3 4 2 AB tie 15+3e B wins 10+6e A wins 20 4 0 2 A wins 20 A wins 20 A wins 20 4 1 2 A wins 20 A wins 20 A wins 20 4 2 2 A wins 20 A wins 20 A wins 20 4 3 2 A wins 20 A wins 20 A wins 20 4 4 2 A wins 20 AB tie 15+3e A wins 20 210+405T+30e+47eT 210+370T+42e+74eT 210+430T+18e+26eT AvgUtil ----------------- ----------------- ----------------- 15 + 25T 15 + 25T 15 + 25T

Depending on the values of e and T, any one of the three voting strategies can be best:

- If e=T=0, all are equal with utility=14.
- If e = +0.5, T=0 then AntiPlurality is best with utility=15.4 (Borda=15.0, Plurality=14.6).
- If e = –0.5 (and any T>0) Plurality is best, with utility=13.4 (Borda=13.0, AntiPlurality=12.6) when T=0.
- If e = +0.8, T=8/9 Borda is best with utility≈16.86 (Plurality=AntiPlurality≈16.79)
- If 0<e<1.25=5/4 and T=2e/(5-4e) then Borda is best. Equivalently, if 0<T and e=5T/(4T+2) then Borda is best.

- If 0<e<25/21 and 12e/(35-27e)<T<12e/(25-21e) then Borda is best.
- If e<0 and T>0, then Plurality is best.
- If 0<e<35/27 and 0<T<12e/(35-27e), then AntiPlurality is best.

**THEOREM:** For any T>0:

- If e < 25T/(12+21T), then Plurality is best strategy.
- If 25T/(12+21T) < e < 35T/(12+27T), then Borda is best.
- If 35T/(12+27T) < e, then AntiPlurality is best.