## Personal Utility vs Strategy Analysis of "Range-2" three-candidate election

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:

UA = 20,    UB = 10+6e,    UC = 0,    for some adjustable e with   –10≤6e≤+10.

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

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

### Conclusion:

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.
Here are some more powerful results. They imply that no matter how small T>0 is, still any and all of the three strategies can be best (depending on e):
• 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.
This can be recast as the following, which seems to be the final word:

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.