[Election-Methods] Reducing 3-cand elections to 8 scenarios

[Kevin Venzke]

Hello,

Lately I've been thinking again about how to adjust a method's incentives in order to encourage a state of affairs where there are three competitive candidates, each of whose strategy is to stand near the median voter.

A couple of drastic measures that appeal to me are only accepting (and requiring) a first and a second preference, and to the extent necessary, discarding ballots that won't cooperate in voting for the top three candidates (according to first preferences).

Another measure occurred to me: Among the supporters of each of the top three candidates, play "winner takes all" for the second preference. In other words, all of the second preferences from the "A-first" voters are considered to be cast for whichever (of the other two candidates B and C) received more. This has a consequence that not giving a second preference (if such were allowed) is never optimal; your second preference is just determined by other voters with the same first preference.

When we play "winner takes all" in this way, there are only 6 possible ballot types, only 3 of which can occur in the same election, and there are only 8 possible elections.

This makes it very easy to describe many methods and then compare their strategic vulnerabilities. First, say that the candidates A B and C name the three candidates in decreasing order of first-preference count. Also, assume that all methods will elect a majority favorite, so that in all 8 scenarios, we know that any two factions are larger than the third.

Here is how I've ordered the scenarios:
The two cycles:
1 ab bc ca
2 ac ba cb
The six with majority coalitions:
3 ab ba ca
4 ab ba cb
5 ab bc cb
6 ac bc cb
7 ac ba ca
8 ac bc ca

I can define methods by which candidate wins in each scenario:

FPP:
AA AAAAAA
IRV:
AB ABBBAA
DSC and (my method) SPST:
AA AABCAA
VFA:
AA AABBAA
Schulze, MMPO, etc.:
AA ABBCAC
Bucklin, MF/Antiplurality:
BA ABBCAC
IRV/DSC combo:
AB ABBCAA

The last method takes the DSC result for scen 6 but otherwise uses the IRV result.

For each method I can mostly summarize the rule:
FPP: Elect A.
IRV: Elect C faction's second preference.
DSC: If there's a majority coalition excluding A, elect A faction's second preference; else elect A.
VFA: If there's a majority coalition excluding A, elect B; else elect A.
Schulze: If a candidate has no last preferences, elect that one; else elect A.
Bucklin: If a candidate has no last preferences, elect that one; else elect C faction's last preference.
IRV/DSC combo: If there's a majority coalition excluding A, elect A faction's second preference; else elect C faction's second preference.

I evaluated two types of strategies from each faction's perspective:
Compromise: The faction tries to improve the result by swapping their first and second preferences, creating a majority favorite (autowinner).
Burial: The faction tries to improve the result by swapping their second and third preferences.

I have a lot of scratchpaper for this task, but I think I'll just show the results.

For each method, the scenarios go across as they do above. The values in a position can be "y"es, the strategy helped; "n"o the strategy did nothing; and "w"orse as in, the strategy made the outcome worse.

Unintuitively the six rows are in this order:
Compromise by C faction
Compromise by B faction
Compromise by A faction
Burial by A faction
Burial by B faction
Burial by C faction

FPP
ny nyyynn
yn nnyyny
ww wwwwww
nn nnnnnn
nn nnnnnn
nn nnnnnn

FPP has no burial strategy, but a lot of potential for compromise strategy by B and C factions. No strategy for A faction.
# of "stable" scenarios: 2
y vs w count: 8 to 8

IRV
nn nnnnnn
yw nwwwny
wy wnnyww
nn nnnnnn
nn nnnnnn
ww wwwwww

IRV has no strategy for C faction, and no burial strategy at all. A and B factions have compromise strategy.
# of "stable" scenarios: 4
y vs w count: 4 to 16

DSC
ny nynwnn
yn nnwnny
ww wwnnww
nn nnwwnn
nw nywwnn
wn nnwwny

DSC has no strategy for A faction.
# of "stable" scenarios: 4
y vs w count: 6 to 16

VFA
ny nynnnn
yn nnwwny
ww wwnyww
nn nnnnnn
ny nywwnn
wn nnwwnw

VFA has no burial strategy for A and C factions.
# of "stable" scenarios: 3
y vs w count: 7 to 14

Schulze etc
ny nnnwnw
yn nwwnnn
ww wnnnwn
ww nywwny
nw nnnwww
wn wwwnnn

Schulze has no compromise strategy for A faction, but (only) they have burial strategy. No compromise strategy at all outside of a cycle.
# of "stable" scenarios: 4
y vs w count: 4 to 20

Bucklin etc
yy nnnwnw
wn nwwnnn
nw wnnnwn
wn nywwnw
ww ynnwww
nn wwwnwn

Similar to Schulze, but the burial strategy is split between A and B factions.
# of "stable" scenarios: 4
y vs w count: 4 to 21

IRV-DSC combo
nn nnnwnn
yw nwwnny
wy wnnnww
nn nnwwnn
nw nnnynn
ww wwwwwy

No compromise strategy for C faction, and no burial strategy for A faction.
# of "stable" scenarios: 4
y vs w count: 5 to 18


Another type of strategy, more complicated to figure out, is where a faction tries to reduce their vote count below another faction's in attempt to gain an advantage. If I implement all of this into a computer, and search for new methods, it would be useful to know whether a method has problems like this.

A useful question, I think, is what types of strategies, beneficial to whom, are most alarming. If I want to have three candidates, then it's quite alarming for C faction to have compromise incentive, and not much less alarming when B faction does (since C aspires to become B). It's not as alarming for A to have compromise incentive, partly because this candidate has already survived the system to become candidate A, and partly because I don't think frontrunners' supporters will be inclined to compromise.

Burial strategy is most alarming when A faction uses it to easily transfer the win to A. I think the incentive is not as bad for the weaker factions, because it should seem less dependable. I also find it's not as alarming in scenarios where the result of the strategy is the election of a candidate who is actually more widely liked.

That's it for now. Hopefully there are no errors in the above.

Kevin Venzke



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