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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