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

Chris Benham cbenhamau at yahoo.com.au
Tue Jun 17 09:03:26 PDT 2008

```"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)."

Kevin,
I have the same question I had the last time you proposed a method focused on 3 candidates:
Instead of  "discarding ballots", why not apply these methods to the ballots modified by eliminations
after all but 3 candidates have been IRV-style one-at-a-time eliminated?
"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."
With this weird  (but I suppose not in principle unacceptable) feature, what is the point of  "requiring a second preference"?
Chris Benham

[Election-Methods] Reducing 3-cand elections to 8 scenarios
Kevin Venzke stepjak at yahoo.fr
Sat Jun 14 10:02:33 PDT 2008

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

Get the name you always wanted with the new y7mail email address.
www.yahoo7.com.au/mail

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