[EM] Random Ballot / Smith

[Kristofer Munsterhjelm]

On 2025-09-21 13:33, Joshua Boehme via Election-Methods wrote:
> 
> 
> On 9/20/25 7:13 AM, Kristofer Munsterhjelm wrote:
> 
>> I think that this property is balanced by that it also goes random if 
>> an innocent cycle occurs or if the buriers' attempts would otherwise 
>> fail. A deterministic method sometimes gets it right (and elects 
>> either the honest winner or the bus) and sometimes is fooled (and 
>> elects the burier's candidate); Smith/RB gets it partially right all 
>> the time instead of fully right or not right at all.
>>
>> I'm not sure what the expectation of getting it right over, say, all 
>> impartial or spatial elections of a particular candidate would be (in 
>> the limit of number of voters going to infinity), so I can't say for 
>> sure if it would be worth it. Maybe it is - maybe even risk aversion 
>> would keep people from trying to engineer a cycle at all. But without 
>> testing, I wouldn't know :-)
>>
>> It does open the possibility to other "nonconventional" methods, 
>> though. For instance, you could have a second round if there's a cycle 
>> (runoff style), or you could draw an assembly at random and have them 
>> decide after deliberation, using some kind of supermajority 
>> requirement or Heitzig-type consensus mechanism. If the burier's 
>> favorite is easy to spot, these should both make burial impractical. 
>> Armytage also suggested a candidate withdrawal option, where the bus 
>> withdraws and thus makes the burial fail.
>>
>> -km
> 
> 
> Where I would quibble is with the characterization of the "right" 
> solution for a cycle. Sure, if you absolutely cannot implement any 
> stochastic solution at all, then yes break the weakest link in a 
> 3-cycle. Then again, what happens if two or three links are tied for 
> weakest? Avoiding stochastic solutions entirely is very challenging.

I meant "right" as in "a way that doesn't reward the buriers" by 
electing the pre-burial winner or punishing the buriers by electing the 
bus (under which the honest candidate is thrown). I guess I could have 
made that clearer before I used the term.

Breaking at the weakest pairwise victory may not be "right" in this 
sense; e.g. fpA-fpC and Smith//IRV resist burial more effectively than 
minmax does.

So my point was more that if you want to ignore or punish buriers who 
set up a cycle, then deterministic methods discourage strategy by making 
burial only work some of the time, and be detrimental to the buriers the 
rest of the time. In contrast, Smith,RB makes burial work partly all of 
the time (as long as the buriers can engineer a cycle), because the 
burier's candidate has a nonzero chance of being elected.

The appeal (as I read it) is that even if burial succeeds, there's a 
chance that the pre-burial honest winner or the bus gets elected 
instead, which makes it risky to bury in the first place. But it's still 
possible that there exist deterministic methods where the expected 
probability of burial succeeding is lower than the same expectation for 
Smith,RB.

> I'd  argue it's also unnecessary. Everyone's comfortable with a random 
> tiebreaker for plurality elections. Cycles are just a more general form 
> of tie.

The few large-scale political Condorcet cycles (Minneapolis and Oakland 
IIRC) seem to be noise, and a random tiebreaker would be okay in that 
case, I think. But adopting Condorcet could lead to more candidates 
running and more issues being considered at once, which in turn could 
lead to "true" Condorcet cycles. Warren Smith used an example like:

Candidate     Domestic                  Foreign         Corruption
A             Improve                   Wage war        Some dirty
               public services                           dealings

B             Maintain status quo       Peace           Clear mob
               with disinterest                          connections

C             Drain maintenance         Vague           Clean and
               budgets                   mumbling        transparent

Suppose the voters who mainly care about services vote A>B>C, the ones 
who care about foreign policy vote B>C>A, and the ones who care about 
corruption vote C>A>B. Then you could get a Condorcet cycle if they're 
mainly focused on their own issue.

If such situations would be common, then just breaking things by a tie 
may not be good enough, because the presence of a cycle indicates a 
society with conflicted opinions rather than one that's indifferent.

> (It's also worth mentioning that avoiding impacts from clones in 
> tiebreakers generally requires a little more care than in the plurality 
> case -- since plurality isn't cloneproof anyway -- but in at least some 
> voting systems it's still straightforward; picking a random ballot as 
> the basis for breaking the tie typically suffices.)

Random ballot is cloneproof, so that should work. Random plurality - 
picking proportional to first preference counts after eliminiating 
non-Smith members - would also work, since the sum of clones' first 
preferences is the same as the first preference count of the cloned 
candidate.

-km