Cycle Cancellation//Condorcet

This is a rankorder voting system motivated by criticisms of the Condorcet criterion present here:
https://www.journals.uchicago.edu/doi/full/10.1086/682019#
A Condorcet component is a block of ballots that form a perfectly balanced Condorcet cycle. For example, a basis for 3candidate Condorcet components is
1: A>B>C
1: B>C>A
1: C>A>BUsing the terminology in the paper, a system is said to “cancel properly” (I would rather say “cancel cycles” to avoid the moral judgment…) if the addition of any number of Condorcet components to the ballot set does not affect the result of the election. It turns out that Condorcet compliance is incompatible with cycle cancellation.
The system being considered is something along the lines as follows: first, remove all possible Condorcet components from consideration (how to address this for cycles of differing lengths?). Second, elect the Condorcet winner from the remaining ballots. This system is not Condorcet compliant because it cancels cycles.
For example, consider
7: A>B>C
6: B>C>A
2: C>A>BThere is no Condorcet winner here. But if we remove the Condorcet component
2: A>B>C
2: B>C>A
2: C>A>Bwe are left with
5: A>B>C
4: B>C>AThere are no more Condorcet components, and the Condorcet winner of these ballots is A.
Correct me if I’m wrong, but I believe cycle cancellation with 3 candidates should guarantee the existence of a Condorcet winner and a Condorcet loser in the remaining ballots unless all of the ballots are canceled. This means that we can do a variant of Tideman’s IRV (Bottom Two Elimination Runoff) with triples of candidates instead of pairs: consider the 3 “worst” candidates (in a noncompensatory manner, such as fewest 1st place rankings), then cancel all 3cycles, and eliminate the Condorcet loser. Repeat until the desired number of candidates is obtained.

@cfrank I suspect you will enjoy Stable Voting
It is a recursive Condorcet method that asks "who would have won if X candidate were not in the election?" and then chooses the winner recursively based on strongest pairwise defeats. It has excellent theoretical properties, although unfortunately it's rather difficult to explain and implement.

@andydienes I’m checking out the paper now (I’ve been doing a lot of driving so I’m mostly having my phone do text to speech lol) it’s interesting but I’ll have to actually read it.
What do you think of the concept of cancelling Condorcet cycles for 3candidate elections?
I’m trying to think about an algebraic structure, where basically ranked ballots are associative, noncommutative products of candidates, and the ballot set is a sum of ranked ballots so that the ranked ballots form a vector space. Also, any two ranked ballots R1 and R2 have a product R1R2 that is 0 if there is some x belonging to both R1 and R2 (this guarantees that candidates don’t appear in more than one place in a ranking).
For example
abc+bca+cab
is a Condorcet component. Cancellation of cycles is the same as imposing that Condorcet components vanish, which is the same as taking the quotient of the ranked ballot vector algebra with the subalgebra generated by Condorcet components. There seems to be a possible connection with Lie theory since saying that
abc+bca+cab=0
is known as Jacobi’s identity and is one of the structural axioms of a Lie algebra. I think reversals are also tieinducing, since
abc+cba
also has no Condorcet winner. On the other hand it seems preferable to select b in this case by symmetry.
Forgive some jargon, I’m just trying to consider this construct mathematically. By taking the quotient with the Condorcet component algebra, we associate a ballot set with its residue modulo Condorcet cycles, and we might then have a “residual” Condorcet winner. However, even if a nonresidual Condorcet winner exists, the residual Condorcet winner is not always the nonresidual Condorcet winner.
This all is only pertaining so far to three candidates.
I’m trying to consider how to extend the notion of a residual Condorcet winner to four candidates. My thought is to operate by analogy with ordinary Condorcet winners, but rather than considering pairs of candidates and finding the majority winner of each pair, instead consider all triples of candidates and find the residual Condorcet winner of each triple. Basically instead of looking at the edges of the candidate graph, the triangular “faces” are looked at (which raises the complexity from quadratic to cubic in the number of candidates).
The issue with this is how to combine all of the 3cyclic residual Condorcet winners into a single winner among 4 candidates. I wonder empirically how often the residual and nonresidual Condorcet winners coincide, my guess is quite often.

I'm skeptical that this will work out the way you want it to.
I see the idea, but the order you choose to cancel orbits in matters. For example if you have
abcdef + cdefab + efabcd + dcbafe
Should you cancel the first three terms (leaving condorcet winner = d) or the last two (leaving condorcet winner = c) ?
If you are going to try to treat a ballot set like a free algebra (modulo some cyclic relations) then now each ballot set is really an equivalence class of ballot sets. and since the condorcet winner could be different in any member of that equivalence class it's really not clear which one to choose. There is not a notion of "pre cancellation" and "post cancellation."

@andydienes Yes indeed, I’ve been running into that breed of issue in my notes. Although I would say that the first three terms are not a Condorcet component, since you would need all six of the terms in the cyclic relation
abcdef+bcdefa+cdefab+defabc+efabcd+fabcde=0
for six candidates. For example, take four candidates and suppose we have something like
9wxyz+8wxzy+15zyxw+16zxwy
This has no 4cyclic Condorcet components. Additionally, we can formally set each candidate to 1 and find ballots corresponding to the other three candidates. Setting w=1 we find
9xyz+8xzy+15zyx+16zxy
Again here there are no 3cyclic Condorcet components. We have the marginal Condorcet victories
x>y, 9+815+16
z>y, 9+8+15+16
z>x, 9+815+16and hence from the triple {x,y,z} we have the residual Condorcet winner as z.
We can do a similar computation excluding each candidate (I.e. looking at each triple). The residual Condorcet winner of {w,y,z} is z, for {w,x,z} it is also z, and of {w,x,y} it is x. These residual Condorcet winners are {x,z}. The residual Condorcet winner of these two is z, and hence z can be chosen as the winner accordingly (and happens to also be the nonresidual Condorcet winner).
This I believe may be a totally welldefined system:
First, for one candidate let that candidate be the residual Condorcet winner, for two candidates let the majority winner be the residual Condorcet winner, and for three candidates take the definition already given with the addition that the residual Condorcet winner of xyz+zyx is y.
To find the residual Condorcet winner among a set of N candidates, take the Ncyclic residue of the ballots, and then from each subset of (N1) candidates, find the corresponding residual Condorcet winner according to that Ncyclic residue. The residual Condorcet winner among the N candidates is then recursively defined to be the residual Condorcet winner among the set of residual Condorcet winners from all subsets of (N1) candidates.
An issue comes about from the possibility that every candidate may be the residual Condorcet winner of some subset. My intuition is that by removing any Ncyclic Condorcet components, it may be impossible for every candidate to appear as one of the residual Condorcet winners. If this is the case, then we can simply iterate the procedure restricted to the set of residual Condorcet winners until a single candidate remains or a tie is reached.
I may be totally wrong about the different aspects that obstruct the existence of a Condorcet winner. Other than cycles, there are reversals, and it isn’t clear how to decide what the residual Condorcet winner of
wxyz+zyxw
should be, for example, or for any reversal of even length. This appears to be a tie between x and y.

@cfrank
Anyway, I'm not sure why you would ever want to use this instead of just a method that operates on a margin graph, like e.g. the Minimax family of Condorcet methods.Any time any ballot contains X > Y (not necessarily adjacent in the ranking), and another ballot contains Y > X, then those two will implicitly cancel each other out in the margin of defeat between X and Y. To me this seems like a strictly better approach to implement the cancellation idea.

@andydienes I’m just trying to incorporate the points I found interesting and reasonable in the linked paper. I do find it odd that the Condorcet winner, which is defined in terms of the pairwise comparisons, firstly does not always exist (which is not that odd), and secondly can change depending on the addition or elimination of Condorcet components. This leads to the notion of a residual Condorcet winner instead, which is invariant under those alterations and has a much better chance of existing than the nonresidual Condorcet winner.
I think that if the residual Condorcet winner among three candidates is defined as above, then they will almost always exist except for when the ballot set is a single perfectly balanced Condorcet cycle. Indeed I think I virtually have a proof of this as a theorem for 3 candidates, so that the residual Condorcet winner always exists for 3 candidates except in the case of a perfectly balanced tie. For this reason especially it seems to me to be a convincing canonical choice of winner, at least among three candidates.
If we handle ties for evenlength reversals appropriately, it may be possible to prove by induction the universal existence at least of a weak residual Condorcet winner for any number of candidates. Possibly there’s some snag, my intuition about excluding at least one candidate may simply be false. It also isn’t totally clear even if existence is assured that the residual Condorcet winner will be a “good” winner other than in the sense of agreement on procedure, which is generally true for any voting system. I think in any case the residual Condorcet winner is at least a reasonable backup for the nonresidual if not a reasonable replacement.
I’m trying to think of arguments in favor of including Condorcet components in the tally. I’m sure some argument would have to do with voter dissatisfaction or regret, and possibly stability. But I don’t see any benefit to voting dishonestly that doesn’t already exist for ordinary Condorcet methods, since in every case one would like his candidate to be a residual Condorcet winner, which is itself a Condorcet winner over a restricted subset of the ballots. I don’t see how a voter could influence which subset of the voters that “decisive” subset would be.
One example is given in the paper where the residual Condorcet winner is different from the ordinary Condorcet winner:
30abc+29bac+10bca+10cab+acb+cba
Here the Condorcet winner is ‘a.’ But when we eliminate Condorcet components, the residual Condorcet winner is that of
20abc+28bac
which is ‘b.’

@AndyDienes A thought occurred to me that makes me believe the sensitivity to Condorcet components is actually not odd at all.
The thought can be best illustrated with three candidates {x,y,z}. In this case, there are two kinds of balanced Condorcet cycles, namely (xyz+yzx+zxy), and (zyx+yxz+xzy). Individually, and even combined, these Condorcet components should indicate a tie by any reasonable deliberation, since the candidates are interrelated by perfect interchangeability.
For the same reason, a ballot set that is a scalar multiple of
(xyz+yzx+zxy)+A(zyx+yxz+xzy),
where A is a nonnegative real number, should equally indicate a tie.
However, consider what would occur if one of the candidates, by new information that has a negligible effect on the relationship between the other two candidates, were discovered to be an invalid winner. Without loss of generality, we can say that candidate is z. It seems reasonable to simply remove z from every ranking in the ballot set, which would transform the ballot above into
(2xy+yx)+A(2yx+xy)=(A+2)xy+(1+2A)yx
At this point, it is clear that the preferable choice between x and y, conditional on the invalidation of z, can be decided from the information available in the combination of the two oppositely oriented Condorcet components. If 1>A we should prefer x, and if A>1 we should prefer y.
In other words, combinations of Condorcet components at least provide conditional information about the preferences between candidates. If this information is considered relevant, which I think it ought to be, then ignoring Condorcet components is improper.
Informally, while Condorcet components should indicate a tie, this can be considered due to a normative balance of potential marginal dissatisfaction among the voters. If some of this dissatisfaction is actualized as a sunk cost, then the margins are changed. In a sense, this is what is being analyzed by considering the pairwise elections—the normative marginal dissatisfaction of the electorate as a whole is being minimized when the Condorcet winner exists and is chosen.
This I think upholds the concept of the Condorcet winner as a canonical choice, given that one exists.