Condorcet Compromise Extension
by Kevin Venzke [Home]

This calculator demonstrates the Condorcet Compromise Extension, or CCE, which I introduce as an approach to extending the Condorcet criterion to scenarios without a Condorcet winner, with the goal of minimizing compromise incentive.

Compromise is when a voter ranks a less-liked compromise choice above some candidate whom they actually prefer, because the preferred candidate is thought to be unable to win the election. An example of this is voting for "the lesser of two evils" out of fear of otherwise spoiling an election.

Many Condorcet advocates recognize that compromise incentive is an important argument for their cause, being a weak point of some other methods currently in use, or proposed. When there is no Condorcet winner, then the Condorcet criterion says nothing and imposes no requirement. Yet some instances of compromise incentive can be foreseen, without knowing any details of the specific Condorcet method under discussion. I suggest that any principle used to motivate Condorcet might also indicate how to resolve situations where Condorcet doesn't apply.

The CCE is based on a concept of voter "claims" (or potential post-election complaints) that, in a situation where there is no Condorcet winner, some candidate Y should not be elected because voters preferring some candidate X to Y demonstrably could have made X into the Condorcet winner if they had ranked X insincerely high (i.e., a compromise strategy). In other words, if Y were to be elected, it would prove that these voters were penalized for submitting their true first preferences.

I suggest two criteria and several methods. The logic proposed may be too unwieldy to use directly. But I hope it will provide food for thought on an interesting topic.

When counting ballots we maintain three matrices for CCE. One is two-dimensional, and two are three-dimensional. Let v(x,y) represent the number of voters ranking some candidate X over Y. (This is the ordinary pairwise matrix.) Let u(x,y,z) represent the number of voters with the relative ranking of X over Y over Z. Let t(x,y,z) represent the number of voters ranking X equal to Y, and both of these over Z.

Then we can say that some candidate X has a CCE claim against candidate Y when v(x,y) > v(y,x) and, for all other candidates Z, v(x,z) + u(z,x,y) + t(z,x,y) > v(z,x) - u(z,x,y). In other words, we check whether X can beat every Z, after applying the changes to the ballots that would result from all X>Y voters raising X to the top position.

If so, then X>Y voters can cause X to become the Condorcet winner by following an insincere strategy. And in that case, by refraining from selecting Y as the winner we may avoid introducing compromise incentive into our method unnecessarily. I admit that every method may have its own peculiar opportunities for compromise incentive to arise, into which we have no visibility given this approach. But often there won't be any: CCE will have found them all, and we may reasonably ask whether ignoring these compromise strategy incentives provides us with anything of value in return.

It's possible that every candidate has a CCE claim against them. This makes it a little unclear how to define a criterion. I offer two possibilities.
Basic CCE is a criterion which says to avoid electing any candidate who has a CCE claim against them. This may leave no possible winners, in which case we are free to elect anyone.
CCE-TopTier (CCE-TT) forms a Schwartz set using the claims, and then the winner must be selected from this top tier. The top tier contains every candidate X, except those for whom there is some candidate Y who has a CCE claim against X while X cannot "trace a path" of claims back to Y.

Depending on how you look at it, CCE-TopTier could be seen as either a natural or unnatural way of adapting the CCE concept. On one hand, it's similar in spirit to Smith or Schwartz, which would be familiar to most Condorcet advocates. On the other hand, the path-tracing concept seems motivated by concerns about clone independence, which aren't obviously relevant if our focus is compromise incentive.

As explained under "More details," I feel we will have to use CCE-TopTier.

More details
Douglas Woodall's Plurality criterion proposes a modest standard, that if some candidate X has more first preferences than another candidate Y has any above-last preferences, then that Y must not be elected. The concern is that Y would look suspiciously unqualified as a winner. Few proposed methods violate this standard, yet it appears any form of CCE sometimes will. The failures involve situations where some voters simultaneously give their first preference to X and a lower preference to Y; Y is prohibited by Plurality due to X's first preferences; and CCE requires that Y win because the aforementioned voters will otherwise regret voting for X. We might say that the Plurality criterion lets these voters stand in their own way. This can be fixed by using a "weak" variant of the Plurality criterion: Remove all Y voters from X's first-preference count when considering whether to disqualify Y.

This fix, however, is still incompatible with Basic CCE. So I am left suggesting the use of methods that satisfy CCE-TT and also the "weak" Plurality criterion. My best current suggestion is rather crude: Compute the pairwise matrix, remove any candidates barred from winning by either CCE-TT or "weak" Plurality, and then perform one of the WV methods on what's left of the matrix. The three WV methods I suggest are Schulze, Tideman's Ranked Pairs method (which is also called MAM in the WV case), and Jobst Heitzig's River method. The resulting methods will be referred to as CCE-Schulze, CCE-RP, and CCE-River.

There are some references to the Schwartz set, a common extension of Condorcet which says that the winner should at least have a path of pairwise wins to any candidate who beats them (or has a path of wins to them). It ensures that if a Condorcet winner were to be cloned into a set of multiple identical candidates, then the new winner would be one of them. That is far from its only effect, but it's the main one I find obviously valuable. If there are no pairwise ties then this is the same as the Smith set.

Enter your ballots like this, one per line:
456: Alice>Bob=Carl>Debra
The number represents the size of the voting bloc. Decimals are OK. The size can also be left off and it will then be randomized.

Candidate names can contain spaces. Each candidate in the list should be separated by > or =. Pipes (i.e. |) and any series of > will be interpreted as single >s. Not every candidate needs to be listed; candidates present on the ballots but missing from one faction's ranking will be interpreted as ranked tied for last, below any explicitly ranked candidates.

Click submit to generate an analysis.

Enter the ballots for an election:

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