## More about the DH3 affliction suffered by both "winning votes" and "margins" Condorcet voting

We here answer some questions about our detailed DH3 analysis. To summarize, I contend that the DH3 scenario is an extremely common and extremely serious problem afflicting Borda and every (meaning: wv or margins) Condorcet voting method.

### Q. Does the DH3 scenario involve Condorcet cycles?

A. Not really. There is no Condorcet cycle (with roughly 50-50 mixtures indicated by the commas) in the honest votes – C is the honest Condorcet winner:

A simple DH3 scenario (honest votes). The notation "X,Y" means a mixture of X>Y and Y>X votes whose composition is not immediately known.
#voters Their Vote
37 C>A,B>D
32 A>B,C>D
31 B>A,C>D

...and there is no Condorcet cycle in the scenario with everybody (or most) voting strategically either (the "dark horse" D is now the Condorcet winner):

DH3 scenario with strategic votes by all voters.
#voters Their Vote
37 C>D>A,B
32 A>D>B,C
31 B>D>A,C

But we admit some thinking about cycles was involved, in a sense, in the decision by the A- and B-voters to use this strategy (which was intended to create a cycle to prevent C from winning and thus cause A or B to win). However, we contend many A- and B-voters would have done that even without ever having heard of Condorcet cycles, since it is a natural attempt to most-hurt their candidates' perceived major rivals.

Check this evidence that real human Australian ranked-ballot voters use this sort of "maximal exaggeration" strategy in massive numbers. Indeed, the Australian parties themselves advocate this kind of voting strategy. Also, it is known from poll data that about 90% of Nader-favorite voters voted, strategically, for somebody else in USA 2000.

I was asked: "why would voters act in this strategic manner if they know it risks the DH3 pathology of electing D?" Well, voters have to assess their risks and benefits. If they raise D, they are more likely to elect their favorite, but also more likely to trigger the pathology and elect D. So a voter might say "ok, I will be altruistic and vote honestly, thus not risking electing D." Trouble is, then the voters on the other side, who are not altruistic, can afford to play the D-raising game and will succeed in their strategic goal! The situation is very similar to the game of "chicken." That isn't a good game for a voting system to resemble.

### Q. Is this really so "dire"?

The critic's question more fully was: To make a dire warning about how "DH3 pathology" could cause "massive destruction" if any of the voting methods that are theoretically susceptible to it are used is little more than a rhetorical ploy... What is most lacking in this and other discussions about strategic voting is empirical data about how people vote in actual public elections in which different voting methods are used... If the point is to make arguments that are logically compelling, such rhetoric is not merely unhelpful but extremely counterproductive.

A. The reason I consider DH3 to be extremely serious and destructive (as opposed to some non-serious random election pathology example) is that it is extremely common and when it happens it is very bad. It is the combination of the two.

Why do I say COMMON?
Because all you need are 3 major rivals and at least one additional "dark horse" candidate voters do not like and do not take seriously as a threat to win. This is very common. Indeed, pretty much the only occasions where this does not happen, are when you only have 2 major rivals, in which case, since plurality is the best system in 2-candidate elections, the whole discussion about improving on plurality would largely be moot. Assuming we are having such a discussion, then you have to regard DH3 as very common. (Also, voter strategic behavior of this sort seems very common too, as we just said.)
Why do I say BAD?
Because it causes the candidate unanimously agreed worst, to get elected. That is as bad as it possibly can be.

I think these are objective criteria, not inflamed rhetoric. (And see above about real-life data from Australian elections.)

### Q. What if the "dark horse" D really was the honest-voter second choice and so D really should be the Condorcet winner?

The question more fully continues: This situation could arise if the candidates ABC are the vertices of an equilateral triangle, D is its center, and all voters are located near A, B, or C and voters prefer candidates closer to them.

A. Yes, in this scenario, each voter honestly would rank D second and D would then be the honest Condorcet winner. Also, D might then be the honest range voting winner (depends how highly the honest voters score D) or that might be one of {A,B,C}.

In this situation the C-voters in Condorcet would be strategically motivated to downgrade D in their votes to bottom (below all others). If enough voters did that, then D would no longer be Condorcet winner and one of {A,B,C} would win. (For concreteness, say C would win.) Of course, in that case some voters (i.e. the A- and B-voters) would then want to upgrade D... with the result

Equilateral triangle scenario after strategizing assuming 37/32/31 split
#voters Their Vote
37 C>D=A=B
32 A=D>B=C
31 B=D>A=C

so D would (fortunately for all) win despite this strategizing and restrategizing (and with either margins or winning votes Condorcet).

Meanwhile with range voting, the C-voters also would strategically want to vote C=99, B=A=D=0 and if enough voters acted that way one of {A,B,C} would win (say C for concreteness); but then the voters for the losers (A and B) would be motivated to upgrade D to co-equal top with 99, and then D would win. So in this situation, range voting and Condorcet voting both (after strategizing) yield the same winner D, who as the question-poser stipulated, is the honest Condorcet winner.

So if the intent of the questioner here was to devise a counter-scenario in which Condorcet looks good and Range Voting looks bad, that attempt failed.

### Q. Why can't the A- and B-voters just agree to vote "A>B>C>D" and "B>A>C>D" to assure either A or B wins (without needing to dishonestly vote D>C?)

The questioner had originally asked why the A- and B-voters could not agree to just all vote for A. The answer to that was "because then B would have zero winning chances, so the B-party and B-voters wouldn't go for that deal." He then asked this. This agreement would work, except in reality, usually what will happen is, it will not be (and cannot be) clear to the voters who is collaborating with whom to defeat whom.

E.g, suppose all the voters know is: there are 3 strong rivals A,B,C and a dark horse D. They do not have precise-enough estimates on who has how many votes exactly.

In that case (the questioner conceded) "I agree. Without a clear frontrunner to try to defeat, I don't see how you could hope for even an incidental alliance with another faction."

### Q. Are you saying I should support IRV and not Condorcet?

The short answer is "no."

My questioner continued: [The CRV] talks of the "DH3 pathology" as a serious problem that should encourage Condorcet backers, such as myself, to abandon ship and switch to IRV.

1. The pathology can occur with certain patterns of votes and plotting.
2. IRV has problems that can occur without plotting.
3. So, NO SALE!

The questioner then went on to point out, e.g, this situation which makes IRV look bad and Condorcet look good:

IRV looks stupid in this 100-voter election by Dave Ketchum.
#voters Their simplified vote Their full (implied) vote
30 A A > B = C = D
25 C > B C > B > A = D
23 D > B D > B > A = C
22 B B > A = C = D

In this example, B is the easy Condorcet winner but IRV elects A; the latter seems obviously silly since the voters prefer B over A by 70-to-30.

A. The questioner misinterpreted the goal of our "salesmanship." It happens to be true that IRV is better than Condorcet in the DH3 particular kind of situation. However, Condorcet seems better than IRV in other situations, including Ketchum's example election above. For more examples: Schulze beatpaths Condorcet is monotonic, never suffers from a winner=loser reversal-pathology, and always elects Condorcet winners if they exist; but IRV enjoys none of those properties.

The correct moral to draw is that IRV and Condorcet both suffer from serious problems and the simpler range voting system, which tends not to suffer those problems, is better. In particular:

1. range voting is immune to DH3 (which afflicts all Condorcet methods);
2. range voting is monotonic (unlike IRV); and range voters are never motivated to betray their favorite by voting him below some opponent (unlike both IRV and every Condorcet method);
3. range voting never suffers from an embarrassing winner-loser reversal failure (unlike both IRV and classic Condorcet);
4. and finally (in the presence of strategic voters) range voting actually is quite likely better than Condorcet methods for the purpose of electing honest-voter Condorcet winners!