Lesser of 2 evils

[Bruce Anderson]

On Jun 4,  1:48am, Bruce Anderson wrote:
> Subject: Briefly why I prefer Reg-Champ to Smith//Con
>
> First, from what I have seen so far, it seems to me that the 
> strategy-protection 
> capabilities of Condorcet and Smith//Condorcet have been vastly oversold.  I 
> have seen no precise statements of such strategy-protection criteria on this 
> list (maybe they were posted before I joined), and have seen no proofs of such 
> criteria anywhere.  When I tried to formulate and prove similar criteria, I 
> found severe weaknesses in this supposed protection.  I have not saying that 
> such criteria could not be precisely stated and formally proven.  I'm just 
> saying them I have seen no proofs at all, and that the statements of such 
> criteria that I have seen had what I considered to be severe flaws.  I will 
> attempt to post an example of such a flaw for the "lesser of two evils" 
> criterion in the near future.
> 
>-- End of excerpt from Bruce Anderson

The discussion of the "lesser of two evils" criterion promised above follows 
below.  I sent an essentially equivalent discussion to Mike last October.

Concerning the lesser of two evils criterion, I think that I might understand 
what it is saying, and, if so, then I think that what it says is not very 
important.  In particular, consider the following situation.  First, either one 
or more pairwise ties are occurring, or no pairwise ties are occurring.  I think 
that it is important to be able to adequately address pairwise ties, but I don't 
think that "two evils" is intended to be important ONLY when such ties are 
occurring.  If I am wrong here, and, like Anderson's voting method, "two evils" 
really is only important when pairwise ties are occurring, then I don't really 
understand it after all.  But if I am right about this, then we can assume that 
no pairwise ties are occurring.  For example, assume that there is an odd number 
of voters, none of whom have any ties or truncations in their individual 
rankings.  Then, given any two alternatives, say A and B, either A beats B or B 
beats A.  Without loss of generally, assume that A beats B.  Then, either there 
exists a third alternative, C, such that A beats B, B beats C, and C beats A, or 
no such third alternative exists.  If no such third alternative exists, then any 
method that satisfies the generalized Condorcet criterion necessarily (I think) 
satisfies "two evils".  Therefore, to be an important addition (over just the 
generalized Condorcet criterion), "two evils" must be important in the case when 
A beats B, B beats C, and C beats A.

In this case, let V(A,B) be the voters who prefer A over B, and let W(B,A) be 
the voters who prefer B over A, so #V(A,B) > #W(B,A).  Define V(B,C), W(C,B), 
V(C,A), and W(A,C) analogously.  Then, as I understand it, "two evils" says that 
there must be some way for the V(A,B) voters to cast their ballots such that:  
1) none of them casts a "partially reversed" ballot, 2) none of them casts a 
ballot that ties A with an alternative ranked below A in the voter's true 
preference (but other false ties are allowed), and 3) B cannot win no matter how 
the W(B,A) voters cast their ballots (the W(B,A) voters are allowed to use 
partial-reversal here).

However, "two evils" necessarily requires that, in this very same election, it 
must also be simultaneously possible for the V(B,C) voters to cast their ballots 
such that:  1) none of them casts a "partially reversed" ballot, 2) none of them 
casts a ballot that ties B with an alternative ranked below B in the voters true 
preferences, and 3) C cannot win no matter how the W(C,B) voters cast their 
ballots.  Further, "two evils" necessarily requires that, in this very same 
election, it must also be simultaneously possible for the V(C,A) voters to cast 
their ballots such that:  1) none of them casts a "partially reversed" ballot, 
2) none of them casts a ballot that ties C with an alternative ranked below C in 
the voters true preferences, and 3) A cannot win no matter how the W(A,C) voters 
cast their ballots.

Thus, for "two evils" to be important, it must be important to guarantee that 
the W(A,B) voters are necessarily able to ensure that B loses, and to 
simultaneously guarantee that the W(B,C) voters are necessarily able to ensure 
that C loses, and to simultaneously guarantee that the W(C,B) voters are 
necessarily able to ensure that A loses, all under the same false-ties and 
partial-reversal conditions in the same election (as stated above).

To me, this analysis raises three issues.  First, is it correct?  If not, then 
why not?  If it is correct, then the second issue is:  Given the analysis above, 
is it important to satisfy the "two evils" criterion in addition to satisfying 
the generalized Condorcet criterion?  Obviously, I don't think so.  In fact, it 
seems to me that the more one thinks about the implications of this criterion 
when a strict majority of the voters prefer A to B, a strict majority prefer B 
to C, and a strict majority prefer C to A, the more ludicrous this criterion 
appears to be.  The reasonably of any criterion is basically a matter of 
personal judgment -- my judgment, based on the above, is that "two evils" is an 
unreasonable addition to the generalized Condorcet criterion.

The third issue is:  If others feel that satisfying "two evils" is important, 
then what methods have been proven to satisfy "two evils," what methods have 
been proven to fail "two evils," and where can these proofs be found?  I know 
that many claims have been made, and it is not wrong to state conjectures 
provided these statements are clearly labeled as conjectures.  I have seen 
neither such proofs nor such labeling.

Bruce