From: "Jobst Heitzig" <[EMAIL PROTECTED]>
Subject: [EM] Sprucing up vs. Condorcet Lottery vs. immunity: The
"twisted prism" exampleJobst wrote:
Dear Forest!
Your sprucing up technique is a very nice idea since it can simplify the tallying of those methods which fulfil beat-clone-proofness and uncoveredness. However, some of which you wrote has confused me completely: Did I understand you right in that you claim that the technique should reduce everything to the 3-candidate case? I wonder how
I have consistently been careful to say that Sprucing Up reduces everthing to the three candidate cycle case in "public elections." I defined public elections to be ballot sets that have no ties and contain no (beat) subgraphs with symmetry as great as the five cycle A0 through A4, in which candidate (i mod 5) beats both candidates (i+1 and i+2 mod 5).
The twisted prism example obviously fails my "public election" definition, since that much symmetry among six candidates is just as unlikely as the five candidate cycle in public elections.
that could possibly be when, in general, there is neither a proper beat clone set nor a covered element! Second, I don't understand how the method I posted yesterday under the name "Condorcet Lottery" could be the same as anything based on Random Ballot???
I thought I said that Lottery was the same as Spruced Up Random Candidate (not Ballot). I did conjecture that Random Ballot Dutta might be better than Lottery.
In order to study these questions I provide the following example which is also interesting with respect to immunity:
The "twisted prism" example:
<snip>
4. No proper beat clone set, hence no possibility to spruce up anything! In particular, spruced up Random Ballot equals Random Ballot here and elects Ai with prob. 4/15 and Bi with prob. 1/15.
The clone collapsing step is never needed in Spruced Up Random Ballot since Random Ballot is already clone independent.
In my three step definition of Spruced Up Random Candidate (in the message to which you are responding) step one was to "cross out all candidates except the members of the Dutta set." Then I mentioned parenthetically that neither of the other two steps was needed for Spruced Up Random Ballot, i.e. Random Ballot Dutta.
5. Banks Set = Tournament Equilibrium Set = Dutta's Minimal Covering Set
= Bipartisan Set = {A1,A2,A3}. Hence all methods which guarantee the
winner to be in one of these sets elect Ai with prob. 1/3, including
ROACC and Condorcet Lottery.
Early in my message I asked if there was any difference between Dutta, Banks, and Iterated Uncovered Set in public elections.
From that point on I adopted Dutta in place of Iterated Uncovered Set inthe Spruce Up Process, just in case there is a difference. In all public election examples that I have seen the Sprucing Up procedure reduces to the same cycle of three, no matter whether Banks, Dutta, or Iterated Uncovered is used in step one.
So to be clear I will repeat:
(1) [Conjecture] Lottery is just Spruced Up Random Candidate. For definition of Spruced Up Random Candidate, see my first response to Jobst's first posting on the Condorcet Lottery method. In particular, Dutta (as opposed to iterated uncovered) is used in step 1. I claim it makes no difference in public elections.
(2) I think that Random Ballot Dutta is apt to be just as resistant to manipulations as Condorcet Lottery, but with higher expected utility.
Finally, a question:
What if we use CR ballots, and say that lottery p beats lottery q iff
p gives higher expectation (i.e. approval cutoff) than q on more ballots than not?
Forest ---- Election-methods mailing list - see http://electorama.com/em for list info
