[EM] Re: twisted prism, etc.

[Forest Simmons]

From: "Jobst Heitzig" <heitzig-j at web.de>
Subject: [EM] Sprucing up vs. Condorcet Lottery vs. immunity: The
 	"twisted	prism" example

Jobst 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 in 
the 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