[EM] Random Ballot Condorcet (Ross Hyman)

Forest Simmons fsimmons at pcc.edu
Thu May 8 16:48:10 PDT 2014

```Ross,

good thinking, but yes this is one of the many random ballot methods that
Jobst considered back before he specialized in proportional probability
lotteries.

Basically, every deterministic method that works off of an approval order
list can be turned into a random ballot method by using random ballots to
develop the list order instead of approval.  If the first drawn ballot is
incomplete, keep drawing additional ballots to resolve ties until a
complete order of the candidates is created.

The original "Condorcet Lottery" was based on game theoretic probabilities
from the mixed strategies that optimized the moves of the two players.  As
in rock paper scissors each player picks a candidate (with full knowledge
of the pairwise win graph).  The player whose candidate loses pairwise pays
the other player a dollar.

Since the game is symmetrical both players have the same optimal strategy.
The probabilities in this optimal strategy form the lottery in question.

Unfortunately the method is not monotonic.  Increasing the strength of a
player can sometimes make it better to choose him less often.  However,
this will not change a positive probability to zero.

Anyway, the candidates with positive probability are from the "Dutta Set,"
which is a subset of the Banks Set, which is a subset of the Landau Set,
which is a subset of the Smith set.

We have monotonic methods (both random and deterministic) that choose from
Landau and even Banks (e.g. MEA and TACC, resp. and their random ballot
versions), but so far no Dutta.

You obviously have talent for this stuff.  Keep up the good work!

Forest

Date: Wed, 7 May 2014 16:51:21 -0700 (PDT)
> From: Ross Hyman <rahyman at sbcglobal.net>
> To: "election-methods at electorama.com"
>         <election-methods at electorama.com>
> Subject: [EM] Random Ballot Condorcet
> Message-ID:
>         <1399506681.89368.YahooMailNeo at web181605.mail.ne1.yahoo.com>
> Content-Type: text/plain; charset="iso-8859-1"
>
> Random Ballot Condorcet: ?Choose a random ballot. ?Elect the lowest ranked
> candidate that pairwise beats all higher ranked candidates.
>
> Has this method been discussed before? ?I believe that the following are
> true: ?It will always elect a Condorcet candidate if there is one.
> ?Otherwise it will elect a member of the Smith set with some nonzero
> probability for each member of the Smith set. ?Non-Smith set candidates
> will have zero probability of being elected. ?It is monotonic in that
> raising a candidate on some ballots cannot decrease its probability of
> being elected. ?It is clone proof in that the probability of electing from
> the clone set is independent of the number of clones in the set. It is
> independent of irrelevant alternatives in that deleting a candidate with
> zero probability of winning cannot effect the probabilities for electing
> other candidates. ?
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