[EM] Democratic Lottery Enhancement

Jobst Heitzig heitzig-j at web.de
Thu Jun 8 16:15:39 PDT 2006

Dear Forest!

That's an interesting game.

Do you agree that the "natural" (and perhaps optimal) behaviour for
voter i would be the following:

Assume that

n is the number of voters,
p(x) is the proportion of original papers naming x,
p(i,x) is the proportion of papers naming x returned by i,
P(i,x,y) means that i prefers x to y.

Then i will return as many papers as possible naming her favourite
candidate, then as many additional papers naming her 2nd and so on until
  the number of papers is reached. Since the maximal possible proportion
of papers naming x i can return is

q(x) := min(2p(x),1),

this means that

p(i,x) = q(x)                 if  q(x) <= 1-sum(q(y):P(i,y,x)),
p(i,x) = 1-sum(q(y):P(i,y,x)) if  q(x) >= 1-sum(q(y):P(i,y,x)) >= 0,
p(i,x) = 0                    otherwise.

Or, equivalently:

p(i,x) = min( q(x), max( 0, 1-sum(q(y):P(i,y,x)) ) ).

Now who can prove that this behaviour will always lead to a converging
sequence of enhancements? And that the limit will be independent from
the initial lottery (as long as there is at least 1 paper for each
candidate)? And that this limit is not simply
p(x) = proportion of voters having x as their favourite?


Simmons, Forest wrote:

> For the purposes of this message, a lottery is a stack of papers of
> standard copier size (8.5 inch by 11 inch) such that each paper has
> the name of exactly one candidate typed on it.
> This stack of papers is a lottery in the sense that a sheet of paper
> can be drawn at random from the stack to determine the election
> winner.
> Democratic Lottery Enhancement (DLE).
> 1.  Each voter is issued two copies of the initial lottery, i.e. the
> lottery that is considered to be in need of democratic enhancement.
> 2.  Each voter selects exactly half of the papers for discard, and
> then returns the other half of the papers.
> 3.  The returned papers are collated or merged to form a new,
> democraticaly enhanced lottery.
> Forest
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