[EM] 0-info approval voting, repeated polling, and adjusting priors

[Simmons, Forest]

You're right Jobst.  So in general, instead of directly adjusting the cutoff value as a weighted average of previous cutoff values, one should take a weighted average of the lotteries, and then (on the basis of this averaged lottery) calculate the cutoff value using the expected value in the case of cardinal ballots and Joe Weinstein's weighted median in the case of ordinal ballots.
 
By the way, your method of adjusting only the relative likelihood of the current winner to the other candidates is growing on me.  Here's a specific version adapted to ordinal ballots:
 
1.  Calculate the random ballot lottery L_0 .
 
2.  Use Joe Weinstein's weighted median (approve X iff L_0 gives more probability to lower ranked than higher ranked candidates) strategy on each ballot to determine the approval winner A_0 .
 
3.  Let L'_0 be the lottery that gives A_0 one hundred percent of the probability, and let L_1 be the average of L_0 and L'_0.
 
4.  Use weighted median based on L_1 to determine A_1.
 
5. etc.
 
6.  In finitely many steps the sequence  A_0, A_1, ... will converge to (i.e. get stuck on) the winning candidate A.
 
Proof of the assertion in step 6 is the same as Jobst's proof in the case of Cardinal ballots, because in Joe Weinstein's strategy the difference in total probability above X and below X has the same sign before and after the averaging step, so A_k's approval doesn't change from stage k to stage (k+1).
 
[Therefore the current approval champ beats the previous one only by setting a new approval record.  This cannot happen more than a finite number of times when there are a finite number of voters.]
 
Forest

________________________________

From: Jobst Heitzig [mailto:heitzig-j at web.de]
Sent: Wed 8/3/2005 11:03 PM
To: Simmons, Forest 
Cc: election-methods-electorama.com at electorama.com
Subject: Re: [EM] 0-info approval voting, repeated polling, and adjusting priors



Dear Forest!

You wrote:
> At each successive stage we would base the new lottery calculation on a
> weighted average of all the old cutoffs.  In other words, the cutoff on
> each ballot is adjusted slightly towards the most recent lottery
> expected value before calculating the new lottery probabilities.

One must know the individual cardinal utility functions for this, it
seems. An alternative would be to switch from expected utility 0-info
strategy to median utility 0-info strategy (aka Weinstein's strategy) so
that only rankings would have to be known.

Jobst



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