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

[Simmons, Forest]

Jobst wrote ...


Dear Forest,

I'm not sure what you mean by the red marble thing or how it clarifies
the meaning of the priors in zero-info strategy.



I reply:

I'm not sure either. I'm stabbing around in the dark looking for some idea to make the probabilities more definitely meaningful, and as a child I enjoyed playing with marbles ;-)

I think that if we want your convergence proof to work, we have to assume that the voters only know who won each round, and not the approval amounts.  At any rate, if we cannot solve the problem with this simplifying assumption, then there is no hope in the general case.

Here's a method where the probabilities have more definite meaning:

A whole number  N > 1 is announced before the process begins, and an empty bag is made available.

The number N is the number of approval polls to be taken before deciding the winner.

After each of the polls, a marble with the name of the candidate with the most approval in that poll is placed into the bag. [I had to get marbles into this somehow.] Approval scores are kept secret. If two candidates tie in approval, the tie is broken by a coin toss before announcing the poll winner.  The occurence of a tie is not mentioned.

After all N of the approval polls are completed, a marble is drawn at random from the bag to determine the grand winner.

Before we treat the case N=100, how about the case N=2 ?

Of course you would use your prior probabilities for the first poll.  And once you knew the winner, A , of the first poll, you would adjust (upward) the probability of candidate A being the winner of the next (and final) poll..

It seems like Bayes should tell us how much to adjust this probability.

And how about the prior probabilities of ties?  Should these be adjusted even though we have no information about ties or relative approval scores from the first poll?

On the one hand we might want to adjust (downward) candidate j's chances of being in a tie.  On the other hand we don't know for sure that the first round wasn't decided by a coin toss.

Can we solve even this simple case?  Throw in independence if it will help.

Forest

 

 

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