Saari's Example
Mike Ositoff
ntk at netcom.com
Sat Jul 11 21:44:19 PDT 1998
I'd like to comment on Saari's example where I gave 100 to all of
a set of candidates, and Saari rated them from 70 to 100.
First of all, I voted as I did because I considered the distinction
between the members of that "preferred set" to be less important than
the distinctions between them & the candidates outside that set.
As I said, the importance of voting between a pair is measured by
their tie-probabililty times their utility difference.
Maybe I considered the utility differences in the preferred set
to be relatively unimportant, reducing that importance product
for pairs among them. In that case why not let Saari choose between
them while I make sure they beat the really worse ones.
Or maybe I mistakenly believed that the members of that set were
unlikely to tie with eachother, in which case I was mistaken in
the event that at least 2 of them would have tied had not Saari
chosen between them. But I stand by my vote, because even if I
counted significant utility difference between them, I believed them
unlikely to tie. I must have believed there'd be more voters, so
I wouldn't be able to influence that pair, believing that the others
wouldn't be evenly divided about them.
But with only 2 voters, I could have influenced _every_ pair
battle, and so maybe here P(i,j) wouldn't mean much, and merit
alone would matter. Had I known that, maybe I should have voted
as one does when P(i,j) values are all completely unknown, and
given 100 to all of the candidates above the mean, and -100 to
all the candidates below the mean. In fact, maybe that's what I
did.
As for the merit of Saari's partial vote strategy, if you agree
with my demonstration, in a previous letter, then you agree that
the partial votes can't be utility-optimal.
***
By the way, let me justify the assumption, which I got from
various authors, that the probability of 2 candidate's vote totals
not differing by N is proportional to N. Say we had a graph of
that probability, graphed vs the difference in the vote totals.
It's a Presidential election with 100,000,000 voters, and so the
"run" of the graph is 100,000,000 and that's what the width of
the graph paper represents. Now, consider how much one person's vote
could change that independent variable, the vote-total-difference.
Now, from zero to that microscopic displacement, how much will the
graph's slope change? Very little. For all practical purposes, within
that range, the probability varies linearly with the difference.
Someone else could demonstrate that more rigorously.
***
But, to settle that particular argument, do we even need the
assumption? If, for candidates A & B, the product of their
tie-probability & their utility difference is greater than
the product for B & C, then you'll gain more by voting a difference
between A & B than for B & C. Say your ranking is A,B,C. Do you
give B something halfway inbetween? No; if A & B have a higher
product, then vote difference given to them is more valuable than
vote difference given to B & C. You give full vote difference to
A & B, though that means giving none to B & C.
So that paragraph alone, without determining the strategies or
assuming anything, seems all that's needed to show that
the utility-maximizing voter gives all or nothing.
***
About the pay-for-vote idea, for measuring preference-intensity,
I probably should have said that you divide your vote difference
between the top 2 by the difference in their vote totals, to determine
your share of the responsibility for the victory. Then multiply
that fraction by your vote difference between those 2, since that's
also assumed to be your utility difference. That gives the
amount by which you improved your utility in the outcome, and
that's what you pay for.
But if you want a reliable description, refer to the book I mentioned.
Mike Ossipoff
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