Dear Forest,
This is indeed astonishing from a slightly different point of view: if I rate the winner well below the benchmark lottery, my action tends to cause other voters'
accounts to lower, but does not cause mine to increase, yet the total is constant. Magic!
Not so astonishing after all, since I am now sure that my claim
concerning the expected value was plain false. The expected value will
be zero. One can see this when writing down the precise sums. I now
believe also that this part cannot be fixed, so what remains is the
strategy-freeness and the conservation of "voting money", while the
third goal (compensating those who liked a Random Ballot lottery better)
probably cannot be achieved.
So, I will post a simpler version tomorrow night, and that we can
analyse further.
-
As to your median idea: I am doubtful whether that could work. Actually,
I think the *only* aggregation function for individual ratings for which
a similar "taxing" mechanism can make the method strategy-free, is the
sum of ratings. Assume the winner is the option X for which
f(R(X,1),...,R(X,N))
is maximal, where f is some (symmetric) aggregation function. When we
try to adjust voter 1's account by some amount determined only from the
other voters' ratings, and hope that the sum of true utility and
adjustment is larger with honest voting, then it seems we end up with a
condition like
R(X,1) + f(R(X,2),...,R(X,N)) > R(Y,1) + f(R(Y,2),...,R(Y,N))
whenever
f(R(X,1),...,R(X,N)) > f(R(Y,1),...,R(Y,N)).
And this most surely implies that f is just the sum plus perhaps some
constant value.
Yours, Jobst
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