<div> Jobst Heitzig wrote:<br>
> Now, I have the impression that a slight modification of the tax
formula may reduce this incentive considerably. Consider this tax:<br>
><br>
> sum { R(W,k) - R(W(i),k)<br>
> + sum { ( R(W(i),k) - R(W(i,j),k) ) / 2<br>
> : j different from i and k }<br>
> : k different from i }<br>
><br>
> where W(i,j) is the winner after removal of both i and j. If I'm right, this formula makes it ineffective to misrepresent ratings for both individual voters and pairs of voters.<br>
><br>
> Please check this!<br>
<br>
This is a 2-d sum, so people could pay way more than they thought.<br>
<br>
The outer loop just calculates the Clarke tax<font face="Arial, Helvetica, sans-serif"> and the inner loop is<br>
an adder</font>.<br>
<br>
Anyway, in simple terms, the system is (I think)<br>
<br>
- count up the totals excluding your ballot<br>
- If the result is different from the actual result<br>
pay to change it to the actual winner<br>
(i.e. the amount the actual winner lost by)<br>
<br>
Your system adds<br>
- If the result is <font face="Arial, Helvetica, sans-serif">further</font> changed by removal of another ballot<br>
in addition to yours, pay half the cost to switch it to<br>
the winner that would result with just the removal of your ballot<br>
<br>
The 2nd cost gets added for every other potential pair that your ballot<br>
can match with. In a large election the tax could be massive.<br>
<br>
For example, if lots of people cast<br>
<br>
A) 10<br>
B) 0<br>
<br>
and A defeated B by 15,<br>
<br>
then the payment would be huge.<br>
<br>
The removal of any one voter's ballot would have no effect, A would still<br>
win by 5. However, the removal of any 2 would cause B to win.<br>
<br>
If there were 1000 voters of that type, then each of them would have to pay<br>
999 times 2.5 in the tax (A loses by 5 999 times and they pay half each<br>
time).<br>
<font face="Arial, Helvetica, sans-serif"><br>
My preference for a solution here is to limit the size that people may bid.<br>
For example, the highest 10% of ratings are assumed to be equal to<br>
the lowest of them.<br>
<br>
I.e. where (Rmax-Rmin) on a ballot is greater than the 90th percentile,<br>
rescale the ballot by <br>
(90th percentile)/(Rmax-Rmin)<br>
</font></div>
<div> <font face="Arial, Helvetica, sans-serif"><br>
This means that bidding a really large number will just be ignored and your<br>
ballot rescaled.<br>
<br>
This does break the optimal utility effect though.<br>
<br>
</font></div>
<div id="sig205" style="clear: both;"><font>Raphfrk<font face="Arial, Helvetica, sans-serif"><br>
</font></font></div>
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