On Revenue Optimal Combinatorial Auctions - IFP Group at the ...
On Revenue Optimal Combinatorial Auctions - IFP Group at the ...
On Revenue Optimal Combinatorial Auctions - IFP Group at the ...
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Simplifying <strong>the</strong> Constraints<br />
A payment rule s<strong>at</strong>isfying <strong>the</strong> constraints exists if and only if<br />
q(x 1 ) ≤ q(x 2 ) ≤ . . . ≤ q(x K ).<br />
The maximum m(.) for a given q(.) is given by:<br />
m(x i ) =<br />
i∑<br />
k=1<br />
[<br />
]<br />
q(x k ) − q(x k−1 ) x k .<br />
For <strong>the</strong> above m(.), E [m(X)] = E [q(X)w(X)].<br />
w(.) is <strong>the</strong> virtual-valu<strong>at</strong>ion function of <strong>the</strong> buyer.<br />
It suffices to find only an optimal alloc<strong>at</strong>ion rule.