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On Revenue Optimal Combinatorial Auctions - IFP Group at the ...

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Worst Case ELR Comput<strong>at</strong>ion - Example<br />

Buyer 1<br />

1‐p p 0 < L < H<br />

r = H/L<br />

L H<br />

If pr < 1, <strong>the</strong>n w(L) > 0. The optimal alloc<strong>at</strong>ions is efficient.<br />

If pr ≥ 1, <strong>the</strong>n w(L) ≤ 0. The buyer wins only if he bids H.<br />

For pr ≥ 1, <strong>the</strong> RSW = pH, while <strong>the</strong> MSW = pH + (1 − p)L.<br />

ELR = (1 − p)/(pr + 1 − p).<br />

ELR is maximized by setting p = 1/r.<br />

Hence, η = (r − 1)/(2r − 1).

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