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By James A. Momoh

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Si? si? si? ; s i / ? si? ; s i // D˛ ? s ; s / c i i ? 37) Let Zi;c D ? si? si? si? 39) c2C? 3, we get that si;c 2 f0; 1g. We now consider two cases: ? • si;c D 0: In this case, si;c sOi;c D si;c D si;c D 0: Therefore, Zi;c 0. 2 Optimal Allocation of Adjustable Channels ? • si;c D 1: In this case, 0 Ä si;c – If si;c – If si;c 31 sOi;c Ä 1. We further distinguish two cases: sOi;c D 0, clearly, Zi;c 0. sOi;c D 1, then si;c 1. si? si? si? ; s i // ? si ; s i / ? si? si ; s i / s is a DSE on channel set C?

22 4 Game-Theoretic Channel Allocation in Clique Fig. 5 shows an example of channel allocation, with 3 players N D fP1 ; P2 ; P3 g and 3 original channels C D fc1 ; c2 ; c3 g. Each player has 2 radio pairs. Channel 2 and channel 3 are combined into a wider channel, resulting in a new channel set C0 D fc10 ; c20 g, where c10 D fc1 g and c20 D fc1 ; c2 g. In Fig. 5, player 1 places its 2 radio pairs on channel c10 and the combined channel c20 , respectively; player 2 only use 1 radio pair and tunes it on the combined channel c20 ; player 3 tune both of its radio pairs to c10 .

A carefully designed winner selection and charging scheme can guarantee the strategy-proofness of the auction. In this section, we analyze possible cheating actions of the buyers, and then strengthen our winner selection and charging scheme step by step to achieve strategy-proofness. Our analysis shows that there are three cheating actions, say, preemptive bidding, depreciated bidding, and retreat for advancing, through which a buyer may improve her utility. We provide a method to prevent each of the cheating actions, respectively.

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