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s.t. using javabean torender quick response code on web,windows application Bar code to 2D Code pi = Uv W . (6.5). A multi-winner cognitive spectrum auction game Proposition 6.3.1 User i has to pay the price pi = max {vi , 0} , for i W, (6.

6). where is chosen s Denso QR Bar Code for Java uch that i W pi = Uv . In particular, if pi = vi W Uv Uv W /. W . 0 for any i, pi = pi applet QR Code JIS X 0510 will be the solution. P ROOF. Letting qi = vi pi for i W and using the fact that i W vi = Uv , the optimization problem (6.

5) is equivalent to the following convex optimization problem:. {qi [0,vi ], i W }. ln qi ,. s.t. qi = Uv Uv W = (6.7). After introducing the L QR Code JIS X 0510 for Java agrange multipliers and i 0, i W , the Lagrangian function is L(q, , ) = . ln qi + qi i (qi vi ),. (6.8). from which Karush Kuhn Tucker (KKT) conditions [41] can be obtained as follows: qi = 1 , + i i 0, i (qi vi ) = 0,. qi = (6.9). De ne = 1/ . For thos QR-Code for Java e i such that qi = vi , qi = 1/( + i ) 1/ = ; for other i such that qi < vi , the third condition implies i = 0, and thus qi = 1/( + i ) = . Therefore, the solution is qi = min(vi , ), (6.

10). with chosen such that the last condition in (6.9) is satis ed. Finally, pi = vi qi yields (6.

6). In particular, if U /. W . mini (vi ), = U /. W . and pi = vi will b e the solution. It can be seen that the payment is split in such a way that the pro ts are shared among the winners as equally as possible. Differently from the VCG pricing strategy, which sometimes may yield low revenue or even zero revenue, such a pricing strategy always guarantees that the seller receives revenue as great as Uv W .

Moreover, if some losers collude to beat the winners by raising their bids, they will have to pay more than Uv W ; however, the payment already exceeds what the band is actually worth to them, and as a result, loser collusion is completely eliminated. Nevertheless, users can still bene t from the sublease collusion, and hence we call the pricing strategy in (6.5) the partially collusion-resistant pricing strategy.

In order to nd a fully collusion-resistant pricing strategy, we have to analyze how sublease collusion takes place, and add more constraints accordingly. It happens when a subset of the winners WC W subleases the band to a subset of the losers L C L, where L = {1, 2, . .

. , N } \ W denotes the set of all losers. The necessary condition for the sublease collusion is i WC pi < i L C vi , so that they can nd a sublease.

6.3 One-band multi-winner auctions price in between that i s acceptable to both parties. Given any colluding-winner subset WC W , the potential users who may be interested in subleasing the band should have no mutual interference with the remaining winners W \ WC ; otherwise, the band turns out to be unusable. Denote the set of all such potential users by L(W \ WC ), i.

e., L(W \ WC ) = {i L. Ci j = 0, j W \ WC QR Code JIS X 0510 for Java }. Therefore, as long as prices are set such that i WC pi max L C L(W \WC ) i L C vi , there will be no sublease collusion. Note that max L C L(W \WC ) i L C vi is the maximum system utility Uv L(W \W ) which can be C obtained by solving the optimal-allocation problem within the user set L(W \ WC ), thus the optimum collusion-resistant pricing strategy is the solution to the following problem:.

{ pi [0,vi ], i W }. (vi pi ),.
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