502 CHAPTER 17 Auction-based spectrum markets in cognitive radio networks assigned to the same channel to a group. Unlike VERITAS, the group formation is independent of the buyer bids to prevent bidders from rigging their bids to manipulate the group size and its members. The group formation can cope with various interference models by using differ- ent spectrum allocation algorithms. If the buyer interference condition is modeled by a conflict graph, the group formation is equivalent to finding the independent sets of the conflict graph [639,640]. If the interference condition is modeled by the physical signal-to-interference and noise ratio (SINR) [641], TRUST finds multiple sets of buyers who can transmit simultaneously and maintain an adequate received SINR [642]. TRUST performs this allocation only to form buyer groups, not to assign specific channels to buyers. 17.5.2 Determining Winners Next, TRUST treats each buyer group as a superbuyer and runs the conventional double spectrum auction algorithm to determine the winning sellers and super- buyers. Let G 1 , G 2 , ..., G L represent the L groups formed. For any group G l with n l = |G l | buyers, the group bid l is l = min B b |n G l · n l . n (17.4) TRUST sorts the seller bids in nondecreasing order and the buyer group bids in nonincreasing order: B : B s B s ... B s , and B : 1 2 ... L . Define M 1 2 k as the last profitable trade: k = argmax lmin{L,M} l B s . l (17.5) Then the auction winners are the first (k - 1) sellers and the first (k - 1) buyer groups. 17.5.3 Pricing To ensure truthfulness, TRUST pays each winning seller m by the kth seller's bid B s and charges each winning buyer group l by the kth buyer group's bid k . This k group price is evenly shared among the buyers in the group l: b P n = k /n l , for all n G l . (17.6) No charges or payments are made to losing buyers and sellers. The uniform pricing is fair because buyers in a winning group obtain the same channel and thus are charged equally. In addition, to ensure individual rationality, a group bid must not exceed the product of the lowest buyer bid in the group and the number of buyers in the group, which is used in the process of determining winning groups. With such pricing mechanism, the auctioneer's profit becomes = (k - 1) · ( k - B s ) k and it is easy to show that 0.