Theoretical and statistic analysis on necessary bidder numbers for combinatorial auctions

The reverse combinatorial auctions are widely applied to the centralized procurements and project tenders of governments and enterprise groups. To guarantee the competition of reverse auctions an enough number of bidders is necessary. Based on the computation method of winner combination numbers in literatures, we deduce the formulas to calculate the average feasible solution numbers in single time of bidding. Two theorems on the feasible solution numbers were proposed and proved. They show us the feasible solution numbers increase with the bidder number raise up, and decrease with object number grows up. The same problems of cases with bidding quota are discussed also. By means of proposed computing approach, the average feasible solution numbers of problems with different sizes are obtained. The results present the necessary bidder numbers to guarantee competition. To verify the correctness of proposed formulas, hypothesis testing method is used to a large number of simulated examples. The results support above conclusions. Above research works provide a theoretic foundation for organization and mechanism design of on-line reverse combinatorial auctions in centralized procurements.