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Aimed at the disadvantage of current research of shill bidding with different online auctions types, in this paper, applying decision analysis method and game theory, firstly, we research the bidderpsilas optimal biding strategy on english auction and first-price sealed-bid auction with shill biding. secondly, we discuss the relations between the sellerpsilas revenue and probability of shill biding and commission rate under two important auctions types. finally, we use honest bidderpsilas expected gain and honest sellerpsilas revenue as a basis to compare these two important auctions types, and interpret some bidding behavior in practice and give some suggestion.

Algorithms in cryptosystem such as RSA and Diffie-Hellman require the large integer multiplication. This paper introduces classical Knuth multiplication, Karatsuba multiplication and their time complexity, on the basis of which a new Karatsuba trick is presented and proved to be available in theory and in practice. The experiment result reveals that the improved Karatsuba multiplication is more efficient for implementation of large integer multiplication.

In this paper, we first analyze the characteristics of online auction and propose a general analysis framework about the design of online auction. Based on the properties of online auction and a general analysis framework related to the design of online auction, then we discuss the multi-unit dynamic programming models of sequential online auction in order to maximize the seller' expected profits whether the reserve price is private or public. It is shown that the seller with private reserve price can gain the bigger profits than the seller with public reserve price. Finally, numerical results for two cases are given, where we compute the maximal expected total revenue. We show that the numerical results are consistent with the theoretical analysis.

In this paper, we first analyze the characteristics of online auction and propose a general analysis framework about the design of online auction. We believe that the auctions rules, the random arrival process of bidders, the auctions time, the lot size and the auctions costs need to be considered on Internet auctions. Based on the properties of online auction and a general analysis framework related to the design of online auction, then we present single-unit and multi-unit optimal auction models in order to maximize the seller' expected profits whether the reserve price is public or private respectively. In single-unit auction, when the seller with public or private reserve price gain the maximal expected profit respectively, the optimal public reserve price is higher than the optimal private reserve price. In multi-unit auction, it is shown that the seller with private reserve price can gain the bigger profits than the seller with public reserve price. Finally, numerical results for two cases are given, where we compute the maximal expected total revenue. We show that the numerical results are consistent with the theoretical analysis.