Approximation Algorithm and Incentive Ratio of the Selling with Preference

Pan Li, Qiang Hua, Zhijun Hu, Hing-Fung Ting, Yong Zhang

Abstract: We consider the market mechanism to sell two types of products, A and B, to a set of buyers $$I=\{1, 2, ..., n\}$$. The amounts of products are $$m_A$$ and $$m_B$$ respectively. Each buyer i has his information including the budget, the preference and the utility function. On collecting the information from all buyers, the market maker determines the price of each product and allocates some amount of product to each buyer. The objective of the market maker is design a mechanism to achieve the semi market equilibrium. In this paper, we show that maximizing the total utility of the buyers in satisfying the semi market equilibrium is NP-hard and give a 1.5-approximation algorithm for this optimization problem. Moreover, in the market, a buyer may get more utility by misreporting his information. We consider the situation that a buyer may misreport his preference and prove that the incentive ratio, the percentage of the improvement by misreporting the information, is upper bounded by 1.618.