带预算约束的广告主均衡报价研究——基于纳什均衡角度

从广告主角度出发,研究基于纳什均衡概念的带有预算约束的关键词拍卖问题。首先推导出纳什均衡下各个广告主报价的上界和下界。其次给出广告主报价向上偏离无利可图的充要条件,并进行了数值分析。此外,通过数值实例分析了广告主报价向下偏离时对其收益的影响。利用各个广告主报价的上下界和预算刻画了带有预算约束的纳什均衡。最后,利用广告主报价在纳什均衡下的上界给出报价向量是带预算约束的纳什均衡的充要条件。对于广告主制定预算策略具有参考作用。

A Study on Advertisers' Equilibrium Bids with Budget-constraints-Based on the Nash Equilibrium

Keyword advertising which has a simple form is increasingly popular, and advertiser can advertise on the search engine at any time. Its several advantages include better targeted, low cost, controllable budget, and a higher rate of return on investment. The budget allocation has a great effect on advertiser's revenue, so it is necessary to study the budget problem of keyword auctions.From the point of view of advertisers, keyword auctions are built with budget-constrained bidders based on the Nash equilibrium. The revenue of advertiser with budget can be expressed as π_j=c_jt_j(v_j-b_j+1), where t_j=(B_j)/((c_jb_j+1)) is the time of advertiser j to take part in the auction. The B_j is defined as the advertiser j's budget. When p_j ≤ B_j means the advertiser j has been always occupying the advertising position(t_j=1), and p_j > B_j means the t_j<1. Whether the bidding vector is the Nash equilibrium partly depends on the budget constraints.
And then, the effect of advertiser's budget on his revenue is discussed. Firstly, the upper and lower bounds of each advertiser's Nash equilibrium bid are derived. Secondly, a sufficient and necessary condition is given under which there is no profitable upward deviation for each advertiser, and the condition is verified by a numerical example. In addition, the effect of advertiser's bid downward deviation on his profit is analyzed. The Nash equilibrium with budget-constraints is characterized by using the upper and lower bounds of each advertiser's Nash equilibrium bid and budget. Finally, the upper bound of each advertiser's Nash equilibrium bid is used to give a sufficient and necessary condition under which any bid vector b∈B^NE is a Nash equilibrium with budget constraints.
The results show that any b∈B^NE is sustainable as the Nash equilibrium with budget constraints if and only if B_j ≥ B_j^** for all j∈N. That is to say, if all bidders have enough budgets, then no deviations are profitable. However, the budgets need not to be extremely large as long as it is larger than B_j^**, which has an important reference for advertisers to make budget strategy.
This paper's results are useful for the advertiser to develop the bidding strategy and set up a budget plan. First, the more budget does not mean the better. The best budget strategy is that the bidding vector reach the Nash equilibrium with budget constraints. Second, advertiser who sets up a bidding strategy cautiously should consider not only the financial situation of enterprise, but also the time of him to take part in the auction. Because the auction time directly affect the advertiser's revenue. At last, the revenue of search engine is also indirectly affected by the advertiser's budget. And it may increase the overall social welfare. Thus, the advertiser need to set up a reasonable budget for the keyword auctions.