We consider budget feasible mechanisms for procurement auctions with additive valuation functions. For the divisible case, where agents can be allocated fractionally, there exists an optimal mechanism with approximation guarantee $e/(e-1)$ under the small bidder assumption. We study the divisible case without the small bidder assumption, but assume that the true costs of the agents are bounded by the budget. This setting lends itself to modeling economic situations in which the goods represent time and the agents' true costs are not necessarily small compared to the budget. Non-trivially, we give a mechanism with an approximation guarantee of 2.62, improving the result of 3 for the indivisible case. Additionally, we give a lower bound on the approximation guarantee of 1.25. We then study the problem in more competitive markets and assume that the agents' value over cost efficiencies are bounded by some $\theta \ge 1$. For $\theta \le 2$, we give a mechanism with an approximation guarantee of 2 and a lower bound of 1.18. Both results can be extended to settings with different agent types with a linear capped valuation function for each type. Finally, if each agent type has a concave valuation, we give a mechanism for which the approximation guarantee grows linearly with the number of agent types.
翻译:我们考虑预算上可行的采购拍卖机制,其估值功能是增加的。对于可以分批分配代理商的分散性案例,我们有一个最佳机制,根据小投标人的假设,提供近似保证$e/(e-1)$(e-美元),在小投标人的假设下,我们研究可分割的情况,但没有小投标人的假设,但假设代理商的真正成本受预算的约束。这种设定有利于模拟货物代表时间和代理商真实成本与预算相比不一定很小的经济情况。非三重性,我们提供近似保证2.62的机制,改善3个不可分割案件的结果。此外,我们对1.25的近似保证有较低的约束。我们接着在更具竞争性的市场上研究这一问题,假设代理商的成本效率价值受一定美元/theta\ge 1美元的约束。对于2美元,我们提供了一种机制,其近似保证为2美元,约束度为1.18。两种结果都可以扩大到具有不同类型代理商类型且具有直线上限估价功能的设置。最后,如果每个代理商类型都拥有一种直线性估价机制,那么我们就会增加一种直线性估价。