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Budget Feasible Mechanisms for Procurement Auctions with Divisible Agents

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 $θ\ge 1$. For $θ\le 2$, 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.