Partial allocations in budget-feasible mechanism design: Bridging multiple levels of service and divisible agents

Budget-feasible procurement has been a major paradigm in mechanism design since its introduction by Singer [28]. An auctioneer (buyer) with a strict budget constraint is interested in buying goods or services from a group of strategic agents (sellers). In many scenarios, it makes sense to allow the auctioneer to only partially buy what an agent offers, e.g., an agent might have multiple copies of an item to sell, might offer multiple levels of a service, or may be available to perform a task for any fraction of a specified time interval. Nevertheless, the focus of the related literature has been on settings in which each agent's services are either fully acquired or not at all. A reason for this is that in settings with partial allocations, such as the ones mentioned, there are strong inapproximability results (see, e.g., Anari et al. [5], Chan and Chen [10]). Under the mild assumption of being able to afford each agent entirely, we are able to circumvent such results. We design a polynomial-time, deterministic, truthful, budget-feasible, (2+√3)-approximation mechanism for the setting in which each agent offers multiple levels of service and the auctioneer has a valuation function that is separable concave, i.e., it is the sum of concave functions. We then use this result to design a deterministic, truthful, and budget-feasible O(1)-approximation mechanism for the setting in which any fraction of a service can be acquired, again for separable concave objectives. For the special case in which the objective is the sum of linear valuation functions, we improve the best known approximation ratio for the problem from (by Klumper and Schäfer [19]) to (3+√5)/2. This establishes a separation between this setting and its indivisible counterpart.

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