Beyond Worst-Case Budget-Feasible Mechanism Design

Motivated by large-market applications such as crowdsourcing, we revisit the problem of budget-feasible mechanism design under a "small-bidder assumption". Anari, Goel, and Nikzad (2018) gave a mechanism that has optimal competitive ratio $1-1/e$ on worst-case instances. However, we observe that on many realistic instances, their mechanism is significantly outperformed by a simpler open clock auction by Ensthaler and Giebe (2014), although the open clock auction only achieves competitive ratio $1/2$ in the worst case. Is there a mechanism that gets the best of both worlds, i.e., a mechanism that is worst-case optimal and performs favorably on realistic instances? Our first main result is the design and the analysis of a natural mechanism that gives an affirmative answer to our question above: (i) We prove that on every instance, our mechanism performs at least as good as all uniform mechanisms, including Anari, Goel, and Nikzad's and Ensthaler and Giebe's mechanisms. (ii) Moreover, we empirically evaluate our mechanism on various realistic instances and observe that it beats the worst-case $1-1/e$ competitive ratio by a large margin and compares favorably to both mechanisms mentioned above. Our second main result is more interesting in theory: We show that in the semi-adversarial model of budget-smoothed analysis, where the adversary designs a single worst-case market for a distribution of budgets, our mechanism is optimal among all (including non-uniform) mechanisms; furthermore our mechanism guarantees a strictly better-than-$(1-1/e)$ expected competitive ratio for any non-trivial budget distribution regardless of the market. We complement the positive result with a characterization of the worst-case markets for any given budget distribution and prove a fairly robust hardness result that holds against any budget distribution and any mechanism.