auction designers

Abstract: Given an incentive-compatible mechanism for a single agent on a restricted domain of valuations, we consider its incentive-compatible extensions to the entire valuation space. We identify two extreme extensions: the "keep-the-menu" extension, which minimizes buyer payoffs and maximizes canonical prices pointwise, and the "keep-the-pricing" extension, which maximizes buyer payoffs and minimizes canonical prices pointwise.

Abstract: We initiate the study of $α$-fair prophet inequalities. This interpolates between utilitarian welfare $(α=0)$, Nash welfare $(α=1)$, and Rawlsian max-min fairness $(α\to\infty)$. Given the non-linearity of the objective, it matters when the expectation is applied. For instance, for the Rawlsian objective, it matters whether we aim to maximize $\min \mathbb{E}[u_i]$ or $\mathbb{E}[\min u_i]$. We refer to the former as the ex-ante model, and the latter as the ex-post model. For ex-ante fairness, full distributional knowledge yields a tight competitive ratio of exactly $1/2$ for every $α\ge 0$. Under sample access, $O(n\log n)$ samples per distribution suffice for a constant competitive ratio when $α\in(0,1]$. In contrast, for every $α>1$, no finite number of samples improves upon the trivial $1/n$ guarantee. Thus, unlike in the utilitarian setting, full-information and sample-access prophet inequalities become fundamentally separated. For ex-post fairness, under full information, we obtain a uniform constant ratio for all $α\in(0,1)$, while for every $α>1$ the competitive ratio collapses to $1/n$. In the sample-access model, one sample per distribution suffices for each fixed $α<1$, but no sample budget depending only on $n$ yields a uniform constant guarantee as $α\to 1$. Beyond these phase transitions for $α$-fairness, our results open the door to a broader theory of prophet inequalities for non-linear welfare objectives.

Abstract: We study prophet inequalities under intersections of $q$ partition matroids, where an online algorithm irrevocably selects elements with independent nonnegative values drawn from known distributions and revealed in an adversarial order. We prove an $Ω(q/\log q)$ lower bound on the competitive ratio. Together with the known $O(q)$ upper bounds, this resolves, up to a logarithmic factor, the optimal dependence on $q$, an open question posed by Correa, Cristi, Fielbaum, Pollner, and Weinberg (IPCO 2022) and Saxena, Velusamy, and Weinberg (ITCS 2023). Our construction also yields an $Ω(d/\log d)$ lower bound for $d$-single-minded auctions, where buyers request fixed bundles of at most $d$ unit-capacity items. Our construction and analysis build on the "big-decisions-first" framework of Rubinstein and Singla (STOC 2026).

Abstract: The round-robin procedure is a simple and well-studied fair division mechanism where agents pick goods in turns. Motivated by draft mechanisms in sports leagues, we investigate strategic behaviour in online round-robin for subadditive agents. This gives rise to an extensive-form game, and we study the computational problem of computing a subgame perfect Nash equilibrium (SPNE). We show that for just two submodular agents, computing an SPNE is $\mathsf{PSPACE}$-hard. Even for the class of $\mathit{OXS}$ utilities, which are a special case of submodular utilities, computing an SPNE remains $\mathsf{NP}$-hard for a small number of agents. We complement our computational results with normative results. We show that for just three additive agents, there exist instances where every equilibrium violates EF1. This separates the online and the direct revelation games. On the positive side, we show that for additive agents every equilibrium allocation is proportional up to one good (PROP1) and for two additive agents it is also EF1. Finally, by showing that round-robin is bossy at equilibrium, we prove that the number of equilibrium allocations can be exponential even if agents have lexicographic preferences.