Hallucinating Flows for Optimal Mechanisms

TL;DR

This work advances multi-item revenue-maximization by introducing a duality-based flow framework that induces hierarchy allocations and virtual values, yielding closed-form optimal mechanisms in several nontrivial discrete settings. The central theorem shows that any flow-induced mechanism that is Bayesian Incentive Compatible and Individually Rational is revenue-optimal, turning mechanism design into constructing suitable dual flows. The authors present three axis-aligned extensions of Yao's bi-valued, multi-agent setting, plus a grand-bundling result for a single agent across general discrete distributions and an approximation guarantee for continuous product distributions. The results deliver interpretable, explicit mechanisms and provide structural insights into how optimal revenue extraction can be achieved, with implications for benchmarks and potential data-driven mechanism design.

Abstract

Myerson's seminal characterization of the revenue-optimal auction for a single item \cite{myerson1981optimal} remains a cornerstone of mechanism design. However, generalizing this framework to multi-item settings has proven exceptionally challenging. Even under restrictive assumptions, closed-form characterizations of optimal mechanisms are rare and are largely confined to the single-agent case \cite{pavlov2011optimal,hart2017approximate, daskalakis2018transport, GIANNAKOPOULOS2018432}, departing from the two-item setting only when prior distributions are uniformly distributed \cite{manelli2006bundling, daskalakis2017strong,giannakopoulos2018sjm}. In this work, we build upon the bi-valued setting introduced by Yao \cite{YAO_BIC_DSIC}, where each item's value has support 2 and lies in $\{a, b\}$. Yao's result provides the only known closed-form optimal mechanism for multiple agents. We extend this line of work along three natural axes, establishing the first closed-form optimal mechanisms in each of the following settings: (i) $n$ i.i.d. agents and $m$ i.i.d. items (ii) $n$ non-i.i.d. agents and two i.i.d. items and (iii) $n$ i.i.d. agents and two non-i.i.d. items. Our results lie at the limit of what is considered possible, since even with a single agent and m bi-valued non-i.i.d. items, finding the optimal mechanism is $\#P$-Hard \cite{daskalakis2014complexity, xi2018soda}. We finally generalize the discrete analog of a result from~\cite{daskalakis2017strong}, showing that for a single agent with $m$ items drawn from arbitrary (non-identical) discrete distributions, grand bundling is optimal when all item values are sufficiently large. We further show that for any continuous product distribution, grand bundling achieves $\mathrm{OPT} - ε$ revenue for large enough values.

Figures (4)

• Figure 1: Flow relationships and layers when $m=4$.
• Figure 2: The Figure depicts the flow graph for a single agent $i$ with parameter $q_i$. The four nodes $(b,b), (b,a), (a,b), (a,a)$ represent the agent’s possible valuations for the two items. We omit the flow from $(a,a)\to\bot$, that is $\mu(a,a)=1$
• Figure 3: Dual flow parametrized by $x\in[0,(1-p)(1-q)]$ and the corresponding virtual values.
• Figure 4: The regions defined for $p\ge q$ and $b>a$. Red: Region 1, Green: Region 2, Orange: Region 3, Purple: Region 4, Brown: Region 5, Black: Region 6, Blue: Region 7. This graph is for $a=1$ and $b=2$ (https://www.desmos.com/calculator/392a84bf6c to the graph parametrized by $a$ and $b$).

Theorems & Definitions (72)

• Definition 2.1: Multi‑Item Hierarchy Allocation
• Definition 3.1: Flow Decomposition
• Theorem 3.1
• proof
• Definition 4.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• proof