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Optimal Auction Design for Dynamic Stochastic Environments: Myerson Meets Naor

Yeon-Koo Che, Andrew B. Choi

TL;DR

In this model, buyers with private valuations and homogeneous goods arrive stochastically and can be held in queues at a cost, and the optimal mechanism pairs allocative efficiency with dynamic admission control: goods are assigned to the highest-value buyer, while entry is restricted by value thresholds.

Abstract

Motivated by applications such as cloud computing, gig platforms, and blockchain auctions, we study optimal selling mechanisms for dynamic markets with stochastic supply and demand. In our model, buyers with private valuations and homogeneous goods arrive stochastically and can be held in queues at a cost. The optimal mechanism pairs allocative efficiency with dynamic admission control: goods are assigned to the highest-value buyer, while entry is restricted by value thresholds that strictly increase with the queue length and decrease with available inventory. This policy smooths competitive pressure across time and is implemented in dominant strategies via auctions with dynamic reserve prices.

Optimal Auction Design for Dynamic Stochastic Environments: Myerson Meets Naor

TL;DR

In this model, buyers with private valuations and homogeneous goods arrive stochastically and can be held in queues at a cost, and the optimal mechanism pairs allocative efficiency with dynamic admission control: goods are assigned to the highest-value buyer, while entry is restricted by value thresholds.

Abstract

Motivated by applications such as cloud computing, gig platforms, and blockchain auctions, we study optimal selling mechanisms for dynamic markets with stochastic supply and demand. In our model, buyers with private valuations and homogeneous goods arrive stochastically and can be held in queues at a cost. The optimal mechanism pairs allocative efficiency with dynamic admission control: goods are assigned to the highest-value buyer, while entry is restricted by value thresholds that strictly increase with the queue length and decrease with available inventory. This policy smooths competitive pressure across time and is implemented in dominant strategies via auctions with dynamic reserve prices.

Paper Structure

This paper contains 40 sections, 15 theorems, 95 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Feasibility Constraints and Relaxed Program
  4. Direct Revelation Mechanism
  5. Reduced-Form Feasibility Constraint
  6. Balance Inequalities
  7. Relaxed Program
  8. Optimal Policy
  9. Storable Goods: $d<\infty$
  10. Robust Implementations of Optimal Policy
  11. Dominant Strategy Implementation
  12. Periodic Ex Post Implementation
  13. Large Market Properties
  14. Welfare maximization
  15. Related Literature
  16. ...and 25 more sections

Key Result

Theorem 1

There exists a triple $(P^*, y^*, X^*)$ that solves the relaxed program $[\mathcal{P}]$. There exist thresholds $\hat{v}_1:=J^{-1}(c/\mu)< \hat{v}_2 < ...< \hat{v}_{K^*}<1$, for some $K^*<\infty$, such that

Figures (8)

  • Figure 1: $S_1(v)$ and $S_2(v)$
  • Figure 2: Stationary CDFs of order statistics ($F=U[0,1]$, $\mu=1$, $\lambda=2$, $c=0.3$)
  • Figure 3: Joint distribution of $v_1$ and $v_2$
  • Figure 4: Three example scenarios
  • Figure 5: Proof idea for \ref{['thm:large-market']}
  • ...and 3 more figures

Theorems & Definitions (21)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Proposition 1: Comparative Statics in $c,\lambda,\mu$
  • Theorem $\mathbf{1'}$
  • Theorem $\mathbf{2'}$
  • Theorem 3
  • Corollary 1
  • Definition 1
  • Corollary 2
  • ...and 11 more