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Optimal Mechanism in a Dynamic Stochastic Knapsack Environment

Jihyeok Jung, Chan-Oi Song, Deok-Joo Lee, Kiho Yoon

Abstract

This study introduces an optimal mechanism in a dynamic stochastic knapsack environment. The model features a single seller who has a fixed quantity of a perfectly divisible item. Impatient buyers with a piece-wise linear utility function arrive randomly and they report the two-dimensional private information: marginal value and demanded quantity. We derive a revenue-maximizing dynamic mechanism in a finite discrete time framework that satisfies incentive compatibility, individual rationality, and feasibility conditions. It is achieved by characterizing buyers' utility and deriving the Bellman equation. Moreover, we propose the essential penalty scheme for incentive compatibility, as well as the allocation and payment policies. Lastly, we propose algorithms to approximate the optimal policy, based on the Monte Carlo simulation-based regression method and reinforcement learning.

Optimal Mechanism in a Dynamic Stochastic Knapsack Environment

Abstract

This study introduces an optimal mechanism in a dynamic stochastic knapsack environment. The model features a single seller who has a fixed quantity of a perfectly divisible item. Impatient buyers with a piece-wise linear utility function arrive randomly and they report the two-dimensional private information: marginal value and demanded quantity. We derive a revenue-maximizing dynamic mechanism in a finite discrete time framework that satisfies incentive compatibility, individual rationality, and feasibility conditions. It is achieved by characterizing buyers' utility and deriving the Bellman equation. Moreover, we propose the essential penalty scheme for incentive compatibility, as well as the allocation and payment policies. Lastly, we propose algorithms to approximate the optimal policy, based on the Monte Carlo simulation-based regression method and reinforcement learning.
Paper Structure (12 sections, 9 theorems, 27 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 12 sections, 9 theorems, 27 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Dynamic Stochastic Knapsack Problem
  4. Optimal Mechanism
  5. Dynamic Mechanism
  6. Framework
  7. Optimal Mechanism
  8. Approximation Algorithms
  9. Approximation Methods
  10. Numerical Experiment
  11. Conclusion
  12. Acknowledgments

Key Result

Lemma 1

($\clubsuit$) Suppose $\Gamma=(a^t,p^t)_{t \in \mathcal{T}}$ is incentive compatible and feasible. Then at any period $t \in \mathcal{T}$,

Figures (1)

  • Figure 1: Cumulative allocation of $(T, \bar{Q}) = (100, 100)$

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Theorem 1
  • Definition 5
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • ...and 6 more