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From Stream to Pool: Pricing Under the Law of Diminishing Marginal Utility

Titing Cui, Su Jia, Thomas Lavastida

TL;DR

A minimax \emph{optimal} algorithm that efficiently computes a non-adaptive policy which guarantees a $1/k$ fraction of the optimal revenue, given any set of $k$ prices is presented.

Abstract

Dynamic pricing models often posit that a $\textbf{stream}$ of customer interactions occur sequentially, where customers' valuations are drawn independently. However, this model is not entirely reflective of the real world, as it overlooks a critical aspect, the law of diminishing marginal utility, which states that a customer's marginal utility from each additional unit declines. This causes the valuation distribution to shift towards the lower end, which is not captured by the stream model. This motivates us to study a pool-based model, where a $\textbf{pool}$ of customers repeatedly interacts with a monopolist seller, each of whose valuation diminishes in the number of purchases made according to a discount function. In particular, when the discount function is constant, our pool model recovers the stream model. We focus on the most fundamental special case, where a customer's valuation becomes zero once a purchase is made. Given $k$ prices, we present a non-adaptive, detail-free (i.e., does not "know" the valuations) policy that achieves a $1/k$ competitive ratio, which is optimal among non-adaptive policies. Furthermore, based on a novel debiasing technique, we propose an adaptive learn-then-earn policy with a $\tilde O(k^{2/3} n^{2/3})$ regret.

From Stream to Pool: Pricing Under the Law of Diminishing Marginal Utility

TL;DR

A minimax \emph{optimal} algorithm that efficiently computes a non-adaptive policy which guarantees a fraction of the optimal revenue, given any set of prices is presented.

Abstract

Dynamic pricing models often posit that a of customer interactions occur sequentially, where customers' valuations are drawn independently. However, this model is not entirely reflective of the real world, as it overlooks a critical aspect, the law of diminishing marginal utility, which states that a customer's marginal utility from each additional unit declines. This causes the valuation distribution to shift towards the lower end, which is not captured by the stream model. This motivates us to study a pool-based model, where a of customers repeatedly interacts with a monopolist seller, each of whose valuation diminishes in the number of purchases made according to a discount function. In particular, when the discount function is constant, our pool model recovers the stream model. We focus on the most fundamental special case, where a customer's valuation becomes zero once a purchase is made. Given prices, we present a non-adaptive, detail-free (i.e., does not "know" the valuations) policy that achieves a competitive ratio, which is optimal among non-adaptive policies. Furthermore, based on a novel debiasing technique, we propose an adaptive learn-then-earn policy with a regret.
Paper Structure (62 sections, 31 theorems, 129 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 62 sections, 31 theorems, 129 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Pool Model: An Informal Description
  3. Our Contributions
  4. Related Work
  5. Dynamic Pricing in the Stream Model
  6. Intertemporal Pricing.
  7. Robust Optimization in Pricing.
  8. Partially Observable Reinforcement Learning.
  9. Formulation
  10. The General Pool Model
  11. Unifying the Pool and Stream Models
  12. Our Focus: The Unit-Demand Pool Model (UDPM)
  13. Detail-Dependent Policy
  14. Monotonicity of the Optimal Non-Adaptive Policy
  15. Two Algorithms
  16. ...and 47 more sections

Key Result

Theorem 1

Let $X$ be any policy. For any $s,t\in [0,1]$, denote by $D_{s,t}$ the (random) demand in the time interval $[s,t]$. Denote by $\mathbb{E}_{\rm GvR}$ and $\mathbb{E}_{\rm Pool}$ the expectation in instances ${\rm GvR}(\mathcal{P}, n\lambda, d_{\bf v})$ and ${\rm Pool}(\mathcal{P},\lambda, {\bf v}, \

Figures (6)

  • Figure 1: Illustration of the UDPM: Red dots represent customer interactions, where they peek at the price. A purchase is made at the first interaction event ($T_3$) where the price is lower than the valuation.
  • Figure 2: Robust non-adaptive policies for a continuous price space $[p_{\min},1]$.
  • Figure 3: Average Revenue for Uniform Price Set. We plot the average revenue for uniform price sets with $k = 2, \dots, 11$. The interaction rate is chosen to be $1$, 3 and 5 in each of these three subfigures.
  • Figure 4: Average Revenue for Geometric Price Set. We plot the average revenue for the geometric price sets for $k = 2, \dots, 11$. The interaction rate is chosen to be $1,3$ and $5$ in each of these three sub-figures.
  • Figure 5: Competitive ratio of RP and RS policy: For each $k = 3, \dots, 12$, we plot the (non-adaptive) competitive ratios of the RP and RS policies as functions under the uniform and geometric price sets.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Theorem 1: Unifying Pool and Stream
  • Definition 1: Non-adaptive Policy
  • Proposition 1: Price Monotonicity
  • Remark 1
  • Proposition 2: Revenues of Non-adaptive Markdown Policies
  • Proposition 3: Concavity of Revenue Function
  • Theorem 2: Computing Optimal Non-Adaptive Policy
  • Definition 2: Competitive ratio
  • Theorem 3: Upper Bound on the Competitive Ratio
  • Proposition 4
  • ...and 36 more