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Assortment Optimization For Conference Goodies With Indifferent Attendees

Fernanda Gutiérrez, Bernardo Subercaseaux

TL;DR

The paper analyzes how to stock $N$ goodies of $K$ types when attendees are indifferent among types and pick uniformly from remaining stock, aiming to minimize the expected number of unhappy attendees. It formalizes the process, proves the conjecture that optimal counts are as equal as possible for $K=2$ and $K=3$ (via simple induction and Wald's equation with a network of inductive lemmas), and develops asymptotic bounds and practical approximations using the time-to-empty events $\tau$. Complementary simulations and an algebraic attempt illustrate the near-optimal equal-distribution principle and highlight the challenge of extending to general $K$. The work contributes a rigorous framework for a simple yet rich assortment problem, provides substantial partial proofs, and offers a reward to spur a complete resolution, with potential implications for practical conference-goody stocking strategies under symmetric preference models.

Abstract

Conferences such as FUN with Algorithms routinely buy goodies (e.g., t-shirts, coffee mugs, etc) for their attendees. Often, said goodies come in different types, varying by color or design, and organizers need to decide how many goodies of each type to buy. We study the problem of buying optimal amounts of each type under a simple model of preferences by the attendees: they are indifferent to the types but want to be able to choose between more than one type of goodies at the time of their arrival. The indifference of attendees suggests that the optimal policy is to buy roughly equal amounts for every goodie type. Despite how intuitive this conjecture sounds, we show that this simple model of assortment optimization is quite rich, and even though we make progress towards proving the conjecture (e.g., we succeed when the number of goodie types is 2 or 3), the general case with K types remains open. We also present asymptotic results and computer simulations, and finally, to motivate further progress, we offer a reward of $100usd for a full proof.

Assortment Optimization For Conference Goodies With Indifferent Attendees

TL;DR

The paper analyzes how to stock goodies of types when attendees are indifferent among types and pick uniformly from remaining stock, aiming to minimize the expected number of unhappy attendees. It formalizes the process, proves the conjecture that optimal counts are as equal as possible for and (via simple induction and Wald's equation with a network of inductive lemmas), and develops asymptotic bounds and practical approximations using the time-to-empty events . Complementary simulations and an algebraic attempt illustrate the near-optimal equal-distribution principle and highlight the challenge of extending to general . The work contributes a rigorous framework for a simple yet rich assortment problem, provides substantial partial proofs, and offers a reward to spur a complete resolution, with potential implications for practical conference-goody stocking strategies under symmetric preference models.

Abstract

Conferences such as FUN with Algorithms routinely buy goodies (e.g., t-shirts, coffee mugs, etc) for their attendees. Often, said goodies come in different types, varying by color or design, and organizers need to decide how many goodies of each type to buy. We study the problem of buying optimal amounts of each type under a simple model of preferences by the attendees: they are indifferent to the types but want to be able to choose between more than one type of goodies at the time of their arrival. The indifference of attendees suggests that the optimal policy is to buy roughly equal amounts for every goodie type. Despite how intuitive this conjecture sounds, we show that this simple model of assortment optimization is quite rich, and even though we make progress towards proving the conjecture (e.g., we succeed when the number of goodie types is 2 or 3), the general case with K types remains open. We also present asymptotic results and computer simulations, and finally, to motivate further progress, we offer a reward of $100usd for a full proof.
Paper Structure (13 sections, 10 theorems, 109 equations, 11 figures, 1 algorithm)

This paper contains 13 sections, 10 theorems, 109 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Context and Related Work
  3. Organization
  4. The simple case, K=2.
  5. A simple proof by induction
  6. Why does this not scale to any $K$?
  7. K=3: Wald and the inductive lemmas.
  8. Asymptotic Bounds and Approximations
  9. Simulations and Experimental Results
  10. An Algebraic Approach
  11. Concluding Remarks
  12. Deferred Proofs
  13. Figures

Key Result

Lemma 1

Let $K = 2$, and let $s \coloneqq \min(n_1, n_2), l \coloneqq \max(n_1, n_2)$We will use the letter $s$ to denote a smaller size, the letter $m$ to denote a medium size (for when we analyze $K=3$), and the letter $l$ to denote a larger size.. If $l > s > 0$, then

Figures (11)

  • Figure 1: An initial assortment of $N = 9$ t-shirts of $K=3$ different types.
  • Figure 2: Illustration of a sequence of attendants over the initial assortment depicted in \ref{['fig:ex1']}. The last two time steps include unhappy attendees.
  • Figure 3: Illustration of the relationship between $\hat{h}$, $h$ computed acocrding to \ref{['eq:general-recursive']}, and the average of 20 simulations of according to Model \ref{['alg:model-M']}.
  • Figure 4: Illustration of the terms of \ref{['eq:main-K-2']} for $N=100$. The global minimum is achieved at $n_1 = 50$ as expected.
  • Figure 5: Measured $\hat{\tau}$ on 30 random samples from $\mathcal{U}^5([1,100])$, $\mathcal{U}^5([1,150])$, $\mathcal{U}^5([1,200])$, $\mathcal{U}^{10}([1,50])$, $\mathcal{U}^{10}([1,80])$, $\mathcal{U}^{10}([1,100])$.
  • ...and 6 more figures

Theorems & Definitions (29)

  • Conjecture 1
  • Lemma 1
  • proof
  • Definition 2
  • Theorem 3
  • proof
  • Definition 4
  • Lemma 5: Consequence of Wald's equation.
  • Theorem 12
  • proof
  • ...and 19 more