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Combinatorial Pen Testing (or Consumer Surplus of Deferred-Acceptance Auctions)

Aadityan Ganesh, Jason Hartline

TL;DR

The paper studies combinatorial pen testing, where one selects high-ink pens under testing costs, and shows how to leverage deferred-acceptance auction theory to obtain near-optimal algorithms. It introduces a black-box reduction that turns surplus-optimal (or near-optimal) deferred-acceptance mechanisms into pen-testing procedures with equivalent performance guarantees, achieving an omniscient approximation of $(1+o(1)) \ln n$ and extending to matroid, knapsack, and online settings. The authors derive environment-specific refinements (e.g., matroids with $\beta(M)$, $k$-identical goods, downward-closed constraints, and online IID) and show substantial improvements over prior results (notably compared to QV23) by connecting mechanism design to pen testing. Overall, the framework provides principled, transferable tools for maximizing residual ink under diverse combinatorial constraints while clarifying the fundamental links between surplus in auctions and residual value in testing.

Abstract

Pen testing is the problem of selecting high-capacity resources when the only way to measure the capacity of a resource expends its capacity. We have a set of $n$ pens with unknown amounts of ink and our goal is to select a feasible subset of pens maximizing the total ink in them. We are allowed to learn about the ink levels by writing with them, but this uses up ink that was previously in the pens. We identify optimal and near optimal pen testing algorithms by drawing analogues to auction theoretic frameworks of deferred-acceptance auctions and virtual values. Our framework allows the conversion of any near optimal deferred-acceptance mechanism into a near optimal pen testing algorithm. Moreover, these algorithms guarantee an additional overhead of at most $(1+o(1)) \ln n$ in the approximation factor to the omniscient algorithm that has access to the ink levels in the pens. We use this framework to give pen testing algorithms for various combinatorial constraints like matroid, knapsack, and general downward-closed constraints, and also for online environments.

Combinatorial Pen Testing (or Consumer Surplus of Deferred-Acceptance Auctions)

TL;DR

The paper studies combinatorial pen testing, where one selects high-ink pens under testing costs, and shows how to leverage deferred-acceptance auction theory to obtain near-optimal algorithms. It introduces a black-box reduction that turns surplus-optimal (or near-optimal) deferred-acceptance mechanisms into pen-testing procedures with equivalent performance guarantees, achieving an omniscient approximation of and extending to matroid, knapsack, and online settings. The authors derive environment-specific refinements (e.g., matroids with , -identical goods, downward-closed constraints, and online IID) and show substantial improvements over prior results (notably compared to QV23) by connecting mechanism design to pen testing. Overall, the framework provides principled, transferable tools for maximizing residual ink under diverse combinatorial constraints while clarifying the fundamental links between surplus in auctions and residual value in testing.

Abstract

Pen testing is the problem of selecting high-capacity resources when the only way to measure the capacity of a resource expends its capacity. We have a set of pens with unknown amounts of ink and our goal is to select a feasible subset of pens maximizing the total ink in them. We are allowed to learn about the ink levels by writing with them, but this uses up ink that was previously in the pens. We identify optimal and near optimal pen testing algorithms by drawing analogues to auction theoretic frameworks of deferred-acceptance auctions and virtual values. Our framework allows the conversion of any near optimal deferred-acceptance mechanism into a near optimal pen testing algorithm. Moreover, these algorithms guarantee an additional overhead of at most in the approximation factor to the omniscient algorithm that has access to the ink levels in the pens. We use this framework to give pen testing algorithms for various combinatorial constraints like matroid, knapsack, and general downward-closed constraints, and also for online environments.
Paper Structure (14 sections, 27 theorems, 41 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 14 sections, 27 theorems, 41 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Pen Testing Problem and Ascending-Price Auctions
  3. Consumer Surplus Optimization and Virtual Valuations
  4. Near-Optimal Deferred-Acceptance Mechanisms for Consumer Surplus
  5. Consumer Surplus versus Surplus
  6. The Single-Agent Problem
  7. Multi-Agent Environments
  8. The $k$-Identical Goods Environment
  9. Pen Testing Corollaries from Deferred-Acceptance Mechanisms
  10. The Online IID Environment
  11. Conclusion
  12. A Discussion on the Surplus of the Single-Agent Optimal Consumer Surplus Auction
  13. Deferred-Acceptance Mechanism for Knapsack Constraints
  14. Lemmas for Theorem 10

Key Result

Theorem 1

Consider a combinatorial pen testing environment with $n$ pens, the analogous auction environment with an optimal omniscient approximation $\zeta(n)$, and a deferred-acceptance mechanism with a standard approximation equal to $\gamma(n)$. Then, there is a pen testing algorithm that is a $\gamma(n)$-

Figures (1)

  • Figure 1: We compare our upper bound against the lower bound and the best known upper bound when $k = \Theta(n)$. Figure (a) plots the ratio between our upper bound from \ref{['thm:TractableTightkID']}, normalized by $(1 + o(1))$ and our lower bound from \ref{['thm:LowerBound']}, $(0.577 + \ln \frac{n}{k})$. The maximum value of the curve is less than $2.27$, and hence, our upper bound can potentially be improved by a factor of at most $2.27 \, (1 + o(1))$. Figure (b) plots the ratio between the current best known bound $\tfrac{2}{\ln 2} \, (\ln 2 + \ln \frac{n}{k})$ and our bound from \ref{['thm:TractableTightkID']}, again normalized by $(1 + o(1))$. The minimum value taken by the ratio is greater than $1.36$, and hence, our bound improves the best known bound by a factor of at least $1.36 (1 - o(1))$.

Theorems & Definitions (49)

  • Theorem 1
  • Definition 1
  • Definition 2: MS14; Deferred-Acceptance Auctions
  • Theorem 2: Myer81
  • Theorem 3
  • Theorem 4: AFHHM12AFHHM19
  • Definition 3: Virtual-Pricing Transformation of a Deferred-Acceptance Mechanism
  • proof : Proof of \ref{['thm:IntroSurplusDA']}
  • Theorem 5
  • Lemma 1
  • ...and 39 more