Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Packing a Knapsack with Items Owned by Strategic Agents

Javier Cembrano, Max Klimm, Martin Knaack

Abstract

This paper considers a scenario within the field of mechanism design without money where a mechanism designer is interested in selecting items with maximum total value under a knapsack constraint. The items, however, are controlled by strategic agents who aim to maximize the total value of their items in the knapsack. This is a natural setting, e.g., when agencies select projects for funding, companies select products for sale in their shops, or hospitals schedule MRI scans for the day. A mechanism governing the packing of the knapsack is strategyproof if no agent can benefit from hiding items controlled by them to the mechanism. We are interested in mechanisms that are strategyproof and $α$-approximate in the sense that they always approximate the maximum value of the knapsack by a factor of $α\in [0,1]$. First, we give a deterministic mechanism that is $\frac{1}{3}$-approximate. For the special case where all items have unit density, we design a $\frac{1}φ$-approximate mechanism where $1/φ\approx 0.618$ is the inverse of the golden ratio. This result is tight as we show that no deterministic strategyproof mechanism with a better approximation exists. We further give randomized mechanisms with approximation guarantees of $1/2$ for the general case and $2/3$ for the case of unit densities. For both cases, no strategyproof mechanism can achieve an approximation guarantee better than $1/(5φ-7)\approx 0.917$.

Packing a Knapsack with Items Owned by Strategic Agents

Abstract

This paper considers a scenario within the field of mechanism design without money where a mechanism designer is interested in selecting items with maximum total value under a knapsack constraint. The items, however, are controlled by strategic agents who aim to maximize the total value of their items in the knapsack. This is a natural setting, e.g., when agencies select projects for funding, companies select products for sale in their shops, or hospitals schedule MRI scans for the day. A mechanism governing the packing of the knapsack is strategyproof if no agent can benefit from hiding items controlled by them to the mechanism. We are interested in mechanisms that are strategyproof and -approximate in the sense that they always approximate the maximum value of the knapsack by a factor of . First, we give a deterministic mechanism that is -approximate. For the special case where all items have unit density, we design a -approximate mechanism where is the inverse of the golden ratio. This result is tight as we show that no deterministic strategyproof mechanism with a better approximation exists. We further give randomized mechanisms with approximation guarantees of for the general case and for the case of unit densities. For both cases, no strategyproof mechanism can achieve an approximation guarantee better than .
Paper Structure (12 sections, 23 theorems, 79 equations, 4 figures, 1 table, 7 algorithms)

This paper contains 12 sections, 23 theorems, 79 equations, 4 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution and Techniques
  3. Further Related Work
  4. Preliminaries
  5. Greedy Solution
  6. General Knapsack
  7. Deterministic Mechanisms
  8. Randomized Mechanisms
  9. Unit-Density Knapsack
  10. Deterministic mechanisms
  11. Randomized mechanisms
  12. Discussion

Key Result

Lemma 2.1

Let $\mathrm{M}$ be a deterministic mechanism that is not strategyproof on $\mathcal{I}' \subseteq \mathcal{I}$ for some $\mathcal{I}'$ closed under inclusion. Then, there exists an instance in $\mathcal{I}'$ with items $E$, an agent $a \in [n]$, and an item $i \in E_a$ such that $v_a(\mathrm{M}(E))

Figures (4)

  • Figure 1: Example of an instance in which computing the integral greedy solution results in an agent benefiting from hiding items. Items of the same color belong to the same agent. The value and size of each item are respectively written inside and below them; the capacity is $10$. If the agent that owns the orange items hides the item with value $15$ in the upper instance, the value of the orange items packed in the integral greedy solution increases from $20$ to $28$.
  • Figure 2: Examples of $a$-dominance of the restricted set of items $R(E)$ over $R(E\setminus \{i\})$ for some $i \in E_a$ when running FitTwo with parameter $\beta=1/\phi \approx 0.618$. Items of the same color belong to the same agent; those of agent $a$ are orange. Items with a background of diagonal lines belong to the restricted set $R$. Four cases are considered as in the proof of \ref{['lem:dominance-deletion']}. In all of them, the values of the orange items in $R$ before the deletion of $i$ are lexicographically larger than those after deletion; the opposite is true for the remaining items.
  • Figure 3: Examples of restricted sets $R(E)$ and $S(E)$ when running FitTwo with parameter $\beta=2/3$ and LargeFit, respectively. Items among $E$ with a background of upward diagonal lines belong to $R(E)$; items among $E$ with a background of downward diagonal lines belong to $S(E)$.
  • Figure : RandomizedFit$(\normalfont\textsc{F}_{\normalfont\text{rnd}})$

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • ...and 34 more