[Paper Review] Optimal Budget-Feasible Mechanisms for Additive Valuations

Nick Gravin, Yaonan Jin+2 more|arXiv (Cornell University)|Feb 12, 2019

Auction Theory and Applications26 references4 citations

TL;DR

This paper presents a new randomized budget-feasible mechanism for additive valuations that achieves a tight 2-approximation guarantee against the fractional Knapsack benchmark, improving the prior best result of 3. It also introduces a deterministic mechanism with a 3-approximation, matching the best-known bound for the integral benchmark, using a novel two-stage design that prunes low-value-per-cost items before applying posted prices.

ABSTRACT

In this paper, we show a tight approximation guarantee for budget-feasible mechanisms with an additive buyer. We propose a new simple randomized mechanism with approximation ratio of $2$, improving the previous best known result of $3$. Our bound is tight with respect to either the optimal offline benchmark, or its fractional relaxation. We also present a simple deterministic mechanism with the tight approximation guarantee of $3$ against the fractional optimum, improving the best known result of $(2+ \sqrt{2})$ for the weaker integral benchmark.

Motivation & Objective

• To close the gap between the best-known upper and lower bounds for budget-feasible mechanisms with additive valuations.
• To design a truthful mechanism that achieves optimal approximation ratio under budget constraints.
• To establish a clear separation in power between randomized and deterministic mechanisms in this setting.
• To improve upon the prior best-known results of 3 for randomized and (2+√2) for deterministic mechanisms.
• To demonstrate that the fractional Knapsack benchmark is a meaningful and tight benchmark for mechanism design in this context.

Proposed method

• Proposes a two-stage mechanism: first, greedily exclude items with low value-per-cost ratios to form a candidate set.
• In the second stage, apply posted-price mechanisms based on the remaining items’ values and costs.
• Uses a randomized mechanism that selects between two price levels with specific probabilities to ensure truthfulness and budget feasibility.
• Employs a probability distribution over budget allocations to balance between high-value and low-cost items.
• Applies a fractional relaxation of the Knapsack problem as the benchmark for approximation analysis.
• Uses probabilistic analysis to bound the probability that any agent accepts the posted price, ensuring individual rationality and truthfulness.

Experimental results

Research questions

• RQ1 Can a randomized budget-feasible mechanism achieve a better approximation ratio than the best-known deterministic mechanism for additive valuations?
• RQ2 Is the 2-approximation ratio tight for randomized mechanisms under the fractional Knapsack benchmark?
• RQ3 Can a deterministic mechanism achieve a 3-approximation guarantee that matches the best-known upper bound for the integral Knapsack benchmark?
• RQ4 What is the optimal trade-off between pruning low-value-per-cost items and preserving high-value options in the second stage?
• RQ5 Does the use of posted prices in the second stage enable better approximation guarantees while maintaining truthfulness and budget feasibility?

Key findings

• The proposed randomized mechanism achieves a 2-approximation ratio against the fractional Knapsack benchmark, which is tight even with respect to the integral Knapsack benchmark.
• The deterministic mechanism achieves a 3-approximation ratio against the fractional optimum, improving upon the previous best-known result of (2+√2).
• The 2-approximation bound for the randomized mechanism is optimal, as no mechanism can achieve better than 2-approximation even against the weaker integral Knapsack benchmark.
• The two-stage design—pruning low-value-per-cost items followed by posted pricing—proves effective and generalizable, with the first stage stopping earlier than in prior work.
• The analysis shows that the probability of any agent accepting a posted price is at least q_T + (v_i - r·c_i)/(2v_i), which ensures truthfulness and individual rationality.
• The results establish a clear separation: no deterministic mechanism can achieve better than (√2+1)-approximation, while the randomized mechanism already achieves 2-approximation, demonstrating the power of randomization.