Mechanism Design for Auctions with Externalities on Budgets

TL;DR

This paper introduces and analyzes auctions where bidders' budgets are externality-affected by others' allocations, formalized via a budget impact factor $\alpha_i$ and budget function $B_i(\mathbf x_{-i})=\alpha_i\sum_{j\neq i} x_j$. It defines liquid welfare $LW(\mathbf x)=\sum_i \min\{ v_i x_i, B_i(\mathbf x_{-i}) \}$ and provides both an optimal allocation algorithm and a truthful, IR $1/3$-approximation mechanism (based on a uniform-price approach with a purchase limit) for maximizing LW. The paper also proves a fundamental upper bound: no truthful mechanism can achieve better than a $1/2$-approximation, establishing a tight gap in the domain. The results advance understanding of budget externalities in multi-agent auctions and offer a framework for designing budget-aware mechanisms with provable performance guarantees, with potential extensions to heterogeneous externalities on budgets and valuations. The approach combines greedy optimization for LW with careful mechanism design to preserve monotonicity and budget feasibility, facilitated by the purchase limit and Myerson’s framework.

Abstract

This paper studies mechanism design for auctions with externalities on budgets, a novel setting where the budgets that bidders commit are adjusted due to the externality of the competitors' allocation outcomes-a departure from traditional auctions with fixed budgets. This setting is motivated by real-world scenarios, for example, participants may increase their budgets in response to competitors' obtained items. We initially propose a general framework with homogeneous externalities to capture the interdependence between budget updates and allocation, formalized through a budget response function that links each bidder's effective budget to the amount of items won by others. The main contribution of this paper is to propose a truthful and individual rational auction mechanism for this novel auction setting, which achieves an approximation ratio of $1/3$ with respect to the liquid welfare. This mechanism is inspired by the uniform-price auction, in which an appropriate uniform price is selected to allocate items, ensuring the monotonicity of the allocation rule while accounting for budget adjustments. Additionally, this mechanism guarantees a constant approximation ratio by setting a purchase limit. Complementing this result, we establish an upper bound: no truthful mechanism can achieve an approximation ratio better than $1/2$. This work offers a new perspective to study the impact of externalities on auctions, providing an approach to handle budget externalities in multi-agent systems.

Theorems & Definitions (14)

• Definition 1.1: Truthfulness
• Definition 1.2: Individual Rationality
• Definition 1.3: Liquid Welfare
• Theorem 1.1
• proof : Proof Sketch
• Theorem 1.2
• Lemma 1.1
• Lemma 1.2: Monotonicity
• Lemma 1.3: Budget Feasibility
• Lemma 1.4: $1/3$-Approximation