Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2107.05690

Worst-Case Welfare of Item Pricing in the Tollbooth Problem

Zihan Tan·Yifeng Teng·Mingfei Zhao
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the worst-case welfare of item pricing in the \emph{tollbooth problem}. The problem was first introduced by Guruswami et al, and is a special case of the combinatorial auction in which (i) each of the $m$ items in the auction is an edge of some underlying graph; and (ii) each of the $n$ buyers is single-minded and only interested in buying all edges of a single path. We consider the competitive ratio between the hindsight optimal welfare and the optimal worst-case welfare among all item-pricing mechanisms, when the order of the arriving buyers is adversarial. We assume that buyers own the \emph{tie-breaking} power, i.e. they can choose whether or not to buy the demand path at 0 utility. We prove a tight competitive ratio of $3/2$ when the underlying graph is a single path (also known as the \emph{highway} problem), whereas item-pricing can achieve the hindsight optimal if the seller is allowed to choose a proper tie-breaking rule to maximize the welfare. Moreover, we prove an $O(1)$ upper bound of competitive ratio when the underlying graph is a tree. For general graphs, we prove an $Ω(m^{1/8})$ lower bound of the competitive ratio. We show that an $m^{Ω(1)}$ competitive ratio is unavoidable even if the graph is a grid, or if the capacity of every edge is augmented by a constant factor $c$. The results hold even if the seller has tie-breaking power.

Explore related subjects

cs.GTcs.DMcs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zihan Tan, Yifeng Teng, Mingfei Zhao. 2022-07-08. Worst-Case Welfare of Item Pricing in the Tollbooth Problem. https://arxiv.org/abs/2107.05690

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ascending Auctions for Combinatorial Markets with Frictions: A Unified Framework via Discrete Convex Analysis

We develop a unified ascending-auction framework for computing Walrasian equilibria in combinatorial markets with strong substitutes valuations and piecewise-linear payment functions. Our auction extends the celebrated ascending auctions of Gul and Stacchetti (2000) and Ausubel (2006) to accommodate payment frictions (e.g., transaction taxes or commission fees). This is achieved by incorporating directional price updates that reflect heterogeneous payment structures. Our framework also generalizes the unit-demand imperfectly transferable utility models of Alkan (1989, 1992) to a fully combinatorial setting, thereby unifying these paradigms. Furthermore, this is the first study to compute the minimum -- also known as the buyer-optimal -- equilibrium in combinatorial markets with such frictions. Our analysis builds upon discrete convex analysis. Our main technical contribution is a characterization of valid price-update directions, together with a strongly polynomial-time algorithm for computing them. Notably, the algorithm uses only demand- and exchange-oracle queries and never requires handling information of exponential size. To compute such a direction, we formulate a lexicographic extension of the polymatroid sum problem and characterize its dual solution via a reduction to a convex flow problem. Exploiting the $\text{L}^\natural$-convexity of the dual objective, we show that the desired direction can be constructed from the minimal dual solution. This convexity also yields transparent economic and potential-based interpretations of the auction dynamics, strengthening the connection between ascending auctions and discrete optimization.

cs.GT↗
arXiv2026

Mechanism Design Is Not Enough: Prosocial Agents for Cooperative AI

Ensuring that AI agents behave safely and beneficially when interacting with other parties has emerged as one of the central challenges of modern AI safety. While mechanism design, as the theory of designing rules to align individual and collective objectives, can incentivize cooperative behavior, it is still an open question whether it alone is sufficient to maximize LLM agents' social welfare. This work proves that the answer is negative: drawing from incomplete contract theory, we formally show that when contracts cannot distinguish all relevant future contingencies, there is a strictly positive welfare loss that no realistic mechanism can eliminate. We show that prosocial agents, who weigh others' welfare alongside their own, can close this gap and achieve outcomes that are socially superior and individually beneficial. Experimentally, we show that in multi-agent resource-allocation environments and canonical social dilemmas where agents are powered by large language models, prosociality is beneficial. The implication for AI safety is clear: to enable cooperative interactions at scale, designing adequate mechanisms is not sufficient; agents must be built to be intrinsically prosocial.

cs.GT↗
arXiv2026

Efficient Nash Equilibrium Computation for Cybersecurity Games

Game-theoretic analyses of cyber defence often compute equilibria of games whose payoffs exist only as the output of a simulator. Iterative equilibrium-finding methods grow a set of attacker and defender policies and need the payoff of every attacker--defender pair, so they are bottlenecked by payoff estimation: each payoff costs many simulator runs. We introduce Regret-Weighted Payoff Sampling (RWPS), which spends a fixed simulation budget on the payoffs the equilibrium actually depends on and predicts the rest with a model trained on every payoff measured so far. Standard error bounds for estimated games are driven by the worst-estimated payoff, so they cannot credit an estimator that is inaccurate only where accuracy does not matter. We prove a bound that weights payoff errors by the opponent's equilibrium strategy, a certificate that can be computed from simulated payoffs alone, and a condition under which errors in the predicted payoffs cannot change either player's regret. On three synthetic general-sum games, one of them a Colonel Blotto game of military resource allocation, the new bounds are four to six times tighter than the standard one, and RWPS finds less exploitable equilibria than minimum-regret-first search, information-gain search and progressive sampling at the same budget. On two cyber-defence simulators, CyGym and a new game whose hosts are LLM agents exposed to prompt injection, it gives the least exploitable equilibria at the smallest budgets.

cs.GT↗