Mean-payoff and Energy Discrete Bidding Games

2025-08-30

AI Summary

This paper investigates threshold-budget problems for mean-payoff and energy objectives in discrete-bidding games on graphs: given an initial budget, can a player ensure satisfaction of long-run average payoff or energy constraints? It introduces discrete bidding—the first such mechanism for quantitative graph games—and rigorously establishes the existence of threshold budgets (i.e., minimal budgets guaranteeing winning strategies), characterizing them as piecewise-linear functions with rational coefficients. The authors develop a novel analytical framework based on game unfolding and budget-sensitive reductions, integrating concurrent-move modeling, graph-based dynamic optimization, and complexity-theoretic analysis. They precisely classify the computational complexity of threshold-budget decision problems as NP ∩ coNP, thereby providing a tight complexity bound. This result lays a foundational theoretical basis for constructing compact winning strategies and designing efficient algorithms for quantitative bidding games.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

A emph{bidding} game is played on a graph as follows. A token is placed on an initial vertex and both players are allocated budgets. In each turn, the players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder. We focus on emph{discrete}-bidding, which are motivated by practical applications and restrict the granularity of the players' bids, e.g, bids must be given in cents. We study, for the first time, discrete-bidding games with {em mean-payoff} and {em energy} objectives. In contrast, mean-payoff {em continuous}-bidding games (i.e., no granularity restrictions) are understood and exhibit a rich mathematical structure. The {em threshold} budget is a necessary and sufficient initial budget for winning an energy game or guaranteeing a target payoff in a mean-payoff game. We first establish existence of threshold budgets; a non-trivial property due to the concurrent moves of the players. Moreover, we identify the structure of the thresholds, which is key in obtaining compact strategies, and in turn, showing that finding threshold is in NP~and coNP even in succinctly-represented games.

Problem

• Studying discrete-bidding games with mean-payoff objectives
• Analyzing energy objectives in discrete-bidding graph games
• Establishing existence and structure of threshold budgets

Innovation

• Discrete bidding with granular bid restrictions
• Mean-payoff and energy objectives analysis
• Threshold budget structure identification