Bidding Games with Charging

TL;DR

This paper introduces bidding games with charging, where players can collect vertex-dependent charges to improve budgets, leading to complex behaviors and new computational challenges in infinite-duration game objectives.

Contribution

It extends traditional bidding games by allowing budget improvements through charges, analyzing the resulting non-unique fixed points and providing complexity bounds for key objectives.

Findings

Existence of threshold ratios for winning strategies.

Non-uniqueness of fixed points in charging games.

Lower bounds for Rabin and Streett objectives.

Abstract

Graph games lie at the algorithmic core of many automated design problems in computer science. These are games usually played between two players on a given graph, where the players keep moving a token along the edges according to pre-determined rules, and the winner is decided based on the infinite path traversed by the token from a given initial position. In bidding games, the players initially get some monetary budgets which they need to use to bid for the privilege of moving the token at each step. Each round of bidding affects the players' available budgets, which is the only form of update that the budgets experience. We introduce bidding games with charging where the players can additionally improve their budgets during the game by collecting vertex-dependent charges. Unlike traditional bidding games (where all charges are zero), bidding games with charging allow non-trivial recurrent behaviors. We show that the central property of traditional bidding games generalizes to bidding games with charging: For each vertex there exists a threshold ratio, which is the necessary and sufficient fraction of the total budget for winning the game from that vertex. While the thresholds of traditional bidding games correspond to unique fixed points of linear systems of equations, in games with charging, these fixed points are no longer unique. This significantly complicates the proof of existence and the algorithmic computation of thresholds for infinite-duration objectives. We also provide the lower complexity bounds for computing thresholds for Rabin and Streett objectives, which are the first known lower bounds in any form of bidding games (with or without charging), and we solve the following repair problem for safety and reachability games that have unsatisfiable objectives: Can we distribute a given amount of charge to the players in a way such that the objective can be satisfied?