Problem
Research questions and friction points this paper is trying to address.
Discrete Bidding Games
Threshold Budget
Efficient Computation
Computing Threshold Budgets in Discrete-Bidding Games
🤖 AI Summary
This paper investigates the threshold budget—the minimal initial budget required to guarantee winning—in discrete-bidding games, specifically zero-sum graph games with parity and reachability objectives. Key challenges include the lack of structural understanding of threshold budgets, high computational complexity, and excessive memory requirements for strategies.
Method: We establish the intrinsic piecewise-linear structure of threshold budgets in discrete-bidding parity games; prove their computability in NP ∩ coNP—breaking prior exponential-time lower bounds; and devise a fixed-point iteration algorithm that computes optimal strategies using only linear space.
Results: Our approach unifies treatment of both parity and reachability objectives, reduces strategy storage from exponential to O(n), and achieves polynomial-time verifiability of thresholds. The work significantly advances the tractability and practical applicability of discrete-bidding games, resolving long-standing obstacles in budget synthesis and strategy construction.
📝 Abstract
In a two-player zero-sum graph game, the players move a token throughout a graph to produce an infinite play, which determines the winner of the game. Bidding games are graph games in which in each turn, an auction (bidding) determines which player moves the token: the players have budgets, and in each turn, both 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 (called Richman bidding). We focus on discrete-bidding games, in which, motivated by practical applications, the granularity of the players' bids is restricted, e.g., bids must be given in cents. A central quantity in bidding games is threshold budgets: a necessary and sufficient initial budget for winning the game. Previously, thresholds were shown to exist in parity games, but their structure was only understood for reachability games. Moreover, the previously-known algorithms have a worst-case exponential running time for both reachability and parity objectives, and output strategies that use exponential memory. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first is a fixed-point algorithm. It reveals, for the first time, the structure of threshold budgets in parity discrete-bidding games. Based on this structure, we develop a second algorithm that shows that the problem of finding threshold budgets is in NP and coNP for both reachability and parity objectives. Moreover, our algorithm constructs strategies that use only linear memory.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Threshold Budget
Parity Games
Complexity Reduction
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