We consider {\em bidding games}, a class of two-player zero-sum {\em graph games}. The game proceeds as follows. Both players have bounded budgets. A token is placed on a vertex of a graph, in each turn the players simultaneously submit bids, and the higher bidder moves the token, where we break bidding ties in favor of Player 1. Player 1 wins the game iff the token visits a designated target vertex. We consider, for the first time, {\em poorman discrete-bidding} in which the granularity of the bids is restricted and the higher bid is paid to the bank. Previous work either did not impose granularity restrictions or considered {\em Richman} bidding (bids are paid to the opponent). While the latter mechanisms are technically more accessible, the former is more appealing from a practical standpoint. Our study focuses on {\em threshold budgets}, which is the necessary and sufficient initial budget required for Player 1 to ensure winning against a given Player 2 budget. We first show existence of thresholds. In DAGs, we show that threshold budgets can be approximated with error bounds by thresholds under continuous-bidding and that they exhibit a periodic behavior. We identify closed-form solutions in special cases. We implement and experiment with an algorithm to find threshold budgets.

翻译：我们研究**竞标博弈**，这是一类双人零和**图博弈**。博弈过程如下：双方玩家均拥有有限预算。一个标记被放置在图的顶点上，每轮中玩家同时提交竞标，出价更高者移动标记，其中竞标平局时以玩家1优先。当且仅当标记访问指定目标顶点时，玩家1获胜。我们首次考虑**穷鬼离散竞标**机制，其中竞标的粒度受到限制，且较高出价支付给银行。先前的工作要么未施加粒度限制，要么考虑**富人竞标**（出价支付给对手）。虽然后一种机制在技术上更易处理，但前一种机制从实践角度更具吸引力。我们的研究聚焦于**阈值预算**，即玩家1在给定玩家2预算情况下确保获胜所需的最小初始预算。我们首先证明阈值的存在性。在有向无环图中，我们证明阈值预算可通过连续竞标下的阈值进行近似并具有误差界，且表现出周期性行为。我们确定了特殊情况下的闭式解。我们实现并实验了一种用于寻找阈值预算的算法。