Reachability Poorman Discrete-Bidding Games
This work addresses a practical problem in game theory for researchers and practitioners by extending bidding game models to include granularity restrictions, though it is incremental as it builds on previous work on bidding games.
The paper tackles the problem of determining threshold budgets for Player 1 to win in poorman discrete-bidding games, a two-player zero-sum graph game with bounded budgets and granularity restrictions, showing existence of thresholds, approximation with error bounds in DAGs, periodic behavior, and closed-form solutions in special cases.
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.