Reachability Poorman Discrete-Bidding Games

TL;DR

This paper introduces and analyzes poorman discrete-bidding in graph games, establishing the existence of threshold budgets, their properties, and practical algorithms for computation, with implications for game theory and practical bidding systems.

Contribution

It is the first to study poorman discrete-bidding in graph games, providing theoretical results, closed-form solutions in special cases, and algorithms for threshold budget computation.

Findings

Threshold budgets exist for poorman discrete-bidding.

Threshold budgets in DAGs can be approximated with error bounds.

Threshold budgets exhibit periodic behavior and have closed-form solutions in special cases.

Abstract

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.