# Determinacy in Discrete-Bidding Infinite-Duration Games

## TL;DR

This paper investigates discrete-bidding infinite-duration games on graphs, establishing their determinacy and analyzing how tie-breaking mechanisms influence the existence of guaranteed winning strategies for players.

## Contribution

It introduces the study of discrete-bidding in infinite-duration games and proves that most natural tie-breaking mechanisms ensure game determinacy.

## Key findings

- Discrete-bidding games form a large determined subclass of concurrent games.
- Tie-breaking mechanisms significantly affect game determinacy.
- Most natural tie-breaking rules guarantee determinacy for all initial budgets.

## Abstract

In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the winner of the game. Such games are central in formal methods since they model the interaction between a non-terminating system and its environment. In bidding games the players bid for the right to move the token: in each round, the players simultaneously submit bids, and the higher bidder moves the token and pays the other player. Bidding games are known to have a clean and elegant mathematical structure that relies on the ability of the players to submit arbitrarily small bids. Many applications, however, require a fixed granularity for the bids, which can represent, for example, the monetary value expressed in cents. We study, for the first time, the combination of discrete-bidding and infinite-duration games. Our most important result proves that these games form a large determined subclass of concurrent games, where determinacy is the strong property that there always exists exactly one player who can guarantee winning the game. In particular, we show that, in contrast to non-discrete bidding games, the mechanism with which tied bids are resolved plays an important role in discrete-bidding games. We study several natural tie-breaking mechanisms and show that, while some do not admit determinacy, most natural mechanisms imply determinacy for every pair of initial budgets.