# Mean-payoff and Energy Discrete Bidding Games

## TL;DR

This paper introduces and analyzes discrete-bidding games with mean-payoff and energy objectives, establishing the existence and structure of threshold budgets, and demonstrating their computational complexity.

## Contribution

It is the first to study discrete-bidding games with mean-payoff and energy objectives, revealing threshold properties and their complexity.

## Key findings

- Threshold budgets exist for energy and mean-payoff games.
- Threshold structures enable compact strategies.
- Computational complexity of finding thresholds is in NP and coNP.

## Abstract

A \emph{bidding} game is played on a graph as follows. A token is placed on an initial vertex and both players are allocated budgets. In each turn, the 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. We focus on \emph{discrete}-bidding, which are motivated by practical applications and restrict the granularity of the players' bids, e.g, bids must be given in cents. We study, for the first time, discrete-bidding games with {\em mean-payoff} and {\em energy} objectives. In contrast, mean-payoff {\em continuous}-bidding games (i.e., no granularity restrictions) are understood and exhibit a rich mathematical structure. The {\em threshold} budget is a necessary and sufficient initial budget for winning an energy game or guaranteeing a target payoff in a mean-payoff game. We first establish existence of threshold budgets; a non-trivial property due to the concurrent moves of the players. Moreover, we identify the structure of the thresholds, which is key in obtaining compact strategies, and in turn, showing that finding threshold is in \NP~and \coNP even in succinctly-represented games.