# Bidding mechanisms in graph games

Guy Avni, Thomas A. Henzinger, Ðorđe Žikelić

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

In two-player games on graphs, the players move a token through a graph to produce a finite or infinite path, which determines the qualitative winner or quantitative payoff of the game. We study bidding games in which the players bid for the right to move the token. Several bidding rules were studied previously. In Richman bidding, in each round, the players simultaneously submit bids, and the higher bidder moves the token and pays the other player. Poorman bidding is similar except that the winner of the bidding pays the “bank” rather than the other player. Taxman bidding spans the spectrum between Richman and poorman bidding. They are parameterized by a constant τ ∈ [0, 1]: portion τ of the winning bid is paid to the other player, and portion 1 − τ to the bank. While finite-duration (reachability) taxman games have been studied before, we present, for the first time, results on infinite-duration taxman games. It was previously shown that both Richman and poorman infinite-duration games with qualitative objectives reduce to reachability games, and we show a similar result here. Our most interesting results concern quantitative taxman games, namely mean-payoff games, where poorman and Richman bidding differ significantly. A central quantity in these games is the ratio between the two players’ initial budgets. While in poorman mean-payoff games, the optimal payoff of a player depends on the initial ratio, in Richman bidding, the payoff depends only on the structure of the game. In both games the optimal payoffs can be found using (different) probabilistic connections with random-turn games in which in each turn, instead of bidding, a coin is tossed to determine which player moves. While the value with Richman bidding equals the value of a random-turn game with an un-biased coin, with poorman bidding, the bias in the coin is the initial ratio of the budgets. We give a complete classification of mean-payoff taxman games that is based on a probabilistic connection: the value of a taxman bidding game with parameter τ and initial ratio r, equals the value of a random-turn game that uses a coin with bias (Formula presented.). Thus, we show that Richman bidding is the exception; namely, for every τ < 1, the value of the game depends on the initial ratio. Our proof technique simplifies and unifies the previous proof techniques for both Richman and poorman bidding.

Original language English 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019 Joost-Pieter Katoen, Pinar Heggernes, Peter Rossmanith Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 9783959771177 https://doi.org/10.4230/LIPIcs.MFCS.2019.11 Published - Aug 2019 Yes 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019 - Aachen, GermanyDuration: 26 Aug 2019 → 30 Aug 2019

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 138 1868-8969

### Conference

Conference 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019 Germany Aachen 26/08/19 → 30/08/19

### Bibliographical note

© Guy Avni, Thomas A. Henzinger, and Ðorđe Žikelić.

## Keywords

• Bidding games
• Mean-payoff games
• Poorman bidding
• Random-turn games
• Richman bidding
• Taxman bidding

## ASJC Scopus subject areas

• Software
• ### Bidding mechanisms in graph games

Avni, G., Henzinger, T. A. & Žikelić, Đ., Aug 2021, 119, p. 133-144 12 p.

Research output: Contribution to journalArticlepeer-review

Open Access