Infinite-duration bidding games

Guy Avni, Thomas A. Henzinger, Ventsislav Chonev

Research output: Contribution to journalArticlepeer-review


Two-player games on graphs are widely studied in formal methods, as they model the interaction between a system and its environment. The game is played by moving a token throughout a graph to produce an infinite path. There are several common modes to determine how the players move the token through the graph; e.g., in turn-based games the players alternate turns in moving the token. We study the bidding mode of moving the token, which, to the best of our knowledge, has never been studied in infinite-duration games. The following bidding rule was previously defined and called Richman bidding. Both players have separate budgets, which sum up to 1. In each turn, a bidding takes place: Both players submit bids simultaneously, where a bid is legal if it does not exceed the available budget, and the higher bidder pays his bid to the other player and moves the token. The central question studied in bidding games is a necessary and sufficient initial budget for winning the game: a threshold budget in a vertex is a value t ∈ [0, 1] such that if Player 1’s budget exceeds t, he can win the game; and if Player 2’s budget exceeds 1 − t, he can win the game. Threshold budgets were previously shown to exist in every vertex of a reachability game, which have an interesting connection with random-turn games—a sub-class of simple stochastic games in which the player who moves is chosen randomly. We show the existence of threshold budgets for a qualitative class of infinite-duration games, namely parity games, and a quantitative class, namely mean-payoff games. The key component of the proof is a quantitative solution to strongly connected mean-payoff bidding games in which we extend the connection with random-turn games to these games, and construct explicit optimal strategies for both players.

Original languageEnglish
Article number31
JournalJournal of the ACM
Issue number4
StatePublished - 16 Jul 2019
Externally publishedYes

Bibliographical note

Funding Information:
This research was supported in part by the Austrian Science Fund (FWF) under grants S11402-N23 (RiSE/SHiNE), Z211-N23 (Wittgenstein Award), and M 2369-N33 (Meitner fellowship). This paper is based on the conference publication of Reference [7]. Authors’ addresses: G. Avni and T. A. Henzinger, IST Austria, Am Campus 1, Klosterneuburg, Austria, 3400; emails: {guy.avni, tah}; V. Chonev, Max Planck Institute for Software Systems (MPI-SWS), Campus E1 5, Saarbrücken, Germany, 66123; email: Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from © 2019 Association for Computing Machinery. 0004-5411/2019/07-ART31 $15.00

Publisher Copyright:
© 2019 Association for Computing Machinery.


  • Bidding games
  • Mean-payoff games
  • Parity games
  • Richman games

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence


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