Abstract
In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the winner or payoff of the game. Such games are central in formal verification since they model the interaction between a non-terminating system and its environment. We study bidding games in which the players bid for the right to move the token. Two bidding rules have been defined. 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. While poorman reachability games have been studied before, we present, for the first time, results on infinite-duration poorman games. A central quantity in these games is the ratio between the two players’ initial budgets. The questions we study concern a necessary and sufficient ratio with which a player can achieve a goal. For reachability objectives, such threshold ratios are known to exist for both bidding rules. We show that the properties of poorman reachability games extend to complex qualitative objectives such as parity, similarly to the Richman case. Our most interesting results concern quantitative poorman games, namely poorman mean-payoff games, where we construct optimal strategies depending on the initial ratio, by showing a connection with random-turn based games. The connection in itself is interesting, because it does not hold for reachability poorman games. We also solve the complexity problems that arise in poorman bidding games.
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 is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
When the initial budget of Player \(1\) is exactly \({\texttt {Th}} (v)\), the winner of the game depends on how we resolve draws in biddings, and our results hold for any tie-breaking mechanism.
References
Almagor, S., Avni, G., Kupferman, O.: Repairing multi-player games. In: Proceedings of the 26th CONCUR, pp. 325–339 (2015)
Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49(5), 672–713 (2002)
Apt, K.R., Grädel, E.: Lectures in Game Theory for Computer Scientists. Cambridge University Press, Cambridge (2011)
Avni, G., Guha, S., Kupferman, O.: An abstraction-refinement methodology for reasoning about network games. In: Proceedings of the 26th IJCAI, pp. 70–76 (2017)
Avni, G., Henzinger, T.A., Chonev, V.: Infinite-duration bidding games. In: Proceedings of the 28th CONCUR, vol. 85 of LIPIcs, pp. 21:1–21:18 (2017)
Avni, G., Henzinger, T.A., Ibsen-Jensen, R.: Infinite-duration poorman-bidding games. CoRR, abs/1804.04372, (2018). arXiv:1804.04372
Avni, G., Henzinger, T.A., Kupferman, O.: Dynamic resource allocation games. In: Gairing, M., Savani, R. (eds.) SAGT 2016. LNCS, vol. 9928, pp. 153–166. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53354-3_13
Avni, G., Kupferman, O., Tamir, T.: Network-formation games with regular objectives. Inf. Comput. 251, 165–178 (2016)
Bhatt, J., Payne, S.: Bidding chess. Math. Intell. 31, 37–39 (2009)
Brihaye, T., Bruyère, V., De Pril, J., Gimbert, H.: On subgame perfection in quantitative reachability games. Log. Methods Comput. Sci. 9(1) (2012). https://doi.org/10.2168/LMCS-9(1:7)2013
Calude, C., Jain, S., Khoussainov, B., Li, W., Stephan, F.: Deciding parity games in quasipolynomial time. In: Proceedings of the 49th STOC (2017)
Canny, J.F.: Some algebraic and geometric computations in PSPACE. In: Proceedings of the 20th STOC, pp. 460–467 (1988)
Chatterjee, K.: Nash equilibrium for upward-closed objectives. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 271–286. Springer, Heidelberg (2006). https://doi.org/10.1007/11874683_18
Chatterjee, K., Henzinger, T.A., Jurdzinski, M.: Games with secure equilibria. Theor. Comput. Sci. 365(1–2), 67–82 (2006)
Chatterjee, K., Henzinger, T.A., Piterman, N.: Strategy logic. Inf. Comput. 208(6), 677–693 (2010)
Chatterjee, K., Majumdar, R., Jurdziński, M.: On nash equilibria in stochastic games. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 26–40. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30124-0_6
Condon, A.: On algorithms for simple stochastic games. In: Proceedings of the DIMACS, pp. 51–72 (1990)
Develin, M., Payne, S.: Discrete bidding games. Electron. J. Combin. 17(1), R85 (2010)
Feldman, M., Snappir, Y., Tamir, T.: The efficiency of best-response dynamics. In: Bilò, V., Flammini, M. (eds.) SAGT 2017. LNCS, vol. 10504, pp. 186–198. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66700-3_15
Fisman, D., Kupferman, O., Lustig, Y.: Rational synthesis. In: Proceedings of the 16th TACAS, pp. 190–204 (2010)
Huang, Z., Devanur, N.R., Malec, D.: Sequential auctions of identical items with budget-constrained bidders. CoRR, abs/1209.1698 (2012)
Jurdzinski, M.: Deciding the winner in parity games is in up \(\cap \) co-up. Inf. Process. Lett. 68(3), 119–124 (1998)
Kalai, G., Meir, R., Tennenholtz, M.: Bidding games and efficient allocations. In: Proceedings of the 16th EC, pp. 113–130 (2015)
Kash, I.A., Friedman, E.J., Halpern, J.Y.: Optimizing scrip systems: crashes, altruists, hoarders, sybils and collusion. Distrib. Comput. 25(5), 335–357 (2012)
Kupferman, O., Tamir, T.: Hierarchical network formation games. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 229–246. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54577-5_13
Larsson, U., Wästlund, J.: Endgames in bidding chess. Games No Chance 5, 70 (2018)
Lazarus, A.J., Loeb, D.E., Propp, J.G., Stromquist, W.R., Ullman, D.H.: Combinatorial games under auction play. Games Econ. Behav. 27(2), 229–264 (1999)
Lazarus, A.J., Loeb, D.E., Propp, J.G., Ullman, D.: Richman games. Games No Chance 29, 439–449 (1996)
Leme, R.P., Syrgkanis, V., Tardos, É.: Sequential auctions and externalities. In: Proceedings of the 23rd SODA, pp. 869–886 (2012)
Mogavero, F., Murano, A., Perelli, G., Vardi, M.Y.: Reasoning about strategies: on the model-checking problem. ACM Trans. Comput. Log. 15(4), 34:1–34:47 (2014)
Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)
Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity laplacian. J. Am. Math. Soc. 22, 167–210 (2009)
Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Proceedings of the 16th POPL, pp. 179–190 (1989)
Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York (2005)
Rabin, M.O.: Decidability of second order theories and automata on infinite trees. Trans. AMS 141, 1–35 (1969)
Winter, E.: Negotiations in multi-issue committees. J. Public Econ. 65(3), 323–342 (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Avni, G., Henzinger, T.A., Ibsen-Jensen, R. (2018). Infinite-Duration Poorman-Bidding Games. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-04612-5_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04611-8
Online ISBN: 978-3-030-04612-5
eBook Packages: Computer ScienceComputer Science (R0)