Advertisement

Infinite-Duration Poorman-Bidding Games

  • Guy Avni
  • Thomas A. Henzinger
  • Rasmus Ibsen-Jensen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

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.

References

  1. 1.
    Almagor, S., Avni, G., Kupferman, O.: Repairing multi-player games. In: Proceedings of the 26th CONCUR, pp. 325–339 (2015)Google Scholar
  2. 2.
    Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49(5), 672–713 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Apt, K.R., Grädel, E.: Lectures in Game Theory for Computer Scientists. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Avni, G., Henzinger, T.A., Ibsen-Jensen, R.: Infinite-duration poorman-bidding games. CoRR, abs/1804.04372, (2018). arXiv:1804.04372
  7. 7.
    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_13CrossRefGoogle Scholar
  8. 8.
    Avni, G., Kupferman, O., Tamir, T.: Network-formation games with regular objectives. Inf. Comput. 251, 165–178 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bhatt, J., Payne, S.: Bidding chess. Math. Intell. 31, 37–39 (2009)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Calude, C., Jain, S., Khoussainov, B., Li, W., Stephan, F.: Deciding parity games in quasipolynomial time. In: Proceedings of the 49th STOC (2017)Google Scholar
  12. 12.
    Canny, J.F.: Some algebraic and geometric computations in PSPACE. In: Proceedings of the 20th STOC, pp. 460–467 (1988)Google Scholar
  13. 13.
    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_18CrossRefGoogle Scholar
  14. 14.
    Chatterjee, K., Henzinger, T.A., Jurdzinski, M.: Games with secure equilibria. Theor. Comput. Sci. 365(1–2), 67–82 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chatterjee, K., Henzinger, T.A., Piterman, N.: Strategy logic. Inf. Comput. 208(6), 677–693 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    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_6CrossRefGoogle Scholar
  17. 17.
    Condon, A.: On algorithms for simple stochastic games. In: Proceedings of the DIMACS, pp. 51–72 (1990)Google Scholar
  18. 18.
    Develin, M., Payne, S.: Discrete bidding games. Electron. J. Combin. 17(1), R85 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    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_15CrossRefGoogle Scholar
  20. 20.
    Fisman, D., Kupferman, O., Lustig, Y.: Rational synthesis. In: Proceedings of the 16th TACAS, pp. 190–204 (2010)CrossRefGoogle Scholar
  21. 21.
    Huang, Z., Devanur, N.R., Malec, D.: Sequential auctions of identical items with budget-constrained bidders. CoRR, abs/1209.1698 (2012)Google Scholar
  22. 22.
    Jurdzinski, M.: Deciding the winner in parity games is in up \(\cap \) co-up. Inf. Process. Lett. 68(3), 119–124 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kalai, G., Meir, R., Tennenholtz, M.: Bidding games and efficient allocations. In: Proceedings of the 16th EC, pp. 113–130 (2015)Google Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    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_13CrossRefGoogle Scholar
  26. 26.
    Larsson, U., Wästlund, J.: Endgames in bidding chess. Games No Chance 5, 70 (2018)Google Scholar
  27. 27.
    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)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lazarus, A.J., Loeb, D.E., Propp, J.G., Ullman, D.: Richman games. Games No Chance 29, 439–449 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Leme, R.P., Syrgkanis, V., Tardos, É.: Sequential auctions and externalities. In: Proceedings of the 23rd SODA, pp. 869–886 (2012)CrossRefGoogle Scholar
  30. 30.
    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)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  32. 32.
    Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity laplacian. J. Am. Math. Soc. 22, 167–210 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Proceedings of the 16th POPL, pp. 179–190 (1989)Google Scholar
  34. 34.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York (2005)zbMATHGoogle Scholar
  35. 35.
    Rabin, M.O.: Decidability of second order theories and automata on infinite trees. Trans. AMS 141, 1–35 (1969)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Winter, E.: Negotiations in multi-issue committees. J. Public Econ. 65(3), 323–342 (1997)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Guy Avni
    • 1
  • Thomas A. Henzinger
    • 1
  • Rasmus Ibsen-Jensen
    • 2
  1. 1.IST AustriaKlosterneuburgAustria
  2. 2.University of LiverpoolLiverpoolEngland

Personalised recommendations