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Bidding Mechanisms in Graph Games

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Guy Avni, Thomas A. Henzinger, and Đorđe Žikelić. Bidding Mechanisms in Graph Games. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.11

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 tau in [0,1]: portion tau of the winning bid is paid to the other player, and portion 1-tau 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 tau and initial ratio r, equals the value of a random-turn game that uses a coin with bias F(tau, r) = (r+tau * (1-r))/(1+tau). Thus, we show that Richman bidding is the exception; namely, for every tau <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.

Subject Classification

ACM Subject Classification
• Theory of computation → Solution concepts in game theory
• Theory of computation → Formal languages and automata theory
Keywords
• Bidding games
• Richman bidding
• poorman bidding
• taxman bidding
• mean-payoff games
• random-turn games

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References

1. M. Aghajohari, G. Avni, and T. A. Henzinger. Determinacy in Discrete-Bidding Infinite-Duration Games. In In Proc. 30th CONCUR, 2019.
2. K.R. Apt and E. Grädel. Lectures in Game Theory for Computer Scientists. Cambridge University Press, 2011.
3. N. Atzei, M. Bartoletti, and T. Cimoli. A survey of attacks on Ethereum smart contracts. IACR Cryptology ePrint Archive, 2016:1007, 2016.
4. G. Avni, T. A. Henzinger, and V. Chonev. Infinite-Duration Bidding Games. J. ACM, 66(4):31:1-31:29, 2019.
5. G. Avni, T. A. Henzinger, and R. Ibsen-Jensen. Infinite-Duration Poorman-Bidding Games. In Proc. 14th WINE, volume 11316 of LNCS, pages 21-36. Springer, 2018.
6. J. Bhatt and S. Payne. Bidding Chess. Math. Intelligencer, 31:37-39, 2009.
7. J. F. Canny. Some Algebraic and Geometric Computations in PSPACE. In Proc. 20th STOC, pages 460-467, 1988.
8. K. Chatterjee. Robustness of Structurally Equivalent Concurrent Parity Games. In Proc. 15th FoSSaCS, pages 270-285, 2012.
9. K. Chatterjee, A. K. Goharshady, and Y. Velner. Quantitative Analysis of Smart Contracts. In Proc. 27th ESOP, pages 739-767, 2018.
10. A. Condon. The Complexity of Stochastic Games. Inf. Comput., 96(2):203-224, 1992.
11. M. Develin and S. Payne. Discrete Bidding Games. The Electronic Journal of Combinatorics, 17(1):R85, 2010.
12. A. R. Howard. Dynamic Programming and Markov Processes. MIT Press, 1960.
13. A. J. Lazarus, D. E. Loeb, J. G. Propp, W. R. Stromquist, and D. H. Ullman. Combinatorial Games under Auction Play. Games and Economic Behavior, 27(2):229-264, 1999.
14. A. J. Lazarus, D. E. Loeb, J. G. Propp, and D. Ullman. Richman Games. Games of No Chance, 29:439-449, 1996.
15. R. Meir, G. Kalai, and M. Tennenholtz. Bidding games and efficient allocations. Games and Economic Behavior, 2018. URL: https://doi.org/10.1016/j.geb.2018.08.005.
16. S. Muthukrishnan. Ad Exchanges: Research Issues. In Proc. 5th WINE, pages 1-12, 2009.
17. N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani. Algorithmic Game Theory. Cambridge University Press, 2007.
18. Y. Peres, O. Schramm, S. Sheffield, and D. B. Wilson. Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc., 22:167-210, 2009.
19. A. Pnueli and R. Rosner. On the Synthesis of a Reactive Module. In Proc. 16th POPL, pages 179-190, 1989.
20. M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc., New York, NY, USA, 2005.
21. M.O. Rabin. Decidability of Second Order Theories and Automata on Infinite Trees. Transaction of the AMS, 141:1-35, 1969.
22. E. Solan. Continuity of the value of competitive Markov decision processes. Journal of Theoretical Probability, 16:831-845, 2003.