Abstract
In game theory, a player’s action is said to be weakly dominated if there exists another action that, with respect to what the other players do, is never worse and sometimes strictly better. We investigate the computational complexity of the process of iteratively eliminating weakly dominated actions (IWD) in two-player constant-sum games, i.e., games in which the interests of both players are diametrically opposed. It turns out that deciding whether an action is eliminable via IWD is feasible in polynomial time whereas deciding whether a given subgame is reachable via IWD is NP-complete. The latter result is quite surprising as we are not aware of other natural computational problems that are intractable in constant-sum games. Furthermore, we slightly improve a result by Conitzer and Sandholm [6] by showing that typical problems associated with IWD in win-lose games with at most one winner are NP-complete.
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References
Apt, K.R.: Uniform proofs of order independence for various strategy elimination procedures. Contributions to Theoretical Economics 4(1) (2004)
Bernheim, B.: Rationalizable strategic behavior. Econometrica 52(4), 1007–1028 (1984)
Brandenburger, A., Friedenberg, A., Keisler, H.J.: Admissibility in games. Econometrica 76(2), 307–352 (2008)
Brandt, F., Brill, M., Fischer, F., Harrenstein, P.: Computational aspects of Shapley’s saddles. In: Proc. of 8th AAMAS Conference, pp. 209–216 (2009)
Brandt, F., Fischer, F., Harrenstein, P., Shoham, Y.: Ranking games. Artificial Intelligence 173(2), 221–239 (2009)
Conitzer, V., Sandholm, T.: Complexity of (iterated) dominance. In: Proc. of 6th ACM-EC Conference, pp. 88–97. ACM Press, New York (2005)
Ewerhart, C.: Iterated weak dominance in strictly competitive games of perfect information. Journal of Economic Theory 107, 474–482 (2002)
Gilboa, I., Kalai, E., Zemel, E.: The complexity of eliminating dominated strategies. Mathematics of Operations Research 18(3), 553–565 (1993)
Knuth, D.E., Papadimitriou, C.H., Tsitsiklis, J.N.: A note on strategy elimination in bimatrix games. Operations Research Letters 7, 103–107 (1988)
Kohlberg, E., Mertens, J.-F.: On the strategic stability of equilibria. Econometrica 54, 1003–1037 (1986)
Koller, D., Megiddo, N.: The complexity of two-person zero-sum games in extensive from. Games and Economic Behavior 4(4), 528–552 (1992)
Luce, R.D., Raiffa, H.: Games and Decisions: Introduction and Critical Survey. John Wiley & Sons Inc., Chichester (1957)
Marx, L.M., Swinkels, J.M.: Order independence for iterated weak dominance. Games and Economic Behavior 18, 219–245 (1997)
Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)
Osborne, M.: An Introduction to Game Theory. Oxford University Press, Oxford (2004)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Pearce, D.: Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4), 1029–1050 (1984)
Samuelson, L.: Dominated strategies and common knowledge. Games and Economic Behavior 4, 284–313 (1992)
Shimoji, M.: On the equivalence of weak dominance and sequential best response. Games and Economic Behavior 48, 385–402 (2004)
Shoham, Y., Leyton-Brown, L.: Multiagent Systems – Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, Cambridge (2009)
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Brandt, F., Brill, M., Fischer, F., Harrenstein, P. (2009). On the Complexity of Iterated Weak Dominance in Constant-Sum Games. In: Mavronicolas, M., Papadopoulou, V.G. (eds) Algorithmic Game Theory. SAGT 2009. Lecture Notes in Computer Science, vol 5814. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04645-2_26
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DOI: https://doi.org/10.1007/978-3-642-04645-2_26
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