Abstract
The computational complexity of a variety of problems from algorithmic game theory is investigated. These are variations on the question whether a strategy in a normal form game survives iterated elimination of dominated strategies. The difficulty of the computational task depends on the notion of dominance involved, on the number of distinct payoffs and whether the game is constant-sum. Most of the open cases are fully classified, and the remaining cases are shown to be equivalent to certain questions regarding elimination orders on graphs. The classifications may serve as the basis for a discussion to what extent iterated dominance could be useful to restrict rationality for computationally bounded agents.
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Notes
The claim of order invariance up to strategy permutation for weak dominance given in [13, Proposition 1] is wrong. A trivial counterexample is the game A=(0,1), B=(0,0) with the two reduced forms A ′=(0), B ′=(0) and A ″=(1) , B ″=(0).
Doing so for strictly dominated strategies does not influence the final result, while doing so for weakly dominated strategies could result in empty strategy sets.
Often the concept of never best responses is considered against mixed strategies instead. Then never best responses and strictly dominated strategies would coincide (compare e.g. [16, Pages 59–60]). Elimination of never best responses against pure strategies however is a distinct concept.
It is straightforward that these cases are equivalent. Technically, our examples will be constant-sum games rather than zero-sum games.
While P-completeness is claimed, the corresponding part of the proof is wrong. However, as we will show later, the P-completeness result is true.
In particular, an And vertex with in-degree 0 always has the value true; thus, we do not need to include designated True vertices. In theory, the same is true for False and Or vertices, however, including False vertices explicitly facilitates our constructions.
I would like to thank Anuj Dawar and Yuguo He for pointing out this result to me.
Or, if x is subsequently eliminated by another strategy z, by (z,y k ), and so on.
Note that approximate Nash equilibria are not necessarily approximations to Nash equilibria, but a far weaker criterion.
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Acknowledgements
I am grateful to Anuj Dawar for his support and advice as my PhD advisor. Furthermore, I would like to thank the anonymous referees of an earlier version for their detailed and helpful comments.
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Pauly, A. The Computational Complexity of Iterated Elimination of Dominated Strategies. Theory Comput Syst 59, 52–75 (2016). https://doi.org/10.1007/s00224-015-9637-1
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DOI: https://doi.org/10.1007/s00224-015-9637-1