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Minimizing Expectation Plus Variance

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Abstract

We consider strategic games in which each player seeks a mixed strategy to minimize her cost evaluated by a concave valuation V (mapping probability distributions to reals); such valuations are used to model risk. In contrast to games with expectation-optimizer players where mixed equilibria always exist (Nash 1950; Nash Ann. Math. 54, 286–295, 1951), a mixed equilibrium for such games, called a V-equilibrium, may fail to exist, even though pure equilibria (if any) transfer over. What is the exact impact of such valuations on the existence, structure and complexity of mixed equilibria? We address this fundamental question in the context of expectation plus variance, a particular concave valuation denoted as RA, which stands for risk-averse; so, variance enters as a measure of risk and it is used as an additive adjustment to expectation. We obtain the following results about RA-equilibria:

  • A collection of general structural properties of RA-equilibria connecting to (i) E-equilibria and Var-equilibria, which correspond to the expectation and variance valuations E and Var, respectively, and to (ii) other weaker or incomparable properties such as Weak Equilibrium and Strong Equilibrium. Some of these structural properties imply quantitative constraints on the existence of mixed RA-equilibria.

  • A second collection of (i) existence, (ii) equivalence and separation (with respect to E-equilibria), and (iii) characterization results for RA-equilibria in the new class of player-specific scheduling games. We provide suitable examples with a mixed RA-equilibrium that is not an E-equilibrium and vice versa.

  • A purification technique to transform a player-specific scheduling game on two identical links into a player-specific scheduling game on two links so that all non-pure RA-equilibria are eliminated while no new pure equilibria are created; so, a particular player-specific scheduling game on two identical links with no pure equilibrium yields a player-specific scheduling game with no RA-equilibrium (whether mixed or pure). As a by-product, the first 𝓟ℒ𝓢-completeness result for the computation of RA-equilibria follows.

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Notes

  1. 1 A 𝓟ℒ𝓢-reduction [10] from a 𝓟ℒ𝓢-problem π to a 𝓟ℒ𝓢-problem π′ is a pair of polynomial time algorithms A 1 and A 2 such that: (C1) A 1 transforms an instance I of π to an instance A 1(I) of π′. (C1) A 2 transforms an instance I of π and a solution s′ to A 1(I) to a solution s of I. (C3) For an instance I of π, if s′ is a local optimum of A 1(I), then s = A 2(I, s′) is a local optimum of I. In our case, the algorithm A 1 is given in the transformation of a player-specific scheduling game G into the modified player-specific scheduling game \(\widehat {{\mathsf {G}}}\) (Section 6.1); the algorithm A 2 is the identity transformation of a local optimum for the modified game \(\widehat {{\mathsf {G}}}\) (viewed as a 𝓟ℒ𝓢-problem) into itself. Theorem 6.1 and Lemma 6.3 prove together Condition (C3).

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Acknowledgments

We would like to thank Martina Eikel for helpful discussions.

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Authors and Affiliations

  1. Department of Computer Science, University of Cyprus, Nicosia, 1678, Cyprus

    Marios Mavronicolas

  2. Faculty of Electrical Engineering, Computer Science and Mathematics, University of Paderborn, 33102, Paderborn, Germany

    Burkhard Monien

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  1. Marios Mavronicolas
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  2. Burkhard Monien
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Correspondence to Marios Mavronicolas.

Additional information

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-the-Fly Computing” (SFB 901) and by funds for the promotion of research at University of Cyprus. A preliminary version of this work appears in the Proceedings of the 5th International Symposium on Algorithmic Game Theory, pp. 239–250, Vol. 7615, Lecture Notes in Computer Science, Springer-Verlag, October 2012.

Part of the work of Marios Mavronicolas was done while visiting the Faculty of Electrical Engineering, Computer Science and Mathematics, University of Paderborn.

Part of the work of Burkhard Monien was done while visiting the Department of Computer Science, University of Cyprus.

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Mavronicolas, M., Monien, B. Minimizing Expectation Plus Variance. Theory Comput Syst 57, 617–654 (2015). https://doi.org/10.1007/s00224-014-9542-z

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