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Computational Aspects of Uncertainty Profiles and Angel-Daemon Games

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Abstract

We analyze the complexity of equilibria problems for a class of strategic zero-sum games, called angel-daemon games. Those games were introduced to asses the performance of the execution of a web orchestration on a moderate faulty or under stress environment. Angel-daemon games are a natural example of zero-sum games whose representation is naturally succinct. We show that the problems of deciding the existence of a pure Nash equilibrium or of a dominant strategy for a given player are \({\Sigma}^{p}_{2}\)-complete. Furthermore, computing the value of an angel-daemon game is EXP-complete. Thus, our results match the already known classification of the corresponding problems for the generic families of succinctly represented games with exponential number of actions.

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Notes

  1. “Most, probably, of our decisions to do something positive, the full consequences of which will be drawn out over many days to come, can only be taken as the result of animal spirits—a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities” [32, pp. 161–162].

  2. The words “publishes”,“returns” and “outputs” are used interchangeably. The terms “site” and “service” are also used interchangeably.

  3. The expression “P(x)<x<Q” is encoded as “P(x) where x:∈Q” in [33].

  4. For instance, if \(C_{\lambda}= x_{1}\vee\overline{x}_{2}\vee x_{3}\) then \(O_{\lambda}=X_{1}\mid \overline {X}_{2}\mid X_{3}\).

  5. Consider \(C_{\lambda}= x_{1}\vee\overline{x}_{2}\vee x_{3}\) then \(O_{\lambda}= (X_{1}\mid\overline{X}_{2}\mid X_{3})\). In this case n=3 and X n+1=X 4. Consider the evaluation α=001 such that \(C_{\lambda}(\alpha)={\tt true}\), the corresponding angelic profile is \(a_{\alpha}=\{X_{4}, X_{1},X_{2},\overline{X}_{3}\}\) then \(\textsf {fail}_{a_{\alpha }}(O_{\lambda})=(0\mid\overline{X}_{2}\mid X_{3})\) and \(\textsf {out}(\textsf{fail} _{a_{\alpha}}(O_{\lambda}))>0\).

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Acknowledgements

The authors want to thank the anonymous referees for a careful reading and useful suggestions that help us to correct some inaccuracies and improve the presentation and readability of the paper.

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  1. ALBCOM Research Group, Universitat Politècnica de Catalunya, Jordi Girona Salgado, 1-3, Barcelona, 08034, Spain

    Joaquim Gabarro, Alina Garcia & Maria Serna

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  1. Joaquim Gabarro
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Correspondence to Maria Serna.

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Part of this work was presented as On the Complexity of Equilibria Problems in Angel-Daemon Games. In Computing and Combinatorics, COCOON 2008. LNCS, vol. 5092, pp. 31–40, 2008.

J. Gabarró and M. Serna are partially supported by TIN2007-66523 (FORMALISM) and SGR2009-1137 (ALBCOM).

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Gabarro, J., Garcia, A. & Serna, M. Computational Aspects of Uncertainty Profiles and Angel-Daemon Games. Theory Comput Syst 54, 83–110 (2014). https://doi.org/10.1007/s00224-013-9481-0

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