The complexity of interacting automata

Abstract

This paper studies the interaction of automata of size m. We characterise statistical properties satisfied by random plays generated by a correlated pair of automata with m states each. We show that in some respect the pair of automata can be identified with a more complex automaton of size comparable to \(m\log m\). We investigate implications of these results on the correlated min–max value of repeated games played by automata.

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Notes

  1. 1.

    Or, equivalently, in the space of oblivious automata of size at most \({\mathcal {O}}(\log m)\), i.e. those that ignore the actions of the other agent.

  2. 2.

    Here and throughout, \(s\hbox { mod }m\) is defined as the number in [m] which is equal to s modulo m.

  3. 3.

    The function D is not the largest possible. Its concavification \(cav\,D\) is also possible as discussed in Sect. 3.8.3.

  4. 4.

    A perfect matching is a matching of size n.

  5. 5.

    Section 3.8.3 suggests that D can be replaced by \(cav\,D\) which implies that \(n-1\) can be attained whenever the two marginals of Q, \(Q^1\) and \(Q^2\), are uniform distributions supported on sets of size n.

  6. 6.

    Note that \(\mathrm {cor\,min\,max}\,G=\min _{x^1\in \Delta (A^1)}\max _{a^3\in A^3}g(x^1,\cdot ,a^3)\) .

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Acknowledgments

We are grateful to three anonymous referees and an editor who contributed valuable suggestions that significantly improved the quality of this paper.

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Correspondence to Ron Peretz.

Additional information

O. Gossner acknowledges financial support from Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047). P. Hernández acknowledges financial support from the Spanish Economy and Competitiveness Ministry (ECO2013-46550-R) and Generalitat Valenciana (PROMETEOII/2014/054).

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Gossner, O., Hernández, P. & Peretz, R. The complexity of interacting automata. Int J Game Theory 45, 461–496 (2016). https://doi.org/10.1007/s00182-015-0521-7

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Keywords

  • Complexity
  • Automata
  • De Bruijn sequences
  • Bounded memory