On the complexity of coordination

  • Authors:

  • Olivier Gossner;Penélope Hernández

  • Affiliations:

  • THEMA, UMR CNRS 7536, Université Paris X-Nanterre, 200 avenue de la Républic, 92001 Nanterre, France, and CORE, 34 Voie du Roman Pays, Université Catholique de Louvain, Louvain, Bel ...;CORE, 34 Voie du Roman Pays, Université Catholique de Louvain, Louvain, Belgique

  • Venue:
  • Mathematics of Operations Research
  • Year:
  • 2003

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Abstract

Many results on repeated games played by finite automata rely on the complexity of the exact implementation of a coordinated play of length n. For a large proportion of sequences, this complexity appears to be no less than n. We study the complexity of a coordinated play when allowing for a few mismatches. We prove the existence of a constant C such that if (m ln m)/n ≥ C, for almost any sequence of length n, there exists an automaton of size m that achieves a coordination ratio close to 1 with it. Moreover, we show that one can take any constant C such that C e|X| ln |X|, where |X| is the size of the alphabet from which the sequence is drawn. Our result contrasts with Neyman (1997) that shows that when (m ln m)/n is close to 0, for almost no sequence of length n there exists an automaton of size m that achieves a coordination ratio significantly larger 1/|X| with it.