Equilibrium Points in Fear of Correlated Threats

  • Authors:

  • Spyros C. Kontogiannis;Paul G. Spirakis

  • Affiliations:

  • Computer Science Dept., University of Ioannina, Greece and R.A. Computer Technology Institute, Patras University Campus, Patras, Greece 26500;R.A. Computer Technology Institute, Patras University Campus, Patras, Greece 26500

  • Venue:
  • WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
  • Year:
  • 2008

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Abstract

The present work considers the following computational problem:Given any finite game in normal form G and the correspondinginfinitely repeated game G  ∞ , determine inpolynomial time (wrt the representation of G) a profile ofstrategies for the players in G  ∞  that is anequilibrium point wrt the limit-of-means payoff. The problem hasbeen solved for two players [10], based mainly on theimplementability of the threats for this case. Nevertheless, [4]demonstrated that the traditional notion of threats is acomputationally hard problem for games with at least 3 players (seealso [8]). Our results are the following: (i) We propose analternative notion of correlated threats, which is polynomial timecomputable (and therefore credible). Our correlated threats arealso more severe than the traditional notion of threats, but notoverwhelming for any individual player. (ii) When for theunderlying game G there is a correlated strategy with payoff vectorstrictly larger than the correlated threats vector, we efficientlycompute a polynomial–size (wrt the description of G)equilibrium point for G  ∞ , for any constantnumber of players. (iii) Otherwise, we demonstrate the constructionof an equilibrium point for an arbitrary number of players and upto 2 concurrently positive payoff coordinates in any payoff vectorof G. This completely resolves the cases of 3 players, and providesa direction towards handling the cases of more than 3 players. Itis mentioned that our construction is not a Nash equilibrium point,because the correlated threats we use are implemented via, not onlyfull synchrony (as in [10]), but also coordination of the otherplayers’ actions. But this seems to be a fair trade-offbetween efficiency of the construction and players’coordination, in particular because it only affects the punishments(which are anticipated never to be used).