Abstract | The presentwork considers the following computational problem:
Given any finite game in normal form G and the corresponding
infinitely repeated game G∞, determine in polynomial time (wrt1 the representation
ofG) a profile of strategies for the players inG∞ that is an equilibrium
point wrt the limit-of-means payoff. The problem has been solved
for two players [10], based mainly on the implementability of the threats
for this case. Nevertheless, [4] demonstrated that the traditional notion of
threats is a computationally hard problem for games with at least 3 players
(see also [8]). Our results are the following: (i) We propose an alternative
notion of correlated threats, which is polynomial time computable
(and therefore credible). Our correlated threats are also more severe than
the traditional notion of threats, but not overwhelming for any individual
player. (ii) When for the underlying game G there is a correlated strategy
with payoff vector strictly larger than the correlated threats vector,
we efficiently compute a polynomial–size (wrt the description of G) equilibrium
point for G∞, for any constant number of players. (iii) Otherwise,
we demonstrate the construction of an equilibrium point for an arbitrary
number of players and up to 2 concurrently positive payoff coordinates in
any payoff vector of G. This completely resolves the cases of 3 players, and
provides a direction towards handling the cases of more than 3 players. It
is mentioned that our construction is not a Nash equilibrium point, because
the correlated threats we use are implemented via, not only full synchrony
(as in [10]), but also coordination of the other players˘ actions. But
this seems to be a fair trade-off between efficiency of the construction and
players˘ coordination, in particular because it only affects the punishments
(which are anticipated never to be used). |