In a landmark paper, Papadimitriou and Roughgarden described a
polynomial-time algorithm ("Ellipsoid Against Hope") for computing sample
correlated equilibria of concisely-represented games. Recently, Stein, Parrilo
and Ozdaglar showed that this algorithm can fail to find an exact correlated
equilibrium, but can be easily modified to efficiently compute approximate
correlated equilibria. Currently, it remains unresolved whether the algorithm
can be modified to compute an exact correlated equilibrium. We show that it
can, presenting a variant of the Ellipsoid Against Hope algorithm that
guarantees the polynomial-time identification of exact correlated equilibrium.
Our new algorithm differs from the original primarily in its use of a
separation oracle that produces cuts corresponding to pure-strategy profiles.
As a result, we no longer face the numerical precision issues encountered by
the original approach, and both the resulting algorithm and its analysis are
considerably simplified. Our new separation oracle can be understood as a
derandomization of Papadimitriou and Roughgarden's original separation oracle
via the method of conditional probabilities. Also, the equilibria returned by
our algorithm are distributions with polynomial-sized supports, which are
simpler (in the sense of being representable in fewer bits) than the mixtures
of product distributions produced previously; no tractable algorithm has
previously been proposed for identifying such equilibria.Comment: 15 page
