Purely Rational Secret Sharing

Rational secret sharing is a fundamental primitive at the intersection of cryptography and game theory. In essence, a dealer wishes to engineer a communication game that, when rationally played, guarantees that each of the players learns the dealer's secret.   Yet, all solutions so far were {\em quite inefficient} and relied on the players' {\em beliefs} and not just on their rationality.
After providing a more complete definition of this problem, we exhibit a very efficient and  purely rational solution to it.  Our solution works in the verifiable trusted third party model.