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We consider a finite state, finite action, zero-sum stochastic games with data defining the game lying in the ordered field of real algebraic numbers. In both the discounted and the limiting average versions of these games, we prove that the value vector also lies in the same field of real algebraic numbers. In the case where the data of the game are rational, the method also provides a way of checking whether the entries of the value vectors are also rational.

(with V. Ejov, J.A. Filar and A. Moghaddam).