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Abstract. We devise an efficient protocol by which a series of twoperson games Gi with unique winning strategies can be combined into a single game G with unique winning strategy, even when the result of G is a non-monotone function of the results of the Gi that is unknown to the players. In computational complexity terms, we show that the class UAP of Niedermeier and Rossmanith [NR98] of languages accepted by unambiguous polynomial-time alternating TMs is self-low, i.e., UAP UAP = UAP. It follows that UAP contains the Graph Isomorphism problem, nominally improving the problem's classification into SPP by Arvind and Kurur [AK02] since UAP is a subclass of SPP [NR98]. We give some other applications, oracle separations, and results on problems related to unique-alternation formulas.

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