We define a new setting related to the evaluation of AND-OR directed acyclic graphs with partially ordered values. Such graphs arise naturally when solving games with incomplete information (e.g. most card games such as Bridge) or games with multiple criteria. In particular, this setting generalises standard AND-OR graph evaluation and computation of optimal strategies in games with complete information. Under this setting, we propose a new algorithm which uses both alpha-beta pruning and cached values. In this paper, we present our algorithm, prove its correctness, and give experimental results on random games and on a card game with incomplete information.