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Motivated by the problems of unknown objectives in strategic bidding in the restructured electricity power markets where in practice, not all participants have full access to competitors' information, this paper investigates a class of noncooperative N-person games with unknown objectives with linear and cross terms in controls. We first formulate and explicitly derive the Nash equilibrium solution for this type of games, then incorporate the concept of fictitious play with the technique of adaptive control to estimate multiple controller strategies, which are not necessarily a Nash solution. We show both theoretically and numerically that the adaptation procedure converges under certain conditions. The proposed scheme is applicable to all systems with similar mathematical structures and objectives.