In this paper we formulate mathematically and solve maximin and minimax detection problems for signals with power constraints. These problems arise whenever it is necessary to distinguish between a genuine signal and a spurious on designed by an adversary with the principal goal of deceiving the detector. The spurious (or deceptive) signal is usually subject to certain constraints, such as limited power, which preclude it from replicating the genuine signal exactly.

The detection problem is formulated as a zero-sum game involving two players: the detector designer and the deceptive signal designer. The payoff is the probability of error of the detector which the detector designer tries to minimize and the deceptive signal designer to maximize. For this detection game, saddle point solutions --- whenever possible --- or otherwise maximin and minimax solutions are derived under three distinct constraints on the deceptive signal power; these distinct constraints involves bounds on (i) the peak power, (ii) the probabilistic average power, and (iii) the time average power. The cases of i.i.d. and correlated signals are both considered.