In this paper, we study a general class of monotone signaling games, in which the support of the signal is limited or the cost of the signal is sufficiently low and as a result, there are multiple pooling equilibria. In those games, when we relax the usual single-crossing condition, the typical restrictions on the out-of-equilibrium beliefs suggested by previous literature cannot discard any of the equilibria obtained. For this reason, we develop a new refinement called the most profitable deviator, which will be useful to select a unique equilibrium in those games. Additionally, when the standard single-crossing condition is satisfied, our criterion also chooses a unique equilibrium, which is the same as that selected by previous literature.
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