Signaling to analogical reasoners who can acquire costly information

We show that separation in signaling games can be obtained without the single crossing condition, in a model where the receiver reasons analogically across a pair of states and can acquire costly information on the sender's type. Beyond ordinary separation (high type sends high signal, low type sends low signal) we find that also reverse separation is sustainable in equilibrium (high type sends low signal, low type sends high signal). Further, reverse separation in one state is obtained only if ordinary separation occurs in the other state. Pooling is possible and can go along with ordinary separation in one state.