Monotone equilibrium in matching markets with signaling

We introduce a notion of competitive signaling equilibrium (CSE) in one-to-one matching markets with a continuum of heterogeneous senders and receivers. We then study monotone CSE where equilibrium outcomes - sender actions, receiver reactions, beliefs, and matching - are all monotone in the stronger set order. We show that if the sender utility is monotone-supermodular and the receiver's utility is weakly monotone-supermodular, a CSE is stronger monotone if and only if it passes Criterion D1 (Cho and Kreps (1987), Banks and Sobel (1987)). Given any interval of feasible reactions that receivers can take, we fully characterize a unique stronger monotone CSE and establishes its existence with quasilinear utility functions.