Signaling Games with Costly Monitoring

If in a signaling game the receiver expects to gain no information by monitoring the signal of the sender, then when a cost to monitor is implemented he will never pay that cost regardless of his off-path beliefs. This is the argument of a recent paper by T. Denti (2021). However, which pooling equilibrium does a receiver anticipate to gain no information through monitoring? This paper seeks to prove that given a sufficiently small cost to monitor any pooling equilibrium with a non-zero index will survive close to the original equilibrium.