The key issue in selecting between equilibria in signalling games is determining how receivers will interpret deviations from the path of play. We develop a foundation for these off-path beliefs, and an associated equilibrium refinement, in a model where equilibrium arises from non-equilibrium learning by long-lived senders and receivers. In our model, non-equilibrium signals are sent by young senders as experiments to learn about receivers' behavior, and different types of senders have different incentives for these various experiments. Using the Gittins index (Gittins, 1979), we characterize which sender types use each signal more often, leading to a constraint we call the "compatibility criterion" on the receiver's off-path beliefs and to the concept of a "type-compatible equilibrium." We compare type-compatible equilibria to signalling-game refinements such as the Intuitive Criterion (Cho and Kreps, 1987) and divine equilibrium (Banks and Sobel, 1987).
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