This paper studies a two-sided search with heterogeneous agents and nontransferable utility in which each individual observes only a noisy signal of the type of a potential partner, as in marriage and entry-level labor markets.
It is shown that the optimal strategy for an agent takes into account not only the signal observed, but also the information contained in the event that the agent is accepted. The latter depends on the characteristics of the strategy followed by the agents on the other side of the market, which complicates the analysis of the model substantially.
An equilibrium in increasing strategies is characterized using some recent results on monotone methods and Bayesian games. In that equilibrium, positive sorting of agents by types holds in a stochastic sense: on average, agents with higher types mate together.
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