Incentivizing Hidden Types in Secretary Problem
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We study a game between N job applicants who incur a cost c (relative to the job value) to reveal their type during interviews and an administrator who seeks to maximize the probability of hiring the best. We define a full learning equilibrium and prove its existence, uniqueness, and optimality. In equilibrium, the administrator accepts the current best applicant n with probability c if n<n^* and with probability 1 if n≥ n^* for a threshold n^* independent of c. In contrast to the case without cost, where the success probability converges to 1/e≈ 0.37 as N tends to infinity, with cost the success probability decays like N^-c.
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