Question
A ferry company pays a forecaster to label its weather instrument standard or premium before deciding whether to run a crossing. The forecaster knows whether the instrument's readings are reliable (type H) or unreliable (type L), but the ferry operator sees only the label. Initially both types are equally likely.
The operator can sail or cancel. Its expected net payoff from sailing is +$4 if the instrument is H and −$6 if it is L; cancellation pays $0. The forecaster receives a $4 fee if the operator sails, regardless of the true type. Sending the standard label costs nothing. Producing a premium-labelled report requires extra checking: it costs the forecaster $1 if H, but $5 if L. Everyone knows these figures; label choice itself is not directly verified.
Treat the interaction as a signaling game. Find a separating perfect Bayesian equilibrium and a pooling perfect Bayesian equilibrium, specifying the operator's beliefs and actions. Can both types pool on the premium label in a pure-strategy equilibrium? Explain the incentive and off-path-belief checks.
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Hint 1
If the operator believes the probability of a reliable instrument is \(\mu\), compute its expected payoff from sailing. Then compare \(\mu\) with the break-even posterior.
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Solution
Let \(\mu\) be the operator's posterior probability of type \(H\) after observing a label. Sailing yields \(4\mu-6(1-\mu)=10\mu-6\) dollars, so the operator sails for \(\mu>\frac35\), cancels for \(\mu<\frac35\), and is indifferent at \(\mu=\frac35\).
Separating equilibrium. Let \(H\) choose premium and \(L\) choose standard. Since both types have prior probability \(\frac12\) and both labels occur, Bayes' rule gives \(\mu(H\mid\text{premium})=1\) and \(\mu(H\mid\text{standard})=0\). The operator therefore sails after premium and cancels after standard. Forecaster \(H\) earns \(\$4-\$1=\$3\) using premium, versus $0 from deviating to standard. Forecaster \(L\) earns $0 on standard, versus \(\$4-\$5=-\$1\) by deviating to premium. Both sender incentive constraints hold. There are no off-path labels in this separating profile.
Pooling equilibrium. Suppose both types choose standard. On that reached path, Bayes' rule requires \(\mu(H\mid\text{standard})=\frac12\); the operator cancels since \(10(\frac12)-6=-1\). Premium is off path, so its posterior is not fixed by Bayes' rule. Assign \(\mu(H\mid\text{premium})=0\) and let the operator cancel there too. Neither sender gains by deviating: premium would produce no fee and cost H $1 or L $5. More generally, any off-path posterior at most \(\frac35\) supports cancellation (at equality, choosing cancellation is optimal); if premium instead induced sailing with certainty, H would deviate for a positive $3 payoff.
No premium pooling. If both types choose premium, the operator's on-path belief must remain \(\frac12\) and therefore it cancels. Each forecaster would then pay a positive premium-label cost for no sailing fee; switching to free standard earns at least $0, regardless of the operator's off-path action. Thus neither type prefers premium pooling. Reverse separation (L premium, H standard) also fails: the costly L report would lead to cancellation, but L could switch to the standard label, trigger sailing and earn $4. These are pure-strategy pooling/separating conclusions; they do not rule out every possible mixed signaling assessment.
Assumptions
Both types are privately known by the forecaster; no third type or direct verification follows a label.
Payoffs are expected monetary payoffs conditional on quality; agents maximise expected money.
The operator observes only the public label; labels are chosen before the sail/cancel decision.
Beliefs at off-path labels are permissible if they make the operator sequentially rational.
Final Answer
Separating PBE: H sends premium, L sends standard; the operator sails only after premium. Pooling PBE: both send standard, the operator cancels after standard and also cancels after an off-path premium label when its belief \(\Pr(H\mid\text{premium})\le\frac35\) (for example, zero). Pooling on premium cannot be a pure-strategy equilibrium.