Question
In a stylized trading game, a claim will pay either $80 or $120 tomorrow, each initially equally likely. Two brokers receive binary inspection notes. A correct note identifies the high or low payoff with probability 75%. With probability 1/2 the brokers have copied the same single note; otherwise they received independently generated notes, each conditionally correct with probability 75%. Whether they shared a note is hidden, and the sharing decision is independent of the payoff.
Each broker is committed to request one claim whenever its note says “high”; a request does not depend further on the dealer's quote. The dealer observes that both brokers request to buy and wants the conditional expected payoff per claim, not a revenue-maximizing offer. What is that fair value? How would the estimate be distorted by treating the two requests as independent confirmations?
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Hint 1
Separate the hidden “copied a note” case from the “two independent notes” case before applying Bayes' rule.
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Solution
Let \(H\) denote the $120 payoff and \(Q\) the event that both notes say high (hence two buy requests). Conditional on H, a shared note is high with probability \(3/4\), while two independent notes are both high with probability \((3/4)^2\). With equal probabilities of sharing and independent generation,
\[
P(Q\mid H)=\tfrac12\cdot\tfrac34+\tfrac12\left(\tfrac34\right)^2
=\frac{21}{32}.
\]
Under the $80 payoff, the probabilities of a false-high note are \(1/4\), so
\[
P(Q\mid L)=\tfrac12\cdot\tfrac14+\tfrac12\left(\tfrac14\right)^2
=\frac5{32}.
\]
With 50–50 prior odds, Bayes' rule gives
\[
P(H\mid Q)=\frac{21/32}{21/32+5/32}=\frac{21}{26}.
\]
The conditional claim value is
\[
E[V\mid Q]=120\frac{21}{26}+80\frac5{26}
=\frac{2920}{26}=\boxed{\frac{1460}{13}\approx\$112.31}.
\]
If the dealer incorrectly assumed independent notes with no chance of sharing, the likelihood ratio would be \((3/4)^2/(1/4)^2=9\), giving posterior \(9/10\) and value \(80+40(0.9)=\$116\). The extra $3.69 is spurious confidence from effectively counting the same inspection twice. The fair value is per claim; if both identical claims fill at a price set from this information, total expected underlying payoff is twice this per-claim value.
Assumptions
The claim payoff is common to both brokers and takes values 80 or 120 with equal prior probability.
The same-note event occurs with probability 1/2 independently of claim quality; conditional on no sharing, notes are independent given quality.
A high note deterministically generates a one-claim buy request; requests reveal no separate information beyond their notes.
Final Answer
\(P(120\mid\text{two buys})=21/26\); conditional fair value \(80+40(21/26)=1460/13\approx\$112.31\). Incorrect independent-signal value: $116.