Bridge Bid Decision: Expected Value and the Price of Imperfect Information
A contractor is deciding whether to bid a $42M design-build bridge. Preparing the bid costs $250,000 and is incurred whether or not the contract is won. If the contractor wins, it expects a net profit of $3.1M after all costs, including the bid cost. The estimated probability of winning is 0.30. A consultant offers a $90,000 pre-bid geotechnical review that would predict the owner's likely award decision. Historical accuracy: when the contractor ultimately wins, the review is favorable 85% of the time; when the contractor loses, the review is unfavorable 70% of the time. The contractor's risk appetite is modest, and a loss of the bid cost would consume 15% of the division's annual discretionary budget.
1. Calculate the expected monetary value of bidding without the review and compare it with not bidding. What is the risk-neutral recommendation?
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Not bidding yields $0. Bidding incurs the $250,000 bid cost in either outcome, and the stated $3.1M is net profit after all costs including the bid cost. So:
EV(bid) = 0.30 × $3.1M + 0.70 × (−$0.25M) = $0.93M − $0.175M = $0.755M. EV(no bid) = $0.
The risk-neutral recommendation is to bid, because the expected value is positive by $755,000. The result is sensitive to the win probability: break-even occurs when p × $3.1M − (1 − p) × $0.25M = 0, which gives $3.35M × p = $0.25M, or p = 7.46%. The estimated 30% is well above that, so the bid decision is robust to a large error in the win probability. However, the expected value ignores the division's modest risk appetite and the fact that a loss would consume 15% of discretionary budget; that is a separate judgement.
2. Compute the posterior probability of winning given a favorable and an unfavorable review, and the expected value of the review. Should the contractor buy it?
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Prior: P(W) = 0.30, P(L) = 0.70. Review accuracy: P(F|W) = 0.85, P(U|L) = 0.70, so P(F|L) = 0.30 and P(U|W) = 0.15.
P(F) = 0.30×0.85 + 0.70×0.30 = 0.255 + 0.210 = 0.465. P(W|F) = 0.255 / 0.465 = 0.5484. P(U) = 1 − 0.465 = 0.535. P(W|U) = (0.30×0.15) / 0.535 = 0.045 / 0.535 = 0.0841.
EV(bid | F) = 0.5484×$3.1M − 0.4516×$0.25M = $1.700M − $0.113M = $1.587M. EV(bid | U) = 0.0841×$3.1M − 0.9159×$0.25M = $0.261M − $0.229M = $0.032M.
With the option to decline after an unfavorable review, EV(with review) = 0.465×$1.587M + 0.535×$0 − $0.090M = $0.738M − $0.090M = $0.648M. This is less than the $0.755M from bidding without the review. Even if the contractor bids after an unfavorable review, the review's value is lower still. The review is not worth $90,000 under these assumptions; the improvement in decision quality does not cover its cost and the residual uncertainty.
3. How should the contractor's modest risk appetite affect the decision, and what conditions would make the review worthwhile?
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Risk appetite affects the decision because the expected-value advantage of bidding is only about $0.755M, while a loss costs $0.25M, or 15% of the division's discretionary budget. A modest appetite might still bid-the probability of loss is 70%, but the loss is bounded and the break-even win probability is only 7.5%. The more relevant question is whether the division can absorb the cash-flow timing and the opportunity cost of tying up bid resources. If the division faces a hard capital constraint, a risk-averse utility function, or a strategic need to avoid a public loss, it might decline or seek a partner to share the bid cost.
The review becomes worthwhile if its cost falls, its accuracy rises, or the prior win probability is closer to the break-even. A useful check is the expected value of perfect information: with perfect information, the contractor would bid only on wins, yielding 0.30×$3.1M = $0.93M before the cost of information. The EV of the current bid is $0.755M, so the maximum value of perfect information is $0.175M before accounting for the cost of information. A $90,000 review with imperfect accuracy cannot be worth more than that, which confirms the calculation.