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bid-price-modeling.md

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+---
+id:
+aliases: []
+tags:
+  - authorship/llm
+  - destiny/fleeting
+  - status/complete
+  - topic/estimating
+  - topic/risk
+title: Modeling Bid Prices Under Intrinsic Cost Uncertainty
+---
+# Modeling Bid Prices Under Intrinsic Cost Uncertainty
+
+> [!warning]
+> This text is almost entirely LLM output.
+> I don't intend to keep or use any significant portions of it.
+
+Consider a construction project characterized by an intrinsic but unknown final cost $C$.
+Prior to project completion, $C$ cannot be observed directly;
+instead the estimator possesses only a probability distribution over feasible outcomes.
+
+Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [probability space](https://en.wikipedia.org/wiki/Probability_space)
+describing the estimator's uncertainty,
+and define the true cost model as a non-negative random variable
+
+$$
+C : \Omega \to [0,\infty)
+$$
+
+with [distribution](https://en.wikipedia.org/wiki/Probability_distribution) $\mu_C$.
+
+The distribution $\mu_C$ summarizes all available information at the time of bidding,
+including quantities, labor productivity uncertainty,
+market conditions, subcontractor pricing variability,
+and correlation structures inherent to the estimator's assumptions.
+
+Although the natural mathematical representation of cost is thus a distribution,
+procurement mechanisms typically require that each bidder submit a single deterministic price.
+
+Denote this bid by
+
+$$
+B \in [0,\infty).
+$$
+
+> Read as
+> "$B$ is an [element](https://en.wikipedia.org/wiki/Element_(mathematics))
+> of the interval from zero to infinity, including zero."
+
+A bid $B$ may be viewed as the output
+of a pricing [functional](https://en.wikipedia.org/wiki/Functional_(mathematics))
+
+$$
+\Phi : \mathcal{P}([0,\infty)) \to [0,\infty),
+$$
+
+mapping a cost distribution $\mu_C$ to a scalar.
+
+Examples of such functionals include:
+
+### 1. Risk-neutral expectation
+
+$$
+\Phi(\mu_C) = \mathbb{E}[C],
+$$
+
+> Read as "Phi of mu sub C equals the expected value of C."
+
+where $\mathbb{E}[\cdot]$ denotes the [expected value](https://en.wikipedia.org/wiki/Expected_value).
+
+### 2. Risk-adjusted expectation
+
+$$
+\Phi(\mu_C) = \mathbb{E}[C] + \lambda\sqrt{\mathrm{Var}[C]},
+$$
+
+> Read as "Phi of mu sub C equals the expected value of C plus lambda times the square root of the variance of C."
+
+where $\mathrm{Var}[C]$ is the [variance](https://en.wikipedia.org/wiki/Variance)
+and $\lambda>0$ is a risk-loading parameter.
+
+> This mirrors mean--variance pricing common in portfolio theory.
+
+### 3. Quantile-based pricing
+
+$$
+\Phi(\mu_C) = Q_p(C),
+$$
+
+> Read as "Phi of mu sub C equals the p-quantile of C."
+
+where $Q_p$ is the $p$-[quantile](https://en.wikipedia.org/wiki/Quantile)
+of the distribution.
+
+### 4. Utility-maximizing bid
+
+Under a bidder [utility](https://en.wikipedia.org/wiki/Utility) function $U$,
+
+$$
+\Phi(\mu_C) = \arg\max_{b\ge0} \; \mathbb{E}[\,U(b - C)\,].
+$$
+
+> Read as "Phi of mu sub C equals the argument b greater than or equal to zero
+> that maximizes the expected value of U of b minus C."
+
+> [$\arg\max$](https://en.wikipedia.org/wiki/Arg_max) is the value of $b$ that maximizes the expression.
+
+***
+
+The central tension is:
+
+* The ontologically correct representation of project cost prior to execution is a **probability distribution**, whereas
+* The procurement mechanism requires a **deterministic scalar**.
+
+The study of such pricing functionals $\Phi$ sits within
+[stochastic optimization](https://en.wikipedia.org/wiki/Stochastic_optimization),
+[risk measures](https://en.wikipedia.org/wiki/Risk_measure),
+and [mechanism design](https://en.wikipedia.org/wiki/Mechanism_design).
+Understanding how different choices of $\Phi$
+compress and distort the underlying uncertainty $\mu_C$
+has direct implications for bidder profitability, competitive strategy,
+and how risk is allocated across the construction market.