vault backup: 2026-01-09 14:57:19

bid-price-modeling.md

@@ -12,8 +12,15 @@ title: Modeling Bid Prices Under Intrinsic Cost Uncertainty
 ---
 # Modeling Bid Prices Under Intrinsic Cost Uncertainty
 
+The cost of a construction project is inherently uncertain until it is completed,
+therefore the most accurate model of cost is a distribution of possible costs.
+Customers request bids as a single cost, however,
+so a contractor must determine some function
+to convert from the true cost model to a single bid price.
+
 > [!warning]
-> This text is almost entirely LLM output.
+> From this point forward,
+> 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 $C$.
@@ -28,9 +35,12 @@ $$
 C : \Omega \to [0,\infty)
 $$
 
+> Read as
+> "C is a function from omega to the interval from zero to infinity, including zero."
+
 with [distribution](https://en.wikipedia.org/wiki/Probability_distribution) $\mu_C$.
 
-The distribution $\mu_C$ summarizes all available information at the time of bidding,
+The distribution $\mu_C$ accounts all available information at the time of bid,
 including quantities, labor productivity uncertainty,
 market conditions, subcontractor pricing variability,
 and correlation structures inherent to the estimator's assumptions.
@@ -59,7 +69,7 @@ mapping a cost distribution $\mu_C$ to a **scalar** (a single value).
 
 Examples of such functionals include:
 
-### 1. Risk-neutral expectation
+## 1. Risk-neutral expectation
 
 $$
 \Phi(\mu_C) = \mathbb{E}[C],
@@ -69,7 +79,7 @@ $$
 
 where $\mathbb{E}[\cdot]$ denotes the [expected value](https://en.wikipedia.org/wiki/Expected_value).
 
-### 2. Risk-adjusted expectation
+## 2. Risk-adjusted expectation
 
 $$
 \Phi(\mu_C) = \mathbb{E}[C] + \lambda\sqrt{\mathrm{Var}[C]},
@@ -80,9 +90,9 @@ $$
 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.
+> This mirrors mean-variance pricing common in portfolio theory.
 
-### 3. Quantile-based pricing
+## 3. Quantile-based pricing
 
 $$
 \Phi(\mu_C) = Q_p(C),
@@ -93,7 +103,7 @@ $$
 where $Q_p$ is the $p$-[quantile](https://en.wikipedia.org/wiki/Quantile)
 of the distribution.
 
-### 4. Utility-maximizing bid
+## 4. Utility-maximizing bid
 
 Under a bidder [utility](https://en.wikipedia.org/wiki/Utility) function $U$,
 
@@ -106,7 +116,7 @@ $$
 
 > [$\arg\max$](https://en.wikipedia.org/wiki/Arg_max) is the value of $b$ that maximizes the expression.
 
-***
+## Conclusion
 
 The central tension is: