zmVault/bid-price-modeling.md

id, aliases, tags, title

id	aliases	tags	title
		authorship/llm destiny/fleeting status/complete topic/estimating topic/risk type/idea	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 describing the estimator's uncertainty, and define the true cost model as a non-negative random variable

C : \Omega \to [0,\infty)

with 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 of the interval from zero to infinity, including zero."

A bid B may be viewed as the output of a pricing functional

\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.

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 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 of the distribution.

4. Utility-maximizing bid

Under a bidder 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 is the value of b that maximizes the expression.

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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, risk measures, and 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.