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