Bid Markup Decision

Friedman-Gates · Bayesian Bid Model · Strategic

Construction · Finance & Bid · Strategic

Every tender confronts the same tension: mark up the bid higher to earn more on winning projects, but lower the probability of winning. The classical Friedman (1956) model and Gates (1967) refinement turn this into a Bayesian decision problem: given the estimated cost $ C $ and a distribution of competitor bid markups learned from historical data, pick the bid $ B $ that maximises expected profit $ E[\pi] = (B - C) \cdot \Pr(\text{win} \mid B) $. Friedman multiplies independent win probabilities; Gates uses a sum-based form that handles correlation better. Ahmad & Minkarah (1988) showed how to update the competitor-bid prior as new bids are observed.

Where This Decision Fits

Tender preparation — the highlighted step is what this page optimises

RFP ReceivedTender documents

→

Cost EstimateBOQ takeoff

→

Competitor PriorHistorical bids

→

Bid MarkupMaximise E[profit]

→

Submit BidWin or lose

The Problem

Expected-profit-maximising bid under competition

We have an estimated direct cost $ C $ for a project. We must submit a bid $ B > C $; our markup is $ m = (B - C) / C $. Competitors $ i = 1, \ldots, n $ will each submit a bid $ B_i $; we win if $ B < B_i $ for all $ i $. Competitor bids are modelled as random with cumulative distribution $ F_i(b) = \Pr(B_i \le b) $ estimated from historical bidding data (same or similar projects, same competitors).

The probability of winning given bid $ B $ depends on the model:

Friedman (1956) — independent competitors $$ \Pr_{\mathrm F}(\text{win} \mid B) \;=\; \prod_{i=1}^{n} \Pr(B_i > B) \;=\; \prod_{i=1}^{n} \bigl(1 - F_i(B)\bigr) $$ Gates (1967) — aggregated win probability $$ \Pr_{\mathrm G}(\text{win} \mid B) \;=\; \frac{1}{1 + \sum_{i=1}^{n} \dfrac{F_i(B)}{1 - F_i(B)}} $$ Expected profit $$ E[\pi \mid B] \;=\; (B - C) \cdot \Pr(\text{win} \mid B) $$ Optimal bid $$ B^{\star} \;=\; \arg\max_{B > C}\; E[\pi \mid B] $$

The Friedman model treats each competitor as placing its bid independently. Gates' adjustment handles the realistic case where competitor bids share a common cost-shock component (same material prices, same labour market). Both models degenerate to the same optimum when $ n = 1 $ but diverge as $ n $ grows — Gates is generally less aggressive (lower optimal markup) because his formulation penalises inflated bids more.

A common parametric assumption is that each competitor's bid is log-normally distributed around the true cost, $ B_i / C \sim \mathrm{LN}(\mu, \sigma^2) $, or equivalently that each competitor's markup is $ \mathcal{N}(\mu, \sigma^2) $. Historical bid-to-estimate ratios on similar projects provide the prior. Ahmad & Minkarah (1988) give the Bayesian update rules as new projects' bids are observed.

Cash flow — the downstream consequence of winning the bid

Try It Yourself

Trace expected profit over markup; compare Friedman vs Gates

Bid Markup Solver

4 competitors · Log-normal

Choose a Scenario

Own cost estimate C (k\u20ac)

\u20ac5000 k

Number of competitors n

Mean competitor markup $ \mu $ (%)

12%

Competitor markup spread $ \sigma $ (pp)

4 pp

Win-probability model

Friedman (1956) — independent

Gates (1967) — aggregated

Competitor bid distribution + your bid position

Expected profit over markup

Peak of the gold curve = optimal markup. Dashed red = Gates alternative.

Adjust terms and click Compute.

The Algorithms

Two classical models and a modern Bayesian update

Friedman (1956)

Independent Competitors

O(M n) \u2014 grid of M bid levels over n competitors

Each competitor's bid is an independent random variable. The probability of beating all of them is the product of individual beat-probabilities: $ \prod_i (1 - F_i(B)) $. Pedagogically clean but optimistic — it ignores the fact that competitors share common cost shocks (same steel prices, same labour market). Implemented on this page as the default.

Gates (1967)

Aggregated Win Probability

O(M n) \u2014 same grid search; different win formula

Gates argued that Friedman's product form systematically overestimates winning probability because it compounds low-probability tails. His alternative: $ \Pr_G = [1 + \sum_i F_i/(1-F_i)]^{-1} $. The two models agree for $ n = 1 $ but Gates is less aggressive at higher $ n $ — empirical tests on historical data (Carr 1982, Skitmore 1987) generally side with Gates.

Bayesian

Ahmad-Minkarah Bayesian Update

Per-bid update: conjugate normal posterior

Ahmad & Minkarah (1988) showed the Friedman and Gates priors can be updated sequentially as new competitor bids are observed. If the markup distribution is normal, the conjugate update gives a closed-form posterior. Wanous, Boussabaine & Lewis (2000)'s extension incorporates competitor-specific win rates and project-type corrections. Practical contractors use a rolling 2–3-year bid history as the prior sample.

Real-World Complexity

Why bidding in practice goes beyond Friedman-Gates

Beyond Textbook Bid Modelling

Unbalanced bids — Front-loading quantities, accelerating early-payment items. Owners penalise this heavily; regulation caps how skewed a bid can be.
Bonding capacity limits — Surety bonds are capped per firm. Bidding aggressively on two large projects simultaneously may exceed capacity; the decision is coupled with portfolio selection.
Pre-qualification effects — Not all competitors are equally qualified; a technical-qualifying score precedes price evaluation in many owners' procedures. Friedman-Gates assumes lowest-bid-wins.
Strategic / game-theoretic competition — At high n and low project variance, behaviour drifts from i.i.d. to game-theoretic equilibrium. Common-value auction theory (Wilson) applies; winner's curse is a real cost.
Collusion & bid rigging — Rare but consequential; Friedman-Gates break down when competitors coordinate.
Owner preferences & relationships — Repeat-owner relationships, prior performance, schedule preference. Bid price is not the only input to award.
Risk-adjusted utility — Risk-averse contractors maximise $ E[U(\pi)] $ not $ E[\pi] $. Mean-variance or CVaR-adjusted formulations apply for bonding-constrained firms.

Related Strategic & Finance Variants

Bid markup is the upstream financial decision

Portfolio Selection. Decide which bids to pursue at all. Joint bid-selection + markup is a two-stage stochastic problem. → Open Portfolio Selection

Cash Flow. Won-bid cash flow is the downstream consequence of markup choice. Higher markup = more profit cushion to reduce borrowing. → Open Cash Flow

Site Selection (UFLP). Owner-side bid markup decisions flip; the owner is selecting a site / contractor, not bidding. → Open Site Selection

Time-Cost Tradeoff. Earlier completion (crashing) lets a contractor bid tighter by reducing financing cost; couples TCTP into bid pricing. → Open Time-Cost Tradeoff

Stochastic RCPSP. Uncertain durations $\to$ uncertain cost $\to$ uncertain optimal markup. Bayesian bid + stochastic schedule is an active research intersection. → Open Stochastic RCPSP

Auction Theory. Common-value auctions (Wilson, Milgrom-Weber) generalise Friedman-Gates; game-theoretic equilibrium bidding is an open research frontier in construction.

Key References

Cited above · DOIs & permanent URLs

Friedman, L. (1956).

“A competitive-bidding strategy.”

Operations Research, 4(1), 104–112. (The foundational model; multiplication of independent win probabilities.) doi:10.1287/opre.4.1.104

Gates, M. (1967).

“Bidding strategies and probabilities.”

Journal of the Construction Division, 93(1), 75–107. (The Gates aggregated-win-probability formula — less aggressive than Friedman.)

Ahmad, I., & Minkarah, I. (1988).

“Questionnaire survey on bidding in construction.”

Journal of Management in Engineering, 4(3), 229–243. (Practitioner survey; Bayesian update rules for competitor priors.) doi:10.1061/(ASCE)9742-597X(1988)4:3(229)

Carr, R. I. (1982).

“General bidding model.”

Journal of the Construction Division, 108(4), 639–650. (Empirical comparison of Friedman vs Gates on historical data; Gates performs better.)

Skitmore, M. (1987).

Construction Prices: The Market Effect.

University of Salford. (Book-length treatment of bidding models and empirical testing.)

Wanous, M., Boussabaine, A. H., & Lewis, J. (2000).

“To bid or not to bid: a parametric solution.”

Construction Management and Economics, 18(4), 457–466. (Parametric bid/no-bid decision with competitor-specific rates.) doi:10.1080/01446190050024879

Li, H., & Love, P. E. D. (1999).

“Combining rule-based expert systems and artificial neural networks for mark-up estimation.”

Construction Management and Economics, 17(2), 169–176. (AI-based markup estimators.) doi:10.1080/014461999371691

Benjamin, N. B. H. (1969).

“Competitive bidding: the probability of winning.”

Journal of the Construction Division, 95(CO2), 313–330. (Early refinement and practical estimation.)

Park, W. R., & Chapin, W. B. (1992).

Construction Bidding: Strategic Pricing for Profit, 2nd ed.

Wiley. (Practitioner textbook covering Friedman, Gates, Carr, and their applications.)

Bidding in a competitive market?

Friedman-Gates turns bid markup from intuition into a Bayesian decision anchored on historical data. Let's discuss how OR-informed bidding can improve your hit rate and profit per win.

Connect on LinkedIn

Disclaimer

Cost estimates, competitor markup distributions, and win probabilities are illustrative. Real bid decisions require project-specific competitor-history data, qualification effects, owner preferences, and risk-adjusted utility not shown here.