Pricing Individual Deals

In many B2B markets, prices are not posted—they are quoted individually. Auto lenders, trucking companies, and telecom providers set a unique price for each customer or deal. The bid-response function replaces the aggregate demand curve as the analytical foundation.

The Problem

A commercial insurance company receives 1,000 quote requests per month. For each, it must set a premium. Too high: the customer goes to a competitor. Too low: the company wins the deal but earns insufficient margin. The challenge is to optimize individual quotes when each customer’s price sensitivity differs.

Unlike posted pricing, where a single price is offered to all customers, customized pricing sets a potentially different price for every transaction. The firm does not observe a demand curve in the traditional sense; instead, each quote is a one-shot decision against a single customer with an unknown reservation price. The analytical tool for this setting is the bid-response function.

Bid-Response Functions

Definition — Bid-Response Function

The bid-response function $w(p)$ gives the probability that the customer accepts a quote of $p$ . It is analogous to the demand function $d(p)$ but applies to a single customer or deal rather than an aggregate market.

$w(p)$ is decreasing in $p$ : higher quotes reduce the chance of winning.
$w(0) \approx 1$ and $w(p) \to 0$ as $p \to \infty$ .

The most common parametric form is the logistic bid-response function:

w(p) = \frac{1}{1 + e^{\alpha + \beta p}}

(1)

where $\alpha$ is an intercept capturing the customer’s baseline willingness to accept (higher $\alpha$ means less willing), and $\beta > 0$ governs price sensitivity. The logistic form ensures $w(p) \in (0, 1)$ and produces a smooth S-shaped curve.

Expected Contribution

The firm’s expected contribution from a single quote is the margin times the win probability:

E[\pi] = (p - c) \cdot w(p)

(2)

where $c$ is the cost to serve (or the reservation price of the seller). This function is bell-shaped: it starts at zero when $p = c$ (no margin), rises as the margin increases, then falls as the win probability drops.

Optimal Quote

Differentiating $E[\pi]$ with respect to $p$ and setting the result to zero yields the first-order condition:

w(p^*) + (p^* - c) \cdot w'(p^*) = 0

(3)

This has the same structure as the posted-pricing FOC $d(p^*) + (p^* - c) \cdot d'(p^*) = 0$ , but applied at the individual level. The optimal quote balances the marginal gain from a higher price against the marginal loss of win probability.

Optimal Win Rate Is Below 50%

At the optimal quote $p^*$ , the win probability $w(p^*)$ is strictly less than 50% whenever the margin $(p^* - c)$ is positive. In the logistic case, $w(p^*) = 1/(1 + e^{\alpha + \beta p^*})$ , and the FOC implies $w(p^*) < 1/2$ for all $p^* > c$ .

Implication: Optimizing for win rate leaves profit on the table. A firm that targets a 50% win rate is systematically underpricing.

This is the central insight of customized pricing. Sales teams that are compensated on win rate, or that feel discomfort at “losing” more than half their quotes, are leaving substantial expected profit unrealized. The interactive explorer below lets you verify this result by observing that the optimal price always lies to the right of the 50% win-rate price.

Customer Scoring

Different customers have different bid-response functions. In the logistic specification (Eq. 1), the parameter $\alpha$ captures baseline willingness to pay: a customer with a low $\alpha$ (or negative) is more likely to accept any given quote, while a high $\alpha$ indicates a customer who is harder to win.

Definition — Customer Score

A customer score is an estimate of the bid-response parameter $\alpha$ derived from observable customer features: industry, company size, relationship length, number of competitive alternatives, and prior purchase history. The score maps each customer to a personalized bid-response function, enabling differentiated quotes.

Firms estimate $\alpha$ using logistic regression or machine learning models trained on historical win/loss data. The features might include:

Industry vertical—some industries are more price-sensitive than others
Deal size—larger deals often have more competitive scrutiny
Relationship tenure—longer relationships reduce switching propensity
Number of competitors—more alternatives push $\alpha$ higher (lower acceptance at any price)

A customer with a low score (price-sensitive) receives a lower optimal quote; a customer with a high score (price-insensitive) can be quoted more aggressively. The interactive explorer below illustrates this by letting you toggle between customer types.

Portfolio Effects

When managing a pipeline of deals, the firm should not optimize each deal in isolation. Consider two extremes: a $500k deal and a $20k deal. If the firm has the same bid-response function for both, the optimal standalone quotes would imply the same win probability. But from a portfolio perspective, it may make sense to accept a slightly lower win rate on the large deal (quoting more aggressively on price) and a higher win rate on the small deal.

The intuition is straightforward: the marginal value of margin is higher on large deals. A 5-percentage-point increase in margin on a $500k deal is worth $25k; on a $20k deal, it is worth only $1k. The portfolio perspective also diversifies risk: winning many small deals at high win rates provides a stable revenue base, while large deals contribute disproportionately to profit.

The portfolio simulator below generates 30 randomly drawn deals and compares independent optimization with uniform win-rate targeting. Independent optimization yields higher total expected profit because it sets different win rates for different deals, pricing each according to its own bid-response function and deal economics.

Interactive Explorer

Use the sliders to change the cost structure, customer type, and price sensitivity. The charts update in real time.

Top: The blue sigmoid curve shows win probability $w(p) = 1/(1 + e^{\alpha + \beta p})$ . The dashed horizontal marks the 50% win rate. The blue dot shows the win probability at the optimal quote—always below 50%. Bottom: The green bell-shaped curve shows expected contribution $E[\pi] = (p - c) \cdot w(p)$ . The dot marks the maximum. Toggle “Overlay all types” to compare customer segments on the same chart.

Key Insights

1. The Optimal Bid Implies a Win Rate Well Below 50%

The first-order condition for the optimal quote guarantees that $w(p^*) < 0.5$ whenever the firm earns a positive margin. A firm that wins more than half its quotes is almost certainly leaving profit on the table. Use the bid-response explorer to verify: at the default parameters, the optimal win rate is around 30–40%.

2. Win-Rate Optimization Destroys Profit

Sellers who optimize for win rate rather than expected contribution leave enormous profit on the table. The metrics panel in the bid-response optimizer shows the expected contribution at a 50% win-rate price versus the optimal price. The gap can be 20–40% of potential profit.

3. Customer Heterogeneity Demands Different Prices

Different customers have different bid-response functions, which means different optimal prices. A price-insensitive customer (high $\alpha$ ) should receive a higher quote; a price-sensitive customer (low $\alpha$ ) should receive a lower one. Uniform pricing across heterogeneous customers is suboptimal.

4. The Bid-Response Function Is the Individual-Level Analog of the Demand Curve

In posted pricing, $d(p)$ maps price to aggregate quantity. In customized pricing, $w(p)$ maps price to the probability of winning a single deal. The first-order conditions have identical structure; the difference is in interpretation (probability versus quantity) and in the heterogeneity across customers.

5. Portfolio Optimization Implies Lower Win Rates on Large Deals

When managing a pipeline, the firm should quote more aggressively (higher prices, lower win rates) on large deals and less aggressively on small deals. The portfolio simulator demonstrates that independent optimization naturally produces this pattern and yields higher total expected profit than uniform win-rate targeting.

Extensions

The single-customer logistic model is a starting point. Practical implementations extend it in several directions:

Machine learning for bid-response estimation—Gradient-boosted trees and neural networks can capture nonlinear interactions between customer features that logistic regression misses. Ensemble methods trained on historical win/loss data are now standard in insurance, lending, and B2B services (Phillips, 2021).
Dynamic customized pricing—When the firm interacts with the same customer repeatedly, it can learn the customer’s bid-response function over time. This creates an explore-exploit tradeoff analogous to the bandit pricing problem, but at the individual level.
Auction mechanisms—When multiple bidders compete for the same contract, the bid-response function must account for competitive dynamics. First-price and second-price auctions induce different bidding strategies and different optimal reserve prices for the seller (Rothkopf and Harstad, 1994).

References

Phillips, R. L. (2021). Pricing and Revenue Optimization, 2nd ed.. Stanford University Press.
Rothkopf, M. H. & Harstad, R. M. (1994). “Modeling Competitive Bidding: A Critical Essay.” Management Science, 40(3), 364–384.
Talluri, K. T. & van Ryzin, G. J. (2004). The Theory and Practice of Revenue Management. Springer.