13

Customized Pricing

13.1 BACKGROUND AND BUSINESS SET TING

The pricing tactics we have studied so far all assume that buyers and sellers interact in the same basic fashion. The seller offers a stock of goods to different customer segments, possibly through different channels. Customers arrive, observe a price, and decide whether to purchase. The seller decides what price to offer for each product to each customer type through each channel. The seller monitors sales of his goods and periodically updates the prices and/or availabilities he is offering. This is the list-pricing modality and it is characterized by the fact that the seller publishes to customers a price or set of prices that are generally final and nonnegotiable. All the methodologies covered so far, including peak load pricing, revenue management, and markdown management, assume list pricing.

List pricing may seem quite general—in fact, it is what first comes to mind when people think about pricing. However, there is another important pricing modality—customized pricing—that is of equal or greater importance in many industries. In customized pricing, potential customers approach the seller, one by one, and the seller quotes each one a price. Often (but not always) each potential buyer wants something different—either a different bundle of products or services, a different quantity, or a different variation on a basic product. In customized pricing, the seller can quote a different price for each request. A price list may exist, but it is usually an upper bound on the price quoted. The price quoted to a buyer can be based on knowledge of the individual buyer, the product(s) requested by the buyer, and other factors, such as general market conditions and competitive offerings. Following are some examples of customized pricing.

• Auto lending. e-Car is an online auto lender. A customer arriving at the e-Car website fills out a loan application with information about the customer’s identity. The loan application also specifies the term and amount of the loan, along with whether the loan would be used to purchase a new car, purchase a used car, or refinance an existing car loan. Using this information, e-Car will pull a credit score for the customer, estimate the riskiness of the loan, and determine whether to extend a loan to the customer.

If an applicant is judged too risky, e-Car will reject her application. If an applicant is judged creditworthy, e-Car will offer her a loan and quote an annual percentage rate (APR) for the loan along with the monthly payment. The customer has 45 days to accept the loan. If the customer does not accept the loan within 45 days, the offer expires and the customer is considered lost.

e-Car can use all of the information in the application including the customer’s credit score, the amount and term of the loan, and whether it is for a used car, a new car, or a refinance in determining the rate to quote to the customer. Since e-Car can quote different rates to different customers, this is a customized-pricing problem.¹

• Trucking services. Businesses that purchase trucking services usually solicit bids from a number of competing carriers, such as Old Dominion and YRC. Each carrier will submit a bid detailing a price schedule at which it will carry freight for the customer for some period of time, usually a year. Each carrier faces the problem of determining what price schedule to bid.

• Telecommunications services. A telecommunication company sells a menu of telecommunications services to corporate, educational, and government customers. It can determine a price to bid for each customer depending on the size and type of customer, the services it requires, and the amount of business it might generate. It has categorized its customers according to the division in Table 13.1. For each customer in each category, the telecommunication company bids annually to provide services for that customer in the following year.

• Automotive fleet sales. A metropolitan police department wants to purchase 50 police cars. It solicits bids from Ford, General Motors, and Chrysler. The bid solicitation describes the performance and options required. Each of the three manufacturers needs to determine what price to bid.

• Medical equipment. A hospital has decided to purchase a magnetic resonance imaging (MRI) machine to improve its diagnostic capabilities. It sends copies of a request for proposal (RFP) to the five leading MRI manufacturing companies. The RFP specifies the required capability as well as the criteria the hospital will choose to determine the winning bidder. Each of the manufacturing companies needs to determine what price to bid.

TABLE 13.1

Global telecommunications company customer categories

The customized-pricing modality has three characteristics that differentiate it from list pricing.

1. Customers approach the seller prior to seeing a final price. In the cases described above, the customer takes the initiative and describes what she wants to purchase prior to seeing the price.

2. The seller can quote a different price to each customer. The customized price can (and should) reflect the best information—including customer-specific information—that the seller has at that time. Of course, the seller will usually not have perfect information about the preferences of any individual customer or about competitive prices. But, as we will see, he can use statistical reasoning to increase his expected profitability for each bid.

3. In customized pricing, the seller can track lost business. In a customized-pricing situation, the seller will either win—that is, get the business—or lose. In some (but not all) situations, a losing seller can find out which of his competitors won the business. However, even when this information is not available, the buyer knows that a potential customer has inquired about his product and decided not to purchase. This is in contrast to list pricing, in which the seller can only observe how many units he sold—not how many customers considered buying his product but decided against it. This lost bid information is important because it can form the basis for statistical estimation of customer bid-response functions.

The existence of a bid or a price quote is a common element in customized pricing. In business-to-business settings, the quote is often in response to an RFP or a request for quote (RFQ) issued by a buyer. In business-to-consumer situations, such as unsecured consumer loans and mortgages, the final price quote usually occurs after a customer has made a phone call or filled out an application. In any case, the price quote occurs only after the seller knows something about the buyer and what she wants. This contrasts with list pricing, in which potential customers find prices by accessing a price list (online or in a catalog) or reading a label in a store and the prices need to be set before the seller has information about any individual customer.

List pricing and customized pricing often coexist, in the sense that a seller will use a list price to set a ceiling, while most customers are quoted a price at a discount from the list. For example, the manufacturer’s suggested retail price (MSRP) is usually an upper limit on the price for, say, a new car, with the selling price ultimately determined by some combination of the customer’s ability to negotiate and the dealer’s willingness to sell at a lower price.² In a similar vein, shipping companies maintain voluminous tariff schedules that specify list prices for shipping different types of freight between every zip code pair in the United States. However, only a small fraction of sales (usually less than 5%) takes place at list price. The rest moves at discounted rates negotiated individually with customers.

In the remainder of this chapter, we formulate the customized-pricing decision as an optimization problem and see how that problem can be solved to maximize the expected contribution margin from each bid. A key element in the customized-pricing problem is the bid-response function, which specifies the seller’s expectation of how each customer will respond to his bid price. We see how bid-response functions can be estimated for different customers who are seeking to purchase different products. We then look at how the customized-pricing model can be enhanced to deal with other pricing settings as well as objectives other than maximizing expected contribution. Finally, we examine how these computations fit into a general process for customized pricing.

13.2 CALCULATING OPTIMAL CUSTOMIZED PRICES

To motivate the formulation of the customized-pricing problem, we put ourselves in the place of a seller who has been asked to bid on a potential deal. Here, deal means a particular piece of business, which may be a single order or a contract to provide future products and services. Bid means the price we offer.³ This is a simplified setting because it assumes the buyer has dictated all elements of the deal and that the only decision on the part of the seller is what price to bid. In the real world, a seller may also have to decide on other elements in his bid, such as delivery date, contract terms and conditions, and even product functionality. Furthermore, we assume initially that the seller is submitting a single take-it-or-leave-it bid. Of course, in many situations, not only price but every element of the deal may well be on the table and there may be more than one round of negotiation.

The first thing we need to determine is our goal for this bid. That is, what is our objective function? For now, we assume that we want to maximize expected contribution from the bid. For a given price, p, expected contribution can be expressed as

The first term on the right-hand side of Equation 13.1 is the deal contribution—the margin we will realize if we win the bid. If the bid is to sell d units of an item with unit cost of c, then our deal contribution would be (p – c)d. The calculation of deal contribution would be more complicated if we were bidding to supply a complex bundle of products and services, each of which has a different unit cost, or if we were negotiating a contract for future services in which the volume and mix of services to be supplied were uncertain.

The second term on the right-hand side of Equation 13.1 is the probability that we will win the deal at a given price. Our uncertainty about winning at a given price derives from two sources.

• Competitive uncertainty. We do not know what our competitors will bid. In fact, in many cases we may not know the identity or even the number of competitors we face.

• Preference uncertainty. We usually do not know exactly what criteria (both conscious and unconscious) the buyer will use to evaluate competitive bids. Nor do we know for sure how the buyer values our brand and our product/service offerings relative to the competition.

In most situations, both preference uncertainty and competitive uncertainty will be present: we will not know exactly what our competitors are going to bid; nor will we know exactly how the buyer will choose among competing bids. Of course, if we knew what our competitors were bidding and how the buyer would choose among bids, our problem would be relatively simple—we would submit the highest-price bid that would win (assuming that we would be profitable at that price). However, in the real world, sellers rarely, if ever, have access to that level of information—at least not legitimately.

There is, however, one important situation in which preference uncertainty is absent—when we know that the buyer is going to choose the lowest bid. Many government agencies are required by law to purchase from the lowest bidder. In these situations, the agency often issues extremely detailed specifications for the goods to be purchased so that price is the only difference among bids. In a seller auction (also called a reverse auction), the buyer initially selects a set of acceptable suppliers and commits to purchase from the lowest bidder. In most seller auctions, each supplier has the opportunity to observe the bids of other suppliers and then to submit a lower bid if he desires to do so. The ability to rebid based on the bids of other suppliers is not addressed here.

13.2.1 Single-Competitor Model

Imagine you are an auto manufacturer bidding against a single competitor to sell 50 pickup trucks to a county park district. The park district will buy all 50 trucks from a single supplier and is committed by law to pick the lowest bidder. Each supplier has been asked to submit a single sealed bid. The bids are final, and the lower of the two bids will win. Your production cost per truck is $10,000. Based on past experience, your belief about what your competitor will bid can be described by a uniform distribution between $9,000 and $14,000, as shown in Figure 13.1. What should you bid to maximize expected profitability?

Call your bid p and the competing bid q. You will win if your bid is less than the competing bid—that is, if p < q. (For simplicity, we ignore the possibility of a tie.) Let ρ(p) be the probability that you would win if you bid price p. Then ρ(p) = 1 – F(p), where F is the cumulative distribution function of the competing bid—that is, F(x) is the probability that the competing bid will be less than x. Your probability of winning the bid as a function of price is shown in Figure 13.2. If you bid below $9,000, then you will win for sure, since you know that the competitor will bid more than $9,000. Your probability of winning the bid decreases linearly between $9,000 and $14,000 and is zero if you bid above $14,000.

Figure 13.1 Uniform probability distribution on a competitor’s bid.

Figure 13.2 Probability of winning the bid as a function of price when our belief about the competitor’s bid follows the uniform distribution in Figure 13.1.

The expression ρ(p) denotes the bid-response function for this deal: Figure 13.2 shows the bid-response function in the case described above. For each deal, the bid-response function specifies the probability of winning the deal as a function of our bid. It is the customized-pricing analog of the price-response function introduced in Chapter 3. Where the price-response function specifies total expected demand as a function of list price, the bid-response function specifies the probability of winning an individual bid as a function of bid. Like the price-response function, the bid-response function is downward sloping; that is, the probability of winning the bid decreases as the bid price increases. The bid-response probability is less than or equal to 1 for all prices and decreases to 0 at high bids. For the example, the bid-response function is given by

and the price that maximizes expected profitability can be found by solving

Figure 13.3 Expected profitability as a function of price bid in the example.

The objective function in Equation 13.3 is maximized at p* = $12,000. At this price, the probability of winning the bid is 2.8 – 12,000/5,000 = .4. The margin per unit if the deal is won is $12,000 – $10,000 = $2,000 and the expected margin is .4 × 50 × $2,000 = $40,000.

Figure 13.3 shows how expected profit varies as a function of bid. The expected profit function has the familiar hill shape. At a bid price below $9,000, we are certain to win the deal, but at a loss. Above $9,000, the probability of winning the bid decreases. Expected profit is positive for any bid above the cost of $10,000. Above $10,000, unit margin increases with increasing price, but this increase is counterbalanced by the decreasing chance of winning the deal. At $12,000, these two effects balance, and expected contribution is maximized. Above $12,000, decreasing chances of winning the deal overwhelm the increased margin. At any bid above $14,000, you are certain to lose the deal, so expected contribution is $0.

The optimization problem in Equation 13.3 is identical to the problem confronting a seller facing a deterministic price-response function of 50(2.8 – p/5,000)⁺ and a unit cost of $10,000. In both cases, the optimal price is $12,000. However, there is an important difference between the two situations. A seller facing a deterministic price-response function of d(p) = 50(2.8 – p/5,000) and charging $12,000 per unit will sell exactly 20 units and will make a profit of $40,000. But in the bid-response case, by bidding $12,000 per unit, the seller will either win the bid and sell 50 units, for a total contribution of $100,000, or the seller will lose the bid and realize nothing. Changing the bid does not change the number of units sold; it changes the probability of selling all the units. At the optimal bid of $12,000, the seller has a 40% chance of winning the deal and making $100,000 and a 60% chance of losing the deal and making nothing.

13.2.2 Multiple Competitors

We would naturally expect the addition of more competitors to reduce the probability of winning a deal. Otherwise, we would be willing to pay additional companies to compete against us—behavior that is never seen in real life. We can see the effect of multiple competitors using the simple example of the county park district. What if we faced two competitors instead of one bidding for the order of 50 pickup trucks? To simplify matters, let us assume we have identical beliefs about how both competitors will bid. That is, we believe that both competitor 1 and competitor 2 will each price according to the uniform distribution shown in Figure 13.1.

We also assume that the two competing bids will be independent—that is, that information about one competitor’s bid would not change our assessment of the bid distribution of the other competitor. While the independence assumption may generally be reasonable, there are situations where it is not appropriate. One case would be when we believe that our two competitors are colluding to coordinate their bids. The second case would be when the two competitors are not necessarily colluding but are basing their bids on joint information that we do not share. If both of the competitors share a common cost (say, from a common supplier) and that cost is unknown to us, learning what one of them planned to bid might enable us to estimate the common cost and therefore to come up with a better estimate of what the other competitor would bid. In that case, the bids would not be independent.

We will win the deal if and only if our bid is below the lower of the bids submitted by competitor 1 and competitor 2. For a given p, the probability that our bid will be lower than the bid from competitor 1 is 1 – F(p) = 2.8 – p/5,000, which, by assumption, is also the probability that our bid will be lower than the bid from competitor 2. Since competitors 1 and 2 are submitting independent bids, the probability that p will be lower than both competing bids is ρ(p) = (2.8 – p/5,000)². This bid-response function is shown in Figure 13.4. You can see that the addition of the second competitor decreases our chance of winning the bid for all realistic prices.

With two competitors, our expected contribution is given by 50(2.8 – p/5,000)² (p – 10,000). The price that maximizes expected revenue is p* = $11,333.33 per unit, with a corresponding probability of winning the bid of 28.4% and an expected total profit of $18,963. Note that the additional competition creates pain for us in two ways: our optimal bid is lower, so total contribution is lower if we win the deal, and our probability of winning the deal is lower, even at our lower bid. The overall effect is dramatic: Our expected contribution from the deal is reduced by more than 50% from $40,000 to less than $19,000. This is a graphic illustration of the power of competition to reduce both the price and the profitability of a deal. Obviously, there is a strong incentive for the buyer to increase the number of bidders—or at least make the bidders believe there are more competitors!

Figure 13.4 Our probability of winning the bid as a function of price with one competitor and with two identical competitors.

As a matter of reference, the probability that a bid p is the lowest when there are n competitors submitting independent bids is given by

where F_i(x) is the cumulative distribution function on the price bid by competitor i—that is, the probability that competitor i will bid less than x.

Equation 13.4 gives the probability that our bid will be the lowest in any situation where competitors are submitting independent bids. If the buyer is choosing a winner based strictly on lowest bid, then this also gives our probability of winning at any price; that is, ρ(p) = g(p). However, if the buyer is taking factors other than price into account when choosing a winner, then our probability of winning the bid will not be the same as our probability that we have submitted the lowest bid.

13.2.3 Nonprice Factors in Bid Selection

So far, we have assumed that the buyer makes her purchasing decision based entirely on price. This is the exception rather than the rule. Price is almost always a factor in bid selection, but it is rarely the only factor. It is much more common for a buyer to make her choice based on a combination of factors. Sometimes the buyer will lay out her decision criteria in an RFP, perhaps like this example:

• Previous experience with supplier—10%

• Quality of proposal—5%

• Technical fit of solution—25%

• Price—60%

In this case, the buyer is declaring that price will be the most important factor in her decision but that other factors will also be considered.⁴ In most cases, the buyer does not provide even that much information. From the sellers’ point of view, the buyer is going to make her decision based on some combination of objective factors, such as price and solution fitness, and subjective factors, such as the brand image of the seller, the buyer’s experience (or lack of it) with the different sellers, and even the personal relationship the buyer might have with a seller’s sales representative. These elements need to be incorporated into the process of optimal customized pricing.

To make this more specific, consider the case of a large insurance company deciding to purchase 150 laptop computers for a new branch office. The information technology (IT) department invites Lenovo and Hewlett-Packard (HP) to bid on this business. Lenovo and HP both know they are the only suppliers invited to bid. However, both of them also know that the company has only purchased Lenovo computers in the past, that its overall experience with Lenovo has been good, and that the purchasing agent has a good relationship with the Lenovo sales representative. In this case, both HP and Lenovo are likely to believe that Lenovo has an edge on this deal. In the extreme case, HP might believe that Lenovo has a lock on the deal and the buyer is only going through a competitive bidding process to satisfy its own internal procedures and to keep Lenovo honest. If HP truly believes this, then it is facing a bid-response function such that ρ(p) = 0 for any value of p ≥ c; that is, it stands no chance of winning the deal. HP might still bid—if only to keep pressure on Lenovo and to signal HP’s willingness to sell to the buyer in the future—but it would need to use these strategic considerations to determine which price to set.

In the more likely scenario, HP might feel that Lenovo has an edge but that HP has a shot at winning the business if it significantly beats Lenovo’s price. How can this situation be addressed? One possibility is for HP to estimate Lenovo’s bid premium on this deal. The HP sales representative might believe that HP needs to bid at least $200 per laptop less than Lenovo to win. This bid premium reflects the edge that Lenovo has won through its strong history and good relationship with the customer.

It is easy to incorporate this type of premium into the calculation of the optimal customized price. Let us assume for a moment that HP’s cumulative distribution on Lenovo’s bid is F(x); that is, for any x, HP believes that the probability that Lenovo will submit a total bid less than x is F(x). If Lenovo and HP were at parity—that is, neither one had a premium relative to the other—then the buyer would be choosing a supplier based strictly on price and, as described in Section 13.2.1, the bid-response function ρ(p) would be equal to 1 – F(p). The effect of a $200-per-unit premium for Lenovo is to shift this bid-response function to the left by 150 × $200 = $30,000, as shown in Figure 13.5. In other words, HP needs to bid $30,000 lower on the deal to achieve the same probability of winning as it would without the premium. For any potential bid p, HP’s probability of winning is the probability that Lenovo’s total bid is greater than p + $30,000 and HP’s bid-response function is ρ(p) = 1 – F(p + 30,000). As shown in Figure 13.5, the net effect is that, for any bid it might make, HP faces a lower probability of winning the deal than if it were at parity with Lenovo. In this situation, HP can calculate its optimal bid by solving Equation 13.4 using the shifted bid-response function (see Exercise 1).

Figure 13.5 Shifted bid-response function for HP when Lenovo enjoys a premium.

Let us pause here to consider several points. First, it is unlikely that HP would know Lenovo’s exact bid premium. In fact, HP might be uncertain about which seller actually enjoys a premium; that is, HP may not know whether the buyer favors HP or Lenovo or is truly neutral between the two vendors. In this case, the bid premium itself is a random variable. Second, the buyer has a strong motivation to mislead the sellers about her preferences—or at least to be selective with her communication. It is in the buyer’s best interest for each bidder to believe that the other bidder is preferred and therefore enjoys a premium. Thus, the buyer may overstate her satisfaction and loyalty to Lenovo when talking to the HP sales representative and to deny any loyalty to Lenovo when talking to the Lenovo sales representative. Of course, both Lenovo and HP should have savvy sales representatives who know enough to take such statements with the requisite grains of salt.

Third, regardless of the posturing and misrepresentation that may take place, at the end of the day each seller must take all the information it has available to it and craft the best bid that it can (or decide not to bid). Information is likely to be asymmetrical among sellers: As the incumbent, Lenovo should know much more about the buyer and her needs than HP does. Since different sellers have different levels of information, the bid-response functions are subjective and specific to each seller. In our example, Lenovo may submit a bid that it believes has a 70% chance of winning, while HP submits a bid that it believes has a 45% chance of winning. Obviously, these estimates are inconsistent with each other, but that is no more unusual than two people with different beliefs or information assessing different probabilities for a given team to win a football game.

13.2.4 The General Problem

The customized-pricing problem for a particular deal is

where ρ(p) is the bid-response function, and m(p) is deal contribution at price p. For our purposes, ρ(p) will be a continuous, downward-sloping function of p, and m(p) will be a continuous, upward-sloping function of p. This means that the expected contribution from each deal will be a hill-shaped function of price as shown in Figure 13.3. Assume that satisfying the order will have an incremental cost of c. Then, we set the derivative of (13.5) with respect to price to 0 and apply some algebra to obtain

The expression on the right-hand side of Equation 13.6 is the bid-response elasticity—that is, the percentage change in probability of winning the bid that will result from a 1% change in price.⁵ It is the customized-pricing version of own-price elasticity. A bid-price elasticity of 2 means that a 1% increase in price will lead to a 2% decrease in the probability of winning the deal. The expression on the left-hand side of Equation 13.6 is the inverse of the contribution margin ratio. In other words, at the optimal customized price, the bid-response elasticity is equal to the inverse of the contribution margin ratio:

Bid-response elasticity at optimal price = 1/(contribution margin ratio)

Equation 13.6 is the customized-price version of the condition for optimal list prices in Equation 5.7, which demonstrates the similarity of the customized-pricing optimization problem to the list price optimization problem. It also means that all of the conditions for existence and uniqueness of an optimal list price described in Section 5.2.1 can be applied to customized pricing.

Note that using probabilities to represent our chances of winning a deal does not imply that either the buyer or our competition is making decisions by flipping a coin or using any other type of random process. Indeed, we would expect the competition to be setting the prices they will bid based on rational analysis of the opportunity—including their best information on what they believe we will bid. The uncertainty in the process arises entirely from our lack of information on the behavior of the other players in the bidding game: the buyer and the competition.

13.3 BID RESPONSE

The most challenging part of optimizing customized prices is determining an appropriate bid-response function. If we had a bid-response function already in hand, finding the price that maximizes expected contribution is relatively simple—just solve the optimization problem in (13.5). So how do we estimate the bid-response function to use for a particular deal? There are at least three possibilities.

• We could use bottom-up modeling to derive a bid-response function from our probability distributions based on how we believe our competitors will bid and the selection process we believe the buyer will use. This is what is done in Sections 13.2.1 and 13.2.2.

• We could convene the people within the company who have knowledge of this particular deal, experience with the customer, understanding of the competition, and experience with similar bidding situations and derive a bid-response function based on their expert judgment. This approach is often used when a deal is particularly large, important, or unique. For example, the telecommunication company with the market segmentation shown in Table 13.1 might use this approach for preparing bids for the relatively small number of customers in the Global market segment.

• We could use statistical estimation based on the historical patterns of wins and losses we have experienced with the same (or similar buyers) in similar past bidding situations. For the Growth and Metro segments in Table 13.1, it would be impossible for the telecommunication company to convene a panel of experts to independently strategize and prepare a unique bid for thousands of bids per month. We describe an approach that uses statistical analysis of prior bids to estimate an optimal price for each bid moving forward.

Customized pricing requires two steps: estimating the bid-response function and then optimizing to find the price that maximizes the expected contribution for each bid. In this section, we consider estimation of the bid-response function.

13.3.1 Estimating the Bid-Response Function

We can use historical information about bids won and lost at different prices to estimate a bid-response function if three conditions hold.

1. The historical record includes bids for similar products to similar customers.

2. The current market conditions and products being offered are similar to those in the historical record.

3. There is a sufficient number of bids in the historical record to estimate a statistically significant bid-response function.

Under these conditions, many different functions can be used to represent bid response, and these functions can be fit to historical data in a number of different ways. This chapter does not try to cover all the different bid-response functions; nor does it cover all the possible estimation approaches, which would involve a long detour into statistical issues that are well outside the scope of this book. Rather, the chapter gives an overview of the most popular approach (maximum likelihood estimation) to estimating the most commonly used bid-response function (the logit function). Along the way, it mentions some of the alternatives and provides references to more extensive discussions of the pros and cons of different approaches.

The starting point for the bid-response estimation process is a bid-history database. A simple example of a bid-history database is shown in Table 13.2, which shows the bid prices and the results of 40 bids for a hypothetical item. Note that the bids range from $9.10 to $9.80. For the first bid, the bid price was $9.20 and the outcome was a win (W). On the other hand, for the third bid, the bid was $9.70 and the outcome was a loss (L).

We can visualize the relationship between price and wins and losses by assigning each win a value of 1 and each loss a value of 0. The outcomes of the 40 bids in Table 13.2 as a function of price are shown as a scatter plot in Figure 13.6. Graph A in Figure 13.6 shows the actual data; graph B in Figure 13.6 shows the same data with the points jittered—that is, a small random number has been added to each value. The sole purpose of jittering is to separate points representing multiple values; in analyses, only the unjittered values are used. It is apparent by looking at the chart that lower bids are more likely to win, but there is not a strict relationship between price and outcome: some bids at $9.50 were won and some bids at $9.30 were lost. We want to find the bid-response function that best captures our probability of winning a bid at each price based on the results of these 40 bids—recognizing before we start that a bid-response function estimated on such a small amount of data is unlikely to have much statistical significance.

TABLE 13.2

Sample bid prices and results

Figure 13.6 Wins and losses as a function of price. (A) The raw data; (B) the same data jittered by adding a small random number to each value.

Fitting a bid-response function to data such as those in Table 13.2 is a problem of statistical estimation similar to the problem of estimating a price-response model discussed in Chapter 4. In Chapter 4, the independent variables in the estimation of the price-response function are price and a set of exogenous variables, and the dependent variable is either total demand or the fraction of potential demand that would convert to actual demand. For customized pricing, the independent variable is price and all of the characteristics of a particular deal, and the dependent variable is the won/lost indicator, which can be either 0 or 1. This means that we want to fit a function that incorporates information about each deal to predict a binary dependent value. This is a problem of binary or Bernoulli regression.

The first question we need to answer is: What kind of function do we want to fit to the data in Table 13.2? One possibility is a linear model, which would correspond to a uniform probability density function on winning the bid, such as the one in Figure 13.1. The linear bid-response function that best fits the data in Table 13.2 can be determined using linear regression: it is ρ(p) = 14.8 – 1.51p, shown in graph A of Figure 13.7 along with the jittered win/loss data.

The linear price-response function in graph A of Figure 13.7 is not too unreasonable. It is continuous and downward sloping. Since it has a constant slope of –1.51, this price-response function implies that each increase of $0.01 in price would result in a decreased probability of winning the bid of about 0.015. However, there are problems with a linear bid-response function. First, for prices above $9.81 it predicts that the probability of winning is less than 0, and for prices less than about $9.14 it predicts win probabilities greater than 1. Since probabilities must be between 0 and 1, a realistic bid-response function would need to be capped at 0 and 1 as shown in graph B of Figure 13.7.

Even when capped, however, a linear price-response function is unrealistic. Inspection of Figure 13.7 indicates that the linear model (capped or uncapped) is not a very good fit to the data. For one thing, the linear model specifies that the change in probability from a $0.01 price is the same for any price between $9.14 and $9.81. However, we would usually expect that the effect of changing our price would be greater when our probability of winning the bid is around 50% than when it is very high or very low. This would imply that we should consider an alternative form for the bid-response function.

Consider the bid-reservation price—the maximum price at which the buyer would accept our bid. (The bid-reservation price is the customized-pricing equivalent of customer willingness to pay.) If the bid-reservation price is $9.50, then we will win the deal if we bid at any price of $9.50 or less, and we will lose the deal if we bid above $9.50. Of course, as a seller we do not know the buyer’s bid-reservation price—if we did, we would simply bid at that price (assuming, of course, that it is above our variable cost). This means we need to treat the bid-reservation price as a random variable, say, x, with a corresponding distribution f(x) and cumulative distribution function F(x). At any price p, our probability of winning the bid is simply the probability that p ≤ x; that is, ρ(p) = 1 – F(p) = F̄(p). Note that the cumulative distribution function incorporates both our preference uncertainty and our competitive uncertainty about the deal.

Figure 13.7 Linear bid-response functions fitted to the win/loss data from Table 13.2. (A) A pure linear bid-response function; (B) the same function capped at 0 and 1. The jittered win/loss values are also shown.

The logit bid-response function. What form would we expect the bid-response function, ρ(p), to have? If f(x) is uniform, then ρ(p) will be linear. But in most cases, we would expect f(x) not to be uniform but to be more or less bell shaped. When f(x) is bell shaped, the corresponding bid-response function ρ(p) has a reverse S shape—similar to the price-response function shown in Figure 3.6. When our bid price is very high or very low (relatively speaking), the effect of a small price change on our chance of winning the bid will be relatively small. However, when our price is in the more competitive intermediate range, we would expect that small changes in our price would have a larger effect on our chances of winning. Among the alternatives, by far the most common and convenient functional form for a bid-response function is the logit.

The logit bid-response function is given by

For a realistic bid-response function in which the probability of the bid is a decreasing function of price, we must have the parameter b > 0. Various properties of the logit are described in Section 3.3.4, and its slope, hazard rate, and elasticity are shown in Table 3.3. The logit provides a smooth, realistic bid-response function, with the probability of winning the bid approaching 0 as price increases indefinitely. Note that Equation 13.7 is identical to Equation 3.11, with A = 1. This highlights that customized pricing is similar to list pricing, but with a sequential series of market-of-one pricing decisions rather than a single price that a population of customers will all see.

Estimating a logit bid-response model. Section 4.2.2 describes how to use linear regression to estimate a logit distribution when the target variable is the fraction of potential demand that converts at a price. In that case, the target variable is the conversion rate for each period, which is between 0 and 1, and we can use the transformation ln[r_t/(1 – r_t)] to convert the underlying model to a linear model. However, in the case of bid-response estimation, the target variable is the win indicator for each bid i, with W_i = 1 indicating that we won the bid and W_i = 0 indicating that we lost the bid. In either case, the transformation ln[W_i/(1 – W_i)] is undefined, which means that we cannot convert the data to a linear model and use linear regression to estimate the parameters as we did with the logit model in Chapter 4. Fortunately, there is a very straightforward approach to the problem of logistic regression called maximum likelihood estimation (MLE).

We specify the problem as follows. We have a sequence of historical bid observations i = 1, 2, . . . , n. For each bid, we have the price that we bid, p_i, and the corresponding outcome, W_i. We want to find the values of the parameters a and b such that Equation 13.7 best predicts the observed values of W_i given the price bid p_i. For any values of the parameters a and b, the probability of winning bid i is ρ(p_i) = e^a–bpi/(1 + e^a–bpi), and the probability of losing bid i is 1 – ρ(p_i) = 1/(1 + e^a–bpi). We want to find the values of a and b that result in predictions of ρ(p_i) that are each as close to the corresponding values W_i as possible.

We can write the probability of realizing the outcome that we actually realized for bid i, given the parameters a and b and the price p_i, as

L_i(a, b) is called the likelihood of observing outcome i if we assume parameters a and b. Note that L_i(a, b) = ρ(p_i) if we won bid i, and L_i(a, b) = 1 – ρ(p_i) if we lost bid i. Now, assume that we have a set of n bids and their outcomes. Assuming that all of the bids are independent, the joint likelihood of observing the particular pattern of wins and losses that we actually achieved for bids i = 1, 2, . . . , n is

where the symbol means that we multiply the n individual elements together. MLE finds the values of a and b that maximize the likelihood of the actual observation. That is, MLE solves the nonlinear optimization problem max_a,b L(a, b), where L(a, b) is given by Equation 13.8.

With a large number of observations, the likelihood associated with even a very good predictor will be very low. For example, assume we had a model that could predict, with 90% accuracy, whether we would win or lose a bid—far higher than we will ever be able to achieve in practice. For a sample of 1,000 bids, the expected likelihood for such an estimator would be 0.9^1,000 = 1.75 × 10 ^{– 46}. Working with such tiny numbers presents all sorts of numerical problems. Therefore, the usual approach is to maximize the natural logarithm of the likelihood. Since the logarithm is an increasing function of the probability, the values of a and b that maximize likelihood will also maximize the logarithm of likelihood (and vice versa). So, in MLE, we find the values of a and b that solve the optimization problem

This is a much easier problem to work with than maximizing the likelihood function itself. Not only does it enable us to work with much more reasonable numeric values; it is additive, which makes it easier to work with in general. Note that, since the likelihood is always less than 1, the log likelihood will always be less than 0. Maximizing the log likelihood means finding the values that bring the value of Equation 13.9 as close to 0—the least negative value—as possible.

Solving Equation 13.9 for the data in Table 13.2 gives us the values of a = – 80.576 and b = 8.506 as the maximum likelihood estimators for the logit parameters. The corresponding logit bid-response function is shown in Figure 13.8. Note that these parameter values imply that we would expect to win 50% of our bids if we set a price of 80.567/8.506 = $9.47. The corresponding total log likelihood is –19.752. We can compare this to the log likelihood that would be obtained using an estimator independent of price—that is, ρ(p_i) = .575 for all values of i. The log likelihood for the constant estimator is –27.274. Using the logit response model has resulted in an improvement in log likelihood of about 30% over a constant estimator. In case this does not seem sufficiently impressive, we can look at it this way: it improves the likelihood by a factor of 1,808!

Figure 13.8 Logit bid-response function fit to the values in Table 13.2 by maximizing log likelihood. The parameters of the function are a = – 80.576 and b = 8.506.

13.3.2 Extension to Multiple Dimensions

We have seen how the parameters of the logit bid-response function can be estimated in the case of a single product being sold to a single customer segment through a single channel. In this case, we only need to estimate values for two parameters: a and b. However, in most cases we would expect there to be differences in customers, channels, and products. Some of the differences that can occur include the following.

Channel Variation

• Internet versus call center

• Retail versus wholesale

• Direct versus indirect

Product Variation

• Size of order

• Products ordered

• Configuration of products

• Ancillary services (e.g., extended warranty)

Customer Variation

• Size of customer

• Location

• Size of account

• Business

• New or repeat

To maximize profitability, a company would like to use as many of these variables as possible in setting its prices. A new, small customer with a large order entering through the internet should get a different price from a repeat large customer with a small order entering through a direct channel, and so on. In the extreme, we want to estimate different bid-response functions—and hence different optimal prices—for every element of the pricing and revenue optimization (PRO) cube in Figure 2.5.

Customized pricing is commonly used for highly configured products, in which case the product dimension of the PRO cube can be quite rich. One example is the heavy truck manufacturing industry:

The 2008 Model 388 Day Cab heavy truck sold by Peterbilt allows the customer to choose his desired options for thirty-four different components of the truck ranging from the rear-wheel mudflap hangers which can be straight, coiled, or tubular to fifteen different choices for rear axles to seven different transmissions to the choice between an aluminum or steel battery box. Multiplying the number of options in the Peterbilt list would imply that they could be combined in 75 × 10¹⁶ possible ways—each corresponding to a different truck. (Phillips 2012a, 467)

The fact that each customer is likely to be ordering a different truck gives wide scope for the price to be customized. The question is how to determine the influence of different combinations of product characteristic, channel, and customer characteristic to determine the best price to offer for each transaction. To do this, we need to have information from historical bids about the customer segment, channel, and product(s) associated with that bid as well as the price and whether we won the bid. This richer set of information is illustrated in Figure 13.9.

We incorporate additional dimensions in the bid-response function by using a multivariate logit function of the form

Here, p_ijk is the price we offer to a customer of type i, purchasing a product of type j, through channel k, and a_ijk and b_ijk are the parameters of the corresponding logit function. Given a sufficient number of bids, we could use MLE to estimate these parameters, establishing a separate bid-response function for each cell in the PRO cube.

The bid-response function specification in Equation 13.10 has a drawback: estimating all the values of a_ijk and b_ijk requires running MLE independently for each cell in the PRO cube. However, reliable estimation of the parameters of the logit bid-response function requires at least 200 historical observations. In general, we cannot expect to have this many observations for every cell. If we had 50 different products being sold to 10 different customer types through five channels, then our PRO cube would have 50 × 10 × 5 = 2,500 cells. For each cell to have 200 historical observations, we would need a historical-bid database with at least 2,500 × 200 = 500,000 entries. This approach would obviously be infeasible for the Peterbilt example above where there are 75 × 10¹⁶ potential products—not even including the customer or channel dimensions. Most companies do not have anywhere near this much historical-bid data. This means that we have to reduce the dimensionality of the problem.

Figure 13.9 Typical bid-history table template.

One way to reduce dimensionality is to define a single value of a and a single value of b for each product, customer, and channel value, rather than for each combination. In this case, changing our notation slightly, our model becomes

where a_i and b_i are the parameter values estimated for customer type i, a_j and b_j are those estimated for a product of type j, and a_k and b_k are those estimated for channel type k. This reduces the number of parameters required for the 50-product, 10-customer-type, 5-channel case from 5,000 to 2 × (50 + 10 + 5) = 130. The historical data required to derive a statistically reliable estimate of this smaller parameter set is much reduced.

To see how this works, consider the case of a company that sells three grades of printer cartridges—silver, gold, and platinum—to three types of customers: government agencies, educational institutions, and retailers. The seller has kept records on the outcome of each bid that it has made historically and the amount of business that it has done in the previous year in a format similar to that shown in Table 13.3.

The seller wishes to use this historical data to estimate a bid-response function. Define the following variables⁶ to represent the information in Table 13.3:

• GRADES = 1 if grade is silver, 0 otherwise.

• GRADEG = 1 if grade is gold, 0 otherwise.

• GRADEP = 1 if grade is platinum, 0 otherwise.

• PRICE = 1 for the price bid.

• SIZE = 1 for the order size.

• NEW = 1 if customer is new, NEW = 0 if it is a previous customer.

TABLE 13.3

Sample historic-bid data available to a printer ink cartridge seller

NOTE: The grade of the product ordered is S for silver, G for gold, or P for platinum. There are three types of customer: Edu is an educational institution, Gov is a government entity, and Ret is a retailer. Business level is the total amount of revenue that the cartridge seller accrued from the customer in the previous year.

• RET = 1 if customer is a retailer, RET = 0 otherwise.

• GOV = 1 if customer is a government agency, GOV = 0 otherwise.

• EDU = 1 if customer is an educational institute, EDU = 0 otherwise.

• BUSLEVEL = 1 for the total amount of business that has been done with the customer in the last 12 months.

The variables specifying the product grade, the type of customer, and whether the customer is new or old are called categorical variables. When there are two or more mutually exclusive categories such as GRADES, GRADEG, and GRADEP, exactly one of the variables will be equal to 1 with the others equaling 0 (assuming that there are no other alternatives). This means that GRADES + GRADEG + GRADEP = 1 and the variables are collinear, which means that if we know the values of two of the variables, we could calculate the value of the other; for example, if we know GRADES and GRADEG we can calculate GRADEP = 1 – GRADES – GRADEG. As a consequence of collinearity, when there are n mutually exclusive alternatives for a feature, we only need to include n – 1 categorical variables. This is obvious in the case when there are only two categories such as new or old: we do not need a variable named OLD; we can simply let NEW = 0 represent the case of an existing customer. When there are more than two alternatives, we can either choose which of the options to leave out or, with most regression software packages, allow the software to choose automatically. For this reason, we exclude GRADEP and OLD from the formulation here.

Define the following linear function:

The corresponding logit bid-response function is given by

The next step is to estimate the values of the coefficients β₀, β₁, . . . , β₈ that best fit the historical data. The same procedure of maximum likelihood estimation discussed in Section 13.3.1 can be applied to Equation 13.11, with the only difference being that, in this case, we are estimating nine coefficients, β₀, β₁, . . . , β₈, instead of two.

Equation 13.11 is not the only possible formulation of a bid-response model given the available data. For example, Equation 13.11 includes the term β₄ × SIZE, which specifies that the effect of order size on winning a bid is linear, all else being equal. It is not unusual for this effect to be significant—customers placing large orders tend to be more price sensitive than those placing smaller orders. However, it may be that this effect is not linear but tends to get weaker the larger the order. To represent this possibility, we might try replacing the term β₄ × SIZE with a term β₄ × ln(SIZE). (Recall that ln(SIZE) denotes the natural logarithm of the variable SIZE.) In addition, we might want to try a model in which the product of order size and price influences the probability of winning a bid, in which case we could add another term of the form β₉ × PRICE × SIZE.

The process of determining the best model to fit the data is discussed in Section 4.3.3—the same process holds for fitting bid-response data as price-response data. Generally, a modeler will use a forward stepwise regression process, which means starting with a simple model such as that in Equation 13.11 and then experimenting with adding transformed variables such as ln(SIZE) and crossed variables such as PRICE × SIZE. For each model, measures of model performance on historical data are calculated, and once multiple models have been tested, the model demonstrating the best performance on these measures is selected.

13.3.3 Binary Model Performance Measures

Section 4.2 describes three measures of model fit: root-mean-square error (RMSE), mean absolute percentage error (MAPE), and weighted MAPE. RMSE can be applied to binary models without any modification; however, neither MAPE nor weighted MAPE can be directly applied to binary models because they both require dividing by the outcome, which is often 0. Fortunately, there are two additional model-fit measures that are applicable to binary regression problems.

• Concordance (also called area under the curve, or AUC) measures the frequency with which a model estimated a higher conversion probability for accepted bids than for rejected bids. A concordance of 1 means the model is a perfect predictor—that is, there is some value of p̂ such that every bid for which the bid-response model estimated ρ(p_i) > p̂ was accepted and every bid for which ρ(p_i) < p̂ was rejected. A concordance of .5 means that the model was no better than random chance and a concordance less than .5 means that the model was worse than chance. The higher the value of the concordance, the better job the model does of predicting success. In practice concordance is typically between .5 and 1.0, with a higher value indicating a better-fitting model.

• Log-likelihood is calculated according to Equation 13.9. It measures the probability of observing the pattern of wins and losses in the test data set if the underlying model were correct. Log-likelihood is always less than 0, with larger values (i.e., less negative) indicating a better-fitting model.

Example 13.1

Assume that our test data set includes only five bids (a woefully inadequate number for any meaningful analysis) and that the predicted win probability ρ(p_i) from our challenger model and the actual outcome W_i are as shown in Table 13.4. The squared error for each bid is e² = [ρ(p_i) – W_i]², and the likelihood for each bid is L_i = ρ(p_i)W_i + [1 – ρ(p_i)](1 – W_i). The mean squared error of the model on the test data set is the average of the squared errors, which is .188, and the RMSE is the square root of this number, which is .44. The likelihood of the observed outcome is the product of the individual likelihoods: L = L₁ × L₂ × . . . × L₅ = .069; the log-likelihood is the natural logarithm of the likelihood, or ln(.069) = − 2.67. To calculate the concordance, we look at all pairs of bids in which one was accepted and the other was not (disparate pairs). There are six disparate pairs in Table 13.4. A disparate pair is concordant if the predicted win probability for the win was greater than or equal to the win probability for the loss. For example, the pair (1, 3) is concordant because the predicted win probability for bid 1 (which was won) is greater than the predicted win probability for bid 3 (which was lost). On the other hand, the pair (3, 4) is discordant because the predicted win probability for bid 3 (which was lost) is higher than the predicted win probability for bid 4 (which was won). Of the six disparate pairs, five were concordant and one was discordant, so the concordance is 5/6 = .833.

TABLE 13.4

Model predictions and actual bid outcomes

Each of these measures can be useful in comparing the performance of different models on the same test data set. In addition, concordance and RMSE can be useful in comparing models across data sets. Log-likelihood is not useful for comparing model performance on different test data sets because it depends strongly on the number of entries in the data set.

13.3.4 Endogeneity

Endogeneity can be an issue in estimation, as seen in Section 4.4.2. Endogeneity arises when unrecorded characteristics influence both the final price and the customer willingness to pay. This is a common problem in customized pricing when the final prices are set by negotiation with a salesperson. In this case, the local salesperson may observe customer or market attributes that are not recorded in the data and use those attributes in negotiating the final price.⁷ For example, a salesperson, noticing that a particular customer seems very eager to purchase, holds the line to achieve a higher price, while another customer seems less eager to purchase and the salesperson drops the price to close the deal. To the extent that the salesperson is correct in his estimation of customers’ willingness to pay, the price offered depends on customer willingness to pay through variables not recorded in the data. In this case, endogeneity will bias the coefficient of price in the estimation process.

The bias introduced by endogeneity can go either way. If sales staff systematically adjust prices toward customers’ willingness to pay (as they should to increase profitability), then a naïve estimation procedure that ignores endogeneity will underestimate customer price sensitivity. If, however, sales staff are using their pricing discretion on the average to adjust prices away from customer willingness to pay, naïve estimation will lead to price sensitivity being overestimated. If sales staff are varying prices in a way that is uncorrelated with customer willingness to pay, then they are just adding noise to the process and the estimate of price sensitivity will be unbiased.

When the variation of prices in historical data is due partially or entirely to sales force discretion, it is prudent to test for the presence of endogeneity and, if necessary, correct for it. One common approach to deal with endogeneity in customized pricing is to use a control-function approach, which proceeds in four steps:

1. Estimate an expected price for each bid. This is the price that would have been bid in the absence of local knowledge of unrecorded characteristics. One approach to estimate this expected price is to average the prices offered for similar bids during the same time period. For example, the expected price might be calculated as the average price of all bids or of similar products to similar customers. An alternative is to build a model that predicts expected price for each bid as a function of customer, product, and channel characteristics.

2. Calculate the differences between the expected prices estimated in step 1 and the actual prices bid.

3. Run a regression with Outcome as the target variable that includes the differences calculated in step 2. If these differences enter the regression as significant, this indicates that endogeneity is present.

4. If endogeneity is present, then the difference between the actual price and the expected price should be included as an explanatory variable in the regression.

Although this may sound complicated, the basic idea is simple. In step 1, we estimate the prices that would have been offered for each bid if the sales staff did not possess unobserved bid-specific information. This expected price is an instrumental variable because it is plausibly correlated with the bid price, but it is not correlated with the customer’s idiosyncratic willingness to pay. If the difference between expected price and the price actually offered is not significant, then endogeneity is not present and the results of a regression using Price as an independent variable will not be biased. If, however, the difference between predicted price and actual offered price is a significant predictor of Outcome, then this difference should be included as an explanatory variable in the regression. The coefficient of this variable will be a better estimate of price sensitivity than the coefficient of Price estimated through simple regression.

13.4 EXTENSIONS AND VARIATIONS

We have seen how a seller can find the customized price that will maximize expected contribution from a prospective deal in a simple bidding situation, assuming that the seller faces no constraints on the price he can bid. There are two obvious variations on this basic model:

• The seller might want to do something other than maximize expected contribution.

• The seller might be constrained in the price he can offer for this deal.

In the first case, the seller is pursuing a strategic goal that involves something other than maximizing expected deal contribution. If it is a goal that can be imposed on a deal-by-deal basis, then he can do this by simply maximizing a different objective function than Equation 13.5 for each deal. For example, if he wants to maximize expected revenue, he could substitute the objective function R(p) = ρ(p)p as an alternative to maximizing expected profit. The properties of revenue- and profit-maximizing prices are similar to those discussed in Chapter 5—the revenue-maximizing price is lower than the profit-maximizing price for each bid. Furthermore, the profit from a bid may be negative at the revenue-maximizing price. It is also possible that the seller is interested in multiple objectives, such as maximizing profit and maximizing sales. In this case, the efficient frontier described in Section 5.7 can be used to understand the trade-off between competing objectives.

The other case occurs when the seller is subject to one or more business rules that limit his choice of prices. These business rules could arise externally (it may be illegal to charge different prices to certain groups of customers) or internally (the company may wish to keep prices through one channel lower than prices through another channel, for example). In both cases, the effect is to make the customized-pricing problem more complex than solving the unconstrained optimization problem in Equation 13.5. In particular, business rules usually require adding constraints to the basic problem.

In addition, there are many cases in which the bid setting itself is more complicated than a simple optimization—bids may involve many different bundled products and services and may involve several rounds of negotiation. These variations require extensions to the simple model, which Section 13.4.2 discusses.

13.4.1 Strategic Goals and Business Rules

Consider the following situations.

• A package express company responds to 100,000 bid requests per year. Each request is for a contract to supply package express services for the next 12 months. The company currently seeks to maximize expected contribution for each bid, and it wins approximately 40% of the time. In 2021, senior management for the company declares that the company has an important new initiative with the goal “to become the dominant provider of shipping services to online retailers.” As part of this initiative, the CEO tells the executive vice president of sales that he now wants the sales force to win at least 75% of its bids to online retailers while maintaining a minimum level of profitability.

• Gammatek is a distributor that purchases various types of electronic equipment from manufacturers and sells the equipment to corporate, government, educational, and retail customers throughout North America. Gammatek has five divisions: CPUs, printers, storage devices, peripherals, and consumer electronics. It has about 23,000 customers and bids on about 400,000 pieces of business a year, with about 25% of the bid requests coming in to the call center and the remainder via the internet. Since Gammatek can set a different price for each bid, it has adopted a customized-pricing process. At the beginning of the 2021 fiscal year, Gammatek’s president announces that each division will have a minimum average margin target that it will be expected to meet for the coming year. These targets were adopted because of the perception that stock analysts will be watching margins particularly closely in the coming year because of a soft economy and are likely to reduce the rating of any company with eroding margins.

• The Bank of Albion in the UK receives about 1 million requests per year for consumer loans. For each loan, the bank quotes an APR based on the size and term of the loan, customer credit score, and channel through which the request was received. Its newest marketing campaign advertises that a “typical” £5,000 loan has an APR of 13.9%. This means that, by British law, the average APR for £5,000 loans issued by the bank must be less than or equal to 13.9%.

In each of these three cases, the seller needs to do something beyond simply determining the prices that maximize expected contribution for each bid. The package express company needs to calculate prices that enable it to hit its market-share target for the online retail market. It can meet this strategic goal by solving the customized-price optimization problem with a different objective function. Gammatek and the Bank of Albion want to maximize expected contribution, but they now face business rules that constrain the prices they can charge. In the case of Gammatek, the constraints arise from the internally imposed margin targets; in the Bank of Albion’s case, the constraint is an externally imposed legal requirement. In both cases, the business rules can be addressed by constraining the basic customized-pricing problem.

Let ρ_i(p) and f_i(p) be the bid-response function and the contribution, respectively, for a particular deal, i. One way for the package express company to meet its market-share target would be to solve the constrained optimization problem

subject to

whenever it is bidding to an online retailer. (Bids to all other customers would still be unconstrained.) Constraint 13.12 guarantees that the company will have at least a 75% probability of winning each bid to an online retailer. If the unconstrained optimal price would result in a probability of winning the bid that was greater than 75%, the constraint will not make any difference—the optimal price will be the same. On the other hand, if the unconstrained optimal price would have resulted in a probability of winning of less than 75%, constraint 13.12 will ensure that the calculated price yields a win probability of 75%. In this case, the constrained price will be lower than the unconstrained price.

Figure 13.10 Applying a minimum market-share constraint.

Figure 13.10 shows expected contribution as a function of the probability of winning the bid in a typical bid-response situation. The unconstrained optimum contribution is, of course, at the top of the hill. The win probability corresponding to the unconstrained optimum price is denoted by ρ*, which is equal to about 0.55 in the figure. If the market-share target is less than ρ*, then the constrained optimum price is the same as the unconstrained optimum price. If the market-share target is greater than ρ*, then the constrained optimum price will be lower than the unconstrained optimum. As shown in Figure 13.10, the expected contribution from this bid will also be lower—we are buying market share by pricing lower than the contribution-maximizing price. Figure 13.10 shows the amount of lost contribution if the seller enforces a target win probability of 0.75 for this bid. This difference is the opportunity cost associated with the constraint that enforces the business rule.

To guarantee that it meets its minimum margin targets, Gammatek will need to apply a constraint to its prices. Gammatek’s unit margin contribution for each sale is m = (p – c)/p. Let m* be the minimum margin required for a particular line of business. (There would be a different value of m* for each line of business.) Then one way to meet the minimum margin requirement is to ensure that (p – c)/p ≥ m* for every bid, or, equivalently, that p ≥ c/(1 – m*) for every bid. This constraint can be applied directly to the customized-pricing problem.

The minimum margin target has an effect opposite to that of the minimum market-share target: For each bid, it will result in prices that are greater than or equal to the unconstrained optimal prices. However, like any constraint, if it is binding, it will result in a lower expected contribution than the unconstrained case.

Portfolio constraints. You may have noticed that we fudged in applying the minimum margin constraint at Gammatek. The corporate goal was for each division to meet a margin target. This target would logically be applied to the total margin achieved by the business—the total contribution for the division divided by the total revenue for the division. However, we applied the constraint that p ≥ c/(1 – m*) to every deal. This is too restrictive! The division could accept a lower-than-target margin on some deals as long as there are sufficient higher-margin deals so that the division average exceeds the target. Specifically, the division could possibly make additional money by pricing below the margin target to some very price-sensitive customers while pricing higher to some other customers who are less price sensitive.

It is clear that the package express company is in a similar situation—it does not need to have a win probability of 75% or higher on every bid in order to win 75% of its bids. It could well be optimal for the package express company to accept a 70% probability of winning a particularly competitive bid if it anticipates that it will be bidding on a less competitive bid in the future with an 80% chance of winning. Finally, it is not clear at all how to incorporate the mean APR constraint into the customized-pricing problem at the Bank of Albion, other than forcing every bid to take place at the mean, which would hardly seem to qualify as PRO.

The upshot is that certain business rules cannot be applied independently to each bid as it arrives. Rather, they need to be treated as portfolio constraints to maximize expected profit over all bids. That is, these rules cannot be applied one bid at a time; they must be applied across the portfolio of all bids. This requires that the seller maintain a forecast of expected business for each combination of customer, product, and channel—in effect, for each cell in the pricing and revenue optimization cube. To see this, let d_i be the number of bids we anticipate from cell i in the PRO cube and p_i be the corresponding optimal price. (In other words, each index i corresponds to a different combination of product, customer type, and channel.) Then the problem faced by the Bank of Albion can be formulated as

subject to

where p* is the advertised typical rate. The constraint guarantees that the average price over all bids will be less than or equal to p*, but it does not constrain every single price to be less than or equal to p*.

From a practical point of view, the existence of one or more portfolio constraints significantly complicates the customized-pricing process. Instead of being able to calculate bid prices for deals one by one as they arrive, prices need to be precalculated for the entire PRO cube. Precise treatment of portfolio constraints also requires the forecasting of anticipated demand for each combination of product, customer type, and channel (the d _i in the earlier example). For these reasons, it is worth carefully considering whether it is important to incorporate portfolio constraints into the estimation of customized prices, or whether bid-by-bid approximations will suffice.

Infeasibility. Business rules are usually applied for good reasons. However, there can be a dangerous tendency for business rules to accrue over time, particularly if obsolete rules are never removed. If unchecked, the accumulation of business rules can lead to the customized-pricing problem becoming overconstrained or even infeasible. For example, a computer equipment wholesaler may want to impose two business rules:

• “We need to win at least 40% of our bids in the small-business segment.”

• “We need to maintain a margin of at least 12% on sales in our printer division.”

Taken individually, each goal might be quite reasonable. However, if printers represent a large fraction of the company’s sales to the small-business segment, the two goals might be mutually incompatible—namely, it might be impossible for the company to win 40% of its bids in the segment while maintaining a margin of 12% or more on its printer sales. This is the problem of infeasibility. The likelihood of infeasibility rises as the number and variety of business rules in place at one time increases.

Infeasibility is often dealt with in customized-pricing software by forcing the user to rank the business rules in the order of importance. In case of infeasibility, the software will start by relaxing the least important rule first and continue to relax rules until it can find a solution. This is a common technical solution. However, the larger issue is that any company using customized pricing needs to actively manage business rules to keep them from being overapplied and accruing over time. As we have seen, adding business rules can never improve expected contribution. Therefore, they should be added only when necessary. In particular, it is important to have a process for periodically reviewing the rules that are in place and replacing those that are obsolete or no longer applicable.

13.4.2 Variations on the Basic Bidding Model

So far, we have looked at customized pricing in its simplest setting—when sellers submit a single bid for a known product to a buyer. Although this setting is realistic in some markets, in many others there are significant variations that have implications for how the sellers should set their prices. This section touches on a few of the most important examples.

Contract versus single purchase. In many cases, a buyer will request bids for a contract for future goods or services, typically for a fixed period, such as the next six months or a year. This is a very common way of purchasing freight, telecommunications, and other services. A five-year information technology (IT) outsourcing or cloud computing deal would also fall into this category. This adds a twist to customized pricing: a seller must make its bid without knowing the precise level or mix of business it will actually realize if it wins the bid.

Consider a video game manufacturer soliciting bids from a trucking company to distribute its video games nationwide for the next 12 months. The trucking company needs to decide what discount from its basic tariff it should offer. If the manufacturer’s newest game is a hit, then the trucker may ship a huge volume of games over the next few months—so much so that it might require additional trucks to be added to the fleet to accommodate the increased business. If, on the other hand, the game is a bust, the additional business from this customer would be simply incremental and not require any additional investment. A trucking company needs to incorporate this uncertainty into its bid.

In this case—when the level or mix of demand from a potential customer is uncertain and the profitability of an account depends on the level or mix of demand—the seller needs to forecast the demand he anticipates from that customer and use that information in calculating the price to bid. Furthermore, if there is a wide range of possible outcomes, he may need to explicitly incorporate the uncertainty in the business level in his bid.

Multiple line items. It is extremely common that a bid includes more than one item. As a very simple example, a park district may submit an RFQ that asks for a bid on 10 medium-size automobiles and 10 pickup trucks as part of the same bid. At the more complex end of the spectrum, a bid to construct a power plant may include thousands of individual line items. As an intermediate example, the heavy truck offered by Peterbilt described in Section 13.3.2 has 75 different options. For each option (e.g., engine type), there are two or more alternatives, each with a different list price. A customer configures the truck by choosing the options she wants, and the manufacturer must determine the overall discount to offer based on the fully configured truck.

When multiple line items are involved in a bid, the tactic a bidder should use to determine his bid price depends on how the buyer is going to select a supplier. Here are the three broad possibilities.

• Single bundled price. The buyer may commit ahead of time to purchase from a single bidder and require only a single, all-inclusive price from each bidder. This is the case with the configurable heavy truck purchase, for example. In this case, the price for the entire bundle can be determined using the standard optimal customized-pricing approach.

• Itemized prices with single supplier. In this case, the purchaser commits to purchase from a single bidder but requires itemized prices. This may seem identical to the previous case, but there are a number of potential pitfalls. First, even though the purchaser will only be paying a single price, his choice of supplier may be influenced by some of the itemized prices. For example, automobile purchase decisions are often more influenced by the base price of the car than by the cost of the options. Some purchasers are more likely to purchase the same car if it is presented as a $19,000 car with $5,000 worth of options than if it is presented as a $20,000 car with $3,800 worth of options. This is an issue of price presentation, which is treated in Chapter 14.

A more important issue with itemized prices is that they give the purchaser more scope and leverage for negotiation, particularly in a competitive bid. When there is only a single price, there is only one dimension for negotiation. When there are many line-item prices, a savvy buyer will negotiate each one along with the total discount. For this reason, sellers prefer to offer a single bundled price, while buyers almost always prefer line-item prices, even when they will only be purchasing from a single supplier.

• Itemized prices and multiple suppliers. In this case, the purchaser requests itemized prices and reserves the right to purchase different items from different bidders. For example, the park district might reserve the right to purchase the cars from one company and the trucks from the other. Obviously, the motivation for the purchaser is to select the cheapest supplier for each item (cherry picking). In this case, the strategy for the seller is to treat each item as an individual bid and to calculate the optimal price for each line item independently of the others. Of course, a supplier might want to provide an extra discount if the purchaser will purchase everything from him.

The tactics for dealing with these different settings are fairly clear: What is most important for a bidder is to understand what game he is playing before he submits his bid. That is, if itemized prices are requested, it is critical to know whether the purchaser is committed to a single supplier or is reserving the option to purchase from more than one bidder.

Negotiation. In many customized-pricing situations there is an element of negotiation—the first deal proposed is not necessarily the final one, and there may be several rounds of negotiation before the final price is determined. Negotiation may take place on price (“Cut the total price by 1%, and I’ll sign right now”), on product bundle (“Throw in the extended warranty and we have a deal”), or both. Clearly, this is a different setting from the single-bid process that we have considered, and recommended prices need to be modified accordingly.

There is a substantial literature on the art and science of negotiation, not to mention a small industry devoted to helping people become better negotiators. However, from the point of view of pricing and revenue optimization, a typical approach is to give the salesperson a range within which to bid. How such a range can be incorporated into sales force guidance is illustrated in Figure 13.11 for the example of the park service vehicle bid. Expected contribution is within 4% of the optimal for any amount bid between $11,600 and $12,400. The salesperson negotiating the deal might be provided with this range of prices and expected to use his negotiating skill to obtain a price as high as possible within the range. The savvy salesperson will choose an initial bid at the high end of the range to give himself room to bargain. This allows the company to provide guidance to a salesperson in the negotiation process while ensuring that the final price achieved is sufficiently close to optimal.

A price range such as the one shown in Figure 13.11 is often used to support a combined centralized/decentralized approach to negotiated pricing. In this approach, headquarters can determine the target price and the upper and lower bounds for each deal. The local salesperson has the freedom to negotiate the best price he can within these bounds. The idea is to gain the best of both worlds: to use all of the data available centrally to calculate bid-response functions and bounds while enabling the salesperson to adjust the price to the characteristics of a specific deal and the characteristics of an individual customer. Ideally, the local salesperson will have immediate knowledge of the local competitive situation or the needs of each customer that can be used to determine a better price than could be generated centrally.

Does setting price ranges centrally improve results over the two alternatives of completely centralized and completely decentralized pricing? In theory, negotiation within bounds should benefit the seller, since a savvy salesperson should be able to evaluate the eagerness of a customer to purchase and adjust the asking price accordingly. This is what a study of auto-loan pricing found—loan officers who had authority to negotiate rates within a range were able to increase the profitability of loans over the price list that had been centrally established (Phillips, Simsek, and van Ryzin 2015). On the other hand, an internal study by Nomis Solutions found that loan officers for a Canadian bank were actually reducing the overall profitability of the portfolio by negotiating prices that were consistently lower than optimal. The difference between the two situations can be best explained by incentives—the auto loan negotiators were incentivized (indirectly) on profitability, while the Canadian loan officers were incentivized in large part by portfolio growth and by a recommender score. This motivated them to offer lower prices than those that would have maximized profit.⁸

Figure 13.11 Range of bids within which expected contribution is within 4% of the optimum for the example with the profitability curve shown in Figure 13.3.

Fulfillment. In many contract bidding situations, a buyer may not want to commit all of her business to a single supplier. For example, a shipper may not want to commit all of her business for the next year to a single trucking company. Instead, the shipper may want to have contracts with two different trucking companies to provide herself with flexibility in case of a dispute with one of the companies or in case one is shut down temporarily by a strike. Furthermore, having contracts with two different trucking companies may save the shipper money if she picks the cheaper trucker for each shipment. In other business-to-business relationships, a purchaser will choose two or more competitors as preferred suppliers. As specific needs arise during the year, the purchaser will choose among the preferred suppliers for each order.

In this case, the determination of the price to bid needs to consider not only the probability of winning—that is, becoming a preferred supplier—but also the amount (and mix) of business that will come his way during the term of the contract. Generally, both the bidder’s probability of becoming a preferred supplier and the amount of business he will receive from the contract will increase as he decreases his price. Both considerations need to be incorporated into the estimated contribution.

13.5 CUSTOMIZED PRICING IN ACTION

Effective customized pricing requires a disciplined business process both to calculate and administer prices and to ensure that the parameters of the bid-response functions are updated to capture changes in the market. A high-level sketch of such a process is shown in Figure 13.12. There are three core steps in the customized-pricing process.

1. Calculate bid-response function. Using available information about the customer, the product (or products) she wishes to purchase, and the channel through which the request was received, the seller estimates how the probability of winning this customer’s business varies as a function of price. This requires retrieving (or calculating) the parameters of the bid-response function that apply to this particular combination of segment, channel, and product.

2. Calculate deal contribution. The seller estimates the deal contribution function relating contribution from this deal to its price.

3. Determine optimal price. Combining the bid-response function and the deal contribution function, the seller calculates the price that maximizes expected contribution (or other goal) subject to applicable business rules.

These three core steps may be executed each time a bid request is received and a price needs to be calculated. Alternatively, the PRO cube might be populated ahead of time with optimal prices that are accessed whenever a bid request is received. This requires executing the three core steps in batch mode prior to processing any bids. Which approach is better depends on the volume of bids to be processed and the time available to respond to each one. A lending company that needs to respond rapidly to thousands of customer inquiries about consumer loans every day is more likely to prepopulate a database with prices. On the other hand, a heavy equipment manufacturer that responds to 40 or 50 RFPs per week, with several days available to respond to each, is more likely to optimize the price for each bid independently.

Figure 13.12 The customized-pricing process.

Whichever approach is used, the core steps need to be supported by three additional steps that are executed less frequently.

1. Segment the market. The seller determines the market segments he will use to differentiate his prices. This establishes the structure of the PRO cube and needs to be done prior to estimating any bid-response parameters. Typically, the market segmentation does not need to be updated often—perhaps annually or semiannually. The exception is when the seller seeks to serve a new market or type of customer. Note that the extent to which a seller will be able to segment his market will be limited by the amount of historical bid-response data that he has.

2. Estimate bid-response parameters. The seller estimates the parameters of the bid-response function when the market is initially segmented. The bid-response parameters need to be reestimated whenever the segmentation itself is changed.

3. Update bid-response parameters. The seller updates the parameters of the bid-response function on a routine basis—usually weekly or monthly—to ensure that changes in the underlying market are captured. Updating the bid-response parameters is not much different from estimating them in the first place. In the simplest approach, the results of the most recent bids (i.e., those since the last update) are appended to the bid-history file in Figure 13.9, and the same approach is used to estimate the updated parameters using the new, larger database of bids. However, this approach would give increasingly old bid results the same weight as much newer bids in estimating the parameters. For this reason, most companies drop old bids—for example, those over a year old—from their bid-history file. Alternatively, other companies use Bayesian updating approaches that incorporate all bid information but weight newer bids more than older bids.

13.6 SUMMARY

• Customized pricing occurs when customers approach a seller individually and describe their desired product (or products) prior to purchasing. Customized pricing is commonplace in business-to-business marketplaces as well as some consumer marketplaces, notably lending and insurance.

• In a customized-pricing setting, the seller knows the desired product for each customer, the channel through which the customer approached, and some of the characteristics of the customer. Furthermore, the seller can use this information to set a customized price for each bid.

• An important aspect of customized pricing is that the seller obtains information about not only bids that he won but also bids that he lost. He can use this information to estimate a bid-response function that determines his probability of winning a bid based on the characteristics of that bid and the price that he charges.

• When a seller quotes a price for a bid, he faces two types of uncertainty: competitive uncertainty regarding competing bids and preference uncertainty regarding the criteria the purchaser will use to choose a winner. In some cases, the buyer is committed to purchasing from the lowest bidder—in this case, the seller can calculate a bid-response function based entirely on his uncertainty regarding competitive bids.

• In the more common case, a seller needs to estimate a bid-response function based on the historical characteristics and prices of bids that he has won and lost. This is a problem of binary regression, and the most popular approach to estimating a bid-response function is to use logistic regression.

• A seller who has estimated a bid-response function can use it to determine the price that maximizes his expected profitability from each bid, which is calculated as the probability of winning the bid at a price times the profitability of the bid at that price if it is won.

• Application of optimal customized prices is complicated by the fact that many sellers are subject to business requirements such as minimum margin requirements and minimum market-share requirements. These are typically imposed as constraints in the customized-pricing optimization problem. Applying inconsistent constraints can lead to infeasibility, in which no set of prices can simultaneously satisfy all the constraints.

• Optimal customized pricing is typically implemented in an ongoing business process in which the optimal price for each bid is determined based on the characteristics of the product, the channel, and the customer at the time of the bid. The parameters of the bid-response function are periodically updated based on the performance of recent bids.

13.7 FURTHER READING

Customized pricing is discussed further in Bodea and Ferguson 2012 and Phillips 2012a. Applications of customized pricing to medical devices are described in Agrawal and Ferguson 2007 and Dubé and Misra 2017. Phillips 2018 describes at length the application of optimized customized pricing to various forms of consumer credit.

A classic text on logistic regression is Hosmer, Lemeshow, and Sturdivant 2013, which includes much of the mathematical details. A shorter introduction is Pampel 2000, and a useful practical guide is Kleinbaum and Klein 2010.

The issue of centralization versus decentralization in customized pricing, including the auto lending study and the Canadian lending study, are discussed in Phillips 2014 and Phillips 2018, chap. 8. A broader discussion of the trade-offs between a centralized and a decentralized pricing organization can be found in Simonetto et al. 2012.

The closely related topics of negotiation and auctions both have a vast literature. Raiffa, Richardson, and Metcalfe 2013 is a great starting point and source for negotiations. Steinberg 2012 surveys the auction literature from the seller’s point of view, and Milgrom 2004 bridges theory and practical applications. Elmaghraby and Keskinocak 2003 reviews the literature on online supplier auctions.

The printer cartridge example was adapted from Phillips 2012a, which also describes the calculation of efficient frontiers in customized pricing. The discussion of endogeneity in customized pricing was adapted in part from Phillips 2018.

13.8 EXERCISES

1. HP is bidding against Lenovo for an order of 150 laptops. HP’s unit cost is $1,000 per laptop, and, based on previous experience, HP’s belief is that Lenovo’s bid will be uniformly distributed between $1,200 and $1,500 per unit.

a. Assuming that HP and Lenovo are at parity (e.g., neither supplier enjoys a premium on this deal), what is HP’s optimal price per unit to bid? What is HP’s corresponding probability of winning the bid and expected contribution?

b. Assume that Lenovo is an incumbent supplier to this customer and, as a result, HP believes that Lenovo enjoys a $200 per unit premium for this deal. Assuming that HP has the same distribution on Lenovo’s bid as previously stated, what is HP’s optimal price per unit to bid? What is HP’s corresponding probability of winning the bid and its expected contribution?

2. The printer cartridge manufacturer with the data shown in Table 13.3 has used historical data to estimate the coefficients in the model in Equation 13.11 and has come up with the model ρ(p, x) = a + bPRICE + cGRADES + dGRADEG + eSIZE + fNEW + gRET + hGOV + iBUSLEVEL. What win probability would this model predict for each of the bids in Table 13.3

TABLE 13.5

Model predictions and actual bid outcomes for telecommunication problem

3. Table 13.5 shows the take-up probability predicted by two different bid-response models and the actual outcome of 10 historical bids for a telecommunications company.

a. What are the RMSE, concordance, and log-likelihood of each model based on the data shown?

b. Which model performed better on these data?

NOTES

1. The e-Car example is adapted from Phillips 2018.

2. The exception is the rare case when a hot new model is introduced and demand at the MSRP far exceeds supply. In this case, dealers may decide to charge more than the MSRP. However, this practice tends to be unpopular with customers. For example, a California dealership quoted a price $10,000 above the MSRP for the extremely popular Corvette Stingray to widespread complaints of price gouging (Automotive News 2013).

3. In this chapter, I occasionally talk about the bid price, by which I simply mean the price associated with the bid. This should not be confused with the bid price used in revenue management, as defined in Section 9.2.3.

4. How these factors might actually translate into a decision between competing suppliers and the extent to which the buyer is being truthful about her selection criteria are separate issues.

5. As in the definition of price elasticity in Equation 3.8, the minus sign guarantees that the elasticity will be positive, since ρ′(p), the derivative of the price-response function, is less than 0.

6. Following common practice, I capitalize and italicize explanatory variables such as GRADEP and PRICE.

7. In this case, the word “negotiation” does not necessarily indicate the back and forth usually associated with the term. It may be that the customer does not know that the price is actually negotiable and therefore treats the quote as fixed.

8. The studies on auto lending and Canadian lending are discussed in Phillips 2014.