Estimating Buyer Willingness-to-Pay and Seller
Reserve Prices from Negotiation Data and the
Implications for Pricing
Robert Phillips
Graduate School of Business, Columbia University, New York, NY 10027 and Nomis Solutions, rp2051@columbia.edu
A. Serdar Şimşek
School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, as2899@cornell.edu
Garrett van Ryzin
Graduate School of Business, Columbia University, New York, NY 10027, gjv1@columbia.edu
We consider a company selling heterogeneous products where the final price is set by negotiation between
sales agents and customers. Negotiation is common in many industries including retail automotive sales,
consumer lending, insurance and many business-to-business markets. We assume that each sales agent has
a minimum reserve price that he/she will accept and each customer has a maximum willingness-to-pay;
however, these values are not directly observable. If the customer’s willingness-to-pay is greater than the
sales agent’s reserve price, then a sale takes places at the Nash bargaining equilibrium price, otherwise
no trade takes place. Given a record of the outcomes of a series of such negotiations, we would like to
estimate the underlying joint distribution of willingness-to-pay and reserve prices. To do so, we propose
and analyze a structural estimation method based on the expectation-maximization (EM) algorithm that
allows for the willingness-to-pay and reserve price distributions to depend on an arbitrary set of covariates
(e.g. characteristics of the customer, product or sales agent) and be potentially correlated. Using a realworld data set, we show, via out-of-sample tests, that our estimation method generates accurate estimates
of customer-price response and the prices of lost deals. We also show how our estimated distributions can
be used to optimize controls on negotiated prices that can significantly increase revenues relative to both
unconstrained negotiations and centrally-optimized fixed prices. These results provide a justification for the
policy commonly observed in practice of providing sales staff limited discretion to change prices relative to
a centrally-established price list.
Key words : negotiation; structural estimation; Nash bargaining solution; limited sales force price
discretion; consumer lending
1
2
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
1.
Introduction
Price negotiations are often associated with markets such as fine art and real estate in which the
items for sale are unique. However, negotiations are also found in many other markets in which the
items sold are identical or similar. Examples include retail automotive sales, insurance, consumer
loans, mortgages, and many business-to-business markets. In these markets, the price paid for
identical or similar items may be quite different depending upon the outcome of the negotiation.
Moreover, the seller and the customer may not be able to agree on a price, resulting in a lost
deal. Whether or not a deal occurs and, if it does, the price that results will depend upon the
willingness-to-pay of the customer, the reserve price of the seller and the negotiating power of each
party.
Understanding the variation of willingness-to-pay among customers is important to most firms
in order to segment customers and intelligently set prices for different products and customer segments. Estimating the willingness-to-pay distribution for a particular product within a particular
customer segment based on historical sales data is always challenging and a variety of approaches
have been developed. (See Breidert et al. 2006, for a good review.) Estimating price response in
cases where negotiation is involved poses two specific challenges. First, price information is typically
unobserved for lost deals, since sales staff may be unwilling or simply unable to accurately estimate the last price under consideration before a customer walked away. Second, when a trade does
take place, the resulting price is correlated with the seller’s reserve price as well as the customer’s
willingness-to-pay.
To address these challenges, we develop a structural estimation method to determine the joint
distribution of customer willingness-to-pay and seller reserve prices in a market with negotiation.
Specifically, we postulate a model of a heterogeneous sellers selling to heterogeneous buyers via
bilateral price negotiations. Each seller has a private reserve price, R, for each deal, and will not
sell at a price lower than R. Similarly, each buyer has a private willingness-to-pay, W , and will not
buy at a price higher than W . Hence, no sale will occur if R > W . If W ≥ R, we assume that the
parties will reach agreement and a sale will occur at the symmetric Nash bargaining equilibrium
price p = (R + W )/2. Given a data set on both completed and lost deals (and the prices at which
deals occur), we address the problem of how to estimate the parameters of the joint distribution of
R and W . We allow for the fact that the distribution of (R, W ) may depend on a set of covariates
describing the product, buyer and seller and that the values R and W may be correlated with each
other.
A novel feature of our model is the assumption of a reserve price distribution among sellers.
This reserve price distribution can reflect sales force heterogeneity; that is, a more skilled sales
agent should be able to extract a higher price from the same customer, which could manifest itself
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
3
as a higher reserve price. Additionally, sales staff might be more or less willing to accept specific
deals at lower prices based on idiosyncratic incentives, e.g. how close they are to meeting a sales
quota. Such differences would also be reflected as differences in reserve prices. If sales staff were
homogeneous in terms of their abilities and incentives to make deals, this would be reflected in
a common reserve price among similar deals – or a reserve price that is largely predictable as a
function of the characteristics of the deal. In our real-world auto lending data, this is not the case:
reserve prices appear to vary significantly among sales representatives for similar deals.
We consider three approaches to estimating the parameters of the joint distribution of R and W
given data on historical deals: direct maximization of the likelihood function and two variations
on the expectation-maximization (EM) algorithm, which is commonly used to estimate model
parameters given censored data. We show that the expectation-maximization algorithm is more
robust and generally provides faster convergence, particularly on large data sets. We therefore use
the EM algorithm to estimate the joint distribution of R and W as a function of customer and
deal characteristics for a large data set of auto lending deals.
Lastly, we illustrate how our estimated joint distribution can be used to improve negotiated
pricing and help compensate for heterogeneity in the sales force. We assume that “headquarters”
wishes to maximize expected profit across all deals and sets the parameters within which local
sales agents must work and compare three policies: centralized deal-specific optimal pricing, negotiated pricing with centrally set deal-specific minimum prices, and optimized seller reserve prices.
Applying these three policies to a real-world auto lending data set we find that optimized seller
reserve prices outperforms centralized deal-specific optimal pricing which outperforms negotiated
pricing with minimum prices. We discuss the implications of these findings in Section 6.
2.
Literature Review
Starting with Weinberg (1975), a number of researchers have considered the question of how much
pricing discretion (if any) should be granted to a sales force. A number of these papers have
assumed that local sales representatives have local market information that they can use to set
better prices than headquarters (Lal 1986). Mishra and Prasad (2004) show that, even when a
customer possesses private information, if the sales agent first observes the private information,
then signs the contract, a company can set prices centrally and capture the full benefits available
from discretionary pricing by using an appropriate contract structure – though compensation may
be a complicated function of quantity and price. They then analyze competitive markets and show
that, under asymmetric information, an equilibrium always exists where all firms use centralized
pricing, regardless of the intensity of competition (Mishra and Prasad 2005).
Joseph (2001) develops a model that explicitly incorporates the sales agent’s superior information
about customer willingness-to-pay and the possibility of sub-optimal trade-offs between price and
4
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
effort (using discounting rather than expending effort on selling). In his model, centralization of
pricing authority is sometimes preferable to full decentralization, depending on the effort cost of
following a high-quality prospecting strategy and the structure of customer segments. Bhardwaj
(2001) reaches a similarly mixed conclusion in the case of competing firms when principals and
agents have the same information. Roth et al. (2006) consider the case of bargaining versus fixed
(or posted) price for customized services and show that, if bargaining costs are low, bargaining
can be a preferred approach for the seller. Kuo et al. (2011) present a model that incorporates the
interactions among dynamic pricing, negotiation, and inventory and show that negotiation is an
effective tool to achieve price discrimination, particularly when the inventory level is high and/or
the remaining selling season is short.
However, all of these models have led to results concluding that either full pricing centralization
or full pricing decentralization is optimal (or showing the equivalence of the two for specified
contract structures). Yet we observe in many markets with decentralized sales forces, that local
sales staff has discretion to negotiate prices for individual deals within pre-set limits. A recent Ernst
& Young Consulting survey showed that about 75% of business-to-business companies surveyed in
the United States allowed their sales staff limited discretion to adjust prices (Williams 2014). One
conclusion of our work is that, in general, neither fully centralized nor fully decentralized pricing
maximizes profits. Fully centralized pricing is not optimal because it does not allow sales staff to
adjust prices toward the willingness-to-pay of individual customers. Fully decentralized pricing is
not optimal because it becomes too dependent on the idiosyncratic reserve prices of sales staff. Sales
staff with very low reservation prices will systematically under-price deals. Rather than relying on
full price centralization or decentralization, our work suggests firms can maximize profits by setting
list prices and giving sales representatives limited discretion to adjust the prices for individual
deals. In this way, our work provides a heretofore missing justification for the common practice of
list pricing with discretion.
In the remainder of the paper, we describe our structural model and how the EM method can be
used to estimate the parameters of the model given deal and lost-deal data. We then use synthetic
data to show that our estimates converge to the underlying true parameters when willingness-topay and reserve prices are drawn from normal distributions. Next, we apply the EM algorithm to
a large data set of negotiated deals and lost deals obtained from a major North American auto
lender and evaluate the predictions produced by our method relative to standard regression-based
predictors. Our method also provides better estimates of the prices associated with lost deals1 .
We then show how the distributions of reserve price and willingness-to-pay can be used to
determine the optimal amounts of pricing discretion to provide to the sales force. Finally, we
1
When a deal is lost, the last price (APR of the loan) offered to the customer is still recorded in this data set.
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
5
discuss the robustness of the model to the specification of the underlying distributions and to the
assumption about the relative bargaining power of sales staff and customers.
3.
Model Description
Consider a setting in which a group of heterogeneous sellers sells a (potentially heterogeneous)
product to a heterogeneous group of buyers. Prices are negotiated bi-laterally between a single
buyer and single seller in each transaction. We assume that in each negotiation, the seller has
a reserve price R and the buyer has a willingness-to-pay W . The values of R and W are both
unobservable, but they both may be correlated with observable attributes of the seller, the buyer
and the product being sold. If the reserve price is greater than the willingness-to-pay, then no
trade takes place. If the willingness-to-pay is greater than the reserve price, then the outcome of
the negotiation is assumed to be the generalized Nash bargaining solution. (See Binmore et al.
1986, Muthoo 1999, for reviews of Nash bargaining.) Nash bargaining is a widely-used model of
negotiations (c.f. Desai and Purohit 2004, Roth et al. 2006, Kuo et al. 2011).
In the Nash solution, both parties divide the total surplus from trade (the difference between
the sum of their utilities before and after the bargaining) according to their relative bargaining
powers. Let s1 and s2 be the utilities to players 1 and 2 derived from the bargaining outcome
and let s01 and s02 be, respectively, the corresponding disagreement utilities (utilities to each party
when no trade takes place). Then the generalized Nash bargaining solution outcome maximizes
(s1 − s01 )α (s2 − s02 )(1−α) , where α is a measure of the relative bargaining power of the buyer and
seller (see Binmore et al. 1986).
We assume that both the seller’s and buyer’s utilities will be zero if no trade takes place, i.e.,
s01
= s02 = 0. If trade takes place at a negotiated price p, then the seller’s utility will be p − R and
the buyer’s utility will be W − p. Therefore, when W ≥ R the generalized Nash bargaining solution
solves:
max
p
(p − R)α (W − p)(1−α)
The solution is
peq = αR + (1 − α)W
which gives the equilibrium price of the bargaining outcome2 . In our model, we assume that the
seller and buyer have equal bargaining powers, i.e., α = 0.5. Hence, if a trade takes place, it does
so at the price determined by the symmetric Nash bargaining equilibrium peq = (R + W )/2. We
show in Section 7.2 that α is in fact not identifiable from the data and moreover the estimation
results are robust to different choices of α. Lastly, in the case where no trade takes place, then we
only know that W < R.
2
Note that we are assuming an efficient bargaining outcome in a full information setting. Modeling an imperfect
bargaining situation in an asymmetric (or private) information setting would require additional assumptions (see, for
example, Chatterjee and Samuelson 1983).
6
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
4.
Structural Estimation Problem
Assume we have a data set D of the outcomes of N negotiations indexed by i, where the values
(Ri , Wi ) are IID draws from a (unknown) bi-variate normal joint distribution F (R, W ). For each
negotiation i in the data set, we observe only i) whether trade takes place or not, ii) if trade takes
place, the outcome price pi of the negotiation, and iii) a set of covariates describing the seller,
buyer and product involved in the negotiation. Given these assumptions and inputs, we want to
estimate the joint distribution F .
Define
• Ri = reserve price of seller in negotiation i (unobservable random variable).
• Wi = willingness-to-pay of buyer in negotiation i (unobservable random variable).
• pi = final price of negotiation i when trade takes place (data).
• Si = vector of seller characteristics for negotiation i (data).
• Bi = vector of buyer characteristics for negotiation i (data).
• Pi = vector of product characteristics for negotiation i (data).
• N = total number of negotiations (data).
We assume that Ri and Wi are joint normally distributed random variables, defined as
Ri = XiT βr + i
Wi = YiT βw + δi
where Xi = (Si , Pi ), Yi = (Bi , Pi ) are the vector of seller and buyer covariates, respectively, βr and
βw are the vectors of coefficients on these covariates, and (i , δi ), i = 1, . . . , N are IID error terms
which are bivariate normal with mean 0 and covariance matrix
σr2 σrw
Σ=
σrw σw2
with corresponding correlation coefficient ρ = σrw /(σr σw ). Note because we assume the error terms
have zero mean, E[Ri ] = XiT βr and E[Wi ] = YiT βw ; hence, the covariates influence only the means
of the willingness-to-pay and reserve price distribution. Without loss of generality, assume that
Xi and Yi have the same dimension M and let Xij and Yij denote the j-th component of Xi
and Yi , respectively. For notational convenience, also define R = (R1 , . . . , RN ), W = (W1 , . . . , WN ),
X = (X1 , . . . , XN ) and Y = (Y1 , . . . , YN ). We denote the marginal cdf of and δ by F and Fδ
with density functions f and fδ , respectively. We denote the marginal cdf of R by Fr (r, θr ) and
the associated marginal pdf fr (r, θr ). Similarly, the marginal cdf of W is denoted Fw (w, θw ) with
associated pdf fw (w, θw ).
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
7
Finally, we let θ = (βr , σr , βw , σw , ρ) denote the vector of parameters to be estimated from the
data set D and define the sets
T = {i : trade takes place}
T = {i : no trade takes place}
We seek to estimate θ from the data D.
4.1.
Non-Quasi-Concavity of the Log-Likelihood Function
Consider the simplest case where there are no covariates and R and W are independent. In this
case, both the reserve price and willingness-to-pay follow independent normal distributions with
θr = (µr , σr ) and θw = (µw , σw ), then θ = (µr , σr , µw , σw ). In this case, the log-likelihood function is
L(D, θ) =
X
ln {P (R + W = 2pi , R ≤ pi ≤ W )} +
i∈T
X
ln {P (W < R)}
i∈T
X
|T |
|T |
1
ln (2π) −
ln (σr2 + σw2 ) −
(2pi − µr − µw )2
2
2
2(σr2 + σw2 ) i∈T
!
!
X
(pi − µr )σw2 − (pi − µw )σr2
µr − µw
p
ln Φ
+
+ |T | ln Φ p
σr σw σr2 + σw2
σr2 + σw2
i∈T
=−
(1)
Proposition 1. The function L(D, θ) in equation (1) is not quasi-concave in general.
Proof:
We prove the proposition by providing a counterexample. Consider a case where there
are nine negotiations and trade happened in the first five of them at prices (60, 70, 75, 80, 88), hence
D = (60, 70, 75, 80, 88, N T , N T , N T , N T , N T ), where N T indicates that no trade took place.
Take the following two points: θ1 = (60, 10, 60, 36) and θ2 = (90, 35, 90, 10). Then, L(D, θ1 ) = −29.36
and L(D, θ2 ) = −29.40. If we take α = 0.5, we get an intermediate point θα = αθ1 + (1 − α)θ2 =
(75, 22.5, 75, 23), with L(D, θα ) = −29.74, which is smaller than the previous two. Therefore, quasiconcavity is violated. Q.E.D.
4.2.
An EM Estimation Method
The expectation-maximization (EM) method is a standard approach to parameter estimation with
censored data. Introduced by Dempster et al. (1977), the EM method is useful when the incomplete
data log-likelihood function is difficult to maximize directly but the complete data log-likelihood
function has a simple form, as we show is the case in our model. The book by McLachlan and
Krishnan (1996) is a good resource for details of the method and its extensions. The EM method is
used widely in the empirical operations management literature for estimation problems in choicebased revenue management (Vulcano et al. 2010), censored sales (Stefanescu 2012), substitutable
8
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
products with stock-outs (Anupindi et al. 1998), and selling with incomplete product availability
(Conlon and Mortimer 2007). Monte Carlo implementations of the EM algorithm have been developed for more complex likelihood functions (see, for example, Levine and Casella 2001, Chan and
Ledolter 1995).
Consider an ideal data set which records every pair (Ri , Wi ) for each negotiation i = 1, . . . , N .
Given these exact observations, as we show below, the log-likelihood function to estimate F is
straightforward. We can consider our data set D to be a partial observation of these complete
data. The idea of the EM method is to work with the conditional expected value of the simpler
complete-data likelihood rather than the more complex incomplete-data likelihood. Specifically,
the EM algorithm starts with initial estimates of the parameters θ (an initial ”guess”) and then
computes the expected value of the complete data log-likelihood function conditional on θ and
D. This is the ”E” (expectation) step. The resulting expected log-likelihood is of the same simple
form as the complete-data log-likelihood function and can be easily maximized to find updated
parameter estimates θ. This is the ”M” (maximization) step. The process is then repeated until
the estimates θ converge. The generalized EM algorithm improves the complete data problem’s
conditional expectation of the log-likelihood rather than maximizing it at every iteration.
In summary, the steps in the algorithm are:
• Initialization: Set θ = θ(0)
On the (k + 1)th iteration:
• E-step: Calculate Q(θ, θ(k) ) = Eθ(k) [Lc |D, θ(k) ]
• M-step: Choose θ(k+1) such that
Q(θ(k+1) , θ(k) ) ≥ Q(θ, θ(k) )
∀θ ∈ Ω
Repeat these E and M steps until convergence (e.g. until θ(k+1) ≈ θ(k) ).
4.2.1.
The EM Method Applied to the Structural Bargaining Model Suppose that
all of the values of Ri and Wi in our negotiating model are known for i = 1, 2, . . . , N . Then, the
maximum likelihood estimates of the model parameters are the solution to the normal equations
for the bivariate normal distribution:
(XT X)β̂r = XT R
(2)
(YT Y)β̂w = YT W
N
1 X
2
σ̂r =
(Ri − XiT β̂r )2
N i=1
(3)
σ̂w2 =
N
1 X
(Wi − YiT β̂w )2
N i=1
(4)
(5)
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
N
1 X
σ̂rw =
(Ri − XiT β̂r )(Wi − YiT β̂w )
N i=1
9
(6)
Consider the E-step on the (k + 1)th iteration of the EM algorithm, where θ(k) =
(βr(k) , σr(k) , βw(k) , σw(k) , ρ(k) ) denotes the incumbent estimate of the parameter values θ after the k th
EM iteration. The conditional expectation of the complete-data log-likelihood is then
1
1
1
Eθ(k) (R − XiT βr(k) )2 |R + W = 2pi , R ≤ pi
− ln(2π) − ln((σr(k) )2 ) −
(k)
2
2
2(σr )2
i∈T
X 1
1
1
Eθ(k) (R − XiT βr(k) )2 |W < R
+
− ln(2π) − ln((σr(k) )2 ) −
(k)
2
2
2(σr )2
i∈T
X 1
1
1
T (k) 2
+
− ln(2π) − ln((σw(k) )2 ) −
E
(k) (W − Yi βw ) |R + W = 2pi , W ≥ pi
θ
(k)
2
2
2(σw )2
i∈T
X 1
1
1
T (k) 2
− ln(2π) − ln((σw(k) )2 ) −
(7)
E
+
(k) (W − Yi βw ) |W < R
θ
(k)
2
2
2(σw )2
Eθ(k) [Lc |D, θ(k) ] =
X
i∈T
The M-step on the (k + 1)th iteration is implemented by replacing Ri , Wi , (Ri − XiT β̂r )2 and
(Wi − YiT β̂w )2 by their current conditional expectations in (2)-(6). Hence, the (k + 1)th estimates
of βr and βw , βr(k+1) and βw(k+1) , are specified by solving the following modified normal equations:
For all j = 1, 2, . . . , M
N X
M
X
(k+1)
Xij Xi` βr`
=
i=1 `=1
N X
M
X
X
Xij Eθ(k) [R|R + W = 2pi , R ≤ pi ] +
i∈T
(k+1)
Yij Yi` βw`
=
i=1 `=1
X
X
Xij Eθ(k) [R|W < R]
i∈T
Yij Eθ(k) [W |R + W = 2pi , W ≥ pi ] +
i∈T
X
Yij Eθ(k) [W |W < R]
i∈T
(k+1)
And, the (k + 1)th estimates of σr , σw , and σrw , σr(k+1) , σw(k+1) and σrw
are
(σr(k+1) )2 =
1 X
{
Eθ(k) (R − XiT βr(k) )2 |R + W = 2pi , R ≤ pi
N i∈T
X
Eθ(k) (R − XiT βr(k) )2 |W < R }
+
i∈T
(σw(k+1) )2 =
1 X
{
Eθ(k) (W − YiT βw(k) )2 |R + W = 2pi , W ≥ pi
N i∈T
X
+
Eθ(k) (W − YiT βw(k) )2 |W < R }
i∈T
(
h
i
X
1
(k+1)
σrw
=
Eθ(k) (R − XiT β̂r(k) )(W − YiT βw(k) )|R + W = 2pi , R ≤ pi
N i∈T

h
i
X
+
Eθ(k) (R − XiT βr(k) )(W − YiT β̂w(k) )|W < R

i∈T
10
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
4.2.2.
Convergence and Theoretical Properties All limit points of the EM algorithm for
our model are provably stationary points of the incomplete log-likelihood function because the
expected log-likelihood function (7) is continuous in βr(k) , σr(k) , βw(k) , σw(k) , and ρ(k) (see Wu 1983).
While the EM algorithm may not converge for a particular instance, if it does converge, it converges
to a stationary point. Moreover, one can easily detect during the course of the algorithm whether
or not it is converging. In practice, the EM algorithm has proved to be a robust and efficient way to
compute maximum likelihood estimates for incomplete data problems. Also, since the EM estimates
are maximum likelihood estimates, the statistical properties of maximum likelihood estimators are
also valid for them, i.e., they are consistent, asymptotically unbiased and asymptotically efficient.
4.3.
Numerical Tests on Synthetic Data
We conducted a series of numeric tests on synthetic data using direct maximization of the maximum
likelihood function in (1) as well as the EM method with and without using Monte Carlo simulation
to estimate the conditional expected values required in the E-Step. Our purpose in performing
these experiments was two-fold: first, to determine whether or not the EM method converged to
reasonable estimates of the underlying parameters and, second to compare the performance and
robustness of the EM method to direct maximization.
We constructed 24 examples by combining three different problem dimensions:
• Number of observations: 100; 1,000; 10,000 and 100,000,
• Percentage of negotiations that resulted in a deal: ∼ 40% and ∼ 80%,
• Number of covariates: 2, 5, and 10.
The mean willingness-to-pay and mean reserve price for each deal were linear in attributes. The
errors were assumed to be independent with zero means and known standard deviations.
We estimated the parameters for each of the 24 instances using the EM method with two different
approaches to calculating the conditional expected values required in the E-Step. In the first
approach (EM), we calculated the conditional expected value numerically. In the second approach
(Monte Carlo EM), we used Monte Carlo simulation to estimate conditional expectation. We
also estimated the parameters by directly maximizing the incomplete log-likelihood function in
MATLAB using its fmincon function. In all cases, we initiated the methods using 10 as the initial
intercept value and 0 as the initial estimates of the other coefficients. We ran the EM algorithm
until either the difference of two consecutive log-likelihood function was less than 10−2 or the
number of iterations exceeded 100. We used the default stopping criteria of the fmincon function.
In all cases, we terminated operations after 20,000 seconds (about 5 and 1/2 hours). The fmincon
function and the Monte Carlo EM method did not converge in some cases in which cases, we
restarted the algorithm using a different randomly-chosen initial point. Table 1 reports the total
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
11
amount of computation time for each method applied to each instance (including multiple starting
points). The times are for a laptop with Intel Core i7-3537U CPU with 2.00 GHz and 2.50 GHz
processor, 8 GB RAM and 64-bit Windows operating system.
Number of
Observations
100
1,000
10,000
100,000
Table 1
Instance
Number
1
2
3
4
5
6
Average
1
2
3
4
5
6
Average
1
2
3
4
5
6
Average
1
2
3
4
5
6
Average
Runtime (sec)
EM Monte Carlo EM Direct Max.
3.15
3.48
1.48
2.33
4.50
0.37
1.18
0.78
0.46
3.08
1.51
0.14
5.34
4.39
51.95
7.31
3.64
0.70
3.73
3.05
9.18
46.38
65.37
0.75
28.09
16.65
9.45
10.12
7.25
0.17
24.92
7.70
0.16
157.79
32.38
1.47
86.01
48.85
73.46
58.89
29.70
14.24
430.89
570.43
123.23
1,102.12
155.71
1,034.20
110.94
83.90
118.62
341.30
92.05
19,659.12
1,548.98
318.14
377.65
1,159.15
540.31
>20,000
782.24
293.42 >6,885.47
4,596.66
5,190.19
>20,000
11,012.59
2,447.16
18.03
2,125.79
768.79
2,076.20
4,194.65
845.05
5.02
15,480.73
4,158.28
>20,000
11,535.06
5,358.41
>20,000
8,150.08
3,127.98 >10,349.88
Computation Time Comparison of the Three Estimation Methods: EM, Monte Carlo EM, and Direct
Maximization of the Log-Likelihood Function
For all problem instances, the Monte Carlo EM algorithm converged to a stationary point in
under 100 iterations. The EM algorithm converged in under 100 iterations for 11 instances, but it
was very close to convergence when we stopped the procedure at 100 iterations for the other seven
instances. For three instances with 100,000 data points and one instance with 10,000 observations,
the direct maximization procedure could not find a solution after 20,000 seconds. When the methods
converged (or stopped in the EM method), they always converged to parameter estimates with
very similar likelihoods. The average deviation from the highest log-likelihood was 0.1% and the
maximum deviation of any method on any instance was 0.79%. Hence, we did not observe a
significant advantage of any procedure in terms of the quality of the parameter estimates. However,
12
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
both implementations of the EM algorithm were more reliable in converging to a solution and, on
average, converged more quickly for the larger instances.
Tables 2, 3, and 4 show the true and estimated parameters of the willingness-to-pay and reserve
price values for 100, 1,000, and 10,000 observations, respectively. They report the EM algorithm
results for 100 and 1,000 observations (since the algorithm converged in a reasonable time) and
the Monte Carlo EM algorithm results for the 10,000 observation cases (since the EM algorithm
required significantly longer computation times). The average mean squared errors between the true
and estimated parameters were 10.4991, 2.8293, and 0.4085 for 100, 1,000, and 10,000 observations,
respectively. For these cases, the EM methods appear to be converging to the true parameter
values.
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
38%
βr,0
5
2.81
βw,0
7
4.71
βr,1
0.5
0.62
βw,1
1
1.13
βr,2
1.5
1.60
βw,2
0.4
0.59
σr
3
1.84
σw
6
7.73
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
40%
βr,0
30
27.75
βw,0
10
10.75
βr,1
5
7.80
βw,1
1.5
6.51
βr,2
-0.1
-0.14
βw,2
-0.05
-0.08
βr,3
-3
-5.08
βw,3
2
-2.28
βr,4
5
5.72
βw,4
4
4.39
βr,5
1
1.36
βw,5
3
3.37
σr
8
10.83
σw
15
10.20
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
35%
βr,0
30
38.51
βw,0
15
15.99
72%
βr,0
12
21.45
βw,0
15
13.82
βr,1
5
4.68
βw,1
1.5
-4.63
βr,2
3
4.31
βw,2
4
8.49
βr,3
-0.1
-0.10
βw,3
-0.05
-0.07
βr,4
-3
-4.39
βw,4
2
0.17
βr,5
5
5.12
βw,5
4
2.67
βr,6
1
0.11
βw,6
3
5.51
βr,7
4
1.21
βw,7
6
9.62
βr,1
6
6.52
βw,1
1.5
1.49
βr,2
3
6.76
βw,2
4
2.21
βr,3
-0.1
-0.12
βw,3
-0.05
-0.05
βr,4
-3
-5.56
βw,4
2
4.19
βr,5
5
5.00
βw,5
4
4.04
βr,6
1
0.27
βw,6
3
2.80
βr,7
4
5.87
βw,7
6
6.64
Table 2
5.
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
78%
βr,0
5
1.78
βw,0
8
15.77
βr,1
0.3
0.21
βw,1
1.2
0.91
βr,2
1
1.65
βw,2
0.4
-0.13
σr
3
5.81
σw
5
1.81
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
82%
βr,0
30
33.28
βw,0
10
15.31
βr,1
5
18.57
βw,1
1
-4.03
βr,2
-0.1
-0.13
βw,2
-0.05
-0.04
βr,3
-3
-1.04
βw,3
2
-1.84
βr,8
-1
-1.29
βw,8
3
2.32
βr,9
10
12.03
βw,9
7
11.18
βr,10
3
2.56
βw,10
1
-0.34
σr
5
3.49
σw
10
12.47
βr,8
-1
-2.75
βw,8
3
4.46
βr,9
10
5.12
βw,9
7
11.41
βr,10
3
3.68
βw,10
1
0.58
σr
5
5.61
σw
10
7.63
βr,4
3
3.85
βw,4
5
4.31
βr,5
1
-0.87
βw,5
3
4.27
σr
8
15.59
σw
15
7.42
True and Estimated Parameters of Willingness-to-Pay (W ) and Reserve Price (R) for 100 Observations
Application to Auto Lending Data
We applied our structural EM method to a data set supplied by a major auto lender operating in
the United States. This data set is also used in Phillips et al. (2014). The lender sells to its end
customers through dealers who are located throughout the country. Although there is a centralized
annual percentage rate (APR) set for each loan, in most cases the dealer has discretion to negotiate
the final interest rate with customers3 . We excluded about 55.5% of the deals in the original data
base from the analysis because they were offered at promotional rates (so-called subvened rates)
over which the dealers had little or no authority to negotiate. Our structural EM method provides
3
We are using the word “negotiation” in a broad sense, since it is possible that some customers do not realize that
the rate is negotiable and may accept (or reject) the lender’s first offer without any “back and forth discussion”.
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
35%
βr,0
5
4.81
βw,0
7
8.31
βr,1
0.5
0.49
βw,1
1
0.88
βr,2
1.5
1.70
βw,2
0.4
0.54
σr
3
3.61
σw
6
5.11
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
38%
βr,0
30
34.40
βw,0
10
11.13
βr,1
5
4.73
βw,1
1.5
0.65
βr,2
-0.1
-0.10
βw,2
-0.05
-0.05
βr,3
-3
-2.50
βw,3
2
1.78
βr,4
5
5.22
βw,4
4
4.04
βr,5
1
0.51
βw,5
3
3.32
σr
8
9.76
σw
15
13.60
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
34%
βr,0
30
26.65
βw,0
15
3.32
73%
βr,0
12
12.52
βw,0
15
12.31
βr,1
5
5.25
βw,1
1.5
1.32
βr,2
3
3.19
βw,2
4
5.04
βr,3
-0.1
-0.10
βw,3
-0.05
-0.04
βr,4
-3
-2.22
βw,4
2
2.39
βr,5
5
5.05
βw,5
4
3.63
βr,6
1
0.98
βw,6
3
4.18
βr,7
4
5.01
βw,7
6
7.06
βr,1
6
6.30
βw,1
1.5
1.38
βr,2
3
3.04
βw,2
4
2.97
βr,3
-0.1
-0.10
βw,3
-0.05
-0.05
βr,4
-3
-2.18
βw,4
2
1.95
βr,5
5
5.05
βw,5
4
3.93
βr,6
1
0.69
βw,6
3
3.32
βr,7
4
4.64
βw,7
6
6.53
Table 3
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
76%
βr,0
5
7.15
βw,0
8
10.21
βr,1
0.3
0.19
βw,1
1.2
0.92
βr,2
1
1.02
βw,2
0.4
0.55
σr
3
5.61
σw
5
2.66
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
79%
βr,0
30
34.15
βw,0
10
7.60
βr,1
5
4.96
βw,1
1
1.01
βr,2
-0.1
-0.10
βw,2
-0.05
-0.04
βr,3
-3
-3.79
βw,3
2
2.94
βr,8
-1
-0.95
βw,8
3
4.03
βr,9
10
10.22
βw,9
7
7.24
βr,10
3
2.88
βw,10
1
0.22
σr
5
4.03
σw
10
12.29
βr,8
-1
-0.76
βw,8
3
3.01
βr,9
10
8.78
βw,9
7
8.44
βr,10
3
3.12
βw,10
1
1.04
σr
5
4.78
σw
10
9.64
βr,4
3
3.04
βw,4
5
4.67
βr,5
1
0.68
βw,5
3
3.33
13
σr
8
9.65
σw
15
12.66
True and Estimated Parameters of Willingness-to-Pay (W ) and Reserve Price (R) for 1,000
Observations
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
38%
βr,0
5
6.96
βw,0
7
7.95
βr,1
0.5
0.41
βw,1
1
0.83
βr,2
1.5
1.62
βw,2
0.4
0.77
σr
3
4.13
σw
6
3.72
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
40%
βr,0
30
29.56
βw,0
10
8.54
βr,1
5
5.08
βw,1
1.5
1.07
βr,2
-0.1
-0.10
βw,2
-0.05
-0.04
βr,3
-3
-2.42
βw,3
2
1.45
βr,4
5
4.97
βw,4
4
3.86
βr,5
1
0.95
βw,5
3
3.09
σr
8
7.61
σw
15
15.88
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
35%
βr,0
30
30.87
βw,0
15
16.86
72%
βr,0
12
12.12
βw,0
15
16.10
βr,1
5
4.96
βw,1
1.5
1.51
βr,2
3
3.08
βw,2
4
4.16
βr,3
-0.1
-0.10
βw,3
-0.05
-0.06
βr,4
-3
-2.87
βw,4
2
2.46
βr,5
5
4.94
βw,5
4
3.95
βr,6
1
0.91
βw,6
3
2.91
βr,7
4
4.63
βw,7
6
6.23
βr,1
6
6.00
βw,1
1.5
1.44
βr,2
3
3.21
βw,2
4
3.97
βr,3
-0.1
-0.10
βw,3
-0.05
-0.05
βr,4
-3
-2.61
βw,4
2
1.94
βr,5
5
5.05
βw,5
4
3.90
βr,6
1
1.00
βw,6
3
3.00
βr,7
4
3.94
βw,7
6
5.76
Table 4
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
78%
βr,0
5
6.60
βw,0
8
8.84
βr,1
0.3
0.17
βw,1
1.2
1.05
βr,2
1
1.05
βw,2
0.4
0.54
σr
3
4.75
σw
5
2.96
Trade pc:
R
true prms
est. prms
W
true prms
est. prms
82%
βr,0
30
28.94
βw,0
10
11.58
βr,1
5
4.81
βw,1
1
1.55
βr,2
-0.1
-0.10
βw,2
-0.05
-0.05
βr,3
-3
-3.20
βw,3
2
1.72
βr,8
-1
-1.10
βw,8
3
2.93
βr,9
10
10.49
βw,9
7
6.92
βr,10
3
2.99
βw,10
1
0.97
σr
5
5.11
σw
10
9.76
βr,8
-1
-1.08
βw,8
3
3.04
βr,9
10
9.47
βw,9
7
6.48
βr,10
3
2.94
βw,10
1
0.99
σr
5
4.97
σw
10
9.99
βr,4
3
2.96
βw,4
5
5.02
βr,5
1
1.18
βw,5
3
2.78
σr
8
7.94
σw
15
14.89
True and Estimated Parameters of Willingness-to-Pay (W ) and Reserve Price (R) for 10,000
Observations
us estimates of the distributions of willingness-to-pay and reserve price as well as estimates of the
last prices (APRs) for lost deals – that is, failed transactions. We evaluate the results of using
the EM approach in two ways. First, we compare the take-up rates and APRs of both successful
deals and lost deals with a hold-out sample. Our benchmark is the set of APRs predicted using
linear regression on deal characteristics. We also compare the take up rates predicted by the EM
method with the take-up rates predicted by standard probit regression. Second, we compute the
price elasticities implied by our estimates and compare these to the endogeneity-corrected estimates
14
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
in Phillips et al. (2014) for the same auto lender data. In both cases, the EM method compares
favorably with the alternatives.
5.1.
Data Set and Model
Our data set includes information on the outcome of all approved applications for auto loans
received by a major North American indirect auto lender data during a multi-year period starting
in January 20094 . As noted previously, we considered only the non-subvened deals – that is those in
which rates were negotiable. There are 950,985 approvals in this data set of which 719,835 (75.69%)
were taken up. All of these loans were offered through the finance and insurance (F&I) departments
of local dealerships and, in each case, the F&I manager had the authority to negotiate the APR
of the loan with the customer. Each record includes the APR at which the loan was offered, the
final disposition of the loan (accepted by the customer or not accepted) and information about the
customer (FICO score), information about the product (make and model of car, size of loan, new
or used car, term of loan, etc.), and the identity and location of the dealer.
The variables in the data and their statistics for the sample that we used are shown in Tables 5
and 6. FICO score is an industry-standard measure of individual default risk that ranges from 300
to 850 with higher values representing lower risk. The term and amount of the loan and whether
or not the car being purchased is new or used are characteristics of the loan being requested.
The customer rate is the final rate charged to the customer in the case of a successful negotiation
and the last rate offered in the case of a lost deal. LIBOR is the interbank lending rate and is
closely correlated with the lender’s cost of funds and with the primer rate. The dealer offered cash
incentives for some deals. The amount of this incentive – if offered – is also recorded in the data.
If the incentive was not offered, this value is 0. The lender applied a proprietary methodology
to classify potential customers into five risk tiers with risk tier 1 being the least risky and risk
tier 5 the most risky. Risk tier is correlated with FICO score but is not fully determined by it
because the lender used additional information to classify customers into risk tiers. To create an
individual risk measure that is not correlated with risk tier, we created a normalized FICO score
for each customer, F\
ICO, defined by F\
ICO = Actual FICO Score - Mean FICO Score for risk
tier.5 . F\
ICO is uncorrelated with risk tier.
Applying the EM method to the entire data set would require an excessive amount of computational time. More specifically, estimating the parameters using the full data set would require 57
hours of CPU time on a laptop with Intel(R) Core(TM) i7-3537U CPU @ 2.00 GHz and 2.50 GHz
processor, 8GB RAM and 64-bit Windows operating system. (We ran the EM algorithm up to
4
The lender has requested anonymity and certain details about the data have been suppressed to conceal its identity.
5
This variable is defined in the same way in Phillips et al. (2014) as well.
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
15
Variable
Mean Std. Dev. Min
Max
FICO Score
740.66
79.35 371
900
F\
ICO
4.6
54.87 -344
231
Term (Months)
59.76
11.40
36
72
Amount ($1,000)
21.21
10.88 1.00 212.77
Customer Rate (%)
8.49
3.95 0.00 33.39
LIBOR (%)
0.27
0.07 0.19
0.56
Cash Incentive ($1,000)
1.32
1.28 0.00
9.00
Table 5
Summary Statistics for Continuous Variables
Variable
Category Frequency Percent
1
114,874
57.44
2
38,146
19.07
Tier
3
29,819
14.91
4
7,163
3.58
5
9,998
5.00
New
135,145
67.57
Vehicle Type
Used
64,855
32.43
Not Take
48,894
24.45
Customer’s Decision
Take
151,106
75.55
Table 6
Summary Statistics for Categorical Variables
1,000 iterations to ensure convergence and computation time for each iteration was 206 seconds.)
Therefore, we randomly selected a sample of 200,000 observed negotiation outcomes from the original data (21.03% of the “non-subvened” deals) for computations (this required 11 hours of CPU
time on the same platform – 1,000 iterations with 40 sec/iteration). We randomly split the data
set into estimation and holdout samples of equal size. As a first step, we estimated a single model
with separate coefficients for each of the variables listed in Tables 5 and 6 allowing R and W to be
correlated using the algorithm described in Section 4.2. To estimate the standard errors, we used
the bootstrapping approach described in Chapter 4.6 of McLachlan and Krishnan (1996). Table 7
shows the results of the estimation.
βr
Variable Est. SE t-stat Est.
Intercept 1.87 0.30
6.27 18.78
Tier2 1.73 0.04 42.02 1.48
Tier3 3.58 0.05 75.60 3.29
Tier4 5.32 0.12 45.92 4.03
Tier5 7.46 0.14 53.20 5.65
Used 5.11 0.08 63.08 -1.17
LIBOR 1.19 0.02 54.61 1.06
Covariance (σRW )
Correlation Coeff. for R and W
Log-Likelihood (for holdout)
Table 7
βw
SE t-stat
Variable Est.
0.33 57.35 Term Class 2 0.23
0.05 32.45 Term Class 3 0.47
0.05 69.04 Term Class 4 0.67
0.09 43.39 F\
ICO (/100) -1.3
0.17 33.48
Log Amount -0.17
0.08 -14.47
Cash Incen. -0.93
0.02 55.34 Std. Dev. (σ) 2.82
0.01 (SE: 0.07; t-stat: 0.14)
0.0014 (SE: 0.01; t-stat: 0.14)
-28,392.9
βr
βw
SE t-stat Est. SE t-stat
0.06
3.96 0.72 0.06 11.33
0.05
9.99 0.37 0.05
7.18
0.05 12.77 0.62 0.06 10.47
0.03 -49.59 -1.25 0.03 -45.16
0.03 -5.53 -1.39 0.03 -44.24
0.04 -22.48 0.98 0.04 22.53
0.03 88.77 2.57 0.03 77.03
Coefficient Estimates for the Structural EM Method. All values are significant at the .01 level.
16
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
The values in Table 7 generally accord with intuition. The coefficient βw increases with risk tier
and decreases with F\
ICO, indicating that customer willingness-to-pay increases with risk. This
is consistent with previous findings that more risky customers are less sensitive to the price of
loans than less risky borrowers (Phillips and Raffard 2011). Average willingness-to-pay decreases
with Log Amount, consistent with the intuition that customers are often more likely to shop for
alternatives if the amount of the loan – and hence the monthly payment – is higher. Term has no
consistent effect on average willingness-to-pay. On the other hand, the reserve price consistently
increases in term. This may be due to the fact that customers preferring longer-term loans tend to
be higher risk than those preferring shorter-term loans. Average seller reserve price increases with
risk tier. This is intuitive because risk represents a cost to the seller and increased cost needs to
be covered by an increased price. Seller reserve price decreases in Log Amount – this is consistent
with the common expectation that sellers should be more willing to discount larger deals. It may
also reflect a revenue-bias on the part of the sales force.
The results in Table 7 indicate that reserve price and willingness-to-pay are not highly correlated
with each other – the correlation coefficient between R and W is .0014 and the covariance is
.01. This suggests an alternative model in which we estimate a different model for each risk tier
assuming that the covariance between R and W is zero. The results for this second approach are
shown in Table 8, which also shows the empirical means and standard deviations of R and W for
each risk tier for the hold-out sample (numbers in parentheses show the bootstrapped standard
errors – all the variables except Term Class 3/4 and Used of Tier 5 are statistically significant at
the .01 significance level). As expected, the mean values of both R and W increase by risk tier.
Henceforth, we use this model as our standard model for comparison.
The results in Table 8 show some of the same patterns as Table 7. Notably, both reserve price
and willingness-to-pay increase with risk. This is evident from the fact that both the intercepts
and the mean values increase with tier and the coefficients for F\
ICO are less than zero for both
R and W . There is no longer any consistent pattern evident in the coefficients for term class for
either R or W suggesting that the pattern in the previous model was indeed due to the correlation
between risk and term. The coefficients for Log Amount are negative for all categories except Tier
1 of R. The coefficients for Used show a striking pattern in that they are all positive for R and
all negative for W . We conjecture that the positive coefficients for R reflect the fact that used-car
APR’s are rarely subvened in the marketplace and hence lenders feel they can extract a higher
APR than for new cars. On the other hand, the negative coefficient for W may reflect the fact
that used car buyers are, in general, more price-sensitive than new car buyers. The pattern for
Cash Incentive is also interesting. Offering a cash incentive increases the willingness of borrowers to
accept a higher APR for the loan, which is intuitive. It is more difficult to explain the consistently
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
Tier 1
Intercept
-1.92
(0.53)
Term Class 2
0.43
(0.09)
Term Class 3
0.73
(0.07)
1.61
Term Class 4
(0.08)
Used
5.15
(0.17)
F\
ICO (/100)
-0.54
(0.04)
Log Amount
0.16
(0.05)
Cash Incentive
-0.96
(0.09)
LIBOR
1.33
(0.04)
Std. Dev.
2.46
(0.06)
Emp. Mean
2.91
Emp. Std. Dev.
2.73
Table 8
Tier 2
14.94
(0.76)
-1.19
(0.19)
-1.76
(0.16)
-1.72
(0.17)
3.6
(0.16)
-1.53
(0.06)
-0.99
(0.07)
-0.81
(0.07)
0.94
(0.04)
2.61
(0.09)
7.05
3.34
R
Tier 3
18.52
(0.9)
-1.35
(0.25)
-1.92
(0.24)
-2.49
(0.23)
3.43
(0.19)
-2.57
(0.07)
-0.98
(0.09)
-0.76
(0.07)
0.68
(0.04)
2.92
(0.11)
10.23
3.46
Tier 4
18.4
(2.44)
-2.03
(0.66)
-3.16
(0.64)
-4.59
(0.69)
4.44
(0.39)
-2.72
(0.18)
-0.75
(0.23)
-1.00
(0.17)
1.01
(0.11)
3.62
(0.21)
12.61
4.58
Tier 5
33.22
(2.52)
-3.58
(0.65)
-3.66
(0.64)
-4.04
(0.68)
5.9
(0.43)
-2.66
(0.2)
-2.23
(0.24)
-1.22
(0.19)
1.58
(0.14)
4.80
(0.23)
15.61
5.82
Tier 1
17.11
(0.48)
0.38
(0.08)
0.39
(0.06)
0.89
(0.08)
-1.61
(0.13)
-0.49
(0.04)
-1.27
(0.05)
1.00
(0.09)
1.18
(0.03)
2.05
(0.06)
9.86
1.85
Tier 2
30.19
(0.84)
-0.95
(0.2)
-2.5
(0.17)
-1.78
(0.17)
-2.03
(0.18)
-1.95
(0.07)
-2.08
(0.08)
0.97
(0.08)
0.84
(0.04)
3.02
(0.10)
9.74
2.61
W
Tier 3
31.69
(0.99)
-0.68
(0.22)
-2.07
(0.2)
-1.65
(0.21)
-1.93
(0.18)
-2.63
(0.07)
-2.03
(0.1)
0.74
(0.08)
0.82
(0.04)
3.11
(0.11)
11.18
2.55
Tier 4
31.46
(2.88)
-0.93
(0.72)
-2.28
(0.7)
-2.56
(0.72)
-1.76
(0.39)
-2.49
(0.15)
-1.85
(0.28)
1.05
(0.16)
0.78
(0.11)
3.75
(0.22)
12.25
2.76
17
Tier 5
40.75
(3.26)
2.29
(0.83)
0.65
(0.71)
-0.56
(0.71)
-0.58
(0.48)
-2.83
(0.14)
-3.04
(0.33)
1.78
(0.19)
1.12
(0.14)
4.85
(0.31)
14.47
3.55
Coefficient Estimates for the Structural EM Method: Each Tier Separately
negative coefficients for R. The coefficients for LIBOR are all positive and are generally close to 1,
reflecting that it is a cost that needs to be reflected in the APR.
5.2.
Quality of Predictions
One of the advantages of our structural estimates is that they predict both take-up and APR.
Hence, the estimates can be used not only to estimate take-up and take-up sensitivity to price, but
also to impute APRs for lost deals (the last APR offered to a loan that a customer rejected) as well
as predict the APRs for future deals. Below, we compare the quality of these predictions to standard
regression-based approaches applied to the same data. In particular, we compare APRs predicted
by our structural EM method to the actual APRs of the deals and to the APRs predicted by linear
regression. We also compare the take-up predictions of our method to the take-up predictions of
a two-stage endogeneity-corrected probit model, as well as to the actual realized value recorded in
the data set.
5.2.1.
APR Predictions Table 9 compares the APR’s predicted for deals in the test data
generated by our method to the actual APRs and to the APRs predicted using linear regression
applied using APR as the target variable and all of the variables listed in Tables 5 and 6 (except
APR) as independent variables. This is the same approach used to generate the first-stage estimates
of APR’s described in more detail in Phillips et al. (2014). The structural EM method more
18
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
Tier
1
2
3
4
5
Total
Table 9
N
57,399
19,000
14,967
3,673
4,937
99,976
All Deals
Won Deals
Lost Deals
Avg.
LR
EM
Avg.
LR
EM
Avg.
LR
EM
APR (%) RMSE RMSE APR (%) RMSE RMSE APR (%) RMSE RMSE
6.36
1.82
1.76
6.23
1.79
1.72
7.45
2.09
2.05
8.52
2.19
2.07
8.22
2.22
2.09
9.07
2.13
2.04
10.80
2.51
2.30
10.44
2.50
2.24
11.21
2.52
2.38
13.37
3.53
3.31
11.49
3.12
2.78
14.76
3.80
3.65
16.90
5.33
4.74
13.82
4.23
3.64
19.22
6.03
5.42
8.21
2.37
2.21
7.33
2.09
1.95
10.94
3.08
2.88
Root Mean Squared Error (RMSE) of the APR’s Predicted for All Deals, Won Deals, and Lost Deals
for Linear Regression (LR) and Expectation-Maximization (EM) Methods
accurately forecasts APRs for all risk tiers for both won and lost deals than linear regression. The
amount of improvement generally increases with risk tier. This probably reflects the fact that the
range of APRs offered for different deals increases in risk tier, providing more room for improvement
in RMSE.
5.2.2.
Take-Up Predictions Table 10 shows the take-up rates estimated by our method
compared to the observed take-up and the take-up forecast by two different probit models. In the
first probit model (Probit A), we assume that prices are not available for lost deals and we use
the prices imputed by linear regression. In the second probit regression (Probit B), we utilized the
prices for all recorded negotiations whether they resulted in transactions or lost deals. A lender who
had access to rates for “lost deals” would typically utilize that information in modeling take-up
and would utilize an approach similar to the second probit model. A lender who did not have rates
for “lost deals” would typically use a method similar to the first probit model. Hence, these serve
as benchmarks for two different possible informational states of a lender.
Table 10 shows the take-up prediction performance of the various methods on the hold-out
sample. Note the two probit approaches did not differ significantly in terms of concordance for risk
tiers 1 - 3, nor for the test data set as a whole. Probit B improved the concordances for tiers 4
and 5 compared to Probit A and had a slightly higher log-likelihood for the sample as a whole.
Our structural EM estimates had higher concordance in total (.928) than either of the alternatives
(.921 and .922 for Probit A and B, respectively). The EM algorithm actually performed worse (in
terms of both concordance and log-likelihood.) in risk tiers 4 and 5 compared to Probit B, but this
was outweighed by its superior performance in the much larger higher risk tiers 1 - 3.
5.3.
Comparison of Elasticities
We can use the coefficients of our structural model to compute customer and seller price-elasticities.
Let FW and FR denote the marginal cumulative distribution functions and fw and fR the marginal
density functions for W and R respectively and let F̄W = 1 − FW . Then, the customer price elasticity
at price p is W (p) = −pfW (p)/F̄W (p). This corresponds to the standard definition of price elasticity;
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
Tier
1
2
3
4
5
Total
Probit A
Probit B
Actual
Pred.
Pred.
Take-up Take-up Concor.
LL
Take-up Concor.
89.73
89.92
.959
-7,748
89.95
.960
64.18
64.57
.862
-7,877
64.57
.864
54.13
53.29
.784
-7,968
53.27
.780
42.42
44.80
.774
-1,925
44.98
.812
42.92
42.21
.759
-2,713
42.04
.815
75.50
75.60
.921
-28,231 75.62
.922
Table 10
EM
Pred.
LL
Take-up Concor.
LL
-7,848
89.57
.961
-7,500
-7,878
64.64
.866
-7,850
-8,002
53.62
.788
-7,839
-1,841
44.85
.781
-1,871
-2,548
42.32
.789
-2,598
-28,117 75.47
.928
-27,659
Take-Up Prediction Performance for Probit A, Probit B and the Structural EM Method (EM)
Tier
Tier
Tier
Tier
Tier
Table 11
19
1
2
3
4
5
Customer
Seller
Price Elasticity Price Elasticity
Probit
EM
EM
-1.10
-1.81
0.16
-0.96
-1.69
1.39
-0.85
-0.89
2.80
-0.80
-0.53
2.68
-0.54
-0.24
2.49
Customer and Seller Price Elasticities by Risk Tier for Structural EM and Two-Stage Probit Estimates
it is the percentage change per unit percentage change in price. By analogy, we can also calculate
a seller price elasticity as R (p) = pfR (p)/FR (p). We computed the elasticity values for both R
and W using the empirical mean and standard deviations of R and W reported in Table 8 and
the mean customer rate reported in Table 5. Table 11 shows both the customer and seller price
elasticities for each risk tier. For comparison, Table 11 also shows the customer price elasticities
estimated via the method in Phillips et al. (2014) using an endogeneity-corrected probit model for
the same auto lender data.
The price-elasticities from Phillips et al. (2014) are not directly comparable because they were
calculated using the entire data set while our elasticities were calculated from the data set excluding
subvened deals. However, our estimates follow the expected pattern of decreasing in magnitude for
larger risk tiers – again reflecting the fact that higher-risk customers are less price-sensitive than
lower-risk customers. Seller price-elasticity increases in tier as well. This would indicate that sellers
are, in general, less-willing to negotiate with higher risk customers, all else being equal.
6.
Implications for Pricing Policy
We next consider how estimated willingness-to-pay and reserve price distributions can be used to
improve profits. We show –first theoretically and then empirically– that, it can be profitable for
headquarters to impose a minimum price by deal category on its sales force. Properly set, these
limits can generate significant profit improvements relative to unconstrained negotiation and even
relative to data-driven optimized fixed prices.
20
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
6.1.
Theoretical Analysis
When the sales force is given full pricing discretion, the expected revenue6 from a negotiation is
R+W
E(Revenue) = E
|W ≥ R P{W ≥ R}
2
We assume that “headquarters” can impose a minimum price for each transaction on the sales
force. That is, for a given negotiation, the sales agent cannot agree to a price less than pmin .
We note that pmin can be a function of observable covariates of a deal. The reserve price of each
individual sales agent is independent of the minimum price imposed by headquarters. In this case,
headquarters wishes to derive the p∗min that will maximize expected revenue. We note that there
are four cases given W ≥ R:
• Case 1: R ≤ W ≤ pmin . In this case no trade will occur.
• Case 2: R ≤ peq ≤ pmin ≤ W . Trade will occur at pmin
• Case 3: R ≤ pmin ≤ peq ≤ W . Trade will occur at peq
• Case 4: pmin ≤ R ≤ W . Trade will occur at peq
Figure 1 illustrates these cases. For cases 3 and 4, the imposition of a minimum price does not
change revenue while revenue may decrease for case 1 and increase for case 2.
R
2pmin
Case 4
pmin
Case 3
pmin
Case 1
Figure 1
2pmin
W
Case 2
Revenue Change Zones Based on the Value or R and W
The expected revenue from a transaction under a minimum price policy (MPP) can be written
as
6
In this section we assume w.l.o.g. that incremental costs are zero so that maximizing revenue is the same as
maximizing profit.
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
21
E(RevenueM P P ) = 0 ∗ P{R ≤ W ≤ pmin }
+ pmin ∗ P{R ≤ peq ≤ pmin ≤ W }
R+W
eq
+E
|R ≤ pmin ≤ p ≤ W ∗ P{R ≤ pmin ≤ peq ≤ W }
2
R+W
+E
|pmin ≤ R ≤ W ∗ P{pmin ≤ R ≤ W }
2
With a little bit of algebra, we can write this expression as
Z pmin
E(RevenueM P P ) = pmin ∗
Fr (x)fw (2pmin − x)dx
−∞
1
+
2
Z
1
−
2
Z
∞
Z
∞
xfw (x)Fr (x)dx −
pmin
xfw (x)Fr (2pmin − x)dx
pmin
∞
Z
pmin
xfr (x)Fw (2pmin − x)dx +
xfr (x)Fw (x)dx +
pmin
−∞
µr
2
(8)
Lemma 1. If Fr and Fw have support on (0, ∞) and if fw is non-increasing in an interval (a, 2a),
then:
∂E(RevenueM P P ) ≤0
∂pmin
pmin =a
Proof:
The derivative of (8) is
Z pmin
∂E(RevenueM P P )
=
Fr (x)fw (2pmin − x)dx − pmin fw (pmin )Fr (pmin )
∂pmin
−∞
Z ∞
=
Fr (2pmin − x)fw (x)dx − pmin fw (pmin )Fr (pmin )
pmin
Because Fr and Fw have support only on (0, ∞),
Z pmin
∂E(RevenueM P P )
=
Fr (x)fw (2pmin − x)dx − pmin fw (pmin )Fr (pmin )
∂pmin
0
By the Mean Value Theorem, there must be some p̂ ∈ (0, pmin ) such that:
Z pmin
Fr (x)fw (2pmin − x)dx = pmin Fr (p̂)fw (2pmin − p̂)
0
So,
∂E(RevenueM P P )
= pmin [Fr (p̂)fw (2pmin − p̂) − fw (pmin )Fr (pmin )]
∂pmin
Since Fr (p̂) ≤ Fr (pmin ) and fw (2pmin − p̂) ≤ fw (pmin ), the partial derivative is less than or equal to
0. Q.E.D.
22
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
Corollary 1. If W is distributed exponentially, then p∗min = 0. If R ∼ U nif orm(ar , br ) and
W ∼ U nif orm(aw , bw ) with some ar , aw ≥ 0, then:
• If aw ≤ ar , then p∗min = 0
• If aw > ar , then p∗min = aw
Proof:
The first part follows immediately from Proposition 1. The second part follows since
MP P )
≤ 0 by Proposition 1 and if pmin < aw then
in case aw > ar , if pmin > aw , then ∂E(Revenue
∂pmin
R
p
∂E(RevenueM P P )
= 0 min Fr (x)fw (2pmin − x)dx ≥ 0 and non-decreasing (zero if pmin < a2w and increas∂pmin
ing if
aw
2
≤ pmin ≤ aw ) in pmin . Finally, in case aw ≤ ar , if pmin > aw , then
Proposition 1 and if pmin < aw then
∂E(RevenueM P P )
∂pmin
∂E(RevenueM P P )
∂pmin
≤ 0 by
= 0. Q.E.D.
Corollary 2. If W follows a normal distribution, then p∗min ≤ µW .
When R and W are both normally distributed, there is no closed-form solution for p∗min but it
can be calculated numerically. For the auto lending data, Figure 2 shows the graph of expected
customer rate from a negotiation given by (8) as a function of pmin when R and W are normally
distributed with the estimated parameters given previously in Table 87 . Note that revenue in each
tier could be increased by imposing a minimum price. The increase is most significant for Tier 1
customers due to the fact that the mean reserve price is lowest for this tier. Note that, as required
by Corollary 2, p∗min ≤ µW in each tier.
6.2.
Policy Comparisons
Using our estimates of the willingness-to-pay and reserve price distribution, we can estimate the
counter-factual change in expected profit that would have occurred from alternative policies. For
a given deal j, we calculate the expected profit at price p by:
Πj (p) = P OPj ∗ (T otalP aymentj (p) − CapitalCostj ) − (1 − P OPj ) ∗ LGD ∗ CapitalCostj
(9)
where POP denotes the Probability of Payment which is the probability that the borrower will repay
the loan in full as estimated by the lender for each prospective deal (this variable was calculated
by the auto lender and recorded in the data), and LGD denotes the loss-given-default ratio – that
is, the expected fraction of the capital cost of the loan that will need to be written off if the
borrower defaults. We set LGD = .5, which is the value used by the lender. The T otalP aymentj
is calculated by multiplying the monthly payments (calculated using the standard equation p =
P r(1 + r)n /[(1 + r)n − 1], where p, P , r, and n are monthly payment, initial principal, monthly
APR, and term in months, respectively) by the term of the loan and CapitalCostj is obtained by
calculating the total payment at the LIBOR, which is an indicator of the lender’s capital cost.
7
In this example, “revenue” corresponds to the customer rate.
23
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
Tier1 (r=2.91%,  r=2.73%, w =9.86%,  w =1.85%)
Tier2 (r=7.05%,  r=3.34%, w =9.74%,  w =2.61%)
7
Tier3 (r=10.23%,  r=3.46%, w =11.18%,  w =2.55%)
6.2
6.1
6
6
6
5.5
Expected Customer Rate (%)
5.8
Expected Customer Rate (%)
Expected Customer Rate (%)
6.5
5.6
5.4
5.2
5
5.9
5.8
5.7
4.8
5
5.6
4.6
4.5
0
1
2
3
4
5
6
7
Minimum Price -- pmin (%)
8
9
10
4.4
0
1
2
3
4
5
6
7
Minimum Price -- pmin (%)
8
Tier4 (r=12.61%,  r=4.58%, w =12.25%,  w =2.76%)
5.92
5.36
5.915
Expected Customer Rate (%)
Expected Customer Rate (%)
5.34
5.32
5.3
5.28
5.26
5.24
5.5
0
1
2
3
4
5
6
7
Minimum Price -- pmin (%)
8
9
10
5.91
5.905
5.9
5.895
5.89
5.885
5.22
Figure 2
10
Tier5 (r=15.61%,  r=5.82%, w =14.47%,  w =3.55%)
5.38
5.2
0
9
1
2
3
4
5
6
7
Minimum Price -- pmin (%)
8
9
10
5.88
0
1
2
3
4
5
6
7
Minimum Price -- pmin (%)
8
9
10
Expected Customer Rate as a Function of the Minimum Price Threshold (pmin ) by Risk Tier
For all three of the policies that we consider, we assume that headquarters wants to maximize
total expected profit – that is, the sum of Πj (p) over all deals – and will set centralized prices and
associated parameters that achieve that goal. Field sales will negotiate individual deals within the
parameters specified by headquarters. We consider the following three policies:
1. Centralized Fixed Pricing (CFP). In this policy, we assume the headquarters does not
allow negotiation and sets a deal-specific rate based on observable customer and product characteristics. The price for each deal is given by:
P
pCF
= arg max
j
Πj (pj ) ∗ P{Wj ≥ pj }.
(10)
pj
2. Negotiated Rate with Floor (NRF). In this policy, headquarters imposes a lower bound
(or floor) on the final rate. Agents negotiate individual deals but cannot commit to a final price
lower than the minimum negotiated rate. The headquarters calculates the optimal lower bound for
each deal, pmin
, to maximize expected profit from each deal:
j
pmin
= arg max 0 ∗ P{R ≤ W ≤ p} + Πj (p) ∗ P{R ≤ peq ≤ p ≤ W }
j
(11)
p
+ E(Πj (peq )|R ≤ p ≤ peq ≤ W )) ∗ P{R ≤ p ≤ peq ≤ W }
+ E(Πj (peq )|p ≤ R ≤ W )) ∗ P{p ≤ R ≤ W },
N RF
where peq
. As before,
j = (R + W )/2. We denote the price for deal j under the NRF policy as pj
RF
min
if R > W , there will be no deal, otherwise, pN
= max[peq
].
j , pj
j
24
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
3. Imposed Reserve Price (IRP) Policy. All of our models so far have assumed that the sales
agent reserve prices are exogenous. Under the IRP policy, we consider the case where headquarters
sets the reserve price for each deal. To the extent that the reserve price incorporates elements
of sales agent negotiating skill, this may not be feasible. However, it is certainly feasible that
headquarters could influence the reserve price by, for example, setting deal-specific commission
structures (e.g., headquarters could provide a bonus to sales agents based on the spread between
the negotiated rate and the imposed reserve price). In this case, headquarters sets an optimal
reserve price `∗j as:
`∗j = arg max
E[Πj ((`j + Wj )/2)|Wj ≥ `j ] ∗ P{Wj ≥ `j }
(12)
`j
The expected price for each deal under the IRP policy is then pIRP
= E[(`∗j + Wj )/2|Wj ≥ `∗j ].
j
For the CFP and NRF policies, we compute the expected counter-factual profit for each deal by
e {Wj ≥ pi }, where pi denotes the price for deal j under policy i where i ∈
E[Πj (pij )] = Πj (pij ) ∗ P
j
j
e is the conditional take-up probability,
{CF P, N RF }, Πj (p) is calculated as in equation (9), and P
e {Wj ≥ pi } = P{Wj ≥ pi | customer’s decision at rate = Customer Rate}. For the IRP policy,
i.e., P
j
j
e {Wj ≥ `∗ }.
the expected counter-factual profit is computed by E[Πj ((`∗ + Wj )/2)|Wj ≥ `∗ ] ∗ P
j
j
j
The results of applying the three policies to the prospective deals in the test data set are shown
in Table 12. “Mean Final Price” is the average APR offered across all prospective deals, the “Mean
Take-up Probability” is the expected fraction of the prospective deals that would be accepted under
each policy, the average profit per negotiation is the average profit realized over all negotiations
(both successful and unsuccessful). All three policies showed significant increases in profit relative
to what was actually achieved. We did not have full information about the actual costs per deal so
some of this increase may be artificial. However, this limitation does not affect the relative results
from the three alternative policies considered.
Mean
Mean
Average
Final
Take-Up
Profit per
Increase
Policy
Price (APR) Probability Negotiation From Actual
Centralized Fixed Pricing (CFP)
9.89%
.76
$3,288
18.10%
Negotiated Rate with Floor (NRF)
9.64%
.70
$3,216
15.52%
Imposed Reserve Price (IRP)
8.82%
.87
$3,973
42.71%
Actual
8.49%
.76
$2,784
–
Table 12
Final prices, take-up probabilities and average profits for the three policies compared with the actual.
The Imposed Reserve Price policy achieved significantly higher revenue than the other two
policies (and the status quo) by a large margin. It was able to both achieve higher average margins
for funded deals as well as increase the expected number of total deals by lowering rates for deals
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
25
that were not accepted by customers. For deals that were actually booked, the mean customer
rate was 7.6%. For the same deals, the mean rate offered under the IRP policy was 9.03% with a
corresponding take-up rate of .96. For these deals, the mean profit under IRP was $4,446 compared
with the actual mean profit of $3,688. IRP also lowered prices on average for deals that were lost
and hence increased the acceptance rate from those deals. IRP offered an average rate of 8.17%
for these deals (the actual mean rate of them was 11.22%) and converted 57.9% of them leading
to additional expected profit of $2,516 for these deals, which contributed no profit in actuality.
The Centralized Fixed Pricing policy performed slightly better than the Negotiated Rate with
Floor policy. This would suggest that, in this case, the benefits of centralized pricing control
outweigh the benefits of allowing sales staff to negotiate prices with a simple lower bound. Still,
it is worth noting that imposing an optimized lower bound on negotiations provides a significant
improvement in profits (15.52%) over the status quo. This shows that the simple change of imposing
a price limit on negotiations can provide significant pricing improvement, provided the negotiating
limits are intelligently computed.
7.
Robustness
The EM method applied to the structural negotiation model described in this paper results in
estimated parameters for the distributions of W and R that meet intuition and lead to APR and
take-up predictions as good or better than competing models. However, the estimation approach
is based on several important structural assumptions. For one, it requires ex ante specification
of the form of the distributions of reserve price and willingness-to-pay. If these distributions are
misspecified, this could lead to incorrect results. Second, it specifies that, when a transaction
takes place, it takes place at the symmetric Nash bargaining equilibrium price – halfway between
the willingness-to-pay and the reserve price. While the symmetric Nash bargaining equilibrium
price has appealing theoretical qualities, it is conceivable that individual deals take place at prices
that are closer to either the reserve price or the customer willingness-to-pay depending upon the
respective bargaining skills of the customer and seller and possibly other factors. In this section
we investigate the robustness of our approach to these assumptions.
7.1.
Misspecification of the Joint Distribution
Our analysis of the auto lender data assumed that R and W were jointly normally distributed.
To assess the robustness of our results with respect to this assumption, we generated synthetic
data from different distributions with known parameters, and applied our EM algorithm to estimate the willingness-to-pay and reserve price distribution parameters, assuming (incorrectly) that
the data was normally distributed. Specifically, we drew 1,000 willingness-to-pay and reserve price
26
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
Dist. of R
Normal
Exp.
Logistic
Normal
Exp.
Exp.
Logistic
Normal
Logistic
Dist. of W
Normal
Exp.
Logistic
Exp.
Normal
Logistic
Exp.
Logistic
Normal
Actual Values
µp
σp Tr. %
94.05 50.93 68.60
87.45 63.12 66.00
89.78 51.44 64.90
88.80 69.95 65.10
93.25 44.58 69.30
88.39 45.80 67.20
88.10 67.20 64.50
89.56 52.21 65.10
94.45 47.74 68.40
Actual Values:
Table 13
EM Parameter Estimations
µˆr
σˆr
µˆw
σˆw
47.48 48.82 102.94
103.51
64.77 93.34 116.66
84.84
37.59 40.69 85.14
118.7
66.61 104.84 120.64
92.52
31.91 17.54 94.57
121.78
24.54 14.11 83.08
127.8
67.19 99.87 117.01
89.88
41.00 46.11 88.22
113.33
47.86 42.72 100.73
101.90
50
50
100
100
Predicted Values
µ̂p
σ̂p Tr. %
94.07 50.94 68.55
87.26 62.89 65.92
89.73 50.94 64.80
88.61 69.57 65.04
92.93 44.74 69.45
87.22 45.34 67.51
88.16 66.98 64.48
89.60 51.94 64.96
94.45 47.71 68.43
EM Method Robustness Check
observations from normal, exponential, and logistic distributions, and, for each distribution combination, estimated the willingness-to-pay and reserve price distribution parameters assuming both
the willingness-to-pay and reserve price are distributed normally. We evaluated the nine instances
shown in Table 13. We fixed the mean and standard deviation of the willingness-to-pay to 100, and
mean and standard deviation of the reserve price to 50, and generated data from the following ran√
dom variables: W1 ∼ N (100, 100), W2 ∼ Exp(λ = 1/100), and W3 ∼ Logistic(µ = 100, s = 100 3/π)
where µ is the location and s is the scale parameter; similarly, R1 ∼ N (50, 50), R2 ∼ Exp(λ = 1/50),
√
and R3 ∼ Logistic(µ = 50, s = 50 3/π). We used the same criteria for the stopping conditions of
the EM algorithm as before.
Table 13 shows the results. In Table 13, µp and σp correspond to the mean and standard deviation of the prices realized among the 1,000 observations (for the deals where the trade occurs), the
column labeled “Tr. %” shows the corresponding trade percentage among these 1,000 observations.
p
µ̂p = E(p|W ≥ R) and σ̂p = V ar(p|W ≥ R) and the last column corresponds to the predicted
trade probabilities when R and W are both normally distributed with mean and standard deviations that are taken as the estimated parameters in columns 6-9. In all cases, the parameters
estimated by the EM method generated price distributions and trade percentages that closely
matched those in the holdout sample. Estimation of the means and standard deviations of the
underlying distributions were less robust to the forms of the distributions. When the actual distributions were normal, the errors in estimating the means of the reserve price and willingness-to-pay
distributions were 5.04% and 2.94% respectively. In the worst cases, the corresponding errors were
50.92% (Exponential/Logistic) and 20.64% (Normal/Exponential) respectively. This degradation
in performance is likely due to the very different form of the exponential distribution relative to
the normal.
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
7.2.
27
Asymmetric Bargaining
In our model, we assumed that the status quo utilities (i.e. the utility obtained if one decides not
to bargain with the other player) of the seller and the customer are equal, so that for cases where
the trade takes place, the Nash bargaining solution is symmetric, i.e.,
R≤W
⇒
p = 0.5R + 0.5W
The generalized Nash bargaining solution (Binmore et al. 1986) specifies that the price for a
particular deal will be given by p = αR + (1 − α)W for some α ∈ (0, 1). If α < 0.5, it can be
interpreted as the sellers’ having systematically more bargaining power than the customers and, if
α > 0.5, the customers would have more bargaining power. Ideally, we would like to estimate the
value of α from the data. Unfortunately, the following proposition shows that α is not identifiable
from data and therefore must be assumed.
Proposition 2. Assume that the true values of the model primitives are R1 , R2 , . . . , RN and
W1 , W2 , . . . , WN and α and that a trade occurs for observations 1, 2, . . . , n at rate pi = (1 − α)Ri +
αWi with Wi ≥ Ri , and no trade occurs for the remaining observations implying that Wi < Ri for
i = n + 1, . . . , N . Then, the model primitives are not jointly identifiable.
Proof:
Fix the W ’s and assume that we consider some α̂ 6= α for some 0 ≤ α̂ ≤ 1. Define
R̂i =
(α − α̂)Wi + (1 − α)Ri
1 − α̂
We will show that the underlying variables R̂1 , R̂2 , . . . , RˆN and W1 , W2 , . . . , WN and α̂ will produce
exactly the same set of observations (prices and no-trade outcomes) as the “real” values. For
i = 1, 2, . . . , n:
(1 − α̂)R̂i + α̂Wi
(α − α̂)Wi + (1 − α)Ri
=(1 − α̂)
+ α̂Wi
1 − α̂
=αWi − α̂Wi + Ri − αRi + α̂Wi
=αWi + (1 − α)Ri
=pi
And for i = n + 1, . . . , N , let γ =
α−α̂
.
1−α̂
Then, γ ≤ 1 and:
R̂i = γWi + (1 − γ)Ri
= Ri + γ(Wi − Ri )
≥ Wi since Wi − Ri < 0 and γ ≤ 1
Q.E.D.
28
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
This raises the question of whether or not an incorrectly specified value of α would lead to
significant misestimation of the parameters of the underlying distribution. To address this question,
we generated synthetic data for three values of α = 0.2, 0.5, and 0.8 and normally distributed values
for R and W . We then estimated the model parameters µr , µw , σr , and σw using the EM algorithm
and assumed values of α̂ from 0.1 to 0.9 at increments of 0.1. The results are shown in Table 14
for true values (µr , σr , µw , σw ) = (58, 6, 55, 10) and (µr , σr , µw , σw ) = (50, 5, 60, 10). The former set
of parameter values corresponds to 40% trade probability and the latter set corresponds to 80%
trade probability.
True Parameters
µr
σr
µw
σw
58
6
55
10
α = 0.2
EM Estimations
α̂
µˆr
σˆr
µˆw
σˆw
0.1
56.95 4.99 54.22 9.27
0.2
56.99 4.81 54.17 10.07
0.3
56.28 4.25 53.21 12.07
0.4
56.46 4.23 52.74 13.71
0.5
56.19 4.04 51.49 17.06
0.6
56.27 4.28 50.97 20.37
0.7
56.21 4.02 49.54 26.34
0.8
56.80 4.47 47.01 36.46
0.9
58.95 5.50 48.02 42.58
α = 0.5
EM Estimations
α̂
µˆr
σˆr
µˆw
σˆw
0.1
57.63 6.46 55.22 6.44
0.2
56.55 5.10 54.14 7.67
0.3
55.55 4.33 52.81 9.57
0.4
55.29 4.08 52.31 11.14
0.5
55.79 4.31 52.53 12.13
0.6
55.48 4.22 51.42 15.31
0.7
55.61 4.20 50.62 19.26
0.8
55.44 3.98 47.98 28.48
0.9
57.57 4.88 49.80 30.40
α = 0.8
EM Estimations
α̂
µˆr
σˆr
µˆw
σˆw
0.1
55.92 7.06 53.69 5.81
0.2
54.49 5.23 52.24 7.15
0.3
54.09 4.80 51.70 8.24
0.4
54.20 4.63 51.62 9.05
0.5
53.69 4.28 50.57 11.35
0.6
54.42 4.52 50.84 12.22
0.7
53.76 4.21 49.35 17.02
0.8
53.91 4.27 47.82 23.47
0.9
54.87 4.57 47.00 31.46
*LL: Log-Likelihood
Table 14
LL*
-6672.10
-6718.01
-6774.54
-6897.76
-7071.26
-7304.71
-7501.00
-7931.36
-8294.12
LL*
-6565.73
-6503.83
-6561.17
-6654.65
-6794.94
-7006.49
-7230.15
-7568.02
-7837.32
LL*
-6550.62
-6459.69
-6515.71
-6573.37
-6721.09
-6850.56
-7110.46
-7445.59
-7806.27
True Parameters
µr
σr
µw
σw
50
5
60
10
EM Estimations
µˆr
σˆr
µˆw
σˆw
40.89 23.70 62.26 7.11
46.92 17.87 63.74 7.33
50.32 15.12 64.95 7.50
52.72 11.60 65.24 8.42
50.32 4.08 64.66 16.07
50.46 3.86 67.99 19.85
50.49 3.79 73.69 26.21
50.83 3.93 84.31 38.03
54.70 6.06 94.44 46.06
EM Estimations
µˆr
σˆr
µˆw
σˆw
36.59 25.08 58.56 4.90
43.81 18.05 59.93 4.70
48.88 11.12 59.53 5.36
50.65 8.00 59.43 6.27
50.26 5.06 59.50 9.20
50.29 4.53 61.32 11.71
50.52 4.34 64.25 15.29
50.88 4.29 69.73 21.24
51.97 4.47 81.56 33.66
EM Estimations
µˆr
σˆr
µˆw
σˆw
30.71 27.48 54.90 3.82
39.96 18.17 55.93 3.54
44.34 13.32 56.42 3.40
47.33 9.40 56.06 3.99
47.45 4.84 54.91 7.12
47.92 4.58 56.14 8.24
47.55 3.93 58.60 11.88
48.02 3.84 62.51 15.95
48.68 3.84 71.96 26.11
Asymmetric Bargaining Synthetic Data Test
LL*
-7964.78
-7712.79
-7568.83
-7419.24
-7021.68
-7175.71
-7437.48
-7845.28
-8468.77
LL*
-7648.84
-7278.44
-6925.84
-6753.32
-6679.13
-6808.68
-7032.34
-7351.11
-7850.62
LL*
-7490.88
-7002.48
-6650.49
-6461.98
-6377.11
-6469.82
-6681.62
-6952.94
-7444.50
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
29
In these cases, assuming α = 0.5 does a reasonable job of estimating the underlying parameters
even when the true value of α is 0.2 or 0.8. In fact, for some cases in which the actual value of α is
not 0.5, using a value of α = 0.5 in the estimation actually results in estimates of parameters that
are closer to their true values than using the true value of α. Additionally, the log-likelihood is not
always the highest for the estimations using the true values of α, reflecting the fact that α is not
identifiable from the data. It also notable that, for each set of estimated parameters, the predicted
p
values of µ̂p = E(p|W ≥ R), σ̂p = V ar(p|W ≥ R), and trade probabilities are very close to their
true values.
8.
Discussion and Conclusions
In this paper, we proposed a structural EM method to estimate the joint distribution of customer
willingness-to-pay and seller reserve prices from a series of bi-lateral price negotiations. The resulting distributions can be used to predict both the take-up probability and the price of negotiations.
We showed that our method successfully uncovers the parameters of willingness-to-pay and reserve
price distributions from synthetic data given a sufficient number of observations. When applied to
a real-world auto-lending data set, our method generally predicted take-up and APRs on a test
data set better than standard regression-based approaches. Furthermore, our estimates provide
interesting insights into the reserve price distribution among sales representatives and how it varied
as a function of deal characteristics.
While our structural EM method has its strengths, the EM algorithm is notoriously computationally intensive and can be slow to converge. Computational considerations limited us to using
only 200,000 observations out of 950,985 available in the estimation process. In this case, the results
using the smaller sample were still an improvement over direct maximization of the likelihood, but
the computational intensity of the approach could be a limitation in other settings. In addition,
our approach requires a priori specification of the bargaining parameter α and the underlying form
of the distributions of R and W . While robustness tests suggest that the results are reasonably
insensitive to the form of the joint distribution and the value of α, severe misspecification (e.g. normal for exponential) can lead to misleading results. Both the development of improved algorithms
and techniques for choosing appropriate distributions and values of the bargaining parameter are
useful directions for future research.
Lastly, we used our estimation method to perform a counter-factual analysis of pricing policies on
a real-world data set of auto loans. The analysis showed if local sales staff are given pricing discretion, significant profit improvements can be achieved by providing an optimized lower bound on the
negotiated price. The lower-bound prevents sales staff with very low reservation prices from dropping the final price too low. These results also provide an explanation for the commonly-observed
30
Phillips, Şimşek, and van Ryzin: Estimating Buyer WTP and Seller RP from Negotiation Data and the Implications
practice of providing sales staff limited discretion to change prices relative to a centrally-established
price list. Yet even with optimized lower bounds, deal-specific optimized fixed prices generated
slightly higher profits than negotiated prices. However, the highest level of profits observed in our
counterfactual analysis occurred when headquarters was allowed to set optimized reserve prices for
each deal and individual sales staff then negotiated deals given those optimized reserve prices. This
last result suggests that a combination of data-driven pricing controls and sales agent negotiation
can produce higher profits than either approach alone. How to achieve such results in practice
through appropriate training, controls, and sales force incentives is an open topic for further study.
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