Investing in Forecast Collaboration

Investing in Forecast Collaboration
Mümin Kurtuluş
Owen School of Management, Vanderbilt University, Nashville, TN 37203
[email protected]
Mike Shor
Owen School of Management, Vanderbilt University, Nashville, TN 37203
[email protected]
Beril Toktay
College of Management, Georgia Institute of Technology, Atlanta, GA 30308
[email protected]
November 2008
Abstract
We study the strategic interaction between supply chain partners involved in collaborative
forecasting. Motivated by the mixed results of collaborative forecasting initiatives in the
consumer goods sector, we analyze the potential of Collaborative Planning, Forecasting,
and Replenishment (CPFR) as well as possible barriers to its implementation. We model
a supplier and retailer who can invest in improving the quality of their demand forecasts.
Strategic interaction takes place at the (1) information acquisition stage and (2) information
sharing stage. Our model shows that CPFR can significantly increase supply chain profits,
provided that management avoids common pitfalls. We suggest reasons for failure of CPFR
initiatives rooted in poor implementation and misdirected managerial expectations.
Key words: supply chain management, demand forecast collaboration, CPFR, information exchange, forecast quality.
1
1
Introduction
Several industry-specific initiatives emphasizing the importance of collaboration across levels
of the supply chain have emerged in recent years. One such initiative, Collaborative Planning,
Forecasting, and Replenishment (CPFR), which originated in the consumer goods sector, creates standards for inter-firm information exchange with a specific focus on sharing and reconciling demand forecasts to improve profits for both the supplier and the retailer. The CPFR
collaboration template (Seifert, 2003) prescribes that the retailer and the supplier each create
a demand forecast and enter it into the collaboration platform. Possibly after some discussion
among managers, a single shared demand forecast is calculated, which serves as the basis for
both production and stocking decisions.
In the absence of CPFR, the parties use their demand forecasts independently—the retailer
to place an order to the supplier, and the supplier to develop an “order forecast” for capacity
planning purposes in advance of receiving the order. As a result, the supplier often incurs
a capacity-order mismatch cost. By relying on a shared demand forecast, CPFR aims to
eliminate this mismatch while enabling the retailer to base orders on a more precise forecast
which incorporates the supplier’s information.
Collaborative forecasting can lead to significant benefits for companies (Aviv, 2001, 2002).
Successful implementations include Procter & Gamble’s experiences with retail partners Metro
and Tesco, and Wal-Mart’s experiences with many of its suppliers such as Sara Lee
(http://www.vics.org/committees/cpfr). However, many other initiatives have failed to
produce results and have been abandoned (GMA, 2002; Supply Chain Digest, 2008). Motivated by mixed results of CPFR in practice, we offer a supply chain modeling framework
that sheds light on the potential value of forecast collaboration, and the barriers that must be
overcome to achieve this potential.
We consider a supply chain in which a single supplier sells to a single retailer who faces
uncertain demand. Both the supplier and the retailer can invest in improving the quality of
their own demand forecasts. The obtained forecasts may be used by the supplier and the
retailer independently, or pooled to form a single shared demand forecast that forms the basis
of the retailer’s ordering and the supplier’s capacity decisions. While acting on a single shared
demand forecast promises benefits to both the supplier and the retailer, it also introduces two
strategic interdependencies among them—incentives to invest in information acquisition and
2
to share information truthfully. Investment into forecasting (to obtain relevant data, improve
data quality, and generate forecasts) is costly and may depend on the anticipated investment
level of the other party. The sharing of information may not be truthful if one partner benefits
from the other misestimating demand (Cachon and Lariviere, 2001; Özer and Wei, 2006).
When the promised benefits of CPFR are not realized by supply chain partners, the high
start-up costs of the collaboration platform and the complexity of the process have often
been cited as culprits (Seifert, 2003; White and Roster, 2004; Supply Chain Digest, 2008).
Our research suggests three alternate reasons for the failure of CPFR initiatives, rooted in
managerial expectations and implementation.
First is the reliance on erroneous measures. Collaborative forecasting partners anticipate
the primary benefit to be improved forecast accuracy (GMA, 2002; Seifert, 2003). Indeed, the
sponsor of the CPFR standard blueprint cites “higher forecast accuracy” as the primary value of
and key metric for evaluating CPFR (VICS, 2002). In contrast, we find that the final forecast
accuracy is often lower with collaborative forecasting even though profit is always higher.
We conclude that forecast accuracy is a misleading measure of the success of collaborative
forecasting practices.
Second is not revising terms of trade. We show that superimposing collaborative forecasting on existing terms of trade (as is often the norm in CPFR pilot studies) is unlikely to
yield sizeable benefits. The effectiveness of CPFR is inexorably linked to the terms of trade
between a supplier and retailer. For example, we show that a simple wholesale-price contract
does not provide sufficient incentives for the supplier to invest in and truthfully share information with the retailer. Thus, while the supplier may enjoy less capacity-order mismatch and
increased profits from using a shared forecast, the retailer need not realize any benefits. This
may be the reason for the prevailing view among retailers that “CPFR primarily benefits the
manufacturer” (Kurt Salmon Associates, 2002) and help explain limited adoption.
Third is misdirected expectations. Equilibrium forecast investments may be highly asymmetric, yet still benefit both parties. It may even be that only one party contributes to the
shared demand forecast in equilibrium. If supply chain partners define success as using both
parties’ forecasts, or as having equal contributions to forecast accuracy, they are likely to
conclude erroneously that CPFR is not working.
Our analysis ends on a positive note. In an extensive numerical study, we show that equi-
3
librium profits with collaborative forecasting are very close to those that can be optimally
obtained by a coordinated supply chain, and far in excess of those obtainable without collaborative forecasting. This is true even for equilibria where only one party’s forecast is used and
forecast accuracy decreases relative to the non-collaborative level. We conclude that as long as
the supply chain partners set appropriate terms of trade and managers track the right metrics,
CPFR holds great promise.
2
Related Literature
There is an extensive literature on incorporating forecast information into inventory management decisions (e.g, Fisher and Raman, 1996; Iyer and Bergen, 1997; Donohue, 2000).
More recently, researchers have focused on the benefits of collaborative forecasting (Aviv,
2001, 2002) and its strategic complexity (Miyaoka, 2003; Cachon and Lariviere, 2001; Özer
and Wei, 2006; Lariviere, 2002; Terwiesch, Ren, Ho and Cohen, 2004). Aviv (2001) models
a collaborative forecasting process among privately-informed supply chain partners, finding
that collaborative forecasting may provide substantial benefits for the supply chain, especially
when the correlation between trading partners’ forecasts is low. Aviv (2002) extends this
model to the case of autocorrelated demand. Both papers assume that forecast accuracy is
exogenous and that partners reveal their local demand forecasts truthfully. Our results complement Aviv’s, as we show that collaborative forecasting, when implemented with the proper
incentives, achieves profits close to the first-best benchmark. In addition, by modeling strategic investments in forecasting and the sharing of forecasts, our research reveals a number of
conditions under which collaborative forecasting may fail to achieve its potential.
The lack of truthful information sharing is one of the potential barriers to the success
of CPFR. Miyaoka (2003) finds that specific forms of buy-back contracts provide incentives
for truthful information sharing, though several other common forms of contracts do not. In
asymmetric contexts in which only the retailer holds proprietary information, authors have
designed mechanisms that induce truthful information sharing through signaling (Cachon and
Lariviere, 2001; Özer and Wei, 2006) and screening (Özer and Wei, 2006; Lariviere, 2002).
Terwiesch et al. (2004) offer empirical evidence that truthful sharing may not always occur.
We generalize the result in Miyaoka (2003) by showing that proportional profit sharing is
truth-inducing and quantify the value of using truth-inducing terms-of-trade.
4
Economists have studied investment in forecasting and information sharing in the context of
oligopolists having private information about uncertain market characteristics. These models
incorporate either an information-sharing stage, where forecast accuracy is exogenous (e.g,
Novshek and Sonnenschein, 1982; Vives, 1984; Gal-Or, 1985), or an investment stage, where
forecasts are not shared (e.g., Li, McKelvey and Page, 1987; Vives, 1988). Main findings include
that firms often wish to distort or fully withhold information, and investment in forecasts
generally fails to achieve the first-best outcome. Also related are applications to R&D joint
ventures, in which investments in cost-reducing technology play a similar role to forecasting
investments in our model (d’Aspremont and Jacquemin, 1988; Kamien, Muller and Zang,
1992). The level of investment depends on the degree of the positive externality one firm’s
investment has on its rivals. In non-cooperative settings, a greater externality decreases the
private incentive to invest.
Our model incorporates both an investment and an information sharing stage, and takes
a supply chain perspective, capturing the strategic aspects of this vertical relationship. The
inefficiencies in this relationship arise not only from reduced incentives to invest in forecasting
as with horizontal competitors, but also from the lack of truthful information sharing and
the capacity-order mismatch. We show that it is possible to exploit the vertical nature of
the relationship to implement terms of trade that eliminate the latter two inefficiencies. In
this case, the gain from collaboration is significant and the inefficiency that derives from the
strategic interaction at the investment stage is negligible. This can be explained by the fact
that with horizontal competition, the firm strategies are strategic substitutes, and a firm that
invests in decreasing its own costs and a competitor’s may find both profits decline, while in
our context, greater accuracy benefits both parties.
The forecasting model that we employ is based on Winkler (1981) and Clemen and Winkler (1985). A decision maker charged with forming an estimate of an uncertain parameter
has access to dependent information sources (e.g., experts) which offer information regarding
the realization of an uncertain event. The authors study the effect of correlation between
information sources on the accuracy of the final estimate. While Clemen and Winkler treat
the accuracy of each information source as exogenous, we allow firms to invest in accuracy,
perhaps by utilizing more (or more expensive) experts and information sources.
5
3
Model
We consider a supply chain in which a single supplier, S, sells to a single retailer, R, who in
turn sells to consumers. Demand is random. The supplier and retailer have identical prior
information about consumer demand but have forecasting capabilities that allow each to invest
in acquiring more accurate demand information.
We analyze three scenarios. Our main focus is the collaborative forecasting scenario where
the parties independently and strategically determine their forecasting investment levels, and
share forecast information with each other.
We also analyze two benchmarks: the non-
collaborative benchmark where the parties again determine their forecasting investment levels
independently, but do not share forecast information, and the first-best benchmark where a central decision maker selects the supplier’s and retailer’s investment levels and pools the demand
information to form a single shared demand forecast. The supplier and retailer (or the central
decision maker in the first-best benchmark) determine capacity and order level based on the
best information available to them after making forecasting investments, obtaining demand
forecasts, and sharing forecast information, where applicable.
The supply chain decisions taken by the retailer and the supplier are the order quantity
Q (determined before the demand is realized), and the capacity K (determined before receipt
of the retailer’s order), respectively. The cost of satisfying order Q with capacity K is cK +
c0 max{Q − K, 0}, where c is the per unit cost of capacity, and c0 > c is the per unit cost of
expediting. In practice, suppliers in the consumer goods industry often fulfill their retailers’
orders in full, even at added expense. We assume a forced-compliance supply chain, where
the supplier must fill the order Q. This assumption is unnecessary when the supplier and
retailer share a common demand forecast, as expediting would never be required. When they
do not share a common demand forecast, forced compliance is not a strong assumption; under
a wholesale price contract, for example, it is sufficient to assume that c0 is smaller than the
wholesale price to make filling the full order optimal for the supplier. With this assumption,
sales revenue is given by r min{Q, D}, where r is the unit sales price of the product and D is
a random variable that denotes demand.
In what follows, we introduce our forecasting model and the sequence of events.
6
3.1
The Forecasting Model
We employ a forecasting model based on Winkler (1981) and Clemen and Winkler (1985). Demand is given by the random variable D which is normally distributed with mean µ. The supplier and the retailer privately observe imperfect signals ψi that are realizations of Ψi = D + Ei ,
i ∈ {S, R}, where Ei ’s are error terms distributed according to the bivariate normal distribution with unconditional mean E[Ei ] = 0 (ensuring that the signals are unbiased), variance
V[Ei ] = σi2 , and correlation ρ, 0 ≤ ρ < 1. Correlation allows for a dependence between the
information that the two signals carry, which captures that the supplier and the retailer might
both utilize some common data, share common assumptions, or have access to some of each
other’s opinions. Observing a signal ψi allows party i to generate a more precise forecast, D|ψi ,
than having only the prior information about demand.
If both signals are utilized for forecasting, the demand forecast is D|ψR , ψS . We assume
that ρ ≤ min{ σσRS , σσRS }, or the covariance between the signals is smaller than the variance. This
condition is always satisfied, for example, if estimates are obtained from sampling independent
normal processes, with some overlap in the samples used by R and S (Winkler, 1981). This
assumption avoids the implausible implication that a joint forecast places negative weight on
one of the two signals, a condition which is likely to produce unstable estimates in practice
(Clemen and Winkler, 1985). We refer to situations where the constraint above is satisfied
with equality (and one party’s signal is given zero weight in the joint forecast D|ψR , ψS ) as sole
forecasting, and to situations where the constraint does not bind (and both parties’ signals are
given positive weight) as joint forecasting. To isolate the role of investment and joint forecasting
from the role of the prior information available, we assume that the distribution of D is diffuse,
but our results can be extended to general normal distributions. We denote the density of the
standard normal distribution by φ(·) and its cumulative distribution by Φ(·).
As a measure of the supplier’s and retailer’s forecast quality, we define the accuracy of a
. 1
, i ∈ {S, R}, which is not contractible but is observable to both partners.
signal as Ai = V[E
i]
These accuracies depend on the supplier’s and retailer’s respective investments into forecasting.
In particular, the cost of achieving accuracy Ai is κAqi , κ > 0, q > 0, where κ is a scaling
parameter and q is the forecasting technology parameter. Values of q above 1 imply costs convex
in accuracy, and values below 1 imply concave costs. In the remainder of the paper, the terms
“investing in forecasting” and “determining the accuracy level” will be used interchangeably.
7
3.2
Sequence of Events
In retailing, forecasting and replenishment generally occur within an established commercial
framework. Terms of trade and mechanisms for information sharing are typically negotiated
for a whole season or year while operational elements such as demand forecasting, ordering and
replenishment are carried out on an ongoing basis. For this reason, we take terms of trade as
given and focus on the sequence of events starting with the forecasting investment and ending
with the realization of demand. In §5.2.1, we discuss how the ability to renegotiate terms of
trade would affect forecasting investments and profit. Our model has three stages:
Stage 1: Invest in forecasting [AR , AS ].
The supplier and the retailer simultaneously determine accuracy levels, AR and AS , and
observe private signals ψS and ψR . We denote the expected profit of each party at stage 1
(under both signal and demand uncertainty) by ΠR (AR ; AS ) and ΠS (AS ; AR ).
Stage 2: Announce public messages [ψ̂S (ψS ), ψ̂R (ψR )] if employing collaborative forecasting.
Update the demand forecast.
With truthful information sharing, ψ̂i (ψi ) = ψi . We do not assume that messages are
necessarily truthful, as the parties may have an incentive to distort their information. The
supplier and retailer form demand forecasts given each party’s own signal, and the message of
the other party, where applicable.
Stage 3: Supplier: set capacity [K]. Retailer: place order [Q].
As discussed earlier, the supplier sets its capacity before receiving the retailer’s order. We
assume that capacity and order quantity are simultaneously set in stage 3 as a convenience; a
sequential game in which the supplier sets capacity and then the retailer places an order leads to
identical results. We denote the expected profit at stage 3 (under demand uncertainty, and excluding stage 1 forecasting investments) by πR (Q; ψR , ψ̂S , AR , AS ) and πS (K; ψS , ψ̂R , AR , AS ).
Note that ΠR (AR ; AS ) = −κAqR + EΨR ,ΨS πR (Q; ψR , ψ̂S (ψS ), AR , AS ), and ΠS is defined similarly.
Finally, the realization d of demand is observed and sales revenue r min(d, Q) is realized.
8
Thus, a strategy for the supplier and retailer are given by the three-tuples
³
´
³
´
AS , ψ̂S (ψS ), K(AR , AS , ψS , ψ̂R ) and AR , ψ̂R (ψR ), Q(AR , AS , ψR , ψ̂S ) .
We solve for a perfect Bayesian equilibrium of each game defined by our assumptions about
information sharing and terms of trade.
4
Analysis of Benchmark Scenarios
In this section, we analyze and compare the non-collaborative benchmark scenario and the
first-best benchmark scenario.
4.1
Non-collaborative Benchmark
Our non-collaborative benchmark scenario mimics the traditional supply chain where the upstream and downstream parties invest in forecasting independently and do not have the ability
to share their forecasts. Formally, we assume that strategies in stage 2 are constrained to be
uninformative, ψ̂i (ψi ) = ∅. The retailer uses only its own signal to forecast demand and the
supplier uses its own signal to forecast the retailer’s order. To determine their respective investment levels, the two parties solve independent optimization problems, weighing forecasting
cost against the benefits of greater accuracy about demand (for the retailer) or about the retailer’s order (for the supplier). They then make ordering and capacity decisions, respectively.
This scenario serves as a benchmark for a supply chain without collaborative forecasting. The
terms of trade consist of a simple wholesale price only contract, which is a common business
model in many supply chains (see Lariviere and Porteus, 2001; Perakis and Roels, 2007).
4.1.1
Retailer’s Strategy
We solve the game via backwards induction, starting with stage 3. For a given accuracy
level AR and signal realization ψR , the retailer decides on an order quantity, Q. The retailer
n (Q; A , ψ ) = E [r min(Q, D|ψ )] − wQ, where w is
maximizes stage 3 expected profit, πR
R
R
R
the wholesale price at which the retailer acquires the product from the supplier, and the
superscript n denotes the non-collaborative benchmark model. The optimal solution is given
by the newsvendor quantity:
9
³
w´p
Qn (ψR ) = E[D|ψR ] + Φ−1 1 −
V[D|ψR ],
r
(1)
where E[D|ψR ] = ψR and V[D|ψR ] = 1/AR . All derivations may be found in the Appendix.
Substituting, the optimal expected stage 3 profit is given by
xR
n
πR
(Qn ; AR , ψR ) = (r − w)ψR − √ ,
AR
¡
¡
.
where xR = rφ Φ−1 1 −
w
r
¢¢
(2)
is the cost of uncertainty and can be interpreted as the retailer’s
loss per unit of standard deviation. The first term in the profit expression is the profit in the
absence of uncertainty and the second term is the loss due to uncertainty.
The above provides the retailer’s optimal expected stage 3 profit for a given signal realization, ψR . Denoting ΠnR (AR ) as the retailer’s expected profit at stage 1, we find
n
ΠnR (AR ) = EΨR [πR
(Qn ; ψR , AR )] − κAqR
xR
= (r − w)µ − √
− κAqR .
AR
(3)
(4)
Denote by AnR the retailer’s optimal accuracy level which maximizes ΠnR and define AnF as
the final demand accuracy on which the order is based in the benchmark model. Since the
demand forecast depends only on the retailer’s accuracy level, AnF = AnR .
Lemma 1. In the benchmark model, AnR = AnF =
³
xR
2qκ
´
2
1+2q
.
For future reference, we note that the retailer’s optimal accuracy depends on the loss per
unit of standard deviation, xR , which is symmetric and obtains its maximum at w = 2r . To see
why, note that the retailer’s order quantity trades off the risk of ordering too much (at a cost
of w per unit) with the opportunity cost of ordering too little (at a cost of r − w per unit).
When w is very low relative to r, inventory is cheap (as reflected in an optimal order quantity
that is much larger than the signal), so the cost of uncertainty is low. When w is close to r,
inventory is expensive and the optimal ordering quantity is much smaller than the signal, so
reducing uncertainty has little value in improving profits.
In sum, the retailer has two levers to manage demand uncertainty: determining the forecasting investment, which reduces the uncertainty, and choosing the order quantity so as to
minimize the impact of uncertainty. At the extremes, the retailer finds it more profitable to
10
adjust the inventory level than to invest in a more precise forecast of demand. The retailer’s
incentive to invest in forecasting is maximized when the costs of over- and under-ordering are
equal (w = 2r ).
4.1.2
Supplier’s Strategy
The supplier decides on capacity prior to receipt of the retailer’s order, which creates an
inefficiency. Since forecasts are not shared, the supplier’s signal has no impact on the retailer’s
order. However, the supplier’s signal provides a basis for estimating the retailer’s signal, and
for anticipating the order quantity Qn (ψR ). The higher the correlation between the signals,
the more information ψS carries about ψR .
The supplier sets capacity K to maximize the expected stage 3 profit πSn (K; AS , ψS ) =
E[wQn (ψR )|ψS − cK − c0 max{Qn (ψR )|ψS − K, 0}]. The optimal capacity is given by
³
c ´p
K n (ψS ) = E[Qn (ψR )|ψS ] + Φ−1 1 − 0
V[Qn (ψR )|ψS ]
c
where
³
w´. p n
E[Qn (ψR )|ψS ] = ψS + Φ−1 1 −
AR ,
r
³
´.
p
V[Qn (ψR )|ψS ] = AnR + AS − 2ρ AnR AS
AnR AS .
(5)
(6)
In stage 1, the supplier selects accuracy AS to maximize its expected profit, which can be
written as follows after substituting K n (ψS ) in the profit expression for stage 3:
Ã
¡
Φ−1 1 −
n
p n
ΠS (AS ) = (w − c) µ +
AR
¡
¡
where xS = c0 φ Φ−1 1 −
c
c0
¢¢
w
r
s
¢!
− xS
p
AnR + AS − 2ρ AnR AS
− κAqS
AnR AS
(7)
is supplier’s loss per unit of standard deviation.
Lemma 2. In the benchmark model, there exists a unique accuracy AnS that maximizes the
supplier’s profit.
We note that AnS cannot be found in closed form, so expected supply chain profit must be
calculated numerically. Final accuracy, AnF , equals AnR , which is given in closed-form.
11
4.2
First-Best Benchmark
In this section, we gauge the first-best level of investment in information from the standpoint
of maximizing overall supply chain profit. Effectively, a central decision maker selects the
supplier’s and retailer’s investment levels and pools the demand information to form a single
shared demand forecast. Formally, ψ̂i (ψi ) = ψi . Based on this forecast, the central decision
maker determines the order quantity Qf that maximizes expected supply chain profit and sets
K f = Qf so that the cost of capacity-order mismatch is eliminated. This serves as a first-best
benchmark to gauge the potential of collaborative forecasting.
We again solve this problem starting in stage 3. For given signal realizations ψS and ψR and
accuracy levels AS and AR , the central decision maker decides on order quantity Q that maximizes the expected stage 3 supply chain profit πSC (Q; ψS , ψR , AS , AR ) = E[r min(Q, D|ψS , ψR )]−
cQ. The optimal order quantity is given by
³
c´p
Qf (ψS , ψR ) = E[D|ψS , ψR ] + Φ−1 1 −
V[D|ψS , ψR ],
r
(8)
where E[D|ψS , ψR ] = v1 ψS +(1−v1 )ψR is a convex combination of the signals and V[D|ψS , ψR ] =
1−ρ2√
.
AR +AS −2ρ AR AS
The superscript f is used to denote the first-best benchmark model. Sub-
stituting, the expected supply chain profit at stage 1 (denoted by ΠSC ) as a function of both
accuracy levels is given by
s
ΠSC (AR , AS ) = (r − c)µ − xJ
1 − ρ2
√
− κAqR − κAqS ,
AR + AS − 2ρ AR AS
(9)
¡
¡
¢¢
where xJ = rφ Φ−1 1 − rc . The central decision maker then solves maxAR ,AS ΠSC (AS , AR )
to obtain AfS and AfR . Let AfF denote the accuracy of the final forecast using AfS and AfR .
Lemma 3. If 1 + ρ2q > 21−q (1 + ρ)q , then joint forecasting is optimal, with AfR = AfS =
³
³ √ ´´ 2
³
³ √ ´´ 2
1+2q
1+2q
f
1+ρ
1+ρ
xJ
xJ
2
√
√
and
A
. Otherwise, sole forecasting is optimal
=
F
2qκ
1+ρ 2qκ
2 2
2 2
³
³
´´ 2
³
³
´´ 2
1+2q
1+2q
f
xJ
xJ
1
1
with AfR or AfS = 2qκ
and
A
=
.
F
2qκ 1+ρ2q
1+ρ2q
Paralleling Lemma 1, the optimal accuracy is a function of xJ , which is maximized at
c = 2r . Depending on the underlying parameters, maximizing the supply chain profit involves
either joint forecasting, characterized by equivalent contributions to accuracy by the supplier
and retailer, or sole forecasting, in which only one party’s investment contributes to the final
12
3
Joint Forecasting
is Optimal
2
q
Sole Forecasting
is Optimal
1
0
0.0
0.2
0.4
0.6
0.8
1.0
Ρ
Figure 1: Optimal forecasting in the first-best benchmark.
demand forecast (Figure 1). When q < 1, sole investment is always optimal. For a concave cost
function, an extra unit of overall accuracy is obtained at lower cost by incrementing whichever
firm’s accuracy level is higher. When q > 1, there exists a threshold ρ̂(q), 0 < ρ̂(q) < 1
such that sole forecasting is optimal when ρ > ρ̂(q) and joint forecasting is optimal otherwise.
A convex cost function implies that an extra unit of forecast accuracy is less expensively
obtained by raising the lower accuracy level, but this is attenuated by the degree of correlation.
When signals are highly correlated, a positive level of investment in both signals is redundant.
Accordingly, ρ̂(q) increases in q, so joint forecasting is optimal for a broader range of ρ as q
increases.
4.3
Comparison of the Non-Collaborative and First-Best Benchmarks
Profit. The first-best benchmark always yields higher expected supply chain profits than the
non-collaborative benchmark. In the non-collaborative benchmark, the supplier’s investment
does not contribute to the accuracy of the demand forecast. Instead, the supplier aims only
to predict the retailer’s forecast, which allows the supplier to infer the order quantity. This
implies three sources of inefficiency. First, the supplier’s information, if known to the retailer,
would lead to a more accurate demand forecast, increasing the retailer’s profit. Second, the
supplier could eliminate the mismatch between capacity and order quantity if the supplier was
privy to the same information as the retailer. Lastly, each party invests in forecasting and
13
makes supply chain decisions to maximize its own profit, rather than the supply chain profit.
The first-best scenario dominates by eliminating these three inefficiencies.
Forecast accuracy. A common expectation among managers adopting collaborative forecasting is that the accuracy of demand forecasts will increase as suppliers and retailers pool
their information. Consequently, forecast accuracy is one of the metrics tracked to judge the
success of CPFR. To evaluate the fruitfulness of relying on this metric, we compare the final
accuracies obtained in the two benchmarks. We find that the final forecast accuracy in the
non-collaborative benchmark exceeds that of the first-best benchmark over a wide range of the
parameter space.
Proposition 1. AnF > AfF iff
¡
¡
¢¢
φ Φ−1 1 − rc
xJ
¢¢ < min{21−q (1 + ρ)q , (1 + ρ2q )}.
= ¡ −1 ¡
xR
φ Φ
1 − wr
Certainly, for signals of a given accuracy, better estimates are obtained by using both
signals rather than one, but this does not account for the change in incentives to undertake
forecasting investment when forecasts will be combined. The proposition indicates that even a
fully-coordinated supply chain may elect to have a poorer demand forecast than that obtained
without CPFR. While the first-best benchmark departs from the reality of CPFR implementations where the supply chain is decentralized, forecasting investments (and hence accuracy)
would tend to be even lower in a decentralized supply chain due to neither party fully accounting for the positive externality of its investment.
The relative accuracy of demand forecasts depends on the ratio of xJ to xR . This condition
is perhaps not very intuitive, and may be better expressed in terms of the cost of production,
c, the retail price, r, and the terms of trade, reflected in the wholesale price, w.
Corollary 1.1.
i. If sole forecasting is optimal in the first-best benchmark and if
w
r
+
c
r
< 1, then AnF >
AfF ∀ρ and q.
ii. If joint forecasting is optimal in the first-best benchmark and if (1 − rc ) rc / β(1 −
where β = 21−q (1 + ρ)q , then AnF > AfF .
iii. If
w
r
+
c
r
'1+
wc
r(2w−c) ,
then AnF < AfF ∀ρ and q.
14
w w
r)r
(a) when sole forecasting is optimal
(b) when joint forecasting is optimal
Figure 2: Comparison of final accuracies in the non-collaborative and first-best benchmarks
as a function of c/r and w/r. The region above the diagonal is not feasible as c ≤ w. When
joint forecasting
is optimal in the first-best solution, the comparison depends on value of
´q
³
.
1+ρ
.
β=2 2
The corollary presents conditions under which the accuracy of the demand forecast is higher
than first-best in the non-collaborative benchmark. The first part of the corollary considers the
case where sole forecasting is optimal in the first-best solution. In this case, both benchmarks
estimate final demand from a single signal. Yet, the retailer overinvests in information in the
non-collaborative benchmark. The left panel of Figure 2 illustrates parameter values under
which this holds. Remember that the optimal accuracy is symmetric around w =
non-collaborative benchmark, and around c =
at these values, respectively. For
w
r
+
c
r
r
2
r
2
in the
in the first-best benchmark, and is maximized
< 1, we have | 2r − w| < | 2r − c|, that is, the relative
costs of under- or over-ordering are more balanced in the non-collaborative benchmark. Hence,
the value of uncertainty reduction is larger, and the optimal accuracy is higher in the noncollaborative benchmark.
The second part of the corollary compares the single forecast obtained by the retailer in
the non-collaborative benchmark to the pooled forecast of the supplier and retailer in the
first-best outcome. Even though the first-best forms a joint forecast from two signals, many
15
cases exist where higher forecast accuracy is obtained from just the retailer’s signal in the
non-collaborative case (right panel of Figure 2).
The third part of the Corollary (using an approximation of the normal distribution of
Strecok, 1968) defines the region where the accuracy of the final demand forecast is always
higher in the first-best benchmark regardless of whether sole or joint forecasting is optimal
(both panels of Figure 2). This is because when w is very close to r, the optimal forecast
accuracy is low in the non-collaborative benchmark.
We conclude that forecast accuracy among information-sharing partners may be lower than
in a non-collaborative supply chain. Yet, the promise of more accurate forecasts remains the
primary motivation for CPFR adoption. For example, two-thirds of Grocery Manufacturers
of America members initiated some level of CPFR by 2002, with 86% citing the expected
improvement in forecast accuracy as the primary reason, but only a minority reported improved
accuracy as a realized benefit (GMA, 2002). The implication of our results is that forecast
accuracy is a misleading measure of the success of collaborative forecasting practices. The
reliance on improved forecast accuracy as a success metric may explain why less than 20% of
these initiatives proceeded beyond pilot studies (GMA, 2002).
We now turn to the analysis of collaborative forecasting in a decentralized supply chain.
5
Collaborative Forecasting
Collaborative forecasting is defined by the existence of the information exchange stage, where
after investing in accuracy and discovering their signals, each player provides a message about
its signal to the other.
5.1
Collaborative Forecasting with a Wholesale Price Contract
We investigate the impact of adding an information-sharing stage to our non-collaborative
benchmark model. In particular, we maintain a wholesale price contract between the supplier
and the retailer, but allow each to send a message after observing their private signal but
before setting capacities and order quantities.
Proposition 2. When collaborative forecasting is implemented in the context of a wholesale
price contract, the supplier does not announce truthfully in equilibrium, but there exist equilibria
in which the retailer announces truthfully.
16
Proposition 2 highlights the importance of terms of trade in the strategic interaction at
the information sharing stage. The supplier has an incentive to inflate its forecast because
the supplier’s profit depends on the order quantity rather than the accuracy of the demand
forecast. Thus, the supplier’s message aims not to reveal the supplier’s forecast truthfully
but to cause the retailer to inflate its expectations about demand. Conversely, the retailer is
indifferent between truthfully and falsely revealing its signal as it always receives its full order
from the supplier at the same wholesale price. In particular, we show that an equilibrium
always exists in which the retailer reveals its signal truthfully, but the supplier reveals no
information at all. Thus, the supplier may benefit by avoiding the capacity-order mismatch,
while the retailer receives the same profit as in the non-collaborative benchmark. In sum,
forecast collaboration (under the wholesale price contract) need not benefit the retailer, but
may benefit the supplier. Indeed, retailers argue that the benefits from CPFR seem to accrue
primarily to suppliers (Kurt Salmon Associates, 2002).
Proposition 2 reveals that CPFR initiatives may not deliver promised results in practice if a
supplier and a retailer agree to share demand forecast information without changing the terms
of trade defining their relationship. While we formally prove this result only for the wholesale
price contract, any contractual arrangement in which the supplier has the incentive to inflate
its forecast to obtain a higher order will similarly fail. While the CPFR template recommends
developing a joint business plan, there is no evidence that the contractual structure is changed
in a typical CPFR implementation. In the next section, we show that a proportional profit
sharing contract where each party receives a fixed and predetermined proportion of the supply
chain profit from sales ensures truthful information sharing, and we analyze collaborative
forecasting under this contractual structure.
5.2
Collaborative Forecasting with Proportional Profit Sharing
For a given order quantity Q, capacity K, and realized demand d, the total supply chain profit
from sales (excluding forecast investments) is given by r min{Q, d} − cK − c0 max{Q − K, 0}.
We show that a proportional sharing of this profit, with the retailer receiving share λ ∈ [0, 1]
and the supplier receiving 1 − λ, leads to truthful information sharing.
Proposition 3. Both the supplier and the retailer share their demand forecasts truthfully
under terms of trade that share supply chain profit from sales proportionally.
17
Various mechanisms can implement proportional profit sharing, including buy-back and
revenue sharing contracts. For example, with a buy-back contract, the supplier charges the
retailer w per unit purchased, but pays the retailer b < w per unit remaining at the end
of the season. If the contract is calibrated so that b = (1 − λ)r and w = b + λc, then the
supplier’s and the retailer’s profit are proportional to the supply chain profit with the retailer
receiving a fraction λ and the supplier receiving the rest (see Cachon, 2003). Miyaoka (2003)
shows that this class of buy-back contracts provides incentives for supply chain partners to
reveal their information truthfully. We generalize this result to all contracts that implement
proportional profit sharing, which allows us to characterize terms of trade by the parameter
λ. This parameter may reflect the relative bargaining power or size of the retailer.
In the collaborative forecasting model, forecasting investment decisions are made independently, with each party incurring the costs of its investment. Since both the supplier
and the retailer truthfully reveal their observed signals and use a common demand forecast,
the supplier anticipates the order and sets its capacity equal to this order. The retailer
chooses the order quantity Qc (superscript c denoting the collaborative forecasting model)
c (Q; A , A , ψ , ψ ) =
to maximize its expected stage 3 profit given ψS and ψR , which is πR
S
R
S
R
λπSC (Q; AS , AR , ψS , ψR ) = λE [r min(Q, D|ψS , ψR ) − cQ]. Note that proportional profit sharing has the additional advantage of coordinating the order quantity in the sense that the
retailer’s decision maximizes the supply chain profit. We find Qc (ψS , ψR ) = E[D|ψS , ψR ] +
¡
¢p
Φ−1 1 − rc
V[D|ψS , ψR ], where E[D|ψS , ψR ] and V[D|ψS , ψR ] are the same as in the firstbest benchmark for given AR and AS . Equilibrium stage 3 expected supply chain profit is
p
c (Qc ; A , A , ψ , ψ ) = (r − c)E[D|ψ , ψ ] − x
πSC
V[D|ψS , ψR ], which is shared between the
S
R
S
R
S
R
J
retailer and the supplier with shares λ and 1 − λ.
The retailer and the supplier decide on their respective signal accuracies strategically to
maximize their own expected profits at stage 1
s
"
ΠR (AS , AR ) =
λ
(r − c)µ − xJ
"
ΠS (AS , AR ) = (1 − λ)
(r − c)µ − xJ
s
1 − ρ2
√
AR + AS − 2ρ AR AS
1 − ρ2
√
AR + AS − 2ρ AR AS
#
− κAqR ,
(10)
− κAqS .
(11)
#
We characterize the equilibrium accuracy levels for the case where λ =
18
1
2
so the post-
investment supply chain profit is shared equally between the supplier and the retailer. The
equilibria of interest are either joint forecasting equilibria where both parties’ signals are given
positive weight in the demand forecast or sole forecasting equilibria where only one party’s
signal is given positive weight. We consider only those sole forecasting equilibria that satisfy
the following property: An equilibrium investment that leads to a signal being the only one that
contributes to the posterior mean should be equal to the investment that would be made if the
other signal simply did not exist. Then, there exists at most one joint forecasting equilibrium
and either zero or two (mirror image) sole forecasting equilibria. Additional discussion is
relegated to §A.6.1 in the Appendix.
Lemma 4. Let λ = 12 . If the joint forecasting equilibrium exists, accuracies in this equilibrium
³
´ 2
√
1+2q
1+ρ
xJ
√
are given by AcR = AcS = 2qκ
, and the accuracy of the final demand forecast is given
4 2
2
³
´
√
1+2q
1+ρ
xJ
2
√
by AcF = 1+ρ
. If the sole forecasting equilibrium exists, the accuracy in this
2qκ 4 2
³ ´ 2
xJ 1+2q
equilibrium is given by AcR or AcS = 4κq
, and the accuracy of the final demand forecast
2
³ ´
xJ 1+2q
is given by AcF = 4κq
.
If λ 6= 12 , closed-form solutions cannot be found but the equilibria are calculated numerically. Figure 3 illustrates the conditions under which each equilibrium exists with equal profit
sharing (λ = 0.5) and disproportionate profit sharing (λ = 0.35). Comparing with the regions
defining when joint and sole forecasting are optimal in the first-best benchmark (Figure 1),
we see that there are cases where there is a sole forecasting equilibrium, but joint forecasting
is optimal, and vice versa. With disproportionate profit sharing, the region where the sole
forecasting equilibrium exists expands and that where the joint forecasting equilibrium exists
shrinks. Disproportionate sharing provides more incentive to the party that obtains a larger
share of the profit while reducing the incentives of the other party.
Comparing the equilibrium accuracies in Lemma 4 to the first-best accuracy in Lemma
3, both types of equilibria result in lower accuracy levels than first-best. The introduction
of strategic interaction between the supplier and the retailer in the investment game leads to
underinvestment and loss of efficiency for the entire supply chain. Loss of efficiency occurs
because neither party fully internalizes the benefit of its forecast investment. In the next
two subsections, we briefly describe how this suboptimality might be overcome in a repeated
context, and then conduct a numerical study to gauge the size of this efficiency loss.
19
3
3
Joint Forecasting
Equilibrium Only
Joint Forecasting
Equilibrium Only
2
Both
Equilibria
2
Both
Equilibria
q
q
1
1
Sole Forecasting
Equilibrium Only
Sole Forecasting
Equilibrium Only
0
0
0.0
0.2
0.4
0.6
0.8
0.0
Ρ
0.2
0.4
0.6
0.8
Ρ
(a) λ = 0.5
(b) λ = 0.35
Figure 3: Equilibrium existence regions in the investment stage. Parameter values: r = 6,
c = 2, µ = 50, and κ = 50.
5.2.1
Repeated Interaction
Our results suggest that collaborative forecasting cannot fully coordinate the supply chain.
Because neither the supplier nor the retailer fully internalizes the benefits of its investment in
forecasting, investments are generally lower than first-best. The resulting game is structurally
akin to a prisoner’s dilemma: If both agree to make optimal investments, each has incentive
to deviate, but deviation by both parties results in lower profits for each. Since the supply
chain interaction occurs repeatedly, profit-improving cooperation may be possible through the
use of informal relationship contracts (Taylor and Plambeck, 2007; Debo and Sun, 2004). If
both players can commit to future actions in our setting, then simple grim trigger strategies
can achieve the optimal payoff (Friedman, 1971; Fudenberg and Maskin, 1986). Firms invest
optimally as long as both have done so in the past, and revert to the investments prescribed
by the static equilibrium if either has ever defected. When the interest rate is sufficiently
low, implying that future profits are valued highly relative to present profits, the benefits from
defecting are offset by the stream of lower future profits.
However, the assumption that firms can commit to future strategies discounts the role
of communication and possible periodic renegotiation among supply chain partners. This as-
20
sumption is fitting for the economic motivation of horizontal competition where communication
transforms implicit cooperation into felonious collusion, but is unreasonable in a supply chain
context where partners regularly meet to discuss terms of trade and coordinate order quantities. The ability to renegotiate future suboptimal actions significantly changes the supply
chain relationship (Plambeck and Taylor, 2007). In particular, punishing a defector hurts both
firms so threats to do so may not be credible if parties can renegotiate, a concept formalized
by the idea of renegotiation-proof equilibria (Farrell and Maskin, 1989).
In our context, since firms interact repeatedly with an independent demand draw in each
period, renegotiation proofness can eliminate all cooperative outcomes. To see why, imagine a
cooperative agreement sustained by the threat of punishment. Since in each period players face
identical continuation games, in any period in which players are to undertake punishment, they
would prefer to renegotiate to the original cooperative agreement. If firms are always tempted
to renegotiate punishments, the threat required to sustain cooperation is absent. For example,
in price competition among two identical firms selling homogeneous products, no level of
cooperation can be sustained using renegotiation-proof strategies (Farrell and Maskin, 1989).
In contrast, in a Markov model where today’s unobservable actions (such as investments in
market research) impact next period’s state so that continuation games are not identical,
Plambeck and Taylor (2006) show that requiring renegotiation proofness need not eliminate
all cooperative outcomes. Yet, they note that first-best outcomes are still not possible as
free-riding is not fully overcome, and threats may be carried out with positive probability.
Another complication arises if investments are not perfectly observable. Deviations from
optimal investment are inferred by ex post comparisons of forecasts to realized demand. A
punishment strategy depends on statistical inference and is triggered whenever the other party’s
forecast error exceeds a predefined threshold (Green and Porter, 1984; Fudenberg, Levine and
Maskin, 1994). Since inaccurate estimates are possible under any investment level, punishment
occurs with positive probability in equilibrium.
We conclude that achieving the first-best outcome in the context of collaborative forecasting is at least challenging, if not impossible. As a practical matter, the development of
reasonable strategies for supporting cooperation is more likely when the gains from cooperation
are great. In the next section, we show that collaborative forecasting, without dynamic cooperation, comes quite close to first-best outcomes, rendering challenging cooperative strategies
unnecessary.
21
f
PT
f
PT
cj
PT
Pcs
T
cj
PT
PiT
PiT
PnT
PnT
Ρ
Κ
(a) ρ ∈ (0, 0.8), κ = 100
(b) κ ∈ (20, 1000), ρ = 0.2
Figure 4: Total supply chain profits in the non-collaborative benchmark (ΠnT ), first-best benchcs
mark (ΠfT ), collaborative forecasting equilibria (Πcj
T for joint forecasting and ΠT for sole forecasting, shown where they exist), and non-collaborative benchmark without costs of ordercapacity mismatch (ΠiT ). Parameter values: r = 6, w = 4, c0 = 3, c = 2, q = 2, µ = 50 and
λ = 0.5.
5.2.2
Quantifying the Loss Due to Strategic Interaction
In extensive numerical simulations, we find that surprisingly little is lost due to the strategic
interaction at the investment stage. All equilibria of the collaborative forecasting model come
quite close to achieving first-best profits.
Figure 4 illustrates the expected supply chain profits in the non-collaborative benchmark
(ΠnT ), in the first-best benchmark (ΠfT ) and in collaborative forecasting with proportional
cs
profit sharing (Πcj
T and ΠT correspond to joint and sole forecasting equilibria, respectively,
where they exist) as a function of the correlation parameter ρ and the forecasting effectiveness
parameter κ.
The value of eliminating strategic interaction at the investment stage equals the gap becj
f
tween the equilibrium profits, Πcs
T or ΠT , and ΠT . Note that this difference is very small in
both examples. We conducted an extensive numerical study to examine the efficiency loss due
to strategic interaction when λ = 0.5.
22
We define the percentage loss as
%Loss =
cs
ΠfT − min(Πcj
T , ΠT )
ΠfT − ΠnT
(12)
which is the portion of potential profit improvement above the non-collaborative benchmark
lost by collaborative forecasting. We simulate 9,639 parameter combinations over the ranges
κ ∈ (20, 520), q ∈ (0.5, 2.5), and ρ ∈ (0, 0.8), in increments of 10, 0.1, and 0.1, respectively.
The average loss is 2.3% with losses smaller than 5% for 97% of parameter combinations. Our
measure of efficiency loss is pessimistic, as we use the worse of the sole and joint forecasting
equilibria when both exist. If parties periodically renegotiate terms of trade, coordination on
the better equilibrium may instead be expected. Additionally, these numbers are for λ = 0.5
but with a more judicious distribution of profits, even higher supply chain profits can be
obtained in equilibrium.
Next, we investigate the sources of profit improvements brought about by collaborative
forecasting. We define ΠiT (in Figure 4) as the expected supply chain profit that would be
achieved in the non-collaborative benchmark if the retailer were to communicate truthfully
its forecast (equivalently, his order quantity) to the supplier. The profit gain from ΠnT to
ΠiT accrues solely from eliminating the capacity-order mismatch for the supplier, but does
not address the other two inefficiencies in the non-collaborative benchmark. In our numerical
simulations, this gain reflects an average of 25.5% (and a maximum of 43%) of the total possible
profit improvement. Clearly, the potential promised by collaborative forecasting cannot be
achieved by simply eliminating the capacity-order mismatch; it requires implementing proper
incentives. Therefore, we conclude that the major benefit of collaborative forecasting comes
from changing the terms of trade.
6
Conclusions
CPFR was developed as a means of reducing the two major supply chain inefficiencies: the
supply-demand mismatch at the retailer, and the capacity-order mismatch at the supplier.
Since it offered benefits to both parties, many retailers and suppliers initiated collaborative
forecasting pilots, with mixed success. The trade literature has offered some evidence supporting and criticizing CPFR, however, this evidence is anecdotal in nature. We offer a supply
23
chain modeling framework that sheds light on the potential value of forecast collaboration, and
the barriers that must be overcome to achieve this potential. We capture two important aspects
of the strategic interaction between the supply chain partners driven by the non-contractible
nature of forecasting investment – incentives to invest in information acquisition and to share
information truthfully. Our model also captures (i) the costly nature of improving demand
forecasts, (ii) the overlap in information obtained by the supply chain partners, and (iii) the
impact of different terms of trade. Our research provides insights on why CPFR might not
be able to deliver its promised benefits and provides managerial recommendations on how to
overcome some of the barriers to implementation.
First, forecast accuracy improvement should neither be the aim of nor be used to judge
the success of CPFR. Yet forecast accuracy improvement is touted as one of the primary goals
of CPFR and is one of the recommended metrics to evaluate its success. This is based on
the belief that two forecasts are better than one, which is certainly true in our model for
given levels of forecast investment. However, forecast accuracy is endogenously determined by
the investment decisions of the parties, which will change from their non-collaborative levels
when forecast collaboration is implemented. We observed that with collaborative forecasting,
final forecast accuracy drops over a large set of parameters, while total profits increase. Thus,
CPFR should be framed not as an initiative that necessarily improves forecasting accuracy,
but as one that improves profits, and should be evaluated as such.
Second, before launching a CPFR initiative, existing terms of trade should be scrutinized for
their truth-inducing potential and be adjusted if necessary. CPFR advocates claim that much
of the added value of CPFR results from the exchange of information. However, implementing
a platform to share information is futile if information is not shared credibly. Unfortunately,
many supplier-retailer relationships are based on wholesale price contracts, which we show are
not conducive to truthful information sharing by the supplier. The CPFR template should
specify the importance of aligning incentives to facilitate truthful information sharing.
Third, the success of collaborative forecasting should not be measured by how much each
party contributes to the final forecast. While underinvestment in forecasting investment (relative to the first-best) is to be expected by both parties, the first-best analysis shows that
the most efficient way of improving profits may require one party be the primary or even sole
contributor to the final forecast accuracy. This is to be expected if there is large overlap in
24
the information available to the parties, or if forecasting cost is concave in accuracy, making
it more efficient to concentrate investment in one location.
Our results demonstrate that once terms of trade are set so that information is shared
truthfully, the gain from collaboration is significant. The inefficiency that derives from the
strategic interaction at the investment stage is negligible. We recommend that the supply chain
partners should focus their efforts on overcoming the strategic interaction at the information
sharing stage, and should be less concerned about overcoming the strategic interaction at the
investment stage.
Our model makes several assumptions that can be relaxed. We assumed symmetric forecasting technology q, scaling parameter κ, and sharing rule λ for both the supplier and the
retailer. Numerically, we verified that introducing asymmetries in these parameters yields
predictable changes to the results. The asymmetric investment game shifts the joint forecasting equilibrium toward greater investment by the advantaged firm. For sole forecasting,
sufficient asymmetries eliminate equilibria where the disadvantaged party is the only one to
invest. Asymmetries instead broaden the space of sole forecasting equilibria where the party
with higher q, higher λ, or lower κ is the sole contributor to the accuracy of the final forecast.
While our model only offers some explanations for the failure of CPFR that are rooted in
managerial expectations and implementation, other explanations are possible. Our model can
be extended in several directions to seek for alternative explanations. First, we assume that the
platform allowing the implementation of CPFR is already in place. However, the high set-up
costs to set up a collaboration platform (e.g., software that allows information exchange) are
often cited as a barrier to CPFR implementation. The benefits we identify must be weighed
against these costs. Second, we consider a model where a single supplier sells to a single
retailer. An extension could consider competing retailers and/or suppliers. In such a setting,
partners involved in collaborative forecasting might fear the leakage of valuable information to
competitors. This might serve as a significant barrier to CPFR implementation. A potentially
fruitful direction for research would be to explore the incentives for information leakages in
collaborative relationships.
25
References
Aviv, Y. (2001), ‘The effect of collaborative forecasting on supply chain performance’, Management Science 47(10), 1326–1343.
Aviv, Y. (2002), ‘Gaining benefits from joint forecasting and replenishment processes: The case
of auto-correlated demand’, Manufacturing & Service Operations Management 4(1), 55–
74.
Cachon, G. P. (2003), Supply chain coordination with contracts, in S. C. Graves and A. G.
de Kok, eds, ‘Supply Chain Management’, Handbooks in Operations Research and Management Science, Amsterdam: North-Holland, pp. 229–339.
Cachon, G. P. and Lariviere, M. (2001), ‘Contracting to assure supply: How to share demand
forecasts in a supply chain’, Management Science 47(5), 629–646.
Clemen, R. T. and Winkler, R. L. (1985), ‘Limits for the precision and value of information
from dependent sources’, Operations Research 33(2), 427–442.
d’Aspremont, C. and Jacquemin, A. (1988), ‘Cooperative and noncooperative r&d in duopoly
with spillovers’, American Economic Review 78(5), 1133–37.
Debo, L. G. and Sun, J. (2004), Repeatedly selling to the newsvendor in fluctuating markets:
The impact of the discount factor on supply chain coordination. Working Paper.
Donohue, K. (2000), ‘Efficient supply contracts for fashion goods with forecast updating and
two production modes’, Management Science 46(11), 1397–1411.
Farrell, J. and Maskin, E. (1989), ‘Renegotiation in repeated games’, Games and Economic
Behavior 1(4), 327–360.
Fisher, M. and Raman, A. (1996), ‘Reducing the cost of demand uncertainty through accurate
response to early sales’, Operations Research 44(1), 87–99.
Friedman, J. W. (1971), ‘A non-cooperative equilibrium for supergames’, Review of Economic
Studies 38(113), 1–12.
Fudenberg, D., Levine, D. I. and Maskin, E. (1994), ‘The folk theorem with imperfect public
information’, Econometrica 62(5), 997–1039.
26
Fudenberg, D. and Maskin, E. (1986), ‘The folk theorem in repeated games with discounting
or with incomplete information’, Econometrica 54(3), 533–54.
Gal-Or, E. (1985), ‘Information sharing in oligopoly’, Econometrica 53(2), 329–43.
GMA (2002), CPFR baseline study: Manufacturer profile. KJR Consulting for the Grocery
Manufacturers of America.
Green, E. J. and Porter, R. H. (1984), ‘Noncooperative collusion under imperfect price information’, Econometrica 52(1), 87–100.
Iyer, A. and Bergen, M. (1997), ‘Quick response in manufacturer-retailer channels’, Management Science 43(4), 559–570.
Kamien, M. I., Muller, E. and Zang, I. (1992), ‘Research joint ventures and r&d cartels’,
American Economic Review 82(5), 1293–306.
Kurt Salmon Associates (2002), Survey of supply chain effectiveness.
Food Distribu-
tors International, Food Marketing Institute and Grocery Manufacturers of America,
www.gmabrands.com/publications/docs/SupplyChain.pdf Accessed 28 August 2008.
Lariviere, M. (2002), Inducing forecast revelation through restricted returns. Working Paper.
Lariviere, M. A. and Porteus, E. L. (2001), ‘Selling to the newsvendor: An analysis of price-only
contracts’, Manufacturing & Service Operations Management 3(4), 293–305.
Li, L., McKelvey, R. D. and Page, T. (1987), ‘Optimal research for Cournot oligopolists’,
Journal of Economic Theory 42(1), 140–166.
Miyaoka, J. (2003), Implementing collaborative forecasting through buyback contracts. Working Paper.
Novshek, W. and Sonnenschein, H. (1982), ‘Fulfilled expectations Cournot duopoly with information acquisition and release’, Bell Journal of Economics 13(1), 214–218.
Özer, O. and Wei, W. (2006), ‘Strategic commitment for optimal capacity decision under
asymmetric forecast information’, Management Science 52(8), 1238–1257.
Perakis, G. and Roels, G. (2007), ‘The price of anarchy in supply chains: Quantifying the
efficiency of price-only contracts’, Management Science 53(8), 1249–1268.
27
Plambeck, E. L. and Taylor, T. A. (2006), ‘Partnership in a dynamic production system with
unobservable actions and noncontractible output’, Management Science 52(10), 1509–
1527.
Plambeck, E. L. and Taylor, T. A. (2007), ‘Implications of renegotiation for optimal contract
flexibility and investment’, Management Science 53(12), 1872–1886.
Seifert, D. (2003), Collaborative Planning, Forecasting, and Replenishment: How to Create a
Supply Chain Advantage, 4 edn, AMACOM, AA.
Strecok, A. J. (1968), ‘On the calculation of the inverse of the error function’, Mathematics of
Computation 22(101), 144–158.
Supply
Chain
Digest
(2008),
The
on-going
battle
over
CPFR.
March
5,
http://www.scdigest.com/assets/on_target/08-03-05-2.php Accessed 24 September 2008.
Taylor, T. A. and Plambeck, E. L. (2007), ‘Supply Chain Relationships and Contracts: The
Impact of Repeated Interaction on Capacity Investment and Procurement’, Management
Science 53(10), 1577–1593.
Terwiesch, C., Ren, Z., Ho, T. and Cohen, M. (2004), ‘An empirical analysis of forecast sharing
in the semiconductor equipment industry’, Management Science 51(2), 208–220.
VICS
(2002),
Collaborative
sion 2.0.
planning,
forecasting
and
replenishment
CPFR
ver-
Voluntary Interindustry Commerce Standards (VICS) Association,
http://www.vics.org/docs/committees/cpfr/CPFR_Tabs_061802.pdf Accessed 28
August 2008.
Vives, X. (1984), ‘Duopoly information equilibrium: Cournot and Bertrand’, Journal of Economic Theory 34(1), 71–94.
Vives, X. (1988), ‘Aggregation of information in large Cournot markets’, Econometrica
56(4), 851–76.
White, A. and Roster, J. (2004), VICS’s CPFR model receives a much-needed makeover.
www.gartner.com Accessed 24 September 2008.
Winkler, R. L. (1981), ‘Combining probability distributions from dependent information
sources’, Management Science 27(4), 479–488.
28
A
Appendix
A.1
Conditional expectations and variances
The variance-covariance matrix for (D, ΨR , ΨS ) is given by:

σ2

2
Σ=
 σ
σ2
σ2
2
σ 2 + σR
σ 2 σ 2 + ρσR σS


0
0
0

 ¡ 2¢


2

σ 2 + ρσR σS 
σR
ρσR σS 
 = σ 3×3 +  0

2
2
2
σ + σS
0 ρσR σS
σS
Throughout, we make use of the properties of three conditional distributions: D|ΨR ,
ΨR |ΨS , and D|ΨR , ΨS . The first two are used in the non-collaborative benchmark model,
and the last in the first-best benchmark and collaborative forecasting models.
Cov[D, ΨR ]
E[D | ΨR = ψR ] = E[D] +
(ψR − E[ΨR ])
V[ΨR ]
µ
¶
µ
¶
2
σR
σ2
=µ
+
ψ
R
2
2
σ 2 + σR
σ 2 + σR
Cov[D, ψR ]2
V[D | ΨR = ψR ] = V[D] −
V[ΨR ]
µ
¶
2
σR
2
=σ
2
σ 2 + σR
Cov[ΨR , ΨS ]
(ψS − E[ΨS ])
E[ΨR |ΨS = ψS ] = E[ΨR ] +
V[ΨS ]
σ 2 − ρσR σS
σ 2 + ρσR σS
= S 2
µ
+
ψS
σ + σS2
σ 2 + σS2
Cov[ΨR , ΨS ]2
V[ΨS ]
¡
¢ σ2
σS2
2
2
2
= σS2 + σR
− 2ρσR σS 2
+
σ
(1
−
ρ
)
R
σ + σS2
σ 2 + σS2

−1 

³
´ Σ
ψ − E[ΨR ]
2,2 Σ2,3

  R
E[D | ΨR = ψR , ΨS = ψS ] = E[D] + Σ1,2 Σ1,3 
Σ3,2 Σ3,3
ψS − E[ΨS ]
V[ΨR |ΨS = ψS ] = V[ΨR ] −
wR ψR + wS ψS + wµ µ
wR + wS + wµ

−1 

³
´ Σ
Σ
Σ
2,2
2,3
  2,1 
V[D | ΨR = ψR , ΨS = ψS ] = V[D] − Σ1,2 Σ1,3 
Σ3,2 Σ3,3
Σ3,1
=
=
wµ σ 2
wR + wS + wµ
2 − ρσ σ ), and w = (1 − ρ2 )σ 2 σ 2 .
where wR = σ 2 (σS2 − ρσR σS ), wS = σ 2 (σR
µ
R S
R S
29
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
As we focus on the case where the prior is diffuse, we take limits of the above as σ → ∞
lim E[D | ΨR = ψR ] = ψR
(25)
σ→∞
2
lim V[D | ΨR = ψR ] = σR
=
σ→∞
1
AR
(26)
(27)
lim E[ΨR |ΨS = ψS ] = ψS
σ→∞
√
µ
¶
AR + AS − 2ρ AR AS
2
lim V[ΨR |ΨS = ψS ] = σS2 + σR
− 2ρσR σS =
σ→∞
AR AS
¡ 2
¢
¡ 2
¢
σS − ρσS σR ψR + σR − ρσS σR ψS
lim E[D | ΨR = ψR , ΨS = ψS ] =
2 − 2ρσ σ
σ→∞
σS2 + σR
S R
lim V[D | ΨR = ψR , ΨS = ψS ] =
σ→∞
A.2
2 σ2
(1 − ρ2 )σR
(1 − ρ2 )
S
√
=
2
2
(σR + σS − 2ρσS σR )
(AR + AS − 2ρ AR AS )
(28)
(29)
(30)
Non-collaborative Benchmark
n (Q; A , ψ )
Proof of Lemma 1. Define DR ≡ D|ψR . At stage 3, the retailer solves maxQ πR
R
R
where
n
πR
(Q; AR , ψR ) = EDR [r min(Q, DR ) − wQ] − κAqr
(31)
= EDR [rQ − r(Q − DR )+ − wQ] − κAqr
(32)
= (r − w)Q − rE[(Q − DR )+ ] − κAqr
(33)
p
The first order condition is given by P (DR ≤ Q) = 1− wr implying Qn (ψ) = E[DR ]+ V[DR ]ϕR
¢
¡
where ϕR = Φ−1 1 − wr . The second-order condition confirms that the solution is unique.
Therefore,
n
πR
(Qn ; AR , ψR ) = (r − w)Qn − rEDR [(Qn − DR )+ ]
(34)
= (r − w)Qn − rQn + rE[DR ] − rE[(DR − Qn )+ ]
Z ∞
p
= (r − w)E[DR ] − w V[DR ]ϕR − r
(x − Qn )fDR (x)dx
(35)
(36)
Qn
By
R∞
n
Qn (x − Q )fDR (x)dx
=
p
V[DR ]
and observing that Φ(ϕR ) = 1 −
w
r,
R∞
ϕR (x − ϕR )φ(x)dx
=
p
V[DR ][ϕR Φ(ϕR ) + φ(ϕR ) − ϕR ],
we can rewrite the profit as
i
h w
p
p
V[DR ]ϕR − r V[DR ] − ϕR + φ(ϕR )
r
p
= (r − w)E[DR ] − rφ(ϕR ) V[DR ]
xR
= (r − w)ψR − √ .
AR
= (r − w)E[DR ] − w
30
(37)
(38)
(39)
In stage 1, the retailer selects accuracy, AR , to maximize ΠnR (AR ), given by
xR
ΠnR (AR ) = (r − w)µ − √
− κAqR .
AR
(40)
³ ´ 2
xR 1+2q
Solving the first-order condition gives AnR = 2qκ
. Checking the second-order condition,
5
¯
³
´
∂ 2 Πn
2κq 1+2q
R (AR ) ¯
= − 1+2q
< 0. For q ≥ 1, the profit function is concave in
¯
4 xR xR
∂A2
n
R
AR =AR
A. When q < 1, the function is initially concave and unimodal, then convex and decreasing.
³
´ 2
1+2q
3xR
The inflection point is given by 4(1−q)qκ
which is greater than AnR . Lastly, AnF =
V[D|ΨR ]−1 = AnR .
Proof of Lemma 2. We follow steps similar to the proof of Lemma 1 by treating the retailer’s
order as a random variable, DS ≡ Qn (ψR )|ψS . The supplier sets capacity by solving
¤
£
max EDS wDS − cK − c0 (DS − K)+ .
K
The first-order condition is given by P (DS ≤ K) = 1 − cc0 and the supplier’s capacity choice
p
¡
¢
is given by K n = E[DS ] + V[DS ]Φ−1 1 − cc0 . Let πSn (K n ; AS , ψS ) denote the supplier’s
expected profit for given investment level AS and signal realization ψS .
£
¤
πSn (K n ; AS , ψS ) = EDS wDS − cK n − c0 (DS − K n )+
p
= (w − c)E[DS ] − c0 φ(ϕS ) V[DS ],
¡
where ϕS = Φ−1 1 −
c
c0
¢
(41)
(42)
and the derivation is similar to the derivation for the retailer’s
expected profit in the benchmark scenario. In addition, the expectation and variance of the
random variable DS are given by
p
E[DS ] = E[Qn (ψR )|ψS ] = E[(ψR + V[DR ]ϕR )|ψS ]
p
= E[ψR |ψS ] + E[ V[DR ]ϕR |ψS ]
p
= E[ΨR |ψS ] + V[DR ]ϕR
(43)
(44)
(45)
and
p
V[DS ] = V[Qn (ψR )|ψS ] = V[(ψR + V[DR ]ϕR )|ψS ]
p
= V[ψR |ψS ] + V[ V[DR ]ϕR |ψS ]
= V[ΨR |ψS ].
(46)
(47)
(48)
31
Thus,
sµ
√
¶
ϕ
AR + AS − 2ρ AR AS
R
n
n
0
πS (K ; AS , ψS ) = (w − c)(ψS + √ ) − c φ(ϕS )
AR AS
AR
(49)
and
ΠnS (AS ) = EΨS [πS (K n ; AS , ψS )] − κAqS
s
√
µ
¶
AR + AS − 2ρ AR AS
ϕR
= (w − c) µ + √
− xS
− κAqS .
AR AS
AR
(50)
(51)
√
The first term is independent of AS . The second term (−xS ·) is unimodal in AS , and the
last term (−κAqS ) is decreasing in AS . Therefore, the profit is quasiconcave in AS and has a
unique solution.
A.3
First-Best Benchmark
Proof of Lemma 3. Define DJ ≡ D|ψS , ψR . At stage 3, the central decision maker solves
maxQ πSC (Q; AR , AS , ψR , ψS ) where
πSC (Q; AR , AS , ψR , ψS ) = EDJ [r min(Q, DJ ) − cQ] .
(52)
Following steps similar to the proof of Lemma 1, the optimal order quantity is given by
Qf (ψS , ψR ) = E[DJ ] +
p
V[DJ ]ϕJ
(53)
¡
¢
where ϕJ = Φ−1 1 − rc and the expected centralized supply chain profit is given by
p
πSC (Qf ; AR , AS , ψR , ψS ) = (r − c)E[DJ ] − rφ(ϕJ ) V[DJ ].
(54)
In stage 1, the central decision maker solves maxAS ,AR ΠSC (AR , AS ) where
h
i
ΠSC (AR , AS ) = EΨR ,ΨS πSC (Qf ; AR , AS , ψR , ψS ) − κAqR − κAqS
s
1 − ρ2
√
= (r − c)µ − xJ
− κAqR − κAqS .
(AR + AS − 2ρ AR AS )
(55)
(56)
For a given Â, define ÂR = Â − δ and ÂS = (2Âq − (Â − δ)q )1/q . Then, total forecasting cost,
κÂqR + κÂqS , is constant for all δ <  and equals 2Âq . Thus, δ defines all pairs of accuracies
32
that can be obtained at the same cost. Substituting into the variance of DJ reveals that
variance is quasiconcave: everywhere nonincreasing, everywhere nondecreasing, or increasing
than decreasing, in δ. Thus, two candidate solutions exist: a symmetric solution satisfying
AR = AS or a corner solution satisfying the constraint AR = ρ2 AS .
Joint Forecasting The first-order conditions are given implicitly by:
´ 2κq(A + A − 2ρ√A A )3/2 √A A
³p
R S
R S
R
S
p
AR AS − ρAR =
2
xJ 1 − ρ
´ 2κq(A + A − 2ρ√A A )3/2 √A A
³p
R
S
R S
R S
1−q
p
AR AS − ρAS =
AR
2
xJ 1 − ρ
A1−q
S
³
When AR = AS , the above reduce to AR = AS =
xJ
2qκ
³√
1+ρ
√
2 2
´´
2
1+2q
(57)
(58)
.
−1/2
Sole Forecasting Evaluating ΠSC when AS = ρ2 AR yields (r − c)µ − xJ AR − (1 +
³
³
´´ 2
1+2q
xJ
1
ρ2q )κAqR , which obtains a maximum at AR = 2qκ
and by construction AS =
2q
1+ρ
ρ2 AR .
Optimal Solution Substituting the joint and sole forecasting solutions into ΠSC yields:
Πjoint
SC
Πsole
SC
µ
¶ 2q
√
xJ 1 + ρ 1+2q
= (r − c)µ − 2
(1 + 2q)
κ
2κq
µ
¶ 2q
¡
¢ 1
xJ 1+2q
2q 1+2q
= (r − c)µ − (1 + 2q) 1 + ρ
κ
2κq
1−q
1+2q
(59)
(60)
Joint forecasting is optimal if
≡
≡
≡
sole
Πjoint
SC > ΠSC
2q
µ √
¶ 1+2q
µ
¶ 2q
1+2q
1−q
¡
¢ 1
x
1
+
ρ
x
J
J
2q
2 1+2q (1 + 2q)
κ < (1 + 2q) 1 + ρ 1+2q
κ
2κq
2κq
1−q
q
¡
¢ 1
2 1+2q (1 + ρ) 1+2q < 1 + ρ2q 1+2q
¡
¢
21−q (1 + ρ)q < 1 + ρ2q .
(61)
(62)
(63)
In the first-best benchmark, joint forecasting is optimal precisely when the final demand
forecast is higher under the joint forecasting solution than the sole forecasting solution.
33
A.4
Comparison of the Non-Collaborative and First-Best Benchmarks
Proof of Proposition 1. From Lemma 1 and 3, the condition AnF > AfF is satisfied when
µ
xR
2qκ
¶
2
1+2q
(
> max
2
1+ρ
µ
xJ
2qκ
2
µ√
¶¶ 1+2q
µ
¶¶ 2 )
µ
1+2q
1+ρ
xJ
1
√
,
2q
2qκ 1 + ρ
2 2
(64)
where the first term on the right-hand side is equal to the final accuracy of the demand forecast
when the optimal is given by joint forecasting and the second term is the final demand accuracy
when the optimal is given by sole forecasting. Simplifying the above condition we get,
¡
¢¢
¡
φ Φ−1 1 − rc
xJ
¢¢ < min{21−q (1 + ρ)q , (1 + ρ2q )}.
= ¡ −1 ¡
w
xR
φ Φ
1− r
Proof of Corollary 1.1. (i). When the first-best solution is given by sole forecasting, AnF > AfF
requires
if
xJ
xR
xJ
xR
< 1 + ρ2q . Since 1 + ρ2q ≥ 1, the condition in Proposition 1 is satisfied for any ρ, q
< 1.
≡
≡
≡
xJ < xR
³
³
´´
³
³
c
w ´´
rφ Φ−1 1 −
< rφ Φ−1 1 −
r ´¯ ¯
r
¯
³
c ¯ ¯ −1 ³
w ´¯¯
¯ −1
1−
1−
¯Φ
¯ > ¯Φ
¯
¯ r ¯ ¯
¯ r
¯1 c¯ ¯1 w¯
¯ − ¯>¯ − ¯
¯2 r¯ ¯2
r¯
(65)
|r − 2c| > |r − 2w|
(69)
≡
(66)
(67)
(68)
When r ≥ 2w, both terms inside the absolute values are nonnegative, so the inequality holds
since w > c. Assume 2w > r > w + c. Then
|r − 2c| = r − 2c > r − 2(r − w) = 2w − r = |r − 2w|.
Thus, xJ < xR if w + c < r which is equivalent to the condition in the corollary.
(ii). When the optimal solution is given by joint forecasting, AnF > AfF requires
21−q (1
+
ρ)q .
xJ
xR
<β≡
This is equivalent to
≡
≡
xJ < βxR
³
³
´´
³
³
c
w ´´
rφ Φ−1 1 −
< βrφ Φ−1 1 −
r
r
³
³
c ´´2 ³ −1 ³
w ´´2
−1
Φ
1−
> Φ
1−
− 2 log β
r
r
34
(70)
(71)
(72)
Using the approximation of the inverse error function given in Strecok (1968),
≈
log
≡
³³
³³
w´ w´
c´ c´
1−
< log 1 −
+ log β
r r
³ r c ´r c
³
´
w w
1−
<β 1−
r r
r r
(73)
(74)
(iii) Note that 1 + ρ2q ≤ 2, implying that min{21−q (1 + ρ)q , (1 + ρ2q )} ≤ 2. Thus, it is
sufficient to show that
xJ
xR
> 2.
xJ > 2xR
(75)
which is the inverse of the condition from part (ii) for β = 2. Thus,
³
≈
Rearranging the above condition yields
A.5
1−
w
r
+
c
r
³
w´ w
c´ c
>2 1−
.
r r
r r
'1+
(76)
wc
r(2w−c) .
Collaborative Forecasting with a Wholesale Price Contract
Proof of Proposition 2. The supplier does not know the signal, and therefore the order, of the
retailer at the time of sending a message. Thus, the supplier’s message induces a probability
distribution on the retailer’s order.
First, consider the supplier always sending a truthful message, ψ̂S (ψS ) = ψS . The retailer’s
optimal order quantity is given by
q
³
w´
Q(ψ̂S , ψR ) = E[D|ψ̂S , ψR ] + Φ−1 1 −
V[D|ψ̂S , ψR ]
r
s
³
´
w
1 − ρ2
√
= vψR + (1 − v)ψ̂S + Φ−1 1 −
r
AR + AS − 2ρ AR AS
(77)
(78)
where v ∈ (0, 1) is a constant that does not depend on ψR , ψS , and ψ̂S . Further, the last
term on the right hand side does not depend on ψR , ψS , or ψ̂S . After the supplier receives the
retailer’s message, the supplier has
E[Q|ψS , ψ̂S , ψ̂R ] = vE[ψR |ψS , ψ̂R ] + (1 − v)ψ̂S + v2
(79)
V[Q|ψS , ψ̂S , ψ̂R ] = v 2 V[ψR |ψS , ψ̂R ]
(80)
where v2 ≥ 0 is a constant. The supplier’s profit is increasing in E[Q] and decreasing in
35
V[Q]. The supplier’s announcement, ψ̂S , does not enter into the variance of the retailer’s order
quantity, and enters positively into its expectation for all ψS . Therefore, the supplier’s profit
is increasing in ψ̂S , so we cannot have truthful revelation by the supplier.
Next, notice that for any supplier signaling strategy, ψ̂S (ψS ), the retailer is indifferent
between any of its own signaling strategies. In particular, ψ̂R may impact the supplier’s
strategy, but this does not change the retailer’s profit. We must exhibit that a Bayes-Nash
equilibrium exists in which the retailer announces truthfully. Consider the following strategies
and beliefs:
i. ψ̂S (ψS ) = v for some constant v. The retailer, upon observing any signal v 0 believes that
all supplier types are equiprobable. Effectively, the retailer treats the signal as uninformative. Clearly, the beliefs are consistent on the equilibrium path, and no deviation by
the supplier is profitable.
ii. ψ̂R (ψR ) = ψR and the supplier believes that a signal ψ̂R is sent only by type ψR = ψ̂R . All
signals occur with positive probability on the equilibrium path, and beliefs are consistent
with the signals. Further, the retailer is indifferent between all signaling strategies, so
the retailer has no incentive to deviate.
Thus, the seller sending an uninformative signal and the retailer revealing truthfully constitutes
an equilibrium. This concludes the proof of Proposition 2.
A.6
Collaborative Forecasting with Proportional Profit Sharing
Proof of Proposition 3. The retailer earns λ [r min{Q, d} − cK − c0 max{Q − K, 0}], not including forecasting costs. The retailer and supplier set
³
Q(ψ̂S , ψR ) = E[D|ψ̂S , ψR ] + Φ−1 1 −
³
K(ψS , ψ̂R ) = E[Q|ψS , ψ̂R ] + Φ−1 1 −
q
c´
V[D|ψ̂S , ψR ],
r q
c´
V[Q|ψS , ψ̂R ]
c0
(81)
(82)
The expected supply chain profit is maximized when Q = Qf and K = Q. Comparing Q to
Qf in Equation (53), the first equality is satisfied only if ψ̂S = ψS . Thus, the supplier shares
truthfully. For the retailer, note that profit is decreasing in |Q − K| for a given Q. Also, the
retailer’s signal does not change Q, but does change K. We have Q(ψS , ψR ) = K(ψS , ψ̂R ) if
and only if ψ̂R = ψR . Thus, the retailer shares truthfully.
36
A.6.1
Characterization of Equilibria
Differentiating equations 10 and 11 when λ = 1/2, we obtain the first order conditions which
are implicitly given by
√
√
4κq(AR + AS − 2ρ AR AS )3/2 AR AS
p
AR AS − ρAR =
xJ 1 − ρ2
´ 4κq(A + A − 2ρ√A A )3/2 √A A
³p
R
S
R S
R S
1−q
p
AR
AR AS − ρAS =
,
2
xJ 1 − ρ
A1−q
S
³p
´
(83)
(84)
where the right-hand side is equivalent in both expressions. Equating the left hand side expressions yields
A1−q
S
A1−q
R
.
where α =
AS
AR .
√
AR AS − ρAS
=√
AR AS − ρAR
√
α − ρα
1−q
α
= √
,
α−ρ
(85)
(86)
Note that α = 1 always solves the above equation, which has an additional zero
or two (reciprocal) strictly positive, real solutions. α = 1 corresponds to a symmetric solution
(AS = AR ) and neither of the two asymmetric solutions (which are identical up to a relabeling
of AS and AR ), satisfy both the first- and second-order conditions simultaneously. Thus, there
is at most one (symmetric) interior equilibrium where both AR and AS are positive.
There may also exist boundary equilibria where one party makes zero investment. As
discussed in §3.1, this leads to several implausible implications, such as the posterior mean
placing negative weight on one of the signals or pooled forecast accuracy improving as the
correlation among signals increases. As a practical matter, these cases are not amenable to
estimation (Clemen and Winkler, 1985). Thus, we maintain the restriction introduced in §3.1
that ρ ≤ min{ σσRS , σσRS }. When this constraint binds, only one signal is used in the calculation of
the posterior mean, with the other signal given zero weight. If this constraint is taken literally,
however, then the strategy space of one partner is constrained by the selected strategy of the
other. A higher investment by the retailer, for example, may force a higher investment by
the supplier simply to satisfy the constraint. Consider a situation where one party invests
Ai and the other prefers an investment close to zero but, to satisfy the minimum bound
implied by our constraint, invests ρ2 Ai . This forced investment also seems implausible, as the
supplier and retailer make investment decisions independently. To avoid this, we consider only
those equilibria that satisfy an additional criterion when the constraint binds: An equilibrium
investment that leads to a signal being the only one that contributes to the posterior mean
37
AS
AS
AR(AS)
AS(AR)
AS(AR)
AR(AS)
AR
A
R
Case A
Case B
A
A
S
S
AR(AS)
A (A )
S
R
AS(AR)
AR(AS)
AR
A
R
Case C
Case D
Figure 5: Possible patterns of best responses.
should be equal to the investment that would be made if the other signal simply did not exist.
We next investigate the best response patterns under the constraint ρ ≤ min{ σσRS , σσRS } and
characterize the equilibria implied by the above criterion. An investigation of the profit function in (10) reveals that (i) there exists an AS so that the retailer’s best response, A∗R > AS /ρ2
whenever AS < AS , and (ii) there exists an AS so that the retailer’s best response, A∗R < ρ2 AS
whenever AS > AS . Thus, there are four possible patterns of best responses, presented in Figure 5. Case A shows an interior solution only, which is the joint forecasting equilibrium. Case B
38
shows also a continuum of sole forecasting equilibria where our constraint binds. However, only
two (at the respective kinks of the best responses) lead to investment levels that would be made
if the other signal did not exist. That is, they maximize ΠR (AR ; ) = ΠR (AR ; AS = ρ2 AR ).
Case C is similar to case B, but without a joint forecasting equilibrium. In Case D, a continuum of equilibria exists, but they do not necessarily correspond to investment levels that
would be made in only one signal if it was the only one available and thus a sole forecasting
equilibrium need not exist. This corresponds to the implausible cases (when ρ is very high)
where better estimates are obtained by decreasing the accuracy of one of the signals. As these
cases have little managerial relevance, we do not analyze them in the paper.
Proof of Lemma 4. Joint Forecasting Equilibrium If a (symmetric) joint forecasting equilibrium exists, the investment levels are given by a pair (AR , AS ) that satisfies equations (83)
³
³ √ ´´ 2
1+2q
1+ρ
xJ
√
and (84). When AR = AS , the above reduce to AR = AS = 2qκ
. Substituting
4 2
these investment levels into the final accuracy expression,
³
´ 2
√
1+2q
1+ρ
xJ
2
√
of the final demand forecast AcF = 1+ρ
.
2qκ 4 2
√
AS +AR −2ρ AS AR
,
1−ρ2
yields the accuracy
Sole Forecasting Equilibria There may exist a continuum of corner equilibria satisfying AR = ρ2 AS (where only the supplier’s signal receives positive weight in the final demand
forecast) or AS = ρ2 AR . By symmetry, we focus on the latter, where only the retailer’s signal
is given positive weight. Substituting AS = ρ2 AR into the retailer’s first order condition, equa³ ´ 2
xJ 1+2q
. We can also confirm that this is equal to the investment
tion (84), reduces to AR = 4qκ
that the retailer would choose if only the retailer’s signal was available. The profit with a
single signal is equal to
r
·
¸
1
1
ΠR (AR , AS = ρ AR ) =
(r − c)µ − xJ
− κAqR .
2
AR
2
(87)
Differentiating yields the first order condition
Aq−1
=
R
xJ
3/2
4qκAR
,
(88)
which yields the same answer as above. The accuracy of the final demand forecast is the same
as the retailer’s signal accuracy as only the retailer’s signal is given weight in the final demand
forecast.
39