Investing in Forecast Collaboration

Transcription

1 Investing in Forecast Collaboration Mümin Kurtuluş Owen School of Management, Vanderbilt University, Nashville, TN Mike Shor Owen School of Management, Vanderbilt University, Nashville, TN Beril Toktay College of Management, Georgia Institute of Technology, Atlanta, GA 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

2 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 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

3 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

4 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

5 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

6 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 noncollaborative 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 + c max{q K, 0}, where c is the per unit cost of capacity, and c > 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 c 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

7 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 ψ i that are realizations of Ψ i = D+E i, i {S, R}, where E i s are error terms distributed according to the bivariate normal distribution with unconditional mean E[E i ] = 0 ensuring that the signals are unbiased), variance V[E i ] = σi 2, 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 ψ i allows party i to generate a more precise forecast, D ψ i, than having only the prior information about demand. If both signals are utilized for forecasting, the demand forecast is D ψ R, ψ S. We assume that ρ min{ σ R σs, σ S σ R }, 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 ψ R, ψ S ) 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. signal as A i = 1 V[E i ], i {S, R}, which is not contractible but is observable to both partners. These accuracies depend on the supplier s and retailer s respective investments into forecasting. In particular, the cost of achieving accuracy A i is κa q i, 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

8 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 [A R, A S ]. The supplier and the retailer simultaneously determine accuracy levels, A R and A S, and observe private signals ψ S and ψ R. We denote the expected profit of each party at stage 1 under both signal and demand uncertainty) by Π R A R ; A S ) and Π S A S ; A R ). Stage 2: Announce public messages [ ˆψ S ψ S ), ˆψ R ψ R )] if employing collaborative forecasting. Update the demand forecast. With truthful information sharing, ˆψi ψ i ) = ψ i. 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 π R Q; ψ R, ˆψ S, A R, A S ) and π S K; ψ S, ˆψ R, A R, A S ). Note that Π R A R ; A S ) = κa q R + E Ψ R,Ψ S π R Q; ψ R, ˆψ S ψ S ), A R, A S ), and Π S is defined similarly. Finally, the realization d of demand is observed and sales revenue r mind, Q) is realized. 8

9 Thus, a strategy for the supplier and retailer are given by the three-tuples A S, ˆψ S ψ S ), KA R, A S, ψ S, ˆψ ) R ) and A R, ˆψ R ψ R ), QA R, A S, ψ R, ˆψ ) S ). 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, ˆψi ψ i ) =. 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) Retailer s Strategy We solve the game via backwards induction, starting with stage 3. For a given accuracy level A R and signal realization ψ R, the retailer decides on an order quantity, Q. The retailer maximizes stage 3 expected profit, πr n Q; A R, ψ R ) = E [r minq, D ψ R )] wq, where w is 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

10 Q n ψ R ) = E[D ψ R ] + Φ 1 1 w r ) V[D ψr ], 1) where E[D ψ R ] = ψ R and V[D ψ R ] = 1/A R. All derivations may be found in the Appendix. Substituting, the optimal expected stage 3 profit is given by π n RQ n ; A R, ψ R ) = r w)ψ R x R AR, 2) where x R. = rφ Φ 1 1 w r )) 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, ψ R. Denoting Π n R A R) as the retailer s expected profit at stage 1, we find Π n RA R ) = E ΨR [π n RQ n ; ψ R, A R )] κa q R 3) = r w)µ x R AR κa q R. 4) Denote by A n R the retailer s optimal accuracy level which maximizes Πn R and define An F 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, A n F = An R. ) 2 Lemma 1. In the benchmark model, A n R = An F = xr 1+2q 2qκ. For future reference, we note that the retailer s optimal accuracy depends on the loss per unit of standard deviation, x R, which is symmetric and obtains its maximum at w = r 2. 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

11 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 = r 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 Q n ψ R ). The higher the correlation between the signals, the more information ψ S carries about ψ R. The supplier sets capacity K to maximize the expected stage 3 profit πs nk; A S, ψ S ) = E[wQ n ψ R ) ψ S ck c max{q n ψ R ) ψ S K, 0}]. The optimal capacity is given by K n ψ S ) = E[Q n ψ R ) ψ S ] + Φ 1 1 c c ) V[Q n ψ R ) ψ S ] where E[Q n ψ R ) ψ S ] = ψ S + Φ 1 1 w ) / A n r R, 5) V[Q n ψ R ) ψ S ] = A n R + A S 2ρ ) / A n R A S A n RA S. 6) In stage 1, the supplier selects accuracy A S to maximize its expected profit, which can be written as follows after substituting K n ψ S ) in the profit expression for stage 3: Π n SA S ) = w c) µ + Φ 1 1 w )) r A n R x S A n R + A S 2ρ A n R A S A n R A S κa q S 7) where x S = c φ Φ 1 1 c c )) is supplier s loss per unit of standard deviation. Lemma 2. In the benchmark model, there exists a unique accuracy A n S supplier s profit. that maximizes the We note that A n S cannot be found in closed form, so expected supply chain profit must be calculated numerically. Final accuracy, A n F, equals An R, which is given in closed-form. 11

12 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, ˆψi ψ i ) = ψ i. Based on this forecast, the central decision maker determines the order quantity Q f that maximizes expected supply chain profit and sets K f = Q f 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 ψ S and ψ R and accuracy levels A S and A R, the central decision maker decides on order quantity Q that maximizes the expected stage 3 supply chain profit π SC Q; ψ S, ψ R, A S, A R ) = E[r minq, D ψ S, ψ R )] cq. The optimal order quantity is given by Q f ψ S, ψ R ) = E[D ψ S, ψ R ] + Φ 1 1 c r ) V[D ψs, ψ R ], 8) where E[D ψ S, ψ R ] = v 1 ψ S +1 v 1 )ψ R is a convex combination of the signals and V[D ψ S, ψ R ] = 1 ρ 2 A R +A S 2ρ A R A S. The superscript f is used to denote the first-best benchmark model. Substituting, the expected supply chain profit at stage 1 denoted by Π SC ) as a function of both accuracy levels is given by 1 ρ Π SC A R, A S ) = r c)µ x 2 J A R + A S 2ρ κa q R A R A κaq S, 9) S where x J = rφ Φ 1 1 c r )). The central decision maker then solves maxar,a S Π SC A S, A R ) to obtain A f S and Af R. Let Af F denote the accuracy of the final forecast using Af S and Af R. Lemma 3. If 1 + ρ 2q > 2 1 q 1 + ρ) q, then joint forecasting is optimal, with A f R = Af S = xj 2qκ 1+ρ 2 2 )) 2 1+2q with A f R or Af S = xj 2qκ and A f F = 2 1+ρ )) q 1+ρ 2q xj 2qκ 1+ρ 2 2 )) 2 1+2q and A f F = xj 2qκ. Otherwise, sole forecasting is optimal )) q. 1+ρ 2q Paralleling Lemma 1, the optimal accuracy is a function of x J, which is maximized at c = r 2. 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

13 3 Joint Forecasting is Optimal 2 q 1 Sole Forecasting is Optimal 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 ˆρq), 0 < ˆρq) < 1 such that sole forecasting is optimal when ρ > ˆρq) 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, ˆρq) 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

14 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. A n F > Af F iff x J = φ Φ 1 1 c )) r x R φ Φ 1 1 w )) < min{2 1 q 1 + ρ) q, 1 + ρ 2q )}. r 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 x J to x R. 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 An F > A f F ρ and q. ii. If joint forecasting is optimal in the first-best benchmark and if 1 c r ) c r β1 w r ) w r where β = 2 1 q 1 + ρ) q, then A n F > Af F. iii. If w r + c r 1 + wc r2w c), then An F < Af F ρ and q. 14

15 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 β. = 2 1+ρ 2 ) q. 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 = r 2 in the non-collaborative benchmark, and around c = r 2 in the first-best benchmark, and is maximized at these values, respectively. For w r + c r < 1, we have r 2 w < r 2 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

16 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

17 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 c 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

18 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 Q c superscript c denoting the collaborative forecasting model) to maximize its expected stage 3 profit given ψ S and ψ R, which is πr c Q; A S, A R, ψ S, ψ R ) = λπ SC Q; A S, A R, ψ S, ψ R ) = λe [r minq, D ψ S, ψ R ) 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 Q c ψ S, ψ R ) = E[D ψ S, ψ R ] + Φ 1 1 c ) r V[D ψs, ψ R ], where E[D ψ S, ψ R ] and V[D ψ S, ψ R ] are the same as in the firstbest benchmark for given A R and A S. Equilibrium stage 3 expected supply chain profit is πsc c Qc ; A S, A R, ψ S, ψ R ) = r c)e[d ψ S, ψ R ] x J V[D ψs, ψ R ], which is shared between the 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 Π R A S, A R ) = λ Π S A S, A R ) = 1 λ) ] 1 ρ [r c)µ x 2 J A R + A S 2ρ κa q R A R A, 10) S ] 1 ρ [r c)µ x 2 J A R + A S 2ρ κa q S A R A. 11) S We characterize the equilibrium accuracy levels for the case where λ = 1 2 so the post- 18

19 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. relegated to A.6.1 in the Appendix. Additional discussion is Lemma 4. Let λ = 1 2. If the joint forecasting equilibrium exists, accuracies in this equilibrium ) 2 are given by A c R = Ac S = xj 1+2q, and the accuracy of the final demand forecast is given by A c F = 2 1+ρ xj 2qκ 1+ρ 4 2 2qκ ) 2 1+2q 1+ρ 4 2 equilibrium is given by A c R or Ac S = xj ) 2 is given by A c F = xj 1+2q 4κq.. If the sole forecasting equilibrium exists, the accuracy in this ) 2 1+2q, and the accuracy of the final demand forecast 4κq If λ 1 2, 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

20 3 3 Joint Forecasting Equilibrium Only Joint Forecasting Equilibrium Only Both Equilibria 2 2 q Both Equilibria q 1 Sole Forecasting Equilibrium Only 1 Sole Forecasting Equilibrium Only Ρ a) λ = Ρ b) λ = 0.35 Figure 3: Equilibrium existence regions in the investment stage. Parameter values: r = 6, c = 2, µ = 50, and κ = 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

21 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

22 T f T cj T cs T f T cj T i T i T n T n Ρ a) ρ 0, 0.8), κ = 100 Κ b) κ 20, 1000), ρ = 0.2 Figure 4: Total supply chain profits in the non-collaborative benchmark Π n T ), first-best benchmark Π f T ), collaborative forecasting equilibria Πcj T for joint forecasting and Πcs T for sole forecasting, shown where they exist), and non-collaborative benchmark without costs of ordercapacity mismatch Π i T ). Parameter values: r = 6, w = 4, c = 3, c = 2, q = 2, µ = 50 and λ = 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 Π n T ), in the first-best benchmark Πf T ) and in collaborative forecasting with proportional profit sharing Π cj T and Πcs T 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 between the equilibrium profits, Π cs T or Πcj T, and Πf T. 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 =

23 We define the percentage loss as %Loss = Πf T minπcj T, Πcs T ) Π f T Πn T 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 Π i T 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 Π n T to Π i T 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