Sharing Forecast Information in a Long-term Supply

Chain Relationship

Z. Justin Ren∗, Morris A. Cohen†, Teck H. Ho‡, Christian Terwiesch†

Revision: August 28 2006

Abstract

Forecast sharing in a supply chain using linear price contracts often leads to inefficiencies

as the buyer has an incentive to inflate demand forecasts to ensure sufficient supply. Recent

research in supply chain contracting has focused on one-shot relationships, and has identified

various contracts that align incentives in the supply chain and induce the buyer to reveal forecast

information truthfully. In this paper, we investigate the effect of having an infinitely repeated

supplier relationship in achieving supply chain coordination. We analyze a capacity game with

forecast sharing under information asymmetry. We establish conditions under which a buyer

operating with a linear price contract reveals demand information truthfully with his supplier,

who in turn allocates system-optimal capacity, both assuming that their business relationship

is long-term. We show that in a repeated forecast sharing game coordination can be achieved

when the industry is stable, both parties value their long-term relationship, and over-forecasting

is easy to detect.

∗Operations and Technology Management Department, Boston University School of Management.†Operations and Information Management Department, The Wharton School, Univ. of Pennsylvania.‡Marketing Group, Haas School of Business, University of California-Berkeley.

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1 Introduction

In November 2004, American Airlines announced that it was deferring the purchase of 54 out

of 56 Boeing aircraft originally scheduled for delivery between 2006 and 2010. The financial

impact for Boeing as the supplier was estimated to be as high as $2.7 billion (Wall Street

Journal, November 22, 2004). This example highlights the inherent problem in sharing forecast

information in a supply chain. Forecasts provide information about what the buyer intends

to do in a given future state of the world. These intentions (often expressed in the form of

preliminary purchase orders or soft orders), however, cannot be easily verified or enforced in

court.

To overcome the problems associated with forecast sharing, previous research has suggested

a set of contracts that align the incentives in the supply chain and induce the buyer to reveal

demand information truthfully. Such contracts include two-part tariff and three-part quadratic

contracts (Desai and Srinivasan 1995, Corbett et al 2004), and capacity commitment and

option contracts (Cachon and Lariviere 2001, Ozer and Wei 2006). Despite their theoretical

appeal of creating a ‘win-win’ situation for buyer and supplier, empirical evidence shows that

they are not being widely used. For example, both the aircraft industry (see Boeing example

above), the semiconductor equipment industry (Cohen et al. 2003), and many other industries

(Desai and Srinivasan 1995, Bajari and Tadelis 2001, Iyer and Villas-Boas 2003) still prefer

the simple linear price contracts over the more complex coordinating contracts advocated by

the supply chain literature. While one could discount this empirical evidence as irrelevant

and simply declare Fortune 500 companies such as Boeing and Intel as laggards with respect

to adopting the latest supply chain research, we offer an alternative explanation.

Most prior treatments of forecast sharing have used static or one-shot games. That is, in

these models buyers and suppliers transact only once as in a spot-buy market. This eliminates

any value of long-term relationships and thereby constrains the involved parties to ignore

reputation. As we observe in practice, many business relationships are long-term. Harmonious

and mutually successful long-term supplier relationships have been observed in many industries

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and documented by researchers in strategy and management (e.g., Hagen and Choe 1998,

McMillan 1990). In these supplier relationships, both the buyer and supplier are concerned

about how their current strategic behavior affects their future interaction, and value long-

term cooperation over one-shot business transactions. In fact, some buyers strongly prefer

to deal with suppliers on a long-term basis, an observation that has been made repeatedly,

especially in the automotive industry (Fujimoto 1999). After studying the long-lasting success

of Toyota, Liker (2004) documented in his book that the number one business principle of

Toyota is that:

“Principle 1: Base Your Management Decisions on a Long-Term Philosophy, Even

at the Expense of Short-term Financial Goals.”1

This is this management principle that this paper aims to model. While multiple forms of

contracts exist that can achieve credible information sharing, we find that having a repeated

relationship itself can also induce truthful information sharing. A long-term relationship

between supply chain parties gives each opportunities to review the credibility of others,

reward truth-telling, and punish otherwise, and therefore provides the right incentive for

truthful information sharing. In other words, repeated sourcing interactions can promote

and form an implicit ‘relational contract’ among supply chain parties, which can be viewed

as an alternative to complex pricing contracts in achieving supply chain coordination.

The present paper analyzes a forecast sharing game between one supplier and one customer

under information asymmetry (The customer has superior information about demand com-

pared to the supplier). There are potentially two sources of inefficiency in this supply chain

setting. The first inefficiency results from “double marginalization” (Spengler 1950). It is

well-know that the supplier operating with a linear wholesale contract would order less than

the system-optimal quantity due to the difference between its profit margin and that of the

1Page 71, The Toyota Way: 14 Management Principles from the World’s Greatest Manufacturer. McGraw-

Hill, 2004.

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supply chain. The other source of inefficiency is information asymmetry. While previous

literature has identified contracting mechanisms that are able to overcome both sources of

inefficiency and achieve supply chain coordination (Cachon and Larievere 2001; Ozer and

Wei 2006), here we restrict the contracting space to the sole usage of linear price contracts

(with no other instruments, such as fixed payment or advance purchase). We show that

under certain conditions an efficient equilibrium can still emerge through the repeated supply

chain relationship. We identify two different types of strategies (trigger strategy and review

strategy) that can both overcome information asymmetry. We show that although the trigger

strategy is the most commonly used strategy in repeated games, it is not the most efficient.

A review strategy is shown to induce both truthful information sharing and system-optimal

capacity for almost every period of the repeated relationship.

This paper makes the following contributions. First, we are — to the best of our knowledge

— the first to establish that truthful information sharing can be achieved in an infinitely

repeated forecasting game setting. Our repeated game model hence complements extant

literature on supply chain contracting by introducing a long-term perspective. Second, we

establish conditions under which an efficient equilibrium is likely to emerge. We show that

coordination can be achieved when the industry is stable, both parties value their long-term

relationship, and defection is easy to detect. Third, we expand the strategy space beyond the

commonly-used trigger strategies and study review strategies in the context of forecast sharing.

We show that in this setting review strategies are more efficient than trigger strategies, in that

the probability of triggering punishment in the review strategy equilibrium can be reduced to

arbitrarily close to zero.

The remainder of this paper is organized as follows. We survey the relevant literature

in §2. After analyzing the one-shot forecasting game (§3), we turn our attention to the

analysis of the repeated game (§4), and establish a trigger strategy of review and punishment

and show that this strategy can induce truthful forecast sharing and optimal allocation of

capacity. In §5, we extend the trigger strategy into a multi-period review strategy and show

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that with an incentive-compatible review strategy, not only can truthful information sharing

be achieved, but the efficiency loss due to punishment can also be fully recovered. In §6, we

discuss comparative static results that offer managerial insight as to when such a long-term

relationship works best in promoting truthful information sharing. In §7, we conclude.

2 Literature Review

In this section, we briefly review supply chain coordination literature in the following order:

one-shot games under perfect information, one-shot games under information asymmetry, and

infinitely repeated games.

Previous literature has focused on studying supply chain coordination as a one-shot game.

When there is perfect and symmetric information, various types of contracts have been iden-

tified that can achieve system coordination. These include buy-back (Pasternack 1985),

quantitiy-flexibility (Tsay 1999; Barnes-Schuster et al. 2002), sales-rebate (Taylor 2002),

quantity-premium contract (Tomlin 2003) and revenue-sharing (Cachon and Lariviere 2005)

contracts.

When there exists information asymmetry in the supply chain, then it matters who offers

the contract. Cachon and Lariviere (2001) formulate such a capacity procurement game,

where the customer who has some private information about demand offers a signalling con-

tract to the supplier. In order to induce the supplier to provide enough capacity for high

demand, a buyer truly expecting high demand has to use more than just a linear price contract

to signal its type. For example, it can offer a fixed side payment to the supplier in order to

separate itself from a low-demand type customer. While Cachon and Lariviere (2001) study

such signalling contracting mechanisms in inducing truthful information sharing in a one-shot

game, we use repeated games and investigate the effectiveness of just having a long-term

relationship on truthful information sharing and supply chain coordination.

When one of the supply chain parties is less informed, it can offer a contract to elicit

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information from the better informed party. Such contracts are called screening contracts.

Porteus and Whang (1999) study capacity procurement with a screening model, where a

supplier, who is unsure about the market size, offers the contract to a buyer who has the market

information. Other screening models include Ha (2001) and Corbett et al. (2004). Ozer

and Wei (2006) study a capacity procurement game with asymmetric information similar to

Cachon and Lariviere (2001), but they consider both signalling and screening contracts. When

the manufacturer can signal her forecast information with an advance purchase, it leads to a

signalling game. On the other hand, the less-informed supplier can induce the truthful forecast

information from the manufacturer by offering a capacity reservation contract, which is a

screening game. Moreover, they show that channel coordination is possible when combining

a capacity commitment contract with a pay-back contract. In this paper, we also consider

channel coordination with information asymmetry. However, we use neither signalling nor

screening devices. Instead, we resort to the repeated nature of supplier relationships, and

show that coordination is achievable in the long-term even with a simple linear contract.

Other papers that study incentives in information sharing include Lariviere and Padmanabhan

(1997), Van Mieghem (1999), Corbett (2001), and Chen (2005). Chen (2002) provides an

extensive literature review on information sharing and supply chain coordination.

During the recent years, building a long-term supplier relationship has been increasingly

recognized as an important aspect of supply chain management. Beth et al. (2003) pointed

out that building a relationship is the key in managing supply chains, instead of investing

in modern technologies. Pyke and Johnson (2003) study a spectrum of spot-buy and long-

term sourcing relationships and identify the key drivers of each type of relationship. Taylor

and Plambeck (2003) study a repeated supply chain relationship, in which production cost is

uncertain and therefore ex ante binding contracting is infeasible. They identify an optimal

‘relationship contract’ that maximizes the customer’s expected profits. Taylor and Plambeck

(2006) further investigate two forms of relational contracts, where the buyer can either promise

to pay a specific price, or promise to purchase a specific quantity. They are both easy

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to implement, and the authors show that which one is optimal to the buyer depends on

the production and capacity cost, as well as the discount factor. In Plambeck and Taylor

(2006), they study a joint production setting with unobservable actions and noncontractible

output. They demonstrate that through repeated interaction an optimal relational contract

can emerge that has a simple form (one that does not depend on past history). They also

find that better information visibility in monitoring unobserved actions can improve system

performance significantly. We develop a qualitatively similar insight in this paper, but in a

different setting.

We use a trigger strategy (§4) to identify one equilibrium where system-optimal capacity can

be supported. Trigger strategies are the most studied strategies in infinitely repeated games

(Friedman 1986), and are commonly adopted in the relational contracting literature (e.g.,

Baker et al. 2002, Levin 2003, Taylor and Plambeck 2003, 2006, Plambeck and Taylor 2006).

They are not restrictive as they do not exclude pareto-dominant equilibria (Abreu 1988).

However, a trigger strategy may result in efficiency loss due to triggered punishment. Hence

we further explore the strategy space and seek alternative strategies that can not only induce

system-optimal capacity, but also minimize unnecessary efficiency loss due to punishment.

We find that using a properly defined review strategy (§5) can achieve this goal. Review

strategies were first studied in economics in the context of repeated principal-agent model

under imperfect monitoring. Radner (1985) studies repeated principal-agent games where the

agent’s action is not perfectly observed. He showed that by constructing a review strategy

the repeated game can achieve asymptotic efficiency. We utilize a similar review strategy in

this paper. However a key difference is that in Rader (1985) , the review length is fixed,

whereas in our model the review length is not fixed. During a review phase, whenever the

supplier finds that the customer does not have an incentive to truthfully share information

any more, the review phase is terminated.

The game we study also belongs to a class of games called cheap-talk games (see, for

example, Crawford and Sobel 1982). Cheap-talk games are different from signaling games in

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that the signals that are being sent in a cheap-talk game are costless, whereas the signals sent

in signaling games must have cost implications (Gibbons 1992, page 210). When cheap-talk

games are played repeatedly, it is generally unclear if efficient outcomes can be sustained.

Morris (1998) considers two types of senders with different preferences over receiver’s actions,

and finds that truth-telling is not an equilibrium even in the infinitely repeated game. On

the other hand, Stocken (2000) studies the credibility of a firm’s manager disclosing private

information to an outside investor, and shows that the manager does not have an incentive to

truthfully reveal his information in a single-shot game, but can be motivated to do so if such

a game is played repeatedly.

3 The Model: One-shot Game

In the supply chain we study, there is one customer (also called the “buyer”, referred to as

‘he’) and one supplier (referred to as ‘she’). The customer wants to acquire a certain amount

of capacity from the supplier before actual demand is realized. We assume that the market

demand the customer faces is a scaled random variable θ·X, where X is non-negative random

variable. Define θ ∈ <+ as the demand size parameter. To capture the essence of the

uncertain market condition, we assume that θ is a random variable with two possible values

“high” and “low”: θi, i = {h, l}. The probability of each happening is P (θl) = α, P (θh) =

1−α, α ∈ (0, 1). The high and low market demand random variables are therefore Dh = θhX,

and Dl = θlX, respectively. It is assumed that θ is independent of X. The distributions of

both θ and X are common knowledge.

The demand distribution that the supplier faces has more random noise. This is because she

is further away from the end market, and does not have the same level of market expertise and

intelligence. This additional level of uncertainty is modeled by a white noise e with mean 0 and

standard deviation σ. That is, the supplier’s demand random variable is D0i = θiX+e, i = h, l.

Denote their CDF Fh (.) and Fl (.) , respectively. It follows that V ar(D0i) > V ar(Di), i = h, l.

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The supplier’s unit capacity cost is c, and she charges the customer a price of r for each

unit allocated and utilized. The customer earns a unit revenue of p for each unit of capacity

allocated and utilized. When the capacity allocated is not enough to satisfy realized demand,

the customer incurs a unit cost of g. This includes potential revenue lost, loss of goodwill, or

the additional cost to find alternative sources of supply. On the other hand, if the supplier

allocates too much capacity, for those units not utilized by the customer, a unit holding cost

of h is incurred. All the cost parameters are public knowledge.

The game begins with a random draw of θ by the customer. The customer observes this

realization of θ (either θh or θl), but the supplier does not. The customer then sends a non-

binding forecast m to the supplier. Here we consider the simplest forecast information set:

either m = H , or m = L. The supplier’s strategy is also binary: to trust, or not to trust the

customer’s forecast. If the supplier trusts the customer’s forecast, then she builds capacity

according to the received forecast. That is, if she receives a forecast of H from the customer

and she trusts the customer, then she would allocate capacityKh that maximizes her expected

profit believing that the market demand random variable would be D0h :

Kh = argmaxKE[rmin(K,D0

h)− h(K −D0h)+ − cK], (1)

Where (x)+ = max (x, 0) . Similarly, if the supplier trusts the customer and receives a forecast

of L, then she would allocate capacity Kl that maximizes her expected profit believing that

the market demand random variable would be D0l :

Kl = argmaxKE[rmin(K,D0

l)− h(K −D0l)+ − cK]. (2)

On the other hand, if the supplier does not trust the customer, then his forecast is considered

meaningless. The supplier then allocates capacity K0 as if there were no meaningful forecast

being shared at all:

K0 = argmaxK

⎧⎨⎩ (1− α)E[(p− r)min(K,D0h)− g(D0

h −K)+]

+αE[(p− r)min(K,D0l)− g(D0

l −K)+]

⎫⎬⎭ . (3)

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After capacity is installed, the true market demand (i.e., a draw from the underlying

distribution of θX) is realized, which we label as d. The customer places an order equal to

the demand realization subject to the supplier’s capacity reservation decision. For each unit

he orders, he pays an unit price of r to the supplier, and gains an unit revenue of p.

In the truthful information sharing situation where the customer shares his forecast truth-

fully and is trusted by the supplier, the customer’s expected profit v∗ and the supplier’s

expected profit u∗ are:

v∗ = (1− α)E[(p− r)min(Kh,Dh)− g(Dh −Kh)+] + αE[(p− r)min(Kl, Dl)− g(Dl −Kl)

+]

(4)

u∗ = (1− α)E[rmin(Kh, D0h)− h(Kh −D0

h)+ − cKh] + αE[rmin(Kl, D

0l)− h(Kl −D0

l)+ − cKl].

We call this the cooperative case. In the non-cooperative case, the forecast is not trusted,

the supplier ignores the customer’s forecasts, and allocates capacity according to her prior

knowledge about θ alone. The expected profits are:

v0 = (1− α)E[(p− r)min(K0,Dh)− g(Dh −K0)+] + αE[(p− r)min(K0,Dl)− g(Dl −K0)

+]

(5)

u0 = (1− α)E[rmin(K0, D0h)− h(K0 −D0

h)+ − cK0] + αE[rmin(K0,D

0l)− h(K0 −D0

l)+ − cK0]

Table 1 summarizes the notation used in the one-shot game.

The following characterizes the equilibrium of the one-shot game: (its proof is provided in

the Appendix)

Proposition 1 In the equilibrium of the one-shot game, the supplier ignores the customer’s

forecast information, and allocates capacity K0 regardless of demand state θ.

Driving the above results is the customer’s underlying incentive to overforecast. While

the customer with a high demand wants to reveal the truth to the supplier, the low-demand

customer also wants to send a high-demand forecast so that he could have more capacity

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θi, i = h, l scale parameter for market demand

α probability of low market demand (θl)

p, r customer’s unit revenue, and transfer price to the supplier for capacity allocated and utilized

c, h supplier’s unit capacity cost, and overage cost when allocated capacity exceeds demand

g customer’s unit capacity-shortage cost when demand exceeds capacity

Di,D0i end-market demand the customer faces, and the demand the supplier faces

Fi(.) CDF for D0i

m = H,L customer’s forecast

d actual demand realization from Di

Ki, v∗, u∗ capacity under cooperation, and the resulting expected profit for customer and supplier

K0, v0, u0 capacity under non-cooperation, and the resulting expected profit for customer and supplier

Table 1: Notation: One-shot game

allocated to him, i.e., he would like to mimic a high-demand customer. Expecting such an

incentive to overforecast from the customer, the supplier cannot distinguish the true demand

state, and is better off ignoring the customer’s forecasts. Such a reaction from the supplier

results in a non-truthful forecast-sharing equilibrium.

Next, we investigate the forecast sharing game under a long-term supply chain relationship.

The mechanism is to let the game play repeatedly. We are interested in answering the

following two questions: (1) Can forecasts be truthfully shared in this setting? (2) Can the

supplier be induced to allocate a system-optimal level of capacity, so that the supply chain

profit as a whole is maximized?

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4 A Long-term Supply Chain Relationship: Truthful

Information Sharing

We model a long term supplier relationship as a repeated game between customer and supplier.

The repeated game consists of the following features. Let time be indexed by t = 1, 2, ...,∞.

At each time t, a stage game is played, starting with a random draw of θ, which we denote

as θt. The stage game is identical to the one-shot game: The customer’s action is to choose

which forecast to provide to the supplier, denoted by mt. The supplier’s action is to install

a capacity level Kt. After capacity Kt has been allocated, true demand dt is realized. The

customer then places the order of dt to fulfill demand. If Kt < dt, only Kt units of the orders

are satisfied, and the remaining unsatisfied orders are lost. If Kt ≥ dt, then all of the demand

is satisfied.

The supplier is paid according to a pre-specified linear pricing contract with a unit rate of

r. This concludes the one stage game. Then time is advanced to t+1, and the game renews,

with another independent draw of θ.

The public information at the start of period t consists of all the past forecasts shared

mt = (m1, ..., mt−1), demand realizations dt = (d1, ...dt−1), and capacity installed Kt =

{K1, ...Kt−1}. Collectively, they are called the public history at time t : ht = mt × dt ×Kt,

which is fully observed by both parties. In contrast, the private history of the repeated game

consists of all the demand signals that the customer privately observed: htprivate = (θ1, ..., θt−1).

The customer chooses a strategy corresponding to a sequence of functions that map the

public history into forecasts: ht → mt, ∀t. The supplier chooses a strategy corresponding to

a sequence of functions that maps the public history into capacity decisions: ht → Kt. Note

that both parties’ strategies do not depend on their private history htprivate, hence they are

public strategies.

Let the discount rate be δ0. To model an uncertain lifetime of the customer-supplier rela-

tionship, suppose at the end of each period, there is a probability p0 that relationship will be

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terminated. Hence δ = δ0(1 − p0) is the effective discount factor. When δ0 is high or p0

is low players discount the future less, which means they are more patient and put a larger

value to the long-term relationship. When δ is low either because of low δ0 or high p0, the

players discount a future gain more relative to short term gain.

A perfect public equilibrium (PPE) is a profile of public strategies that starting from any

period t, given any public history ht, constitutes a Nash equilibrium from that point onward

(Fudenberg and Tirole 1991, Osborne and Rubinstein 1994).

We first focus our attention on the trigger strategy. Trigger strategies are the most used

strategies in repeated games (Friedman 1986), and are commonly adopted in the relational

contracting literature (e.g., Baker et al. 2002, Levin 2003, Taylor and Plambeck 2003, 2006,

Plambeck and Taylor 2006). They are not restrictive as they do not exclude pareto-dominant

equilibria (Abreu 1988). Consider the following strategy profile and its associated equilibrium

for the repeated game. Time is divided into cooperative (truthful-sharing) phases and non-

cooperative (non-truthful-sharing) punishment phases. First, through communication, both

parties are made aware of what the system-optimal capacity levels Kci are:

Kci = argmax

KE[pmin(K,Di)− g(Di −K)+ − h(K −Di)

+ − cK], i = h, l. (6)

The supplier’s strategy is as follows. In each period in the cooperative phase, the supplier

distinguishes between two cases. If the customer forecasts low demand (m = L), the supplier

trusts the forecast and simply allocates the system-optimal level capacity Kcl ; If the customer

reports demand to be high (m = H), however, the supplier takes the following two actions.

First, she still allocates the system-optimal level Kch. Moreover, after demand for that period

dt is realized and the customer has placed the order, the supplier performs a ‘credibility

check’ by comparing dt with a threshold d. If dt ≥d, the game continues in the cooperative

phase; If dt <d, the demand is deemed as too low to support the customer’s claim of a high

demand. Consequently, the cooperative phase is terminated, and the game enters into the

non-cooperative punishment phase. This non-cooperative phase lasts for M periods.

The customer’s strategy is as follows: During each period in the cooperative phase, the

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customer shares forecasts truthfully. If the supplier fails to reserve system-optimal Kci , i = h, l

according to his forecast, then the customer stops sharing his forecasts truthfully, and the game

enters into the non-cooperative punishment phase.

During each period in the non-cooperative phase, the two parties play the one-shot game

equilibrium: forecasts are not shared truthfully and therefore are ignored by the supplier,

who allocates capacity K0 regardless of what forecasts she receives.

To identify the long run communicative equilibria, and their associated conditions, we use

dynamic programming to formulate the system of equations that characterize the equilibrium.

But first we introduce some additional notation for the repeated game.

V customer’s expected long-run profit in equilibrium

Vi customer’s expected long-run profit, given current period demand parameter θi.

vi(m) customer’s one-period expected profit when demand parameter is θi, he sends forecast of m,

and the supplier builds the channel-optimal capacity Kci .

vc customer’s one-period expected profit under supply chain coordination (= (1− α)vh (H)+αvl (L))

uc supplier’s one-period expected profit under supply chain coordination

Table 2: Notation: Repeated Game

With the trigger strategy profile identified above, we have the following recursive equilib-

rium equations for the customer:

V = (1− α)Vh + αVl (7)

Vh = (1− δ)vh(H) + δ©Fh (d)

£(1− δM)v0 + δMV

¤+ (1− Fh (d))V )

ªVl = (1− δ)vl(L) + δV

The above system of equations defines the customer’s equilibrium payoff. The first equa-

tion states that the total expected payoff for the customer is the averaged sum of what he

is expected to get under the high and low demand conditions. The second equation above

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describes the customer’s expected payoff when demand is high (θ = θh). Under the trigger

strategy, the customer would share forecasts truthfully and sends a forecast m = H. But with

probability of Fh (d) , the realized demand would fall below d, and thus trigger punishment

by the supplier. Such punishments last for M periods, during which K0 units of capacity are

allocated and the customer receives an expected payoff of v0. After the punishment phase

ends, the game renews with the customer’s total expected payoff of V . If on the other hand,

demand is greater than d and punishment is not triggered (with probability 1 − Fh (d)), the

games renews, again with the customer’s total expected payoff of V. The third equation says

that when the true demand state is θl, the customer would tell the supplier truthfully and

send m = L. According to the trigger strategy, punishment would not occur, and the game

renews. Solving this system of equations yields:

V =(1−α)[(1−δ)vh(H)+δβ(1−δM )v0]+α(1−δ)vl(L)

1−δ+δ(1−α)β(1−δM ) (8)

Vl =(1−δ)[1−(1−α)δ(1−β(1−δM))]vl(L)+δ(1−α)[(1−δ)vh(H)+δβ(1−δM )v0]

1−δ+δ(1−α)β(1−δM )

Vh =(1−δ)(1−δα)vh(H)+δ[(1−δα)β(1−δM )v0+α(1−δ)(1−β(1−δM))vl(L)]

1−δ+δ(1−α)β(1−δM )

In order for the above equations to hold in a truth-sharing equilibrium, the customer

should find no incentive to lie about his private information about demand. That is, the

following incentive compatibility constraints have to be satisfied:

Vh ≥ (1− δ)vh(L) + δV (9)

Vl ≥ (1− δ)vl(H) + δ©Fl (d)

£(1− δM)v0 + δMV

¤+ (1− Fl (d))V )

ª(10)

Equation (9) states that when demand is high (θ = θh), the customer would prefer to tell

the truth (m = H) rather than to forecast a low demand (m = L) . Equation (10) states that

when demand is low (θ = θl), the customer would prefer to tell the truth (m = L) rather than

to over-forecast (m = H) .

One can verify that the incentive constraint for Vh, i.e. equation (9) is always satisfied.

Next, we solve the incentive constraint for Vl. Substituting the solution for V and Vl from

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equation (8) into (10), and after some simplification, we get

Fl (d) ≥∙

1− δ

δ(1− δM)+ (1− α)Fh (d)

¸vl(H)− vl(L)

vc − v0, (11)

Condition (11) is the key to sustaining a truth-telling equilibrium with a trigger strategy.

First, it states that Fl (d) , the probability of correctly catching and punishing a customer

overforecasting should be sufficiently large, relative to Fh (d) , the probability of incorrectly

punishing a customer reporting high demand truthfully. Essentially, this condition requires

that the two demand distributions are sufficiently distinguishable (“far apart”) from each

other, so that the credibility checking system, governed by the trigger threshold d, is sufficiently

informative.

Second, notice that the right-hand side of (11) is monotonically decreasing in both punish-

ment length M and the discount factor δ. This is because 1−δδ(1−δM ) is monotonically decreas-

ing in δ, and that limδ→01−δ

δ(1−δM ) = ∞.The same holds true for the punishment length, as

limM→01−δ

δ(1−δM ) = ∞. They imply that in order for the inequality to hold, M and δ have to

be large enough.

Lastly, vl(H)−vl(L)vc−v0 is the ratio of the immediate one-period gain to the consumer from

inflating his demand forecast vl(H) − vl(L) to the long-term benefit vc − v0. Therefore it

captures the “normalized” temptation of lying by the customer. Condition (11) captures the

essential trade-off of the customer’s immediate temptation of over-forecasting and getting a

large allocation of capacity, against the long-run cost of not being trusted and receiving lower

capacity. For the truth-telling equilibrium to hold, it requires that such a temptation cannot

be too great.

For the supplier, the long-term relationship cannot leave her worse off by reserving the

system-optimal capacity during the cooperative phase, compared to the reservation level profit

she would get from not cooperating with the customer. That is,

uc ≥ u0. (12)

With conditions (11) and (12) satisfied, a communicative equilibrium can be supported in

16

which the customer shares forecasts truthfully and the supplier allocates capacities that are

optimal to the whole supply chain.

Theorem 2 There exists a discount rate δ, a punishment length M , such that for ∀δ >

δ, M > M, there is a perfect public equilibrium of the repeated game in which the cus-

tomer shares forecasts truthfully and the supplier allocates system-optimal capacity levels dur-

ing the cooperative phases, provided that the customer’s credibility test is powerful enough

(equation 11 is satisfied), and the system-optimal capacities yield the supplier no worse off

than her expected profit with no forecast sharing at all (equation 12 is satisfied).

Our results indicate that when there is information asymmetry, we need both a high dis-

count rate and a strong credibility checking system to check the incentive of the customer for

lying about his private information. The key for the above result to hold is that the supplier

can evaluate the customer’s credibility at the end of each stage game. The actual demand

realization at the end of each stage game provides an effective, albeit imperfect, monitoring

and credibility checking mechanism. The repeated supply chain relationship itself provides

the supplier with a credible threat to punish the customer if he is not truth-telling. Hence,

if the credibility checking system is powerful enough in terms of detecting lies and deviations,

and the corresponding punishment severe enough, the customer would find it in his own best

interest to share forecasts truthfully, and the supplier in her best interest to allocate the

system-optimal capacity level.

The existence of such a communicative equilibrium depends on a multitude of factors, such

as the discount factor, length of the cooperative phase and non-cooperative phases, and the

credibility assessment system. The above theorem prescribes a feasible set for those parameter

values under which a communicative equilibrium is permissible. When the parameters of the

game do not permit a communicative equilibrium as described above, having a repeated supply

relationship alone may not achieve truthful forecast sharing, and other mechanisms such as

non-linear contracting are desirable.

17

5 A Long-term Relationship: Asymptotic Full Efficiency

An important feature of the long-run communicative equilibrium identified above is that

punishment is part of the equilibrium path, even though the customer, in equilibrium, would

be sharing forecasts truthfully. This punishment phase reduces both parties’ expected payoff,

compared to the cooperative phase. Therefore it is important to quantify this efficiency

loss. Below we show that the switching between the normal cooperative phase and the non-

cooperative punishment phase can be modelled as a Markov process with stationary transition

probabilities. From the transition probabilities we can derive an exact bound on the loss due

to punishment.

Define state space S = {0, 1, ...M}, with 0 being the cooperative normal phase, and 1, ...,M

being the M periods of punishment phase. Let [Zn, n = 0, 1, ...∞] be a sequence of random

variables with state space S. This Markov Chain {Zn} has finite states and is aperiodic,

therefore its equilibrium distribution πn, i.e., the percentage of time that the Markov Chain

stays in any one state, can be solved with the following steady state equations:⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

π0 = πM + (1− (1− α)Fh (d))π0

π1 = (1− α)Fh (d)π0

πi = πi−1, i = 2, ...MPj πj = 1, ∀j

(13)

Solving these equations, we have

π0 =1

1 + (1− α)Fh (d)M(14)

πi =(1− α)Fh (d)

1 + (1− α)Fh (d)M, i = 1, ...,M.

Hence we can state the following:

Proposition 3 In the trigger-strategy equilibrium, the supply chain is in the cooperative phase

11+(1−α)Fh(d)M of the time, and in the punishment phase (1−α)Fh(d)M

1+(1−α)Fh(d)M .

18

A direct consequence of the above result is a net loss of efficiency to the whole supply chain

due to the punishment phase, even though we can see from the above result that the supply

chain in such an equilibrium could be highly efficient, especially when the trigger probability

(1− α)Fh (d) is low. This leads to the question how such an efficiency loss can be reduced.

The root cause of the efficiency loss is the trigger strategy profile. Under information

asymmetry, the supplier relies on the trigger strategy to punish the customer if he is believed to

be inflating his forecasts. So, in a sense efficiency loss is indispensable with trigger strategies.

Below we show that such a loss can be asymptotically restored with a different strategy

profile called a multi-period review strategy. This is accomplished with the supplier setting

up a score system and evaluating the customer’s credibility over a number of consecutive

periods. Review strategies have been studied in the economics literature (Radner 1985,

Stocken 2000). In practice, score systems have been widely used by companies such as

Chrysler, Intel (Minahan 1998a,b), Sun (Farlow et al. 1996), Toyota and Mitsubishi (Hagen

and Choe 1998) as a tool for evaluating the performance of supply chain players. Compared

to the trigger strategy, the advantage of a review strategy in the context of a repeated game

with imperfect monitoring is that the reviewer does not immediately commence punishment

once the reviewee fails the reviewer’s credibility test. Instead, the reviewee is given more

chances. As a result, the likelihood of mistakenly punishing a truth-telling player is smaller.

The challenge, however, is that the review strategy needs to be carefully designed so as not

to give the reviewee opportunities of dynamically adjusting his strategies according to his

current score. Below we derive a review strategy that achieves both truth-telling and better

efficiency.

Our review strategy is based on the results of Radner (1985) and Stocken (2000), and works

as follows. Time is divided into a sequence of review phases, each having a maximum length

of R periods. During each review phase, the customer reveals information truthfully, and the

supplier maintains an index (i.e., a ‘score’) It and a credibility threshold γt. At the end of

each period t ∈ [1, R], after demand is realized, the supplier evaluates the truthfulness of the

19

customer with a credibility test, and updates the score It in the following way. If the customer

reports demand to be low (mt = L) , then the customer’s forecast is believed to be truthful,

and the supplier score is incremented by one, It = It−1 + 1. If the customer reports demand

to be high (mt = H), then the customer’s score is incremented only if demand at that period

is over a threshold, i.e., Dt ≥d. Otherwise if demand falls below the threshold, the supplier

score is not updated, i.e., It = It−1. The overall probability that the customer’s forecast is

evaluated as truthful is S = (1− α) [1 − Fh (d)]+α. S hence measures the strength of the

credibility checking system. The credibility threshold γt is defined as γt =RS−JR

t, where J

is some allowance for measurement error. When J = 0, the credibility threshold is simply

γt = St.

At the end of each period, the supplier also evaluates whether the customer still has an

incentive for truth-telling. If yes, the review phase is continued. If she finds that the customer

no longer has an incentive to report forecasts truthfully, the review phase is terminated. Upon

termination of the review phase, if the supplier score It ≥ γt, the customer passes the review

and the games restarts with a new review phase. If It < γt, the customer fails the review and

the games enters a non-truthful-sharing (punishment) phase that lasts for M = qR periods,

where q > 0.

During the review phase, the supplier always trusts the customer, and builds capacity

according to the customer’s forecasts. During the punishment phase, the supplier ignores

the customer’s forecasts, and the customer does not share forecasts truthfully. Therefore

the game reverts to the one-shot uncommunicative Nash equilibrium for M periods. The

repeated game begins with a review phase, with supplier score I0 = 0. Let the customer’s

total discounted expected payoff at the beginning of the game be denoted as V (I0).

Theorem 4 There exists a discount rate δ, a review phase of length R, a punishment length

M, and a credibility assessment value S, such that for ∀δ > δ,R > R,M > M,S > S, there

is a perfect public equilibrium of the repeated game in which V (I0) > vc − ε,∀ε > 0, provided

that the system-optimal capacities yield the supplier no worse off than her expected profit with

20

no forecast sharing. In this equilibrium, the customer always shares information truthfully,

and the supplier allocates the system-optimal capacity.

Proof for this Theorem is provided in the Appendix.

The above theorem not only states that there exists parameters of the game that can

support a truthful-sharing equilibrium, it also states that in such an equilibrium, the non-

truthful-sharing punishment phase would almost never be observed. This is in contrast with

the truthful-sharing equilibrium supported by a trigger strategy in section 4, which punishes

a truth-telling customer in (1−α)β1+(1−α)βM percent of periods of the times.

The reason for such an improvement in supply chain efficiency is that with a review strategy

defined as above, the customer is given an extended period of time to prove his intention of

sharing information truthfully, instead of being given no opportunity at all in the trigger-

strategy equilibrium. If, for example, the customer had bad luck and had a very low demand

realization in one period but had forecasted high demand in good faith, punishment would

ensue if trigger strategy profile has been adopted. In contrast, if the supply chain parties had

adopted the review strategy, then the customer would not have been punished immediately.

Instead, his credibility score would suffer but he would still be regarded as trust-worthy and be

given more periods to vindicate himself. In the long run, the probability that a truth-telling

customer fails the review goes to zero as the review periods R becomes large (provided also that

the discount factor δ is sufficiently high, and the credibility check system S is effective enough).

Also note that as R becomes large, the length of punishment phase is also longer. This also

increases the customer’s incentive to share forecast information truthfully. The effect of the

inefficiency brought by the lengthening punishment phase is mitigated by a reduction in the

probability of inappropriately punishing a manager who has been sharing forecast information

truthfully. As is shown, for sufficiently large review length, the probability of the customer

failing the review when he shares his forecasts truthfully can be bounded arbitrarily close to

zero. Hence, this supply chain converges to a fully efficient coordinated equilibrium.

21

6 Managerial Implications

In this section, we provide a set of managerial implications of our findings. Specifically, we

study how time horizon and length of the punishment, the discount factor, and the nature of

demand uncertainty will make truthful forecast sharing more or less likely.

First, consider the length of the non-cooperative phase lengthM in both the trigger-strategy

equilibrium and the review-strategy equilibrium. As the length of the non-cooperative phase

M increases, the threshold for punishing untruthful forecasting increases, which increases the

customer’s incentive to share forecasts truthfully. This implies that in order for a supply chain

relationship to be communicative and cooperative, there has to be a long enough horizon so

that non-cooperative strategies can be carried out. If a supply relationship tends to be

short-term, such as the one in a spot-market purchase, then there simply may not be enough

transactional opportunities for a credible threat of punishment to sustain. In such cases,

alternative coordinating mechanisms may be more desirable, such as explicit contracting.

Second, as the discount factor δ increases, future long-term gains from sharing the forecast

truthfully, as well as the punishment threat from deviating become more important. Therefore

the customer is less inclined to behave myopically and opportunistically.

There are a number of factors that affect the discount factor δ. For example, the customer’s

and supplier’s dominance in their own industries determines their ‘walk-away position’, and

in turn their evaluation of continuing the relationship. If the customer is the dominant player

in the industry, whereas its suppliers are highly competitive, then the customer has significant

bargaining power, and is free to switch among suppliers. Therefore the customer is likely to

care less about maintaining a long-term relationship with any particular supplier. This would

lead to a low discount factor. For example, GE has developed its Trading Process Network

(TPN) for its component procurement and works with over 2000 suppliers, who compete for

each contract (Reinhardt 1998). If the supplier base is relatively small, and the product they

provide are of strategic importance, then a long-term relationship is more likely to emerge.

22

For example, Boeing has over the years developed very close long-term partnership with its

small set of suppliers for airplane engines. Such a partnership include sharing information on

demand as well as product design (Pyke and Johnson 2003; Kulp et al 2003). On the other

hand, if the supplier’s bargaining power grows large relative to that of the customer, then a

cooperative relationship is also difficult to maintain. For example, due to Intel’s ‘copy exactly’

policy, some of its large suppliers became the sole source provider for a large proportion of

Intel’s key production equipment. Such a ‘lock-in’ created difficulty in forming a cooperative

relationship (Cohen et al. 2003).

From a management point of view, the employment horizon of senior executives, and their

incentive structure also affect the discount rate. When management is rewarded based on

relatively short-term gains, they tend to reduce the value given to future payoffs from long-run

relationships. As a result, communicative equilibrium conditions are difficult to meet. Other

factors such as corporate management culture play similar roles in the discount rate factor δ.

Recall that δ = δ0(1 − p0) is also a decreasing function of p0, the probability that the

repeated game may stop at the end of each stage game. Some industries are highly risky,

with firms entering and exiting at a fast rate. In such industries, p0 is high, and therefore

firms tend to focus more on near-term payoff. As a result they tend to ignore the value of

long-term relationships. In such environments, our model suggests that truthful information

sharing is more difficult to obtain, and a long-term relationship is less likely to emerge.

Third, consider the relationship between demand uncertainty and the power of the cred-

ibility test system in the trigger-strategy equilibrium. How reliably and accurately can the

supplier detect untruthful forecasts depends upon how informative the credibility checking is

(equation (11)). It is in turn determined by how distinguishable (“far apart”) the two demand

distributions are from each other. To formally show this, recall from equation (11) that

Fl (d) ≥∙

1− δ

δ(1− δM)+ (1− α)Fh (d)

¸vl(H)− vl(L)

vc − v0. (15)

23

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

NH4,1LNH4.4,1.1L

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

1

d

NH4,1L

NH4.4,1.1L

Fl HdL¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡Fh HdL =1.3

0 2 4 6 8 10 12

0

0.1

0.2

0.3

0.4

NH4,1L

NH8,2L

0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

d

NH4,1L

NH8,2L

Fl HdL¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡Fh HdL =17.2

(a) A strong test: PDF (left panel) and CDF (right panel)

(b) A weak test: PDF (left panel) and CDF (right panel)

Figure 1: Power of Credibility Test

Let δ → 1 and M →∞. Then the above inequality is simplified to be

Fl (d)

Fh (d)≥ (1− α) [vl(H)− vl(L)]

vc − v0(16)

This “CDF ratio” captures the strength of the credibility test. A larger value of Fl (d) /Fh (d)

indicates a greater probability of catching a customer lying, while not triggering punishment

for a truth-telling customer too frequently. For any fixed d, when the high and low demand

distributions are far apart from each other, the ratio Fl (d) /Fh (d) is large. The other ex-

treme is when two demand distribution are very close to each other, then Fl (d) /Fh (d)→ 1.

This in turn is driven by the differences in the means and variance of the distributions. For

example, consider the following two scenarios: (a) Normal(4,1) vs. Normal(8,2) (two demand

distributions are distinguishable); and (b) Normal(4, 1) and Normal(4.4, 1.1) (two demand

distributions are not distinguishable). Their distribution plots are shown in Figure 1.

One can see that (a) provides a stronger test, as the test statistic Fl (d) /Fh (d) = 17.2

(when d = 4.5) has a larger value. On the contrary, (b) is the uninformative case, where

24

Normal(4, 1) and Normal(4.4, 1.1) distributions are plotted. As is demonstrated, the two

scaled demand scenarios are too close to each other to be distinguishable. As a result, the

test statistic Fl (d) /Fh (d) =1.3 is much smaller, therefore significantly reducing the power of

the credibility test.

Another driver of the test statistic is the underlying variability in the supplier’s demand

estimate. In our model the supplier’s demand distribution has more variability than that of

the customer, because of the additional noise term e (Recall that D0i = θiX + e, i = h, l) that

has zero mean and standard deviation σ. This reflects the inaccuracy in the supplier’s market

intelligence. It is easy to see that as such an error becomes larger (σ → ∞), Fl (d) /Fh (d)

monotonically decreases. This confirms our intuition that in order to effectively detect the

customer’s over-forecasting, the supplier has to have an accurate estimate of the underlying

demand distribution. If, for instance, the supplier does not have an accurate estimate about

the end market demand distribution (due to insufficient market research, for example), then

she is less likely to distinguish from the customer’s order quantity if it is from a high-demand

distribution, or a low-demand one. Knowing this, the customer then has an stronger incentive

to inflate his forecast when he faces a low-demand market. As a result, a communicative

equilibrium is less likely to be sustained. A similar observation can be made for the review

strategy equilibrium as well. As the error becomes larger (σ →∞), the customer’s probability

of not meeting the demand threshold Fh (d) is greater. Therefore the strength of the credi-

bility system measured by S = (1− α) [1 − Fh (d)]+α is weaker. As a result, a cooperative

equilibrium is less likely to emerge. In the semiconductor equipment industry, for example,

demand is highly volatile and forecasts are known to be unreliable (Cohen et al. 2003). This

may explains why in this industry a cooperative supply chain relationship is difficult to be

forged (Terwiesch et al. 2005).

Finally, consider the effect of having a lengthened review period R in the review-strategy

equilibrium on supply chain coordination. With a review-strategy, customers are given an

extended period of time to prove his intention of sharing information truthfully, instead of

25

being given no chance at all in the trigger-strategy equilibrium. If, for example, the cus-

tomer had bad luck in one period and had a very low demand realization but had reported

high demand in good faith, punishment would immediately ensue if trigger strategy profile

has been adopted. In contrast, if the supply chain parties had adopted the review strategy,

the customer still has chances to ‘acquit himself’ during later time periods. Also, as the

review length R is extended, so is the the length of punishment phase. This increases the

customer’s incentive to share forecast information truthfully. A prolonged punishment phase,

triggered by errors that are inherent in the credibility system under imperfect monitoring, is

not pareto-efficient. However, such inefficiency is mitigated by a reduction in the probability

of inappropriately punishing a manager who has been sharing forecast information truthfully.

As is shown, for sufficiently large review length R, the probability of the customer failing the

review when he shares forecasts truthfully can be bounded arbitrarily close to zero. There-

fore, a sufficiently long review length R is indispensable to the asymptotically fully efficient

supply chain. Such a review strategy can indeed be observed in practice. For example,

Sun Microsystems rates each of its suppliers’ performance with a ‘scorecard’ system, and re-

view the scores with the supplier each quarter. Suppliers are given multiple opportunities

to meet the company’s requirements. The review system helps Sun to achieve its goal of

building long-term relationship with suppliers (Farlow et al. 1996). Companies such as Intel

(Minahan 1998a) , Chrysler (Minahan 1998b), and Toyota and Mitsubishi (Hagen and Choe

1998) all have similar review practices.

7 Conclusion

In this analysis, we study the practice of forecast sharing and supply chain coordination with

a game-theoretical model. We find that in a one-shot version of the game, forecasts are

not shared truthfully by the customer. The supplier will rationally discount the forecast

information in her capacity allocation. This results in Pareto sub-optimality for both supply

26

chain parties. However, we show that a more efficient, truth-sharing outcome could emerge

as an equilibrium from a long term relationship. In this equilibrium, forecast information is

transmitted truthfully, and trusted by the supplier who in turn allocates the system-optimal

equilibrium. This renders both customer and supplier better off, compared to the non-

truthful-sharing equilibrium.

We identify two different types of strategies that can support the truthful-sharing equilib-

rium. We first show that a trigger-strategy profile can induce the customer to truthfully share

his forecast information, and supplier to allocate system-optimal capacity. In the equilibrium,

the supplier uses the actual demand realization to gauge the customer’s private demand in-

formation, and to test if the customer is truth-telling. If the customer fails the test, then

a punishment phase is triggered where lower levels of capacity are allocated. Such a trigger

strategy is intuitive and relatively easy to implement. The downside, however, is that it nec-

essarily triggers punishment if the supplier believes the customer is not sharing information

truthfully, even though in fact he is, which introduces a loss to supply chain efficiency. Next,

we illustrate that by using a multi-period review strategy, such an inefficiency can be reduced

asymptotically to zero. The key element of this strategy is that the supplier computes a

scoring index of the customer’s behavior that is updated over time and used to evaluate if the

customer has sufficient incentive to share his private information truthfully in each transaction

of the repeated game. Compared to trigger strategies, review strategies are more tolerant,

but require diligence and more monitoring effort. This is also consistent with what we observe

in a coordinated supply chain. For example, Toyota when working with its suppliers, do not

immediately sever the business relationship with any of its suppliers when it has quality issues

or delivery problems, which fail to meet Toyota’s standards. Instead, Toyota dispatches its

own staff to the supplier’s site and works with their suppliers for extended periods of time

to help them correct the problem. Only under extreme circumstances does Toyota drop any

supplier from its supplier base (Liker 2004).

The implication of our findings is that when supply chain parties have a long-term ongoing

27

relationship, or expect to do so, they could achieve coordination even though incentives are

not aligned in the short run. In the case of forecast sharing, the customer has an incentive to

overforecast if the supplier relationship is of short term nature (spot market, etc.). However,

when they interact repeatedly, then the threats of retaliation in response to bad behavior

and the promise of cooperation in response to good behavior in the current period can have

significant impact on future payoffs. Concerns about credibility and payoff in the long run

can thus yield cooperation and coordination. This ‘relational contract’ is desirable especially

when actions are not perfectly observable and cannot be contracted or enforced.

We observe truthful forecast sharing in some long-term supplier relationships in the indus-

try, with non-coordinating linear contracts. However, we also observed some other supply

chain dyads that are not truthfully sharing forecasts. Indeed, having a repeated relationship

may not work everywhere. As we show, in order to achieve credible information sharing,

certain environmental conditions have to be met, such as a high enough discount factor, and

an effective enough credibility checking system. Where such conditions cannot be met in a

supply chain, mechanisms other than a repeated relationship have to be in place to achieve

truthful information sharing, such as auctions. For example, GE has developed its Trading

Process Network (TPN) for its component procurement with its over 2,000 suppliers. For

each procurement, specifications are posted and shared electronically. Suppliers submit their

bids for each contract, with very little face-to-face interaction (Reinhardt 1998, Pyke and

Johnson 2003).

Our model has its limitations. There are only two demand states (θh and θl) in our model.

An extension to multiple states will be more complex, as the tool for distinguishing each state

from multiple potential states has to be more sophisticated. For example, consider the three-

state case: θh > θm > θl. The credibility test for the middle case (θm) will become a two-way

test, instead of the one-way test in the current model. The trigger strategy associated with

θm would also become a two-directional trigger. One can also generalize other aspects of

the model. For example, the supplier’s capacity choice can be not perfectly observable or

28

verifiable. As a result, the customer has to use some monitoring mechanism for detecting

if the supplier has allocated the right amount of capacity. Currently, our demand state

is drawn independently each period. It may be more realistic to consider a repeated model

setting where demand states are correlated over time, and see how that affects the equilibrium.

Alternatively, one can model the customer’s type to be fixed over time (e.g., a honest type

who always tells the truth, or a strategic type), and the supplier has to learn the customer’s

type over some periods of time. It would also be interesting to generalize the supply chain

structure to be one supplier with multiple customer, and see how the dynamics changes in

a long-term relationship, especially how competition among customer affects the equilibrium

outcome. We hope to address at least some of the modeling aspects in our future research.

Acknowledgement

This research was partially funded by two National Science Foundation grants (DMI01-

21395 and DMI-0075391). The authors thank Gerard Cachon, Jiri Chod, Noah Gans, Ser-

guei Netessine, Fuqiang Zhang and seminar participants at the Wharton School and MIT

Sloan reading group for their helpful comments. Steve Graves provided detailed comments

to an early version of the paper. This paper has benefited greatly from helpful discus-

sions with Phillip Stocken. A copy of this paper is available from the first author’s website:

http://smgpublish.bu.edu/ren/.

Appendix: Proofs

Proof of Theorem 1: Suppose the supplier trusts the customer and always reserves capacity accord-

ing to his forecast. The high-type customer (θ = θh) will sure send H to reserve more capacity. However,

the low-type customer also has an incentive to forecast H even though he receives low demand θ = θl,because

his profit is strictly increasing in capacity reserved. The supplier’s expected profit would be higher if she just

29

build K0 ∈ [Kl,Kh], because

(1− α)E[(p− r)min(Kh,D0h)− g(D0

h −Kh)+] + αE[(p− r)min(Kh,D

0l)− g(D0

l −Kh)+]

≤ (1− α)E[(p− r)min(K0,D0h)− g(D0

h −K0)+] + αE[(p− r)min(K0, D

0l)− g(D0

l −K0)+]

by the definition of K0(see equation 3). Therefore, trusting the customer’s forecast is not a best response for

the supplier. Given that the supplier’s best response is to ignore the forecasts, then it does not matter what

the customer forecasts, because his assigned capacity level is fixed at K0.

Proofs of Theorem 4: The proof follows Stocken (2000), and involves the following steps:

1. We show that players do not have an incentive to deviate from the equilibrium strategies;

2. The customer will only fail the review when It < max(0, t− (R− q)),i.e., when It is such that the

customer will fail the review for sure. A sufficient condition of the above is to show that the customer

has an incentive to share forecasts truthfully when IR−n = q − n.

3. There exist parameters of the game such that the customer will share forecasts truthfully at the begin-

ning of the game.

4. There exist parameters for the game that would almost always yield truthful information sharing.

First, we define the following:

v(I0) is the customer’s normalized discounted expected profit at the beginning of the game;

v(IR−n = x) is the customer’s normalized discounted expected profit when there are nperiods remaining

in the review phase and the value of the trust index at period t = R− n is x;

β = 1−Fl (d) is the conditional probability that the supplier will assess the customer’s truthful forecast

of m = L to be truthful;

η = 1− Fh (d) is the conditional probability that the supplier will assess the customer’s over-forecast of

m = H to be truthful when the customer actually observes θ = θl;

λ denotes the probability of failing the review conditional on being at the beginning of the game;

30

q = R.S − J ; and,

∆ = vl(H)− vl(L).

Now we establish the main result step by step with a series of lemmas.

(1) Step 1:

First, neither the customer nor the supplier has an incentive to deviate. Given that the customer would

share forecasts truthfully, the supplier’s best response is to trust and build Kci . Therefore the supplier does

not have a strict incentive to deviate from the characterized equilibrium strategies.

For the customer, the equilibrium is constructed so that if the truth-telling constraints are no longer

satisfied, the review will be terminated and if It ≥ γt, the customer passes the review, or if It < γt, the

customer fails the review. Thus the customer follows a sequentially rational strategy and has no incentive to

deviate.

(2) Step 2:

Lemma 1 Suppose the customer is truthful-telling always. For all n such that 1 ≤ n ≤ q,

v (IR−n = q − n) = (1− δ)vcn−1Xi=0

(Sδ)i + v(I0)(Sδ)n

+£(1− δK)v0 + δKv(I0)

¤ nXi=1

(1− S) δiSi−1.

Proof. By induction. When n = 1, the manager either passes the review or the players enter the

punishment phase at the end of the period ; hence,

v (IR−1 = q − 1) = (1− δ)vc + δSv(I0)

+ δ(1− S)£(1− δK)v0 + δKv(I0)

¤.

Suppose the induction hypothesis holds for n. Now show that it holds for n+ 1.

v¡IR−(n+1) = q − (n+ 1)

¢= (1− δ)vc + δSv(IR−n = q − n)

+ δ(1− S)£(1− δK)v0 + δKv(I0)

¤.

31

Substituting in the induction hypothesis yields:

v¡IR−(n+1) = q − (n+ 1)

¢= (1− δ)vc + δS{(1− δ) vc

n−1Xi=0

(Sδ)i + v(I0) (Sδ)n+

¡(1− δK)v0 + δKv(I0)

¢ nXi=1

(1− S) δiSi−1}+ δ(1− S)£(1− δK)v0 + δKv(I0)

¤= (1− δ)vc

n−1Xi=0

(Sδ)i + v(I0) (Sδ)n+1 +

£(1− δK)v0 + δKv(I0)

¤ n+1Xi=1

(1− S) δiSi−1.

Lemma 2 The customer will only fail the review when It < max(0, t− (R − q)), i.e. when It is such

that the customer will fail the review with probability 1.

Proof. To show that the customer will truthfully share his private information when It is such that

It ≥ max(0, t− (R− q)), it is sufficient to show that the customer has an incentive to release his private

information truthfully when IR−n = q − n. This result is established in the following claim.

Claim: Let IR−n = q − n and consider n such that 1 ≤ n ≤ q. There exists δ0 such that for all

δ ∈ [δ0, 1), there exists a S and R0 such that for all S > S and R > R0 the customer truthfully reveals his

private information.

Proof of the claim is as follows. Because the customer has a dominant strategy to truthfully share his

private information when he observes θh, it is sufficient to check that the truth-telling constraint is satisfied

when the customer observes θl.

Consider a one-shot deviation. For any R, IR−n = q−n and any n such that 1 ≤ n ≤ q, the customer

would share forecast truthfully if:

(1− δ)v(Kh, θl) + δβv¡IR−(n−1) = q − (n− 1)

¢+ (1− β)

£δ(1− δK)v0 + δK+1v(I0)

¤≤ (1− δ)v(Kl, θl) + δηv

¡IR−(n−1) = q − (n− 1)

¢+ (1− η)

£δ(1− δK)v0 + δK+1v(I0)

¤.

Rearranging terms, we have

∆ ≤ (β − η)

1− δδ£v¡IR−(n−1) = q − (n− 1)

¢−¡(1− δK)v0 + δKv (I0)

¢¤32

Using Lemma 1, which gives the customer’s expected profit when IR−n = q − n for all 1 ≤ n ≤ q, and

substituting for v¡IR−(n−1) = q − (n− 1)

¢yields the condition:

(β − η)

1− δδ (1− δ) vc

n−2Xi=0

(Sδ)i + v (I0) (Sδ)n−1

+¡(1− δK)v0 + δKv (I0)

¢Ãn−1Xi=1

(1− S) δiSi−1 − 1!≥ ∆.

Using L’Hospital’s rule , we have as δ → 1,

(β − η)

−1

(v(I0)S

n−1 + v(I0)

Ãn−1Xi=1

(1− S)Si−1 − 1!

− vcn−2Xi=0

Si + (n− 1)v(I0))Sn−1 + U 0(I0)Sn−1

+¡−Kv0 +Kv(I0) + U 0(I0)

¢Ãn−1Xi=1

(1− S)Si−1 − 1!+ v(I0)(1− S)

n−1Xi=1

iSi−1

)≥ ∆

Using the fact thatn−1Pi=1

iSi−1 = 1−Sn−nSn−1(1−S)(1−S)2 ,the above is simplified to be

(β − η)

½(vc − v(I0))

µ1− Sn−1

1− S

¶+ Sn−1K

¡v(I0)− v0

¢¾≥ ∆

As prescribed in the review strategy, fix K = µR, µ > 0and note that v(I0) ∈ [v0, vc].

For large n, Sn−1K → 0 and 1−Sn−11−S → 1

1−S . Thus for all δ ∈ [δ0, 1), there exists a Sand a R0such

that for all S > Sand R > R0, β−η1−S [v

c − v(I0)] will exceed ∆ for all IR−n = q−n. Hence the customer’s

incentive constraint is violated and consequently fails the review only if It < max(0, t− (R− q)).

(3) Step 3:

Lemma 3 Let I0 = 0. There exists a δ00 such that for all δ ∈ [δ00, 1), there exists a R00 such that

for R >R00, the customer will truthfully share his private information and the supplier will respond to the

customer’s forecast for at least the first period of the review phase.

Proof. We only need to check if the customer would have an incentive to deviate from a good faith

strategy of sharing forecasts truthfully when he observes θ = θl.

33

Consider the following sufficient condition for the customer’s truth-telling constraint for hold: the passing

of the review phase is deferred until Itis such that IR−n ≥ q. Because v(I0) ∈ [v0, vc], the truth-telling

constraint will hold if the sufficient condition is imposed and the customer still forecasts truthfully.

Recall from claim proved above that the failing of the review occurs only when It < max(0, t−(R−q)).

Therefore, as δ → 1, if,

(β − η)

⎧⎪⎨⎪⎩ (vc − v(I0))

∙q−2Pi=0

¡R−q+iR−q

¢Si(1− S)R−q −

R−q−1Pi=0

¡q−1+iq−1

¢(1− S)i Sq−1

¸+¡R−1R−q¢Sq−1(1− S)R−qK (v(I0)− v0)

⎫⎪⎬⎪⎭> ∆,

then the customer will truthfully share his private information at the start of the review phase.

The left-hand side of the inequality has three major components. The first one, (β − η) (vc − v(I0))q−2Pi=0

¡R−q+iR−q

¢Si(1− S)R−q, represents the marginal profit from avoiding failing the review before period R.

The review is terminated before period Rif the customer has had R− q assessments where the trust index is

not incremented. In this case, for each assessment where the trust index is incremented the customer avoids

termination for an additional period up to the beginning of the final review period, i.e. R− 1. The histories

Siand (1 − S)R−q can be arranged in¡R−q+iR−q

¢combinations. Notice that for i = q − 2, the summand

becomes¡R−2R−q¢Sq−2(1− S)R−q; this product multiplied by (β − η)represents the probability of failing the

review phase in period R− 1.

Second, (β − η)(vc − v(I0))R−q−1Pi=0

¡q−1+iq−1

¢(1− S)i Sq−1 represents the marginal loss from passing

the review before period R. The review is passed before period R if the customer has had q assessments

where the trust index has been incremented before period R. If It < q, then the customer is assessed to

have reported dishonestly and the passing of the review is delayed for an additional period. The passing of

the review can be delayed for at most R − q − 1 periods prior to the beginning of the final period. These

histories (1− S)iand Sq−1can be arranged in¡q−1+iq−1

¢combinations.

And finally, (β−η)¡R−1R−q¢Sq−1(1−S)R−qK (v(I0)− v0) represents the marginal benefit from avoiding

the punishment phase in period R. This event occurs for a history of R − q assessments where the trust

index is not incremented and q assessments where it is incremented. The histories Sq−1and (1− S)R−qcan

34

be arranged in¡R−1R−q¢combinations.

We now show that the constraint can always be satisfied if R is sufficiently large and if K = µR, where

µ > 0. Fix K = µRand let J = τRρ where τ > 0, 12< ρ < 1for the remainder of the analysis. Recall

that q = RS − J . Then let R→∞ and after applying Stirling’s formula

limR→∞

R!√2πR(R+

12)e−n

= 1

we get

(β − η)

µR− 1R− q

¶Sq−1(1− S)R−qK

¡v(I0)− v0

¢=

√R (β − η)µ (v(I0)− v0)p

2πS(1− S)→∞.

Similarly, it can be shown that¡R−q+iR−q

¢Si(1 − S)R−q <

¡R−q+i+1

R−q¢Si+1(1 − S)R−q for all i =

0, 1, 2, ..., q − 3, and also that, as R→∞,µR− 2R− q

¶Sq−2(1− S)R−q =

RS − 1S(R− 1)

1p2πRS(1− S)

→ 0.

Thus as R→∞,q−2Pi=0

¡R−q+iR−q

¢Si(1− S)R−q→ 0, and

R−q−1Xi=0

µq − 1 + i

q − 1

¶(1− S)i Sq−1 → 0.

Since ∆ is bounded and the inequality is strict, it follows that for R > R00and δ ∈ [δ00, 1), the customer

will truthfully share his private information and the supplier will trust the customer’s forecast for at least the

first period of the review phase.

(4) Step 4:

Lemma 4 The probability of failing the review when the players use the specified review strategies is less

than or equal to the probability of failing the review when the customer reports in good faith and is evaluated

at the end of period R only and is said to fail the review if IR < q and pass the review if IR ≥ q.

Proof. Since once the customer passes the review, the review phase is terminated, it follows that the

probability of failing the review under the specified review strategies (i.e. the customer can pass the review

35

before the end of period R and fail the review if for any t < R, It < max(0, t− (R− q)); see claim 3.1) is

less than or equal to the probability of failing the review if the customer can pass the review only at the end

of period R but can fail the review before period R if for any t < R, It < max(0, t − (R − q)). To see

this, choose some history that results in the review been passed in the former review specification but failed

in the latter specification.

If the customer discloses in good faith then the probability of failing the review if the customer can

pass the review only at the of period R but can fail the review before period R if for any t < R, It <

max(0, t− (R − q)) equals the probability of failing the review when the customer is evaluated at period

R only and is said to fail the review if IR < qand pass the review if IR ≥ q. To prove this consider:

if a history occurs such that IR < q, then it follows that there exists a history for the game such that

It < max(0, t− (R− q)). Conversely, if there exists a history such that It < max(0, t− (R− q))), then

It+(R−t) < L(t) + (R− t) = q. However, It+R−t ≤ It + (R− t) < q. Thus IR < q.

We now show that the characterized equilibrium yields almost full revelation of the customer’s private

information. Since the review phase may terminate earlier than at the end of period R, the following sufficient

conditions are imposed to obtain a lower bound on the customer’s normalized discounted expected profit:

assume that during the review phase the customer reports in good faith but if he fails, then he fails the review

at the earliest possible period, i.e. t = R− q; and assume that the customer reports in good faith and if he

passes, then he passes the review at the earliest possible time period denoted as t0; recall that v(I0) ∈ [v0, vc]

and t0 > 0(as shown in claim above). The Markovian Property of this repeated game yields a lower bound

on the customer’s normalized discounted expected profit, v(I0):

v (I0) ≥"(1− δ)

R−qXi=1

δi−1vc + δR−q¡1− δK

¢v0 + δR−q+Kv (I0)

#λ

+

"(1− δ)

t0Xi=1

δi−1vc + δt0v (I0)

#(1− λ) .

Solving for v(I0)yields:

36

v (I0)≥¡1− δR−q

¢vcλ+ δR−q

¡1− δK

¢v0λ+

¡1− δt

0¢vc (1− λ)

1− δR−q+Kλ− δt0 (1− λ). (17)

Substitute for K = µRand observe that limR→∞

KR−q =

µ1−S . Since t0 > 0, it follows that the left-hand

side of the inequality in expression (17) converges uniformly in R as δ approaches 1 to

³vc + K

R−qv0´λ+ t0

R−qvc (1− λ)

R−q+KR−q λ+ t0

R−q (1− λ).

Lemma 4 shows that the probability of failing the review when the players use the specified review strategies

is less than or equal to the probability of failing the review when the customer reports in good faith and is

evaluated at period R only and is said to fail the review if IR < q and pass the review if IR ≥ q; this upper

bound is denoted λ (IR < R.S − J). If the customer reports in good faith, then IRis a binomial random

variable. Invoking Chebychev’s inequality implies that λ (IR < R.S − J) ≤ RS(1−S)J2

. If J = βRρ, where

β > 0and 12< ρ < 1, then λ (IR < R.S − J) approaches zero as R increases with out limit; since λis

bounded from above by λ (IR < R.S − J), λ→ 0.

Hence, for any ε > 0, there exists a discount factor δ ≥ max (δ0, δ00)(defined as above) such that for

all δ ∈ [δ, 1), there exists a R where R≥ max (R0, R00) (also defined above) such that for all R > R, it

follows that U (I0) > vc − ε. This provides a lower bound on the customer’s expected profits. Moreover,

λcan be made arbitrarily close to zero by choosing R sufficiently large.

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