14.1 Introduction

Supply chains are typically composed of multiple players, each with competing goals. For example, the newsvendor wants to pay a small wholesale cost per unit to the supplier, but the supplier wants a large wholesale cost. If each player acts selfishly, the resulting solution is generally suboptimal for the supply chain as a whole—the total profit earned by the supply chain is smaller than if the players could somehow bring their actions in line with one another. (By “selfishly” we don't mean they're behaving meanly or inappropriately—just that each player naturally acts in his or her own best interest, making decisions to maximize his or her own profit.)

In the past few decades, a great deal of research has studied contracting mechanisms for achieving supply chain coordination—for enticing each player to act in such a way that the total supply chain profit is maximized. The basic idea is that the players agree on a certain contract that specifies a payment, called a transfer payment, made from one party to another. The size of the transfer payment can be determined in any number of ways (and identifying these ways are the focus of much of the research). Many are quite intuitive: For example, the retailer might pay a wholesale price to the supplier but receive a credit for unsold merchandise at the end of the period (like the newsvendor's wholesale price and salvage value). If these mechanisms are designed correctly, then even when each player acts in his or her own best interest, the supply chain profit is maximized.

In this chapter, we return to the newsvendor problem, now considering the newsvendor's supplier as an active player in the game. We will show that under the assumptions studied previously, the newsvendor (whom we'll now refer to as the retailer) does not order enough inventory to maximize the total supply chain profit. We will then introduce a few contract types that coordinate the newsvendor model. The material in this chapter originates from Pasternack (1985) and other sources cited below, as well as Cachon (2003), who reviews many of the basic ideas of supply chain coordination. We first review some important concepts from game theory.

14.2 Introduction to Game Theory

The literature on supply chain coordination draws heavily from game theory. There are many textbooks on game theory, e.g., Osborne (2003); see also the review of game theory as it applies to supply chain analysis by Cachon and Netessine (2004). We will not cover game theory formally here, but it is worth introducing a few terms. A game consists of two or more players (we will assume exactly two). Each player may choose from a set of strategies, and a choice of strategies (one for each player) is called an outcome. For each outcome, there is a payoff to each player.

For example, if there are two players (A and B), each with two strategies (1 and 2), the payoffs might be as given in Table 14.1. Player A's payoff is the first number in the pair, player B's is the second number. If player A chooses strategy 1 and player B chooses strategy 2, the payoff is to player A (a loss) and 2 to player B.

			Player B
			1	2
Player A	1
	2

Note that there is no randomness in this game. The term “outcome” refers to the deterministic result of choices that the players make, not to the result of some random experiment. In the games we will consider, there is also some randomness that determines the payoffs, in which case the “outcome” represents the expected payoffs to the players.

An outcome is called Pareto optimal if there is no other outcome in which both players have higher payoffs, or in which one player has a higher payoff and the other player has the same payoff. For example, in Table 14.1, the outcome in which both players choose strategy 1 is Pareto optimal, since one player can't be made better off without making the other worse off. Pareto optimal outcomes are considered to be “fair” in some sense.

A Nash equilibrium is an outcome such that neither player can change strategies unilaterally and improve his or her own payoff. (Nash equilibrium is named after the mathematician and economist John Nash.) If the players act selfishly, the game will move to a Nash equilibrium. There is one Nash equilibrium in the game depicted above: Each player chooses strategy 2. (You should verify that this is the only Nash equilibrium in the game.) However, this outcome is not Pareto optimal, since both players would be better off if they each chose strategy 1.

(The game in Table 14.1 is an example of a prisoner's dilemma. In a prisoner's dilemma, the Nash equilibrium is different from the Pareto optimal solution, so the players will always find themselves at an undesirable solution (the Nash equilibrium) even though a mutually better solution (the Pareto optimal solution) is available.)

Now suppose that the players entered into the following simplistic contract: At the end of the game, the players will equally split any profit or loss. The resulting payoff structure is given in Table 14.2.

			Player B
			1	2
Player A	1
	2

Now the Nash equilibrium is for both players to choose strategy 1 (neither player has any incentive to change strategies), and this strategy is also Pareto optimal. This is the outcome the players would have preferred in the original game, but acting individually they would never have arrived at that outcome. By introducing a simple contract, the players choose the best solution, even when they act in their own interest.

Notice that the contract does not force any player to choose a strategy other than the one that maximizes his or her outcome. That is, it does not force the players to choose the outcome . It simply restructures the payoffs so that the players want to choose that outcome.

In the supply chain context, we will see that the Nash equilibrium outcome, to which the players would gravitate if acting in their own interest, is generally not Pareto optimal—there are other outcomes that would improve the payoff to both players. The goal of supply chain coordination is to change the structure of the payoffs so that the Nash equilibrium is also Pareto optimal. One important question will be whether, in the resulting Nash equilibrium, both players earn more than they did without the contract. (If not, one party may refuse to enter into the contract.) The goal of supply chain contracts is not to force one player to earn a smaller piece of the pie so that the other player can earn a bigger piece. Rather, it's to make the pie bigger so that both players can get bigger pieces than they hadbefore.

The games presented in Tables 14.1 and 14.2 are called static games because the two players choose their strategies simultaneously (though a player may alter his or her strategy in response to the other player's strategy). Supply chain contracts, however, are a different type of game, namely, a Stackelberg game, in which one player chooses a strategy first and then the other player chooses one. (Stackelberg games are also known as leader–follower games.) The models presented below are based on the newsvendor model, and in these models, the supplier is the leader, setting the parameters of the contract, and the newsvendor is the follower, setting the order quantity.

14.3 Notation

As in the classical newsvendor model, we consider a single‐period model with stochastic demand. Let D be the demand during the period, with mean , pdf f, and cdf F. The retail price (i.e., revenue per unit sold by the retailer) is r per unit. The supplier's production cost is per unit and the retailer's cost is per unit. Note that does not get paid to the supplier—it represents the cost of processing, shipping, marketing, etc., at the retailer. It is incurred when the unit is procured from the supplier, not when it is sold. We assume (otherwise the system cannot make any profit).

Unsatisfied demands are lost (since this is a one‐period model), incurring a stockout penalty of at the retailer and at the supplier. These costs reflect the loss‐of‐goodwill that the parties incur; they do not include the lost profit resulting from a lost sale. This is because the profit is already explicitly calculated in this model, so including lost profit in and would double‐count this penalty. (Similarly, see the explicit formulation of the newsvendor problem in Section 4.3.2.4.) For convenience, we let and . Each unsold unit at the retailer at the end of the season can be salvaged for a salvage value of v per unit, with . The retailer's order size is denoted Q.

The notation is summarized in Table 14.3.

D
r	selling price
,	supplier's, retailer's per‐unit cost
c
,	supplier's, retailer's loss‐of‐goodwill cost
p
v	salvage value
Q	retailer's order size

14.4 Preliminary Analysis

The following sequence of events occurs in the game:

1. The supplier chooses her ¹ contract parameters. (Each contract type has its own parameters.)
2. The retailer chooses his order quantity Q and places his order to the supplier; the order arrives immediately.
3. Demand occurs; as much as possible is satisfied from inventory, and the rest is lost.
4. Costs are assessed and transfer payments are made between the players.

The transfer payment depends on the type of contract, several of which will be explored below. Note that we are assuming that the supplier offers the contract to the retailer—that the supplier is the powerful player in the market. This is not necessarily the case, and other models have explored the newsvendor problem when the retailer is the powerful player.

Our first goal is to formulate the supplier's and retailer's expected cost as functions of Q. To that end, let be the expected sales as a function of Q:

where is the complementary loss function. The second equality follows from (C.5), while the third follows from (C.13). Then letting ,

(14.2)

by (C.16). Let be the expected inventory on hand at the end of the period:

(14.3)

(The third equality follows from (C.14).) Finally, let T be the expected transfer payment (whose size is yet to be determined).

The retailer's expected profit function is then

is basically just a newsvendor cost function, written in a very different way—maximizing profit rather than minimizing cost (but the two are mathematically equivalent) and writing the expectations using the functions , , and . The supplier's expected profit function is

The supply chain's total expected profit function is therefore

(14.7)

Let's find the order quantity that maximizes the total supply chain profit.

14.8

is a maximizer, not a minimizer, because

so is concave.

Equation 14.8 agrees with our previous results from the newsvendor model. In particular, if we think of the supply chain as a whole acting as the newsvendor, with per‐unit cost c, sales price r, penalty cost p, and salvage value v, then the newsvendor has costs and , and . From (4.17), the optimal newsvendor order quantity satisfies

or

The question now is, does the retailer choose as his order quantity? And, is this also the order quantity that the supplier prefers? That is, if and maximize 14.5 and 14.6 (respectively), then does ?

The supply chain is considered coordinated if . A contract type is said to coordinate the supply chain if there exist contract parameters such that and the players each earn positive profit. If the optimal order quantities coincide but one player earns a negative profit, the player's willingness to enter into the contract depends on a number of factors, such as the player's profit under the status quo (which could, after all, be even more negative), the other business relationships the players may jointly have, the players' relative levels of power, and so on. We ignore these rather messy issues and focus below on determining which contract types are guaranteed to have parameters such that the supply chain is coordinated and the players both earn positive profits.

14.5 The Wholesale Price Contract

The simplest possible contract is the wholesale price contract, in which the retailer pays the supplier a given cost w per unit ordered. This is identical to settings we've discussed previously, in which the retailer pays a per‐unit purchase cost that goes to the supplier, except now the purchase cost is the supplier's decision variable. For a given wholesale cost w, the transfer payment is given by

The subscript w identifies the type of contract, while the arguments specify the two decision variables—order quantity and wholesale cost—one per player.

The retailer's and supplier's expected profits are both functions of w and Q:

14.9

(14.10)

The supply chain is coordinated if there exists a value of w such that , where , , and are the order quantities that maximize , , and , respectively.

It is straightforward to show that and are both concave functions of Q (assuming that w is fixed). Therefore, and satisfy:

(14.11)

(14.12)

The next theorem demonstrates that there exists a value of w such that . However, for this value of w, the supplier earns a negative expected profit.

From 14.12 one can show (see Problem 14.1) that if , then is strictly decreasing in Q; if , then is first increasing and then decreasing; and if , then is strictly increasing. Thus, for sufficiently large w, since the supplier earns a positive margin on items sold to the retailer, and she pays no penalty for overage at the retailer. For moderate values of w ( ), is finite: Although the supplier earns a negative margin on each unit sold to the retailer, she still prefers a nonzero order quantity since small order quantities cause stockouts, for which the supplier incurs a goodwill cost. Her expected profit is still negative, but minimizes the losses. Finally, for small values of w, the supplier's margin is so negative that it more than offsets the goodwill cost, and she sets .

The phenomenon evident in Theorem 14.1 is known as double marginalization (Spengler, 1950): When both players add their own margin (markup) to their costs, the supply chain is not coordinated since the players ignore the total supply chain profit when making their individual decisions. If, on the other hand, the retailer has a positive margin but the supplier has a negative one, the supply chain is coordinated. However, the supplier clearly would not enter into this arrangement, so the wholesale price contract is not considered to be a coordinating one. Nevertheless, there are still several interesting things to say about it.

Suppose that , so that the supplier earns positive profit but the supply chain is not coordinated. We first examine the retailer's and supplier's optimization problems and then discuss how close the wholesale price contract comes to coordinating the supply chain.

Theorem 14.2 says that, assuming the supplier earns a positive profit, the retailer will under‐order. This happens because the retailer is absorbing all of the risk of overage, but only part of the risk of underage (since the supplier pays a stockout penalty). Therefore, the retailer orders less than the supplier (and the supply chain as a whole) wants him to. In the contracting mechanisms discussed in later sections, the supplier absorbs some of the risk of overage, thus giving the retailer the flexibility to increase his order quantity. If the contract parameters are set correctly, he'll increase it so that it equals .

Now let's turn our attention to the supplier's optimization problem. For given w, 14.12 determines the supplier's optimal order quantity. But what if the supplier could choose whatever w she wants? In order to choose w, she must anticipate the Q that the retailer will choose for each value of w. Put another way, the supplier can entice the retailer to choose whatever Q she wishes by selecting the unique wholesale price, call it , that makes Q optimal for the retailer. In particular,

(14.14)

(from 14.11).

Which Q does the supplier want the retailer to choose? The supplier's profit function is now a function of Q and the corresponding :

(14.15)

The supplier wishes to maximize this function. The question is, does it have a unique maximum? If the function is strictly concave, it does, but in general is not concave. But it is usually close enough. Before we explain further, we need to introduce a new property that is important in contract analysis: Demand distributions for which is increasing are called increasing generalized failure rate (IGFR) distributions (Lariviere and Porteus, 2001; Lariviere, 2006; Banciu and Mirchandani, 2013). Many common distributions are IGFR, including normal, exponential, and gamma. Only a few families of distributions are not IGFR; these include Gumbel and generalized logistic (Paul, 2005).

In the next theorem, we show that if the demand distribution is IGFR, then is unimodal—that is, there is a value such that the derivative of is positive to the left of and negative to the right. This implies that the function has a unique maximum. Unimodality (sometimes called quasiconcavity) is similar to strict concavity but is weaker: Every strictly concave function is unimodal, but not every unimodal function is concave.

In what follows, we will assume that the demand distribution is IGFR. Hence is unimodal, and there is a unique order quantity that maximizes the supplier's profit. (Note that this is the order quantity the supplier would choose if she can also choose w. This is not the same as as defined in 14.12, which is the supplier's optimal quantity for fixed w.) Of course, the supplier does not set this order quantity directly; she sets w to and waits for the retailer to set the order quantity to .

For contracts such as the wholesale price contract that do not coordinate the supply chain, we'd like to know how close they come. This is measured by the efficiency of the contract: the proportion of the optimal supply chain profit attained by the Nash equilibrium order quantity, or . The greater the efficiency, the closer the contract is to achieving coordination. Another important measure is the supplier's profit share: the percentage of the total profit captured by the supplier, or . Both players would like the efficiency to be high, but only the supplier would like the supplier's profit share to be high. Experimental tests using the power distribution show the efficiency to be around 75% and the supplier's profit share to be in the range of 55–80% (Cachon, 2003). Perakis and Roels (2007) examine the efficiency of the wholesale price contract in a variety of settings.

It is worth noting that the wholesale price contract is considered to be noncoordinating because there is no value of w that (1) makes and (2) guarantees both players positive profits. In contrast, the contract types we discuss in the next sections are considered to be coordinating because there always exist some values of the contract parameters for which both (1) and (2) hold, even though not all parameter values do the trick.

14.6 The Buyback Contract

We now examine a contract type that does coordinate the supply chain. In the buyback contract (Pasternack, 1985), the supplier charges the retailer per unit purchased, but pays the retailer b for every unit of unsold inventory at the end of the period:

We assume

(14.18)

otherwise it is better for the retailer not to sell an on‐hand item to satisfy a demand (thereby earning but paying ) than to sell it (earning r). We also assume

(14.19)

otherwise, the retailer incurs no overage risk since his revenue for salvaging an item ( ) is more than what he paid for it ( ).

The name “buyback” is a little misleading, because usually the retailer does not physically return the products, he just receives a credit from the supplier. The supplier is offering to share some of the risk of overage with the retailer in exchange for higher supply chain profits.

Many suppliers offer buyback credits as a way to prevent the unsold goods from being sold at a steep discount. For example, high‐fashion clothing makers don't want to see their products on the bargain rack at Marshall's at the end of the season, so they give high‐end retailers a credit to prevent them from unloading unsold merchandise to discounters.

Letting in 14.5, the retailer's profit function becomes

The Q that maximizes the retailer's profit satisfies

Now, the supplier can, of course, choose any values for and b (subject to 14.18 and 14.19). However, it turns out that for a given b, the “correct” value of (i.e., the one that will coordinate the supply chain) is given by

(14.22)

(It should not yet be obvious how we get this value of .) Note that is increasing in b (since ). Therefore, in exchange for receiving a more generous buyback credit, the retailer must pay a higher wholesale price.

If w and b satisfy 14.18 and 14.22, then they also satisfy 14.19. (See Problem 14.7.) Therefore, we can ignore 14.19 and assume only that the feasible region for b is .

Theorem 14.4 establishes that, if the parameters are set appropriately, then the optimal order quantities coincide. Moreover, by Theorem 14.5 below, there exists a b such that both parties earn positive profit. Therefore, the buyback contract coordinates the supply chain.

We'll present two proofs of this theorem. The first is more straightforward than the second, but the second is quite elegant and can also be applied in more general settings.

Before introducing the second proof, we can answer the question of how to determine (if it wasn't already given by 14.22): It's the only value of that makes the conditions and both reduce to . The value of can be “backed out” from these conditions.

In general, we can use the approach from Proof #1—setting and to 0 and showing that —to prove that a given contracting mechanism coordinates the supply chain. However, there's another elegant way to accomplish this, and this approach is taken by Proof #2.

As increases, the retailer's profit increases and the supplier's profit decreases, so in a sense represents the division of profit between the players. One would like to know whether any division is possible—that is, is there some value of such that the supplier captures all of the profit and another value such that the retailer captures all of the profit? (Keep in mind that is not a parameter of the contract—the supplier does not actually choose . But by choosing b, the supplier automatically chooses given 14.25.) If so, then there is also a value that gives any desired mix. As the next theorem demonstrates, this is indeed possible.

At first it may seem surprising that the supplier's profit is increasing in b, since b is a payment made to the retailer. However, is increasing in b, and increases in the buyback credits paid to the retailer are more than offset by increased revenue from the wholesale price.

One consequence of Theorem 14.5 is that, for any noncoordinating contract, there exist b and such that neither player has a lower profit under the buyback contract, and at least one player has a strictly higher profit. Therefore, the supplier can always choose b and such that both players prefer the buyback contract to the status quo if the supply chain is not currently coordinated.

Which value of b will she choose? We can't solve this as an optimization problem as we did for the wholesale price contract because the supplier's profit is an increasing function of b. Left to her own devices, she would choose a large b that gives the retailer negative profit. Instead, the choice of b is the result of some sort of negotiation process that reflects the relative power of the two players as well as other factors, which we ignore. The contract types discussed in the following sections are similar in this regard.

14.7 The Revenue Sharing Contract

In the revenue sharing contract (Cachon and Lariviere, 2005), the supplier charges the retailer a wholesale price of per unit and the retailer gives the supplier a percentage of his revenue. All revenue is shared, including both sales revenue and salvage value. Let be the fraction of revenue the retailer keeps and the fraction he gives to the supplier. The transfer payment is then

14.33

Again, we magically determine the “correct” value of one contract parameter ( ) for a given value of the other ( ):

(14.34)

If we define in this way, then the supply chain is coordinated. The next theorem demonstrates this; its proof uses a method similar to Proof #2 in Section 14.6, but it could also be proven using a method similar to Proof #1 (see Problem 14.13).

The retailer's profit is increasing in and the supplier's profit is decreasing in . Since is increasing in , the retailer's profit increases and the supplier's profit decreases as the retailer's revenue fraction increases. One can prove a theorem that is analogous to Theorem 14.5 demonstrating that any allocation of profits is possible under the revenue sharing contract. In particular, the retailer earns the entire profit if

(14.40)

and the supplier earns the entire profit if

(14.41)

(Note the similarity to 14.31 and 14.32.)

Revenue sharing and buyback contracts are actually quite similar. For the sake of clarity, denote the wholesale price under the buyback contract and the revenue sharing contract as and , respectively. We can think of a buyback contract as requiring the retailer to pay per unit purchased and an additional b per unit sold. (This is equivalent to paying per unit purchased and receiving b per unit unsold.) In a revenue sharing contract, the retailer pays per unit purchased and per unit sold. Then a revenue sharing contract with parameters and is equivalent (in the sense that it generates the same profits no matter what the demand is) to a buyback contract if the parameters of the buyback contract satisfy

that is,

(14.42)

(14.43)

However, in more complicated settings (for example, with more than one retailer), the two contracts are not equivalent.

14.8 The Quantity Flexibility Contract

We next introduce the quantity flexibility contract (Tsay, 1999). The quantity flexibility contract is similar to the buyback contract in that the retailer pays a wholesale price per unit purchased and the supplier reimburses the retailer for unsold goods. The difference is that, in the buyback contract, the supplier partially reimburses the retailer for every unsold item, whereas in the quantity flexibility contract, she fully reimburses the retailer for a portion of his unsold items.

In particular, the quantity flexibility contract has two parameters, and ( ). The retailer pays the supplier per unit ordered, and the supplier pays the retailer , where I is the on‐hand inventory at the end of the period. Thus, the supplier agrees to protect the retailer against only a portion of his order: She will reimburse him for his losses on unsold merchandise (which equal per unit), but only up to a maximum of units.

The quantity flexibility contract coordinates the supply chain from the retailer's end (his optimal order quantity is also the supply chain's optimal order quantity). However, unlike the contracts in Sections 14.6 and 14.7, the quantity flexibility contract only coordinates the supplier's decision for certain values of the parameters.

The transfer payment in the quantity flexibility contract is

(14.44)

(14.45)

(See Problem 14.10).

Let be defined as

(14.46)

We also need to check whether is optimal for the supplier; if it is not, the contract does not coordinate the supply chain. The supplier's profit function is

(14.50)

One can show that does equal 0 when . (See Problem 14.11.) However, for to be a maximizer, the second partial derivative must be nonpositive. This derivative is

(14.51)

Unfortunately, this expression is not always nonpositive. For example, suppose , , , and . If , then 14.51 equals (so is a local max), but if , then 14.51 equals (so is a local min).

All of this means that the quantity flexibility contract coordinates the supply chain from the retailer's point of view but not necessarily from the supplier's. In other words, when the retailer places an order of size , the supplier might wish to deliver an order of a different size . The supplier can certainly not force the retailer to accept a larger order than he placed, so if there's nothing the supplier can do about it. But if , the supplier wants to deliver an order smaller than the order placed by the retailer.

The attitude of the model toward this behavior is called the compliance regime. If the supplier is allowed to deliver less than the order size, the regime is called voluntary compliance. If the supplier is forced to deliver the entire order (because failing to do so would expose the supplier to a court action or to too much loss of goodwill, for example), it's called forced compliance. Since the supplier's optimal decision was also supply chain optimal in the coordinating contracts we've studied so far, the two regimes have been equivalent—the supplier wants to comply, even if she's not forced to do so. In the quantity flexibility contract, the supplier may not voluntarily comply.

Assuming that the supplier complies (either because she is forced to or because the parameters are such that her profit function is concave), the quantity flexibility contract, like the others, can allocate the profits in any way we like. (See Problem 14.12.)

PROBLEMS