1 Introduction
When demand is uncertain, the characteristics of the supply chain and the contract arrangements will determine investment decisions. In capitalintensive industries, such as petrochemical, electronics and semiconductors, manufacturers need to invest heavily in building capacity and the lead times are long (Kleindorfer and Wu, 2003), so that it remains a critical challenge to incentivize and manage capacity investment. When manufacturers are competing against each other and a buying firm may switch to a different manufacturer, the problems are intensified. If the financial risks from investment are carried by the firms who build capacity, it is difficult to find good solutions as there is no certainty on how much capacity will be required in the future (Wu et al., 2005a). The result may be that capacity is only built for the part of the demand that can be more or less guaranteed: this conservative approach will reduce the buyer’s ability to meet customers’ demand. From a supply chain perspective, it is usually preferable for buyers to reserve capacity from suppliers in advance, with a payment made to the suppliers associated with this capacity reservation. Then the buyer also shares the risk that demand is low and not all the purchased capacity is required.
We model capacity reservation in a supply option framework. In the first stage, before knowing the actual demand, a buyer reserves a certain amount of capacity by paying a reservation price. In the second stage, after discovering the actual demand, the buyer asks for supply up to the lesser of the reserved amount and the observed demand. At this stage, the buyer pays an execution price only for the amount of capacity that is used. The underlying assumption of this model is that suppliers have to install capacity, or place orders with their own suppliers, before demand materializes because of the lead times involved.
A distinctive feature of our model is that marginal costs (of capacity and production) may depend on the volume, which is often the case in practical situations. For example, capacity investment often involves a oneoff setup cost (Luss, 1982; Van Mieghem, 2003), there may be (dis)economies of scale in production (Haldi and Whitcomb, 1967; Ha et al., 2011), or in cases where a supplier manages a portfolio of facilities with heterogeneous technologies the overall cost is far from being linear (especially in the electricity industry). In our model, each supplier has a total reservation cost (incurred before demand realization), which is an arbitrary increasing function of the capacity reserved, and also a total execution cost (incurred after demand realization), which is an arbitrary increasing function of the amount actually supplied. This framework includes constant marginal costs as a special case, so our model is flexible enough to capture many practical settings in terms of cost modelling. We believe this is an important theoretical advancement as most literature on supply chain contracts assumes constant marginal costs (Cachon, 2003). The linear treatment may help simplify the analysis and develop highlevel intuitions especially for competitive models, but may not align with realworld practices.
With constant marginal costs, it is plausible to focus on simple contract forms, but with general cost functions, more sophisticated contract formats may be worthwhile, such as bids which specify a schedule of different prices for different quantities (i.e., a supply function). This type of supply function bid often occurs in practice through the application of some form of quantity discount contract.
In this paper, we consider a supply chain with a single buyer and multiple suppliers, who supply a homogeneous item to the buyer. The suppliers compete by offering price functions for both reservation and execution. Given the suppliers’ bids, the buyer first reserves capacity before knowing the actual demand, and then decides how much capacity to use after observing the demand. The buyer may choose not to enter into a contract with a specific supplier in which case no costs are incurred by that supplier. Competition with function bids fits well with the situation where a buyer does not stipulate the specific bidding format in its Request for Quote (RFQ), so that suppliers can bid in whatever format they like.
We suppose that the buyer faces a random demand, with a known distribution. There are suppliers, and each has a total reservation cost, which is expressed as a general increasing function of the amount reserved, as well as a total execution cost which is a general increasing function of the amount supplied. We model this game in a Stackelberg framework, where the suppliers are leaders and the buyer is a follower. Each supplier has complete information about the buyer’s demand distribution and all supplier costs, but the buyer may not know the suppliers’ costs. This assumption has been widely used in supplier competition models and is appropriate for industries where the operating environments are more transparent and/or the production technologies are more mature such as electricity, electronics and semiconductors (Wu and Kleindorfer, 2005; Martínezde Albéniz and SimchiLevi, 2009; Anderson et al., 2017).
The sequence of events is depicted in Figure 1. First each supplier, with the aim of maximizing their own expected profits, offers a function bid consisting of a (marginal) reservation price function and a (marginal) execution price function to the buyer. Second, prior to knowing the actual demand, the buyer determines how much to reserve from each supplier and pays the reservation price. After demand is observed, the buyer chooses what capacity to use and pays the execution price for the amount of capacity that is used. If the demand exceeds the total amount of reserved capacity, there will be lost sales. Finally, the buyer sells the product to the consumer market at the retail price . We will call this the capacity game with function bids. This game can be considered as a special case of a general capacity game setting, in which a first stage choice by the buyer implies a constraint on the decisions to be made by the buyer at a second stage when demand information becomes known. Thus, our discussion will be carried out for the general capacity game, which in turn provides a theoretical foundation for the capacity game with function bids.
The framework we consider also applies to capacity mechanisms in electricity markets. In this context the buyer and customers are replaced by a set of consumers and a system operator who aims to maximize system welfare. Problems of this form are usually formulated with a value of lost load (VoLL). In the case where demand is stochastic but independent of price, the system operator’s problem, of minimizing the expected cost of procurement less VoLL for demand not met, becomes equivalent to maximizing VoLL times (demand met) minus the cost of procurement, which is the same as the buyer’s problem in our supply chain setting when we set the retail price to be VoLL.
It is not uncommon to have a supply chain where multiple suppliers act as Stackelberg leaders and a buyer must select amongst the available suppliers on the basis of contracts that the suppliers offer. This framework will lead to suppliers setting as high a price as they can while still being chosen. The consequence is that each supplier makes a bid that leads to the buyer being indifferent between choosing or not choosing the supplier. From this it can be shown that the profit available to each supplier is simply its contribution to the overall supply chain profit, i.e., the difference in the supply chain profits when it is chosen and when it is not.
A major contribution of this paper is to study submodularity in this setting. We will demonstrate that a natural sufficient condition for a wellbehaved equilibrium is that the supply chain profits are submodular in terms of the set of suppliers available. This property is equivalent to a requirement that the profit available to a supplier cannot increase if a new supplier is added to those available. We are therefore led to considering whether or not the supply chain profits achieved are submodular. This is not always the case, but can be established in commonly studied settings. Specifically, the property holds if either: (i) there are only two suppliers; or (ii) each supplier’s marginal reservation cost is nondecreasing and marginal execution cost is constant.
A second contribution of this paper is to study how suppliers compete with each other by using supply functions rather than scalar prices. In other words, the strategy space of each supplier is extended to a function space, and as a result, the suppliers have more flexibility in their bidding decisions. On the other hand, the buyer can determine how much to reserve from each supplier (i.e., pick any point from a supply function) rather than being restricted to a fixed quantitypayment contract. Our aim is to develop an understanding of the players’ strategic behavior in such a market setting. Previous models have focused on competition problems when suppliers are restricted to a simple strategy space (i.e., scalar decision variables, including prices, quantities, lead times, quality, or specific contract types). To the best of our knowledge, this paper is among the first to study supply function competition in a capacity reservation setting.
We find that given knowledge of the other suppliers’ bids, it is optimal for each supplier to set execution prices at execution costs and make profits only through the buyer’s reservation payment. Our findings also show that, under some reasonable assumptions, there exists a continuum of equilibria in which the suppliers charge costs only for the execution prices but add a margin to the reservation costs. Despite the multiplicity of equilibrium bidding strategies, the equilibrium outcome is essentially unique: in equilibrium the buyer makes a reservation choice that maximizes the overall supply chain profit, each supplier makes a profit that is equal to their marginal contribution to the supply chain, and the buyer takes the remaining profit.
The rest of this paper is organized as follows. After a review of the relevant literature in Section 2, we present our general capacity game model in Section 3, and study the best response and equilibrium strategies for suppliers in Sections 4. Then we show how these results apply to the more specific case we have introduced above in Section 5. We discuss in Section 6 two commonly studied settings and show that the supply chain optimal profit is submodular in a broad class of capacity games with function bids. Finally, we conclude in Section 7. All technical proofs are presented in the appendix.
2 Literature review
Supply options have been extensively studied in the operations management literature (see, e.g., BarnesSchuster et al., 2002; Burnetas and Ritchken, 2005; Wu et al., 2005b; Fu et al., 2010; Secomandi and Wang, 2012). An initial step is to investigate a buyer’s optimal purchasing decision given a fixed set of supply options (see, e.g., Martínezde Albéniz and SimchiLevi, 2005). As an extension of this, several papers examine option contract design problems in a Stackelberg game between a buyer and a supplier with a focus on the interaction between option markets and spot markets (see, e.g., Wu et al., 2002; Pei et al., 2011). Further extensions have been made to consider supplier competition in an option market (Wu and Kleindorfer, 2005; Martínezde Albéniz and SimchiLevi, 2009; Anderson et al., 2017). Wu and Kleindorfer (2005) study a commodity procurement problem where an industrial buyer purchases supply options from a small set of suppliers, while having access to an open spot market. No reservation cost at the suppliers is considered, and the buyer’s downwardsloping demand (arising from utilization maximization) is determined by the spot price and the execution prices. Such a demand modelling approach allows the buyer to rank suppliers by using a single price index combining the reservation and execution price of supply options, which leads to a Bertrand type of equilibrium for the suppliers. We study a similar situation but consider suppliers having a twodimensional cost and a newsvendor type buyer facing uncertain demand.
The closest papers to ours within this literature are Martínezde Albéniz and SimchiLevi (2009) (hereafter, “MS”) and Anderson et al. (2017) (hereafter, “ACS”). MS studies a setting where marginal costs are constants and each supplier chooses a reservation price and an execution price for their limitless capacity. They show that there may be efficiency loss in equilibrium, which is up to of the overall supply chain optimal profit. Our paper differs from MS by considering general cost functions and allowing suppliers to submit function bids. By enlarging the strategy space of suppliers, we find a relatively clean and intuitive result as discussed earlier. In particular, when the supply chain optimal profit is submodular, there is no efficiency loss in equilibrium and each supplier makes a profit equal to their marginal contributions. We further show that this condition is not restrictive as it holds under some commonly studied settings. Comparing with MS, we also find that allowing suppliers to offer a function bid makes the buyer worse off, thereby highlighting the significant impact of the strategy space on equilibrium outcomes.
ACS studies a similar problem where each supplier owns a block of capacity with the same size and the buyer must reserve all of a block or none. Our general framework of capacity games covers the model by ACS as a special case since we can recover ACS by restricting the buyer’s capacity choice from each supplier to be either zero or the block size. In this respect, our paper generalizes ACS by studying a situation where the buyer can reserve any amount from each supplier and suppliers can charge different prices for different quantities.
This paper studies a situation where suppliers compete with each other by offering a function bid, and this resembles what is studied in the supply function equilibrium (SFE) literature (Klemperer and Meyer, 1989; Anderson and Philpott, 2002; Anderson and Hu, 2008; Johari and Tsitsiklis, 2011). However, the game type is different. Our model is a Stackelberg game with multiple leaders (i.e., suppliers) and one follower (i.e., buyer), and the buyer strategically responds to the supplier offers. In contrast, the SFE literature focuses on Nash games that aim to study the strategic interaction between multiple firms, and this literature does not involve a profitmaximizing buyer. In other words, we examine the buyer’s optimization problem explicitly; while in the SFE literature, the buyer’s problem is to choose a clearing price to equate the total supply with the demand, and each supplier’s best response is characterized by a differential equation. In addition, our capacity games involve the buyer making a twostage decision and each supplier submits a twodimensional bid consisting of a reservation price function and execution price function, which is not considered in the SFE literature.
This work is also related to the multiunit auction literature. Problems of decentralized resource allocation (Klagnanam and Parkes, 2004) are central to auctions. A subset of this literature examines the efficiency and profit allocation of a given type of auction, for example, share auction (Wilson, 1979), menu auction (Bernheim and Whinston, 1986), split award auction (Anton and Yao, 1989, 1992), discriminatory price auction (Menezes and Monteiro, 1995), and uniform price auction (Bresky, 2013). Those papers generally assume that the total purchase amount of a buyer is exogenously given, while in our model it is endogenously determined by the buyer based on the supplier bids. Furthermore, we study capacity reservation games where the buyer’s capacity reservation problem is a twostage stochastic program, which is new to this literature.
Finally, the equilibrium profit allocation obtained in this paper is of VickreyClarkeGroves (VCG) style, so it is instructive to discuss the key point of departure from VCG models (Vickrey, 1961; Rothkopf et al., 1990). Those models study profit allocation from a mechanism design perspective: the VCG profit allocation is an outcome that arises from the payment rule. In essence the VCG approach selects bids that maximize system welfare, and pays according to each bidder’s contribution (i.e., the difference between system welfare with and without that bidder). With this setup the equilibrium result is for bidders to bid truthfully and the auction result is efficient. The model we propose uses the same bid selection approach, but pays as bid. It is then an equilibrium (in the case with submodularity) for the bidders to mark up by the exact VCG amounts. We still get an efficient auction result, but without truthful bids. This is a different approach and has some advantages over the usual VCG mechanism. Hobbs et al. (2000) and Rothkopf (2007) amongst others have discussed some of the difficulties of using a VCG auction in practice. Many of these problems are resolved with payasbid prices. For example, in the auction format we discuss, there is not the same incentive for the buyer to encourage bids that will not be accepted with the aim of reducing the payments to the successful bidders. Note that a submodularity condition has also been discussed in the VCG framework by Ausubel and Milgrom (2006), who show that this condition reduces the problems ordinarily associated with VCG. Within the mechanism design literature, several papers study multidimensional bidding environments in an electricity context, where the focus is on designing scoring and payment rules that incentivize bidders to truthfully reveal their costs (Bushnell and Oren, 1994; Chao and Wilson, 2002; Schummer and Vohra, 2003).
3 Model setup
The problem we study is an example of a broader class of capacity games, in which the suppliers offer bids, then a buyer selects a capacity amount to buy from each supplier, then depending on the outcome of a random variable (the demand) in the second stage, the choice of amounts to buy is made by the buyer (with amounts constrained by the capacity already bought). The revenue to the buyer is a function of demand and the amounts bought. We are not considering risk aversion, so we can use the expected payoff as the objective for the buyer and the suppliers. Supplier
faces costs that are the sum of a capacity cost and an execution cost. We will establish our results in this general framework and then apply this to the model described in the introduction section.The buyer makes capacity decisions , where capacity is restricted to lie in a set , and hence . We assume that each is compact containing and is specified as part of the capacity game. Typically, is a closed interval or , which corresponds to the case where the buyer either does not use this supplier, or uses the full amount of the supplier.
The capacity payment to be made by the buyer to each supplier can depend on the complete set of capacity decisions , which may occur in practice, for example, when the suppliers have some exclusivity clauses. Note that, we are making our model as general as possible, although in the specific applications of our model considered in Section 5, the capacity payment made to a supplier depends only on the buyer’s choice from that supplier. In the second stage, after demand becomes known, there is a set of amounts (execution quantities) chosen by the buyer, where depends on the observed demand and satisfies constraints imposed by the capacity decisions of the first stage. Thus we have for some set . We require that if then . The case that is most natural is to allow any selection at the second stage that is no more than the capacity purchase at the first stage, so that and . In general, however, we only assume is compact. Then supplier decides on two payment functions and that the buyer will be offered, where is the payment made to supplier when the buyer makes a capacity reservation of , and is the payment made to supplier given execution amounts . In the formulation of the game there may be restrictions on the bids allowed, which we capture by specifying an allowed set of functions , , and restrict the supplier’s choice to and . Since the buyer can decide not to include supplier , in which case no payment is made, the feasible bids have the property that for all , whenever (similarly for ). For maximal generality of our model, we specify only the minimal further restrictions on at the end of this section.
Besides payments made by the buyer to the suppliers, we also have costs and incurred by supplier that may depend on the complete set of capacities purchased and amounts executed. This occurs when competing players can partially collaborate (e.g., they share warehouse facilities or transport links). However, in the capacity game, a supplier’s costs may be independent of other suppliers’ quantities, as in our specific applications considered in Section 5. We assume that these costs are zero if supplier is not used at either stage and hence when , and when .
Finally, we have the expected revenue to the buyer from an external source, which depends on the demand and the execution amounts . Rather than include a restriction that the total execution amounts are no more than the demand (i.e., ), we will assume that this is achieved through setting appropriate values for the revenue function .
Example 1.
We consider an example that illustrates the range of modelling applications for the general capacity game. Consider a buyer facing uncertain demand:
with probability
and with probability . There are two suppliers with identical cost structures who produce products that are fully substitutable. We take . Each supplier has costs of reservation for an amount of capacity given by with an additional fixed cost of $20 associated with building a transport link. This fixed cost is split between any suppliers who have capacity reserved, so it is $20 if just one supplier is used, and otherwise $10 each. Thus the reservation costs arewhere the indicator function if , and otherwise. Execution costs are $1 per unit so that . Customers pay an amount $3 per unit and in addition there are fixed costs at the buyer of for each product that apply only if that product is supplied. Thus, . We have the standard form for so there is a restriction that .
Since execution amounts will depend on , we can write a policy for the buyer as , where and is a function from the demand set to , taking the value at . For simplicity, we will write for hereafter. The total expected buyer profit with this policy choice is
given a set of supplier bids . The buyer’s problem is to maximize her expected profit by choosing an optimal reservation choice and a set of execution amounts for each possible demand , i.e., to solve
(1) 
With these bids and buyer choice , supplier has a profit of
(2) 
The supply chain profit is
(3) 
which is independent of the bids made.
We write
for the support of a vector
. We use , , to denote the optimal supply chain profit when the capacity reserved is restricted to be zero outside the set . Thus(4) 
Similarly, given a set of supplier bids , we use , , to denote the optimal buyer profit when the capacity reserved is restricted to be zero outside the set . Thus
The maximizations above are taken over (a vector) and (a function). To ensure that the maxima exist and are attained, we make the following (additional) assumptions: (a) the setvalued function is upper hemicontinuous; (b) the revenue function is upper semicontinuous in both arguments; and (c) the set of possible bids is such that are lower semicontinuous.
4 Best response and equilibrium behavior
We now look at each supplier’s best response problem. As a Stackelberg leader, each supplier is able to anticipate the buyer’s reservation choice provided that the competitors’ bids are observed. Since the buyer’s optimization problem is embedded in the suppliers’ best responses, we need to specify the buyer’s choice when there are different options with the same expected value to the buyer. We say that supplier has preferred status when a solution with is chosen by the buyer even if other options give the same value.
Theorem 1 (Best Response).
For any fixed supplier , we have the following statements for the supplier’s best response:

Given bids , the profit for supplier is no more than
where .

If and supplier has preferred status, then the profit is achieved by the offer defined by:

If , then for any , an offer of will achieve within of the maximum possible supplier profit, where .
Theorem 1 shows the maximum profit for a supplier , which is the increase in profit available to the buyer from including the bids of supplier when these are made at cost. Moreover, when optimizing for the supplier, it is sufficient to consider supplier bids that are at cost for the execution component and make money only from the reservation payments. However, we should note that the expected profit to the supplier is unaffected by parts of the function bids that are never selected by the buyer whatever demand occurs. The consequence is that there are a continuum of other best response function offers available.
Now let us consider the equilibrium strategies for suppliers. In general, the optimal solution to the buyer’s maximization problem (1) is not unique at an equilibrium as we shall show later. Since our focus is on supplier competition, and each supplier’s payoff depends on the buyer’s reservation choice, we need a tiebreaking rule to pin down the buyer’s optimal choice in case of multiple solutions. The suppliers have an interest in raising prices to the point where the buyer is about to drop them from consideration. Therefore, a tiebreaking assumption is critical, otherwise, we can have a difficulty in defining optimal behavior for the suppliers, as a type of optimality could occur when a supplier sets his price just below a benchmark value at which the buyer no longer wishes to select the supplier. Consequently, we make the following assumption.
Assumption 1 (TieBreaking Rule).
In case of multiple optimal solutions, the buyer will select the one that gives the largest supply chain profit.
This assumption is in line with that for the classic Bertrand competition model, where the firm with the lowest cost wins when all the firms charge the same price. The rationale of this assumption is as follows: if the buyer selects an alternative that gives a lower supply chain profit, then keeping the other suppliers’ bids unchanged, any supplier in the supply chain optimal set (with respect to the buyer’s equally good choices) will have incentives to raise his price by an amount that is smaller than the difference between the supply chain optimal profit and the one corresponding to the buyer’s choice. This would make both the buyer and the supplier better off.
We need to rule out cases where two suppliers are identical and the optimal supply chain solution can use either one or the other, so we introduce the following assumption.
Assumption 2 (Uniqueness of Support).
All the supply chain optimal capacity choices use the same set of suppliers.
We will write to denote an optimal solution to the supply chain optimization problem (4). So Assumption 2 amounts to a statement that the support of the supply chain optimal capacity choices is unique. Let us identify the suppliers who make a contribution to the optimal supply chain profit by defining
It is easy to observe that, under Assumption 2, we have
(5) 
To establish an equilibrium, we will need to make use of submodularity of the optimal supply chain profit as a set function of the suppliers available, where the submodularity property states that, for any and ,
(6) 
Inequalities (6) are satisfied for a large class of problems of interest, as we shall discuss in detail in Section 6. It is convenient to make this property an assumption:
Assumption 3 (Submodularity).
The supply chain optimal profit is submodular as a function of the set of available suppliers.
This submodularity assumption implies two properties of the set function , which are established in the following two lemmas.
Lemma 1.
Under the Submodularity Assumption, the following inequality holds for any :
(7) 
The following result shows that all the suppliers in will still be included in the optimal supply chain choice when supplier is unavailable.
Lemma 2.
Under the Submodularity Assumption, for any , we have .
We will consider the possibility of varying the offers that a supplier makes in a way that does not change the supply chain optimal solutions. We say that a set of functions from to is consistent with the cost functions if adding to the cost does not change the optimal solution or the optimal value, i.e., for every ,
(8) 
and the maximization is attained at . It is easy to see that this is equivalent to the conditions of (a) and (b) if and for all , then
(9) 
Note that when is zero for any , then it is trivially consistent with the costs. Now we are ready to characterize the equilibrium that occurs when is submodular as a function of .
Theorem 2 (Nash Equilibrium).
We can see immediately from Theorem 2 that the following set of bids is a Nash equilibrium (by setting all ):
and when .
Several points about this theorem are worth mentioning. First, the profit allocation in equilibrium is a VCG result. However, this is not from a mechanism design that sets prices paid in a particular way, but arises as an equilibrium from our payasbid capacity game. It is straightforward that each supplier makes a nonnegative profit. Suppliers in each make a strictly positive profit while the other suppliers each make zero profits. On the buyer’s side, the profit is nonnegative, which is a direct result of the submodularity property of and can also be implied by the fact that the buyer will not purchase from any supplier should it make a negative profit.
The profit allocation is primarily driven by the level of competition between suppliers. In the extreme case where there is a perfectly competitive market, then each supplier’s contribution is zero, since in an individual supplier’s absence another supplier will step in with no overall reduction in supply chain profits. Thus the entire supply chain profit will go to the buyer.
Another property of the equilibrium is that the buyer’s profit will remain the same if any single supplier is dropped from the set of available suppliers. We establish this in the proof of Theorem 2 (see equation (A3)). The intuition is that if the buyer makes a lower profit when any supplier is removed, then this supplier would have an incentive to increase its bid by a positive amount while still ensuring it is chosen by the buyer.
Finally, in equilibrium the buyer makes a choice that maximizes the supply chain profit, i.e., the supply chain is coordinated in equilibrium. It is instructive to relate our findings with those of the supply chain coordination literature (Cachon, 2003). In general, this literature focuses on designing sophisticated contracts (e.g., revenue sharing or buy back) to achieve the chain optimal profit in a dyadic supply chain. In contrast, we have a different supply chain structure (i.e., many to one), and the supply chain optimality arises as a result of supplier competition (rather than by way of design).
Example 1 (continued).
Let us consider again the capacity game given in Example 1. If just one of the suppliers, say , is available, then it can be shown that the optimal reservation amount for the supply chain as a whole is and the supply chain profit is . In this case, the amount supplied is less than the demand in the high demand case. On the other hand, if both suppliers are available, then it is optimal (to the supply chain) to reserve from each, so . In this case no demand is lost, but the buyer finds it worthwhile to purchase from only one of the suppliers in the event that demand is low (thus saving the additional $20 cost in this case). The supply chain profit is . Thus the supply chain profits are submodular and there is an equilibrium where each supplier offers and for and otherwise, where .
5 Application to capacity games with function bids
We will apply our results for the general capacity game to the specific case we described in the introduction section. The buyer faces a random demand
, which follows a probability distribution
with density over where . For each unit of items sold, the buyer collects a revenue . Supplier has a reservation cost, which is expressed as a general increasing function of the amount reserved, with a marginal reservation cost for an amount given by (so that total reservation costs are given by ). In addition supplier has an execution cost which is a general increasing function of the amount supplied, with a marginal execution cost of . We have switched to lower case letters here as an indication that these are marginal costs rather than the total costs as occur in the general capacity game. Note that and could be constants. Suppliers each maximize their own expected profits by offering a function bid consisting of a (marginal) reservation price function and a (marginal) execution price function for . Without loss of generality, we assume since the buyer will never reserve more than units. In addition, if a supplier does not want to offer that much, he may simply set a very high price for the quantities beyond the desired amount so that the buyer will never choose to reserve more than the supplier wishes to offer.Suppose the bids offered by suppliers are , where the functions and are defined on and may be constants. The buyer makes a reservation decision in the first stage and makes an execution decision in the second stage. Since the reservation payment becomes sunk at the time when the buyer makes the execution decision, with any realized demand, the buyer simply chooses the cheapest way to meet the demand based on execution prices up to the amount reserved from each supplier. If prices are continuous and less than , then it is not hard to see that the buyer will meet the demand through choosing amounts from supplier where and the prices are all equal so that we may write , , except where is the reserved capacity for , in which case . However in the general case, when prices can decrease, there may be more than one solution having these properties even with continuous prices. In this case the buyer will select the solution with the lowest total payment.
To avoid these complications and allow us to develop explicit expressions for and , we make the following simplifying assumption.
Assumption 4.
The execution price function is nondecreasing for every .
In order to put this problem into the general form of a capacity game, we need to switch from marginal costs to the full costs given by their integrals. Also we note that for the capacity game with function bids, , , and only have dependence on and and not on the full vectors and (so that we need to capture the restriction that bids and are of this form through the specification of the sets ). Thus by setting , , , , , we obtain the problem in the required form.
For the capacity game with function bids we can write the vector of second stage choices explicitly in terms of by recognising that the optimal choice for the buyer will use the low cost suppliers first. As we will demonstrate below, this allows an explicit expression for
where and we write for the th component of . Similarly we define . Let
(10) 
Then is the amount purchased from supplier if an amount is purchased from supplier . When is strictly increasing and continuous, then . In the special case where marginal execution costs are constant (i.e., for all where is a constant), we have if , and if .
Let denote the vector of reservation choices from suppliers other than . We now write the cumulative amount of capacity with execution prices less than as follows,
(11) 
where is defined in (10), the term indicates the dispatched amount from supplier , and the second term represents the dispatched amount from suppliers other than . If the buyer chooses optimally then this is the total amount that the buyer purchases conditional on her purchasing amount from supplier . Hence with optimal buyer choices, the amount purchased from supplier is greater than if and only if the demand is greater than .
Thus, with the reservation choice , the buyer’s optimal expected profit is
(12) 
where , is given in (11) and . We know that represents the probability that the buyer will use the th unit of the reserved capacity from supplier , and thus each term within the summation gives the buyer’s profit of reserving from supplier . Similarly for the supply chain as a whole we define
(13) 
and we have
(14) 
Now we apply Theorem 1 to get the following corollary in this case.
Corollary 1.
Given bids of the other suppliers, if supplier has preferred status, then it is optimal for supplier to set for and choose the function such that
where and provided is chosen in such a way that the optimal buyer choice is . An optimal choice for supplier is to choose for all and in addition charge a fixed amount of for any nonzero amount reserved. In the case where supplier does not have preferred status, then it is possible for supplier to achieve within of the best profit by choosing the same bid of and a bid of that is reduced by .
This result shows that for the capacity game with function bids there is an optimal solution with for the supplier ’s best response problem, but there will be multiple possible solutions for such that the buyer’s optimal choice is and the supplier makes a profit of (which will imply that the buyer makes the same profit as ). In all of these solutions, the optimal buyer choice is , which maximizes the buyer’s profit when supplier charges only his costs.
For the capacity game with function bids we also have the following immediate corollary from Theorem 2.
Corollary 2 (Nash equilibrium with a lump sum payment).
One property of the equilibrium involving a lump sum payment is that it cannot be represented simply by defining marginal prices for capacity: to do so would require an infinite price for the first of capacity. However, we can use the functions occurring in Theorem 2 to construct an equilibrium solution with finite marginal costs. This is done by adjusting the capacity offer from the lump sum bid by smoothing out the beginning of the offer. We will make use of power functions to do this.
Proposition 1 (Nash equilibrium with power functions).
This construction involves adding the margin to the reservation costs, which is a decreasing function of . Choosing a high value for will imply that decreases more steeply at the beginning and becomes flatter at the end of this interval.
From Proposition 1, we see that there is a continuum of Nash equilibria with power functions, each associated with a possible set of values. We can show that when approaches positive infinity, the power function bid reduces to the lump sum bid. Thus, Corollary 2 can be considered as a limiting example of Proposition 1. Moreover, one may easily construct an equilibrium where some suppliers use lump sum bids (with infinitely large values) while others use power function bids. Despite the multiplicity of equilibrium bidding strategies, all these equilibria lead to the same profit allocation and thus the equilibrium outcome characterized in Proposition 1 is essentially unique.
Comparison with MS
As discussed earlier, existing models have focused on the case where the marginal costs are constant. As we will show in Section 6, the supply chain optimal profit is submodular for this special case, and hence the equilibria characterized in Corollary 2 and Proposition 1 apply. It is interesting to draw a parallel between our model and the existing ones for this case. To this end, we draw on an example from MS to demonstrate the difference in equilibrium outcomes.
Example 2 (Example 1 in MS).
The buyer’s demand is uniformly distributed over
, so for . There are two suppliers and their marginal costs are and . The retail price is . We now examine the supply chain optimal problems.If the buyer reserves from two suppliers, the supply chain problem is:
The optimal solution is . The supply chain optimal profit is .
If the buyer chooses supplier only, the supply chain problem is:
The optimal solution is and the supply chain optimal profit is .
If the buyer chooses supplier only, the supply chain problem is:
The optimal solution is and the supply chain optimal profit is .
If suppliers are restricted to each offering a pair of reservation price and execution price, in equilibrium these two suppliers bid infinitesimally close to each other. MS show that the following is a continuum of equilibria, which are parameterized with :
In equilibria, the buyer’s reservation choice is
The profit split amongst players is given by
(15) 
and the supply chain profit is given by
Note that all the above equilibria are inefficient (i.e., not supply chain optimal) except the one with . At this efficient equilibrium, each supplier offers a bid which is identical to the supplier ’s cost. The supplier ’s profit is , the supplier ’s profit is , and the buyer’s profit equals .
We now demonstrate that the above strategies do not form an equilibrium if we allow suppliers to offer function bids. Suppose supplier chooses the proposed bid , and we now examine supplier ’s best response in choosing a function bid.
First, if supplier is the sole supplier, the buyer’s reserved amount will be the sum of the two components of , and the buyer’s profit is equal to . Therefore, we have
Second, we show that the following strategy for supplier improves his profit: setting prices to be costs and charging a lumpsum payment of . Given this offer (and the supplier ’s offer ), we can show that the solution for the buyer’s problem is
and the buyer’s profit from choosing is equal to . Also if the buyer purchases from only supplier , the buyer makes a profit of as well. According to the tiebreaking rule, the buyer will select . Then supplier 2’s profit becomes
which is strictly greater than for any . This shows that the equilibria in MS do not hold if we allow suppliers to offer a function bid.
Under our model setup, the problem is well behaved and there is an equilibrium with lumpsum payments (and there is also an equilibrium with power functions). The following bids form a Nash equilibrium (where and are lump sum payments):
At this equilibria, the buyer’s optimal reservation choice is . The profit split amongst players is given by
(16) 
In this equilibria, each supplier’s profit equals his contribution to the supply chain system and the buyer takes the remaining profit. Moreover, the reservation choice by the buyer is supply chain optimal (in distinction to the case with constant prices).
The key message of Example 2 is that imposing the restriction that each supplier submits a pair of reservation price and an execution price rather than a function bid leads to a higher buyer profit. On the other hand, the suppliers are better off if they submit function bids. This can be easily seen by comparing the profit splits at (15) and (16).
Comparison with ACS
In our setting, each supplier offers a function bid with a marginal reservation price function and a marginal execution price function, and the buyer has the freedom to reserve whatever amount she likes. This setting resembles the two extensions to the setting with blocks of capacity discussed in Anderson et al. (2017). The two extensions they discussed were:
(a) Partial Reservation, in which the buyer is not restricted to reserving a whole block. Each supplier owns a single block and the blocks can be of different sizes. Every supplier chooses an execution price and a reservation price that apply to all elements of his block.
(b) Multiple Blocks with Common Owner, in which each supplier owns multiple unitblocks and can choose different prices for different unit blocks. The buyer can freely choose among the offered blocks.
In both extensions, the authors demonstrate that one of their key results in the baseline model, for supplier ’s best response, does not hold. In contrast, we show that the result holds in our setting as shown in Corollary 1.
The key differences are as follows. In the case (a) of partial reservation, the buyer can freely reserve any amount as in our setting. However, suppliers are restricted to offering just two scalar prices rather than general function bids. In the case (b) of multiple blocks with common owner, each supplier can choose different prices for different blocks of capacity as in our setting. However, the buyer can freely choose any of the blocks, whereas in our function bidding setting, the buyer does not have as much flexibility, because the buyer will need to accept the prices for the earlier blocks in order to enjoy the prices for the next blocks. These differences in results highlight the importance of each player’s choice flexibility on the equilibrium outcomes.
6 Discussion of submodularity assumption
In this section we examine two commonly studied settings in supply chains: (a) there are only two suppliers competing for the buyer’s procurement business, and (b) there are more than two suppliers and the marginal execution costs are constants. In each case we are interested in establishing conditions that will ensure submodularity of .
6.1 Case with two suppliers
When there are only two suppliers (denoted by and ) we can find conditions that are enough to ensure that submodularity occurs, and it turns out that in the special case of the capacity game with function bids submodularity will hold without any further restrictions.
Proposition 2 (Subadditivity).
If is subadditive in and the cost functions are defined separately on each component, so is determined by the th component of , and is determined by the th component of , then is submodular in . In the case of the capacity game with function bids, these conditions will hold.
6.2 Case with constant marginal execution costs
In the case of more than two suppliers, the submodularity of supply chain profit may not hold in general, as we show in Example 4 below. The following theorem identifies some sufficient conditions for the submodularity property and these conditions are satisfied for a large class of the capacity games with function bids. The notion of laminar convexity stated in the theorem is defined in the Appendix.
Theorem 3 (Submodularity).
Let and for . If is laminar concave in and each cost function () is a convex function of (only) the th component of , then is submodular in . In the case of the capacity game with function bids, these conditions are satisfied if for each supplier the marginal execution cost is constant and the marginal reservation cost is nondecreasing.
The complexity of the analysis arises from the fact that the supply chain profit function cannot be easily decomposed into components related to individual suppliers. The profit made from supplier is related to the capacity reservations from all the suppliers with lower execution prices.
Theorem 3 implies that, in settings considered by the existing studies where the marginal costs are constant, the supply chain optimal profit is submodular. We use the following example to illustrate this point.
Example 3.
Suppose the buyer demand follows a uniform distribution over . There are three suppliers whose costs are: ; ; and . The retail price is . We know from Theorem 3 that the problem is submodular, and we can carry out the detailed calculations to find the supply chain optimal solutions for different sets of available suppliers. One way to do these calculations is to use the screening curve approach that is common in calculation of optimal generation mix in electricity markets (see Green, 2005). We summarize the optimal reservation choices and profits in Table 1.
Available Suppliers  

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