Mechanism design for capacity allocation with price competition

Mechanism Design for Capacity Allocation with PriceCompetition

Masabumi FuruhataIntelligent Systems LaboratoryUniversity of Western Sydney,

Australiaand

IRIT-Université de Toulouse,France

[email protected]

Laurent PerrusselIRIT-Université de Toulouse,

Francelaurent.perrussel@univ-

tlse1.fr

Dongmo ZhangIntelligent Systems LaboratoryUniversity of Western Sydney,

[email protected]

ABSTRACTStudies on mechanism design mostly focus on a single mar-ket where sellers and buyers trade. This paper examinesthe problem of mechanism design for capacity allocation intwo connected markets where a supplier allocates productsto a set of retailers and the retailers resale the products toend-users in price competition. We consider the problemsof how allocation mechanisms in the upstream market de-termine the behaviors of markets in the downstream marketand how pricing policy in the downstream market influencesthe properties of allocation mechanisms. We classify an ef-fective range of capacity that influences pricing strategies inthe downstream market according to allocated quantities.Within the effective capacity range, we show that the re-tailers tend to inflate orders under proportional allocation,but submit truthful orders under uniform allocation. We ob-serve that heterogeneous allocations results in greater totalretailer profit which is a unique phenomenon in our model.The results would be applied to the design and analysis ofBusiness-to-Business (B2B) marketplaces and supply chainmanagement.

Categories and Subject DescriptorsK.4.4 [Computers and Society]: Electronic Commerce;I.2.1 [Artificial Intelligence]: Applications and ExpertSystems—Games

General TermsDesign, Economics, Management

KeywordsAllocation mechanism design, supply chain management,oligopoly

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1. INTRODUCTIONMechanism design (MD) has been one of the most promis-

ing research topics in the fields of e-business and artificialintelligence in recent years [8, 26, 22, 13]. The main concernof MD is to induce truthful preferences from self-interestedagents. Studies on MD mostly focus on a single marketwhere sellers and buyers trade. However, in many situations,traders’ preferences are restricted to market situations. Ifthe traders are located in a supply chain, the informationabout the traders’ preferences can then be partially deducedfrom the analysis of the market situations. Particularly, it isinteresting to show how such information could be appliedto mechanism design. In this paper we explore the problemin the context of capacity allocation mechanisms.

Consider a supply chain where a supplier sells products toa set of retailers. Suppose all orders from the retailers exceedthe capacity of the supplier. To solve unbalance of supplyand demand, allocation instead of an adjustment by price iscommonly observed in many supply chains. This is particu-lar in the upper stream of supply chains such as component,raw material and natural resource markets. The reasonsthat allocation is favored in the upstream is changing priceare time consuming and costly for the customers. A changeof price of raw material involves the product cost controllingof the customers and, in the worst case, the customer mustgo back to a product design phase. In fact, a long-termprice contract is common and a spot price contract is ob-served for a small amount of trades in such markets. Hence,the price adjustment is not always proper method. When acommodity supplier determines to allocate products, a num-ber of working staffs in a sales and operations departmentask customers about the truthful demand. This activity isvery common in industry and a typical procedure when thesupplier employs proportional allocation mechanism whichis the most popular mechanism. We show a reason why thisprocedure is required under the mechanism. Furthermore,we present how to induce truthful orders from the retailersby an allocation mechanism.

There are two reasons for us to choose the subject. First,capacity allocation is one of the most important issues incomputing-related applications, such as resource allocation [9,1, 15], task allocation [17, 23]. Moreover, capacity alloca-tion deals with the problem of scarce resources. A smallchange of allocation in the upstream market would affectthe downstream markets significantly. Therefore the design

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of allocation mechanism has to take the whole supply chaininto account.

In this paper, we consider a supply chain with two-connectedmarkets where a supplier allocates products to a set of retail-ers (upstream market) and the retailers resale the productsto end-users (downstream market). We assume that the up-stream market applies a certain allocation mechanism andthe downstream market applies a price competition. Ourinterests are how the allocation mechanism applied in theupstream market influences the competition among the re-tailers in the downstream market and how a variation ofprices in the downstream market affects the behavior of theallocation mechanism in the upstream market. We restrictour analysis to two most popular and widely used allocationmechanisms in industry, proportional allocation and uniformallocation [7, 6]. We investigate the behavior of these twoallocation mechanisms by calculating the equilibria of theretailers’ order quantities and the price equilibria.

Our model is differed from a typical single-market modelsuch as [12, 31, 19, 27] with respect to the orders from theretailers. In our model, the truthful orders are not sim-ply determined from a given preference and an employedmechanism. Due to a market competition, we take into ac-count strategic interactions among the retailers for the orderdetermination. In our setting, a comprehension of marketrules in the downstream market is important to design al-location mechanisms in the upstream market, because theorder quantities of retailer are restricted to the competitionin the downstream market. In other words, the order quan-tity does not only represent private information, but also itreflects the effect of the price competition. We present thatthe properties of allocation mechanisms are highly relatedto market rules in the successive market. Cachon and Lar-iviere [6] have made a contribution to the study of allocationmechanism design in supply chain network based on the as-sumption that the retailers in the downstream market enjoylocal monopolies. With such an assumption, each retailerdoes not face in a direct competition. Therefore the orderquantities of the retailers become purely private informationsimilarly to the single-market model. It implies that no ex-tra information from the downstream market is required todesign truth-inducing mechanisms.

Furuhata and Zhang [11] consider the problem by intro-ducing competition into the downstream market. They con-sider a simple case where the downstream market is in quan-tity competition. However, in the real-world, price competi-tion is more commonly applied market mechanism and muchmore complicated in conjunction with allocation mechanism.The complexity comes from the fact that retail price is deter-mined by the market supply, i.e., the supplier’s capacity, inquantity competition, while individual sellers do determinetheir selling prices in price competition. In fact, we observethat the largest retailer can increase its selling price in spiteof having unsold products which even leads a greater overallprofit (see Theorem 7). This phenomenon diverges not onlyfrom Cournot quantity competition but also from Bertrandprice competition.

This paper is organized as follows. Section 2 presentsour model. In Section 3, we classify effective ranges of thecapacity that influence pricing strategies in the downstreammarket according to allocation quantities. In Section 4, weshow how the retailers place orders according to allocationmechanisms and how retailers determine the retail prices.

Then, in Section 5, we focus on how allocation mechanismsaffect the total retailer profit. Section 6 briefly concludesthis paper.

2. MODELWe consider a supply chain model with two connected

markets: a monopolistic upstream market (wholesale mar-ket) and an oligopolistic downstream market (retail mar-ket) as shown in Figure 1. In the upstream market, a sup-plier sells products to intermediaries, called retailers. Whenorders from the retailers exceed the capacity size of thesupplier, the supplier allocates products according to a se-lected allocation mechanism. In the downstream market,the retailers resale products to a range of end-users in pricecompetition. We investigate how a competitive model inthe downstream market affects properties of the allocationmechanisms in the upstream market and how the allocationmechanisms affect behaviors of the successive downstreammarket. With such a supply chain model, the capacity allo-

Figure 1: Supply Chain Model

cation problem consists of two stages: order placement andallocation in the upstream market; price setting and resaleto end-users in the downstream market.

2.1 Upstream MarketIn the first stage, the supplier sets its capacity exogenously

denoted by K based on its own demand forecast, i.e. thesupplier does not know either a demand function of end-user or how many quantities the retailers will order. Oncecapacity K is determined, it is not able to change after-wards. Then, the supplier selects an allocation mechanismdenoted by g by which the supplier allocates product wherethe capacity is bound. The supplier notifies the selectedallocation mechanism to all retailers. Let N = {1, . . . , n}be the set of retailers usually noted i, j or k. Let us denotej 6= i as all retailers except for retailer i. Let mi be an or-der quantity of retailer i and let us denote −i as sum of allquantities except for the quantity of retailer i, for instance,m−i =

∑j 6=i

mj . Retailer i determines order quantity mi with

respect to market demand, allocation mechanism, and otherretailers’ orders to maximize its profit denoted by πi. Let

us denote revenue of retailer i as Πi. All retailers submittheir orders, m = (m1, . . . , mn), simultaneously and inde-pendently. We assume a wholesale price denoted by w isfixed, same for all retailers and determined exogenously. Ifthe total order quantity exceeds K, the supplier allocatesproducts according to the adopted allocation mechanism.

Let A = {a ∈ <n : a ≥ 0 &n∑

i=1

ai ≤ K}, where a vector

a ≥ 0 means for any component ai of the vector, ai ≥ 0. Wecall each a ∈ A a feasible allocation.

Definition 1. An allocation mechanism is a function g :<n → A which assigns a feasible allocation to each vector oforders such that for any retailers’ order vector m, gi(m) ≤mi for each i = 1, · · · , n.

Let gi(m∗) be the allocation quantity of retailer i under

allocation mechanism g with respect to the vector of theequilibrium order quantity m∗. An allocation g is said to

be efficient ifn∑

i=1

gi(m) = K whenevern∑

i=1

mi ≥ K. For the

supplier, the efficient allocation is the preferable one sincethe capacity has been fully used.

The main concerns of mechanism design are efficiency andstability. For a supplier, capacity utilization is a key perfor-mance index to evaluate its performance. If a mechanismmakes the capacity utilization increase, it is a preferablemechanism for the supplier. A typical mechanism design cri-terion in capacity allocation is individually responsive (IR)which contributes to increase the capacity utilization.

Definition 2. An allocation mechanism g is said to beIR if for any i,

m′i > mi and gi(m) < K imply gi(m

′i, mj 6=i) > gi(mi, mj 6=i)

where m is a vector of retailers’ orders, mi is the i’s com-ponent of m, mj 6=i is a vector of the other retailers’ orders,and m′

i is a variation of mi.

Under IR mechanisms, a retailer receives more allocationsif it orders more. Consequently, retailers frequently placemore orders than they actually need. On the other hand,the inflated orders prevent a right evaluation of the capacityinvestment. When demand is unstable, the supplier oftenfails to make decision on capacity planning due to lack oftruthful order information. Hence, a mechanism inducingtruthful order information from retailers is desirable mech-anism design criteria. This criterion is formally representedas follows:

Definition 3. An allocation mechanism g is said to beincentive compatible (IC) or truth-inducing if all retailersplacing orders truthfully at their optimal sales quantities isa Nash equilibrium of g, formally for all i:

πi(gi(m∗), gj 6=i(m

∗)) ≥ πi(gi(mi, m∗j 6=i), gj 6=i(mi, m

∗j 6=i)).

2.2 Downstream MarketIn the second stage, the retailers are in the price com-

petition in the downstream market. Let D(p) be the de-mand of the end-users at price p and P (q) be its inversefunction where q stands for the total supply quantity. Weassume that the function P (q) is strictly positive on somebounded interval (0, q̂), on which it is twice-continuously dif-ferentiable, strictly decreasing and concave. For q ≥ q̂, we

simply assume P (q) = 0. In order to focus on interestingcases, we assume retailers only have purchase costs w wherew < P (K). Let ai be the allocated quantity for retailer iwhich has been determined in the first stage. Once retailersare allocated by the supplier, the retailers determine pricespi simultaneously and independently. Like Levitan and Shu-bik [18] and Kreps and Scheinkman [16], we assume surplusmaximizing rule where the end-users choose from the lowestprice-offering retailers.

Here, we introduce following notations based on Cournotquantity competition in order to describe a price competi-tion. Let q−i stands for

∑j 6=i∈N

qj . We define the best re-

sponse function for retailer i at cost w in Cournot quantitycompetition as:

rw(q−i) = arg maxqi

{qiP (qi + q−i)− wqi}. (1)

We assume that qiP (qi + q−i)− wqi is concave in qi for allq−i. Based on this assumption, rw(q−i) is a unique solution

of P (qi + q−i) + qidP (qi+q−i)

dqi− w = 0 and satisfies

−1 <∂rw(q−i)

∂q−i< 0. (2)

Hence, rw(q−i) + q−i is increasing in q−i. Let us denoteCournot equilibrium at cost w as qcw and the total Cournot

quantity as qCw =N∑

i=1

qcw. Note that rw(qcw−i ) = qcw. At the

special case where cost is zero, we denote qc as the Cournotequilibrium at zero cost and qC as the total Cournot quan-tity.

From now on, we show how retailers set the price. Letus consider from the case where the supplier allocates theproduct exclusively. Let i be the exclusively allocated re-tailer enjoying the benefits of the monopoly price pM . SinceqP (q) is concave in q, where q ∈ (0, q̂), we have

qM = argmaxq

qP (q). (3)

Therefore, the selling quantity is xi = min{qM , ai}. Henceretailer i sets the monopoly price pM = P (xi) to maximizeits profit πi = xiP (xi)− wai.

Now let us consider the case where the capacity is allo-cated to several different retailers. The remaining part of theproblem is how the retailers determine retail prices accord-ing to the allocation. This pricing problem is similar to amodel proposed by Francesco [10] where several manufactur-ers compete in price in an oligopolistic market with capac-ity pre-commitments. When the pre-commitment of the ca-pacity exceeds the best response quantity, the manufacturerconsiders two options which are: (i) selling all products at alower price or (ii) selling a limited quantity at a higher price.Francesco shows how the manufacturers choose the options.Based on the option, Francesco shows price equilibria thatare dependent on the pre-committed capacity sizes. Noticethat the manufacturers in Francesco correspond to the re-tailers in our model and the capacity pre-commitments cor-respond to the allocation quantities. Before we describe therelationship between allocation quantity and price equilib-ria, we show the relationship between allocation and bestresponse (the lemma is based on Boccard and Wauthy [4,5]).

Lemma 1. Given an efficient allocation mechanism and

let a be the allocated quantities. Suppose ai > aj . If ai ≤r(a−i) then aj < r(a−j).

Proof. According to suppositions, we have a−i < a−j .According to Equation (2), we have r(a−i) < r(a−j). Hencewe have aj < ai ≤ r(a−i) < r(a−j).

According to lemma 1, if allocation ai is less than or equalto its best response, the smaller allocation quantities sat-isfy the same relationships. Furthermore, it implies, if thelargest allocation al satisfies al ≤ r(a−l), then all the otherallocations have the same characteristic.

In [10], Francesco investigates price equilibria in both purestrategy and mixed strategy. Now let us denote, p̄ and pas an upper and a lower bound of the price equilibriumin the mixed strategy. The following lemma characterizesthe price equilibria that are a straightforward conversionfrom the capacity sizes of the manufacturers in the modelof Francesco [10] to the allocated quantities of retailers interms of our model,

Lemma 2. [10] Given an allocation mechanism g, and or-der m, let ai = gi(m). Let the largest allocated retailerl = argmax

i∈Nai. Then

1. if for all i, ai ≤ r(a−i), P (ai + a−i) is a unique equi-librium.

2. if al > r(a−l) and D(0) > a−l, thenp̄l = P (r(a−l) + a−l) ,

pi =P (r(a−l)+a−l)r(a−l)

alfor all i, and

Πl = r(a−l)P (r(a−l) + a−l),

3. if D(0) ≤ a−l, p∗ = 0 is the unique price equilibrium.

Proof. If we view i and ai as a certain manufacturer andits respective capacity choice in [10], the pricing problem inthe downstream market can be seen as the same problem asFrancesco [10]. See proofs of Propositions 1 and 2 in [10].

According to Lemma 2, Francesco shows that there are threetypes of price equilibria according to the patterns of alloca-tions in our context. First, if an allocation quantity for eachretailer is less than or equals to its best response quantity,the retailers set the price P (ai + a−i). Second, if the allo-cation quantity for the largest allocated retailer is greaterthan its best response quantity and the sum of the alloca-tions for the rest of retailers does not exceed the maximumdemand, the retailers set the price between p and p̄ in mixedstrategy. Third, if the sum of the allocations for the retailersexcept for the largest allocated retailer is greater or equalsto the maximum demand, the retailers set price zero. Oncewe have the allocation quantities for the retailers, accord-ing to Lemma 2, we know how the retailers choose pricingstrategies and the price equilibria in the downstream market.

Here, let us describe the difference between Bertrand pricecompetition and our price competition having capacity allo-cation as constraints for the sales. In Bertrand competition,the sellers do not have any upper bound for the selling quan-tities. Hence, the equilibrium price becomes zero. For exam-ple, if there is one seller sets the price lower than the others,the lowest price offering seller dominates the whole market.This is an incentive for the sellers to set lower prices. Con-trary, the retailers have upper bounds (allocations) for the

selling quantities in our model. Therefore, there are someresidual demands for the retailers offering higher price un-der certain circumstances. Hence, the price equilibria arenot always zero in our model as shown in Lemma 2.

Allocation quantity is one of the most influential factorsto determine selling prices. We address an issue: how manyallocation quantities are desirable in an oligopolistic pricecompetition with constraints? We follow the result of Krepsand Scheinkman [16] to deal with the issue. The model ofKreps and Scheinkman is a duopoly version of Francesco’smodel. Kreps and Scheinkman show that the equilibrium ofthe capacity pre-commitment is the Cournot quantity qcw.They assume that the manufacturers are able to determinetheir amounts of capacity pre-commitments independently.This assumption means that we do not have allocation con-straints in our model. It implies that we are able to viewthe Cournot quantity as a truthful demand in our model.

In the next section, we extend Lemma 2 in order to estab-lish a link between capacity sizes and price equilibria whichis useful to know the effective range of capacity for pricingstrategies.

3. EFFECTS OF CAPACITY ALLOCATIONTO THE DOWNSTREAM MARKET

Our interests are how capacity allocation in the upstreammarket influences the market behaviors in the downstreammarket and how the market rules influence the propertiesof allocation mechanisms. Since it is not simple to find anorder equilibrium and a price equilibrium under certain al-location mechanisms, this section aims to shrink a problem.In this section, we find a price equilibrium according to thecapacity size. In other words, we aim to comprehend howthe retailers determine their prices under different capacitysettings. In this section, we assume that the order quantityis given and the total order quantity exceeds the capacity,i.e. g is efficient. Since the order quantity is given, we treatthe purchase cost as sunk cost.

Prior to investigation of the relationship between the sup-plier’s capacity and the retailers’ pricing strategies, we wouldlike to emphasize some typical economical indicators whichhave been shown in Section 2, such as qM : the monopolyquantity of the downstream market, qC : the total Cournotquantity in the oligopoly market, q̂: the maximum demandof the downstream market, and P (K): the market price cor-responding to the quantity of K.

We divide the capacity ranges in four ranges for the anal-ysis.

• Strictly scarce capacity (K ≤ qM ): the supplier’s ca-pacity K is less than the monopoly quantity qM of thedownstream market.

• Relatively scarce capacity (qM < K ≤ qC): the ca-pacity is greater than the monopoly quantity and lessthan or equal to the total Cournot quantity qC .

• Enough capacity (qC < K < q̂): the capacity is greaterthan the total Cournot quantity and less than the max-imum demand of the downstream market.

• Excessive capacity ( q̂ ≤ K): the capacity is greaterthan the maximum demand.

The following theorem deals with the first situation wherethe capacity of the supplier is very limited (i.e. its capacitysize is less than the monopoly quantity).

Theorem 1. Given an allocation mechanism g. Supposethat K ≤ qM and g is efficient. If g is feasible, then P (K)is an equilibrium, i.e., p∗ = P (K)

Proof. According to Lemma 2, it is sufficient to showthat for any i, gi(m) ≤ r(g−i(m)). Since g is efficient andK ≤ qM , we have gi(m)+g−i(m) = K ≤ qM = argmax

qqP (q),

with respect to Equation (3).Case 1: If r(g−i(m)) + g−i(m) ≥ qM , we have

gi(m) + g−i(m) ≤ r(g−i(m)) + g−i(m).

It follows that gi(m) ≤ r(g−i(m)) as desired.Case 2: Assume that r(g−i(m)) + g−i(m) < qM . Ac-

cording to the definition of Cournot best response function,we have r(g−i(m)) = argmax

qi

qiP (qi+g−i(m)) = argmaxqi

((qi+

g−i(m))P (qi + g−i(m)) − g−i(m)P (qi + g−i(m))). Let y =qi+g−i(m) and y∗ = argmax

y(yP (y)−g−i(m)P (y)); we have:

r(g−i(m)) = argmaxy

(yP (y)− g−i(m)P (y))− g−i(m)

= y∗ − g−i(m) (4)

According to the assumption of Case 2, Equation (4) impliesy∗ < qM . It follows that P (y∗) > P (qM ) because P isstrictly decreasing in quantity. It turns out that

−g−i(m)P (y∗) < −g−i(m)P (qM )

Notice that the case g−i(m) ≤ 0 is ruled out since y∗ < qM .On the other hand, y∗P (y∗) ≤ qMP (qM ) because qM =argmax

qqP (q). Therefore, we have,

y∗P (y∗)− g−i(m)P (y∗) < qMP (qM )− g−i(m)P (qM )

This contradicts the definition of y∗. That is r(g−i(m)) +g−i(m) ≥ qM and we have gi(m) ≤ r(g−i(m)).

This theorem shows that the equilibrium price is P (K),when the capacity size of the supplier is less than the marketmonopoly quantity qM and allocation mechanism is efficient.In this case, no retailers can make greater profit by charg-ing higher price than P (K), which is the monopoly pricewhen K ≤ qM . Therefore, the allocation mechanism doesnot affect the market price within the capacity range. Inother words, in this context, the supplier has no interests tochoose a specific allocation mechanism as long as the mech-anism is efficient.

Next, we consider the case where the capacity is relativelyscarce that is qM < K ≤ qC . We have the following result:

Theorem 2. Suppose qM < K ≤ qC . Under any effi-cient allocation mechanism g, price equilibria are,

1. p∗ = P (K), if for all i gi(m) ≤ r(g−i(m));

2. p∗ > P (K), if there exists i such that gi(m) > r(g−i(m)).

Proof. Since g is efficient, we have Kn≤ gl(m) ≤ K.

It implies n−1n

K ≥ g−l(m) ≥ 0. Since K < qC , we haven−1

nK ≤ qc

−i. According to Equation (2), we obtain r(qc−i) ≤

r(n−1n

K) ≤ r(g−l(m)) ≤ r(0) = qM . Therefore, we have ei-ther gl(m) ≤ r(g−l(m)) or gl(m) > r(g−l(m)). In the formercase, according to Lemma 1 and Lemma 2 case 1, we havep∗ = P (K). In the later case, according to Lemma 2 case 2,we have p∗ > P (K).

According to Theorem 2, there are two types of pricingstrategies for the retailers, when the capacity is relativelyscarce. If allocation quantities for each retailer do not ex-ceed the best response quantities, the market price is stableand the equilibrium price reaches the price at the capacitysize, which is the statement 1. Otherwise, the market pricebecomes unstable and higher than P (K), which is the state-ment 2. This is a very interesting phenomenon. If there is aretailer enjoying privilege from unbalance of allocation, theequilibrium price in the downstream market is higher thanthe balanced case. The following two extreme cases help tounderstand the difference. If the supplier allocates the ca-pacity exclusively to a retailer which is one case of the state-ment 2, the retailer sets the resale price as P (qM ) > P (K)to maximize its profit, even if the retailer is not able to sellall the products. On the other hand, if the supplier allocatesthe capacity to all retailers equally, which is one case of thestatement 1, the mechanism of the price competition worksproperly and the retailers are not able to increase their prof-its by charging higher prices than P (K).

We turn to investigate a case where qC < K < q̂ in thefollowing theorem,

Theorem 3. Suppose qC < K < q̂. For any efficientallocation mechanism g, p∗ > P (K).

Proof. Since K < q̂, for any g, we have gl(m) ≤ K < q̂.Suppose, for all i, we have gi(m) ≤ r(g−i(m)). Accord-ing to the definition of Cournot best response function, we

obtain gi(m) ≤ qci . It implies

n∑i=1

gi(m) ≤ qC which is a

contradiction of the supposition of an efficient allocationn∑

i=1

gi(m) = K. Hence, for retailer l, we have gl(m) >

r(g−l(m)). According to Lemma 1 and Lemma 2 case 2,we have p∗ > P (K).

According to Theorem 3, the retailers set the resale pricegreater than P (K) if the capacity is in qC < K < q̂. In thiscase, even if the allocation is equal to all retailers, the allo-cation quantity is greater than the best response quantity.Hence, the retailer sets the price higher than P (K) similarlyto the Statement 2 in Theorem 2.

Finally, we show the case K ≥ q̂.

Theorem 4. Suppose K ≥ q̂. For any efficient allocationmechanism,

1. p∗ = 0, if g−l(m) ≥ q̂.

2. p∗ > 0, otherwise.

Proof. Since K ≥ q̂ and g is efficient, g satisfies eitherg−l(m) ≥ q̂ or 0 ≤ g−l(m) < q̂. The first case is the con-dition of Lemma 2 case 3. Hence, we have p∗ = 0. In thelater case, since K

n> qc, we obtain p∗ > P (K) as same as

Theorem 3.

According to Theorem 4, if q̂ ≤ K and if the sum of theall retailers’ allocations excepts for the largest allocation ex-ceeds the maximum demand, the equilibrium price is zero.This is similar to the result of Bertrand price competition.On the other hand, if the sum of the all retailers’ allocationsexcepts for the largest allocation that does not exceed themaximum demand, the retailers are able to earn some prof-its by setting the price higher than zero, but the profits arevery limited.

q̂CqMq0 Capacity

)(* KPp =

0* =p

)(* KPp >Price Equlibria

Effective range on Pricing Strategy Selection

Figure 2: Capacity and Corresponding Pricing

Now, let us summarize the results of this section. We haveclassified how capacity allocation in the upstream marketaffects the pricing strategy and the price equilibria in thedownstream market as shown in Figure 2. The effectiverange of the capacity for the pricing strategy selection isqM < K ≤ qC and a special case q̂ < K and g−l(m) < q̂.A crucial range of capacity that affects the pricing strategyselection is qM < K ≤ qC . Within the capacity range, theallocation mechanism selection is remarkably sensitive to thedownstream market. If K ≤ qM , the equilibrium price isthe monopoly price. If qM < K ≤ qC , allocations gives asignificant impact on the price equilibria. If qC < K < q̂,the price equilibria are sensitive to allocations. In the nextsection, we relax the assumptions that are the given orderquantities and the total order quantity exceeds the capacity.

4. EFFECTS OF ALLOCATION MECHANISMIN SUPPLY CHAIN

In the previous section, we have classified how capacityallocation affects the pricing strategies and the price equi-libria in the downstream market corresponding to the ca-pacity size. In this section, we consider how the retailers setorder quantities based on allocation mechanisms and howthe retailers determine selling prices according to allocatedquantities in the supply chain. Particularly, we focus on twopopular allocation mechanisms in industry, uniform alloca-tion and proportional allocation. Notice that the truthfulorder quantity is qCw

i as shown in Section 2.A main goal of this model with respect to mechanism de-

sign aspect is to obtain a truth-inducing mechanism. Weinvestigate market behaviors under uniform allocation pre-sented in [28], which is a truth-inducing mechanism in [6]and [11]. Under uniform allocation, the retailers are indexedin ascending order of their order quantity, i.e., m1 ≤ m2 ≤. . . ≤ mn. Let

λ = max

{i : K − nm1 −

i∑j=2

(n− l)(mj −mj−1) > 0

}and

Figure 3: Uniform Allocation Mechanism

uniform allocation is,

gi(m) =

K/n, if nm1 > K,mi, if i ≤ λ,

mλ+

(K − (n− λ + 1)mλ −

λ−1∑j=1

mj

)/(n− λ),

otherwise.

Under uniform allocation mechanism, the retailers with or-ders less than a threshold mλ receive the same quantitiesas respective orders, and the rest of retailers receive mλ

and the rest of capacity divided by the number of retail-ers ordered greater than mλ. The threshold of mλ is ledby the following procedure. If m1 × n is greater than thecapacity, all retailers receive K

n, otherwise, there is a thresh-

old mλ where λ is greater than or equal to 1. In case ofm1×n ≤ K, the supplier counts up the number of retailers,

whilei∑

j=1

mj + mj × (n − i) ≤ K (the sum of i-th smallest

orders and the quantity of the i-th order times the numberof the rest of retailers is less than the capacity size of thesupplier). If the sum of orders is greater than or equal tothe capacity, the allocation quantity is equal to the capacity,which is the area below the horizontal dashed line in Figure3.

We have the following order quantity equilibria and theprice equilibria under uniform allocation.

Theorem 5. Under uniform allocation, there is a uniqueequilibrium order quantity m∗

i = qcw in the upstream market,which induces price equilibria p∗i = max{P (K), P (qCw)} inthe downstream market.

Proof. If m∗i = qcw

i for all i, according to the definitionof uniform allocation, we have gi(m

∗) = min{qcwi , K/n} ≤

qcw. Since gi(m∗) ≤ rw(g−i(m

∗)) < r(g−i(m∗)), the profit

of retailer i is

πi = P (gi(m∗) + g−i(m

∗))gi(m∗)− wgi(m

∗). (5)

If m′i > m∗

i and m∗j 6=i = qcw, we have gi(m

′) = gj 6=i(m′) =

K/n, which is the same allocation quantity to the case at m∗.

Hence, by increasing order m′i, retailer i cannot increase its

profit. If m′i < m∗

i , we have gi(m′) ≤ K

n≤ gj 6=i(m

′) ≤ qcw.We check whether this case fits to the condition of case 2 ofLemma 2. According to Equation (2), we have r(g−i(m

′)) ≤r(g−j(m

′)). Since gj 6=i(m′) ≤ qcw

j = rw(qcw−j) < r(qcw

−j) ≤r(g−i(m

′)) ≤ r(g−j(m′)), this case does not satisfies the

condition of case 2 of Lemma 2. Hence, we only consider thecase of the pure strategy. The profit of retailer i is same asEquation (5). Recall that Equation (5) is concave in gi(m)and maximized at rw(g−i(m)). Since gi(m

′) ≤ rw(qcw−i ), by

decreasing m′i, πi is not increased. Therefore, we have an

equilibrium order quantity m∗i = qcw

i , an allocation quantityat gi(m

∗) = min{K/n, qcwi }, and an equilibrium price p∗i =

max{P (K), P (qCw)}.

According to Theorem 5, all retailers place truthful orderquantities qcw under uniform allocation, since no retailersare able to increase their profits by increasing orders or de-creasing orders from the truthful order quantity. An inter-esting property of uniform allocation is its robustness of allo-cation at the truthful order quantity. Even if competitors in-crease their order quantities, the allocation quantity for theretailer that submits the truthful order is not decreased. Atthe equilibrium order quantity m∗, we have allocation quan-tities gi(m

∗) = min{K/n, qcwi }. Since the allocation quan-

tity for each retailer does not exceed the best response quan-tities, we have the price equilibria p∗i = max{P (K), P (qCw)}under uniform allocation.

Uniform allocation satisfies the truth-inducing property.This is a very important criterion to choose an allocationmechanism for mechanism designers. However, in most cases,the supplier distributes products for several different mar-kets. Hence, proportions to fulfill the demands by alloca-tions are different under uniform allocation. An undesirablepoint for the supplier is that the larger market is allocatedless proportion compared to the smaller market under uni-form allocation.

In industry, the most commonly used allocation is pro-portional allocation, which is a representative IR allocationmechanism. An allocation mechanism g is proportional al-location if

gi(m) = min

{mi, Kmi/

N∑j=1

mj

}. (6)

In other words, whenever capacity binds, allocated quan-tity to each retailer is the same fraction of its order underthe proportional allocation. Similarly to Cachon and Lariv-iere [6], we show that the retailers inflate orders and there isno equilibrium under proportional allocation in our model.

Theorem 6. There does not exist an order equilibriumm∗ under proportional allocation, if K ≤ qCw.

Proof. Suppose there exists symmetric m∗. Let us de-note πi(g(m)) = πi(gi(m), gj 6=i(m)). Since K ≤ qCw, wehave gi(m

∗) ≤ qcw and g−i(m∗) ≤ qcw

−i . It turns out r(g−i(m∗)) ≥

r(qcw−i ). Since gi(m

∗) ≤ qcwi = rw(qcw

−i ) < r(qcw−i ) ≤ r(g−i(m

∗)),the profit of retailer i is πi(g(m∗)) = P (gi(m

∗)+g−i(m∗))gi(m

∗)−wgi(m

∗), according to Lemma 2 case 1. Let m′ = (m′i, m

∗j 6=i).

Equation (6) implies gi(m′) > gi(m

∗) where m′i > m∗

i .If

∑i∈N

m∗i < K, we have gl(m

′) such that gl(m′) < qcw

l .

Equation (6) implies g−l(m′) ≤ g−l(m

∗) and r(g−l(m′)) ≥

r(g−l(m∗)). It turns out that gl(m

′) < qcwl = rw(qcw

−l ) <r(g−l(m

∗)) ≤ r(g−l(m′)). According to Lemma 2 case 1, we

have πl(g(m′)) = P (gl(m′)+g−l(m

′))gl(m′)−wgl(m

′). Ac-cording to the concavity of the profit function and gi(m

∗) <gl(m

′) ≤ r(g−l(m′)), we have πl(g(m′)) > πi(g(m∗)). Hence,

symmetric m∗ does not exist in the case of∑

i∈N

m∗i < K.

If∑

i∈N

m∗i ≥ K, Equation (6) implies gl(m

′) − gl(m∗) =

−(g−l(m′) − g−l(m

∗)) where m′l > m∗

l . It follows gl(m′) <

r(g−l(m′)) and πl(g(m′)) = P (gl(m

′) + g−l(m′))gl(m

′) −wgl(m

′). The difference between πl(g(m′)) and πi(g(m∗))is (gl(m

′)−gi(m∗))(P (K)−w). According to the assumption

of w, we have πl(g(m′)) > πi(g(m∗)). Hence, symmetric m∗

does not exist in the case of∑

i∈N

m∗i ≥ K.

Suppose there exists asymmetric m∗ such that gl(m∗) <

r(g−l(m∗)). The supposition is contradicted similarly to the

case of symmetric m∗ and∑

i∈N

m∗i < K.

Suppose there exists asymmetric m∗ such that gl(m∗) >

r(g−l(m∗)). The profit of retailer l is

πl(g(m∗)) = P (r(g−l(m∗))+g−l(m

∗))r(g−l(m∗))−wgl(m

∗),

according to Lemma 2 case 2. Similarly, the profit of retaileri where m∗

i < m∗l is

πi(g(m∗)) =P (r(g−l(m

∗)) + g−l(m∗))r(g−l(m

∗))gl(m∗)

gi(m∗)

− wgi(m∗).

The difference of profits between retailer l and i is

πl(g(m∗))− πi(g(m∗)) =gl(m

∗)− gi(m∗)

gl(m∗)

(P (r(g−l(m∗)) + g−l(m

∗))r(g−l(m∗))− wgl(m

∗)).

At the equilibrium, πl(g(m∗)) must be positive. Notice thatπl(g(m∗)) = P (r(g−l(m

∗))+g−l(m∗))r(g−l(m

∗))−wgl(m∗).

Hence, we have πl(g(m∗)) > πi(g(m∗)). It follows thatasymmetric m∗ such that gl(m

∗) > r(g−l(m∗)) does not

exist.

According to Theorem 6, there does not exist an equilib-rium in order quantity under proportional allocation wherethe retailers tends to increase their order quantities morethan they need. Under proportional allocation, each retaileris able to decrease the competitors’ allocations by increas-ing its order that makes greater profits for each retailer.Thus, proportional allocation is not robust and the supplierreceives more order quantities than actual needs.

One way to obtain an order equilibrium under propor-tional allocation is to assume a maximum order quantitydenoted by m̃. In reality, we frequently encounter the casewhere there exists a maximum order quantity, which is deter-mined by either the supplier side or the retailer side. In thiscase, according to Theorem 6, we are easily able to obtainthe equilibrium order quantity m∗

i = m̃i. The interesting

case of the capacity range is K ≤n∑

i=1

m̃i. If the maximum

order quantity is symmetric, we have allocation gi(m∗) = K

nand the equilibrium price p∗ = P (K), which is the sameprice as the quantity competition shown in [11]. If it is

asymmetric, we obtain allocation gi(m∗) = Km∗

i /n∑

i=1

m∗i .

The equilibrium price is either p∗ = P (K) or p∗ > P (K),

which is dependent on a relationship between the largestallocation gl(m

∗) and its best response r(g−l(m∗). If the

maximum order quantities are heterogeneous and the allo-cation for the largest allocated retailer exceeds its best re-sponse quantity, the retailers set their resale price greaterthan P (K) which is a unique phenomenon in this modelcompared to the quantity competition in [11].

5. HETEROGENEOUS ALLOCATIONSIn industry, the allocation quantities are not always equal.

For example, under proportional allocation with maximumorder quantity shown in the previous section, it is commonlyobserved that some prioritized customers have the greatermaximum order quantities. As we have seen in Theorem 2,the difference of allocation quantities to the retailers affectsthe pricing strategies and the largest allocated retailer in-creases the selling price to enjoy the benefit of the privilegedallocation. This is a unique phenomenon in our model. Inthis section, we further investigate how heterogeneous allo-cations affect the total retailer profit.

In the exclusive distribution model in [6], the most im-portant design criteria is Pareto optimality. Here, Paretooptimality means that under allocation mechanism that sat-isfies Pareto optimal criteria the total retailer profit is max-imized, if all retailers submit orders truthfully. However, inour model, to allocate all the products to a single retaileris a way to maximize the total retailer profit. Therefore,Pareto optimality is not a significant criterion in our model.Meanwhile, it does not mean the total retailer profit is notimportant. The supplier frequently encounters a situationwhere the supplier needs to give some privileges to some re-tailers on allocation. We investigate how heterogeneousnessof allocation make an impact for the total retailer profit.

We focus on the interesting capacity range qM < K ≤ qC

as shown in Section 3. Let us call gP as a strong hetero-geneous allocation which satisfies the condition of case 1 inLemma 2 and let us call gM as a non strong heterogeneousallocation for the condition of case 2. According to theseconditions, mechanisms gM reflects the case where some re-tailers have been privileged. Notice that a symmetric allo-cation is included in gP , since K ≤ qC .

Let us remind that strong heterogeneous allocation corre-sponds to statement 2 in Theorem 2 and non strong hetero-geneous allocation corresponds to statement 1 in Theorem 2.Within the capacity range qM < K ≤ qC , the retailers havetwo types of pricing strategies and price equilibria. It is cer-tain that the largest allocated retailer earns greater profitunder strong heterogeneous allocation than the profit un-der non strong heterogeneous allocation. However, we haveone question whether the higher equilibrium price under nonstrong heterogeneous allocation results in the total retailerprofit greater. The following theorem corresponds to thisquestion.

Theorem 7. Suppose that qM < K ≤ qC . For any effi-cient gP(m) and gM(m),

∑i∈N

πi(gM(m)) >

∑i∈N

πi(gP(m)).

Proof. Let πi(g(m)) = πi(gi(m), gj 6=i(m)). First weshow the total retailers’ profit under gP . According to the

case 1 of Lemma 2, the total retailers’ profit is∑i∈N

πi(gP(m)) =

∑i∈N

(P (K)gPi (m)− wgPi (m)

)

=P (K)K − wK. (7)

Now we show the case of gM. If gMl (m) 6= K, according tothe case 2 of Lemma 2, the total retailers’ profit is

∑i∈n

πi(gM(m)) =

∑i∈n

(pig

Mi (m)− wgMi (m)

).

We have p∗ > P (K) under gM according to Theorem 2.Hence, we have

∑i∈n

πi(gM(m)) = pK − wK > P (K)K − wK. (8)

According to Equation (7) and (8), we obtain∑i∈n

πi(gM(m)) >

∑i∈N

πi(gP(m)).

If gMl (m) = K, retailer l is a monopolist. Hence, we haveπl >

∑i∈n

πi(gP(m)) according to Equation (7) and K >

qM .

According to Theorem 7, under strong heterogeneous allo-cation gM, the total retailer profit is greater than the oneunder non strong heterogeneous allocation gP , when the ca-pacity of the supplier is relatively scarce. Since there isno difference with respect to the total cost of all retailersbetween gM and gP , we focus on the revenue. The totalrevenue is P (K)K under gP . Meanwhile, the total revenueis pK under gM. At first glance, it seems inconsistent, be-cause the market demand cannot be K if the retail price isp > P (K) under gM. However, pK consists of the revenuesof all retailers and they do not set price p at once. In fact,the total selling quantity under gM is less than K, sincethe selling quantity of retailer i at price p̄ is less than gMi .Notice that the profit of the mixed strategy is equal at anyprice between p and p̄. Therefore, it implies that even ifthe less prioritized retailers under gM decrease their profitscompared to the ones under gP , the increasing amount ofthe profits of the prioritized retailers exceeds the decreasingamount of the less prioritized retailers.

The phenomenon that the heterogeneousness of allocationaffects the total retailer profit shown above is not observed inthe quantity competition model in [11]. In quantity compe-tition, the price is determined by the total supply quantity.Hence, the individual supply quantities do not affect themarket price. Hence, the heterogeneousness of allocation isnot a significant point to consider the total retailer profit inFuruhata and Zhang’s model [11].

6. CONCLUSION AND RELATED WORKIn this paper we studied capacity allocation problems in

a supply chain where a supplier allocates capacity to a setof retailers and the retailers compete in price competition ina same market. We showed how the capacity allocations inthe upstream market affect the pricing strategy selection andthe price equilibria in the downstream market. According tothe classification of the effective range of the capacity thataffects the market behaviors in the downstream market, we

are able to know the crucial capacity ranges to choose theallocation mechanisms.

With the equilibrium analysis on purchasing and pricing,we investigated the effects of allocation mechanisms in sup-ply chain, especially for two popular allocation mechanisms,uniform allocation and proportional allocation. We foundthat the equilibrium order quantity would not always beCournot quantity in our model. This is a significant differ-ence from the result of the Kreps and Scheinkman’s model.The difference is observed in a situation where the retailersare able to decrease their competitors’ allocation quantitiesunder certain allocation mechanisms such as proportional al-location. Under proportional allocation, the retailers inflateorders to be allocated more than competitors. Contrary,we showed that uniform allocation induces truthful orderquantity from the retailers under which we have the sameequilibrium price as Furuhata and Zhang [11].

We observed a unique phenomenon in our model that isheterogeneousness of allocations results in higher marketprice if capacity is relatively scarce. Furthermore, the to-tal retailer profit is increased in this case, even though someretailers decrease their profits. In quantity competition, thetotal market supply determines the market price, thus theprice is not affected by heterogeneousness of allocations. Onthe other hand, each retailer determines the selling pricein price competition with allocation constrains. Therefore,the privileged retailers are able take an opportunity to sethigher price for the residual demand, because less prioritizedretailers are not able to fulfill all the demand. In our model,the total retailer profit is maximized, where the supplier ex-clusively allocates to a single retailer. It means that oneimportant criteria, Pareto optimality, in Cachon and Lar-iviere’s model [6] is not a significant criteria in our model.Hence, truth-inducing property enhances the significance ofmechanism design criteria.

In this model, choosing an allocation is a trade-off betweenthe efficiency goal and the stability goal. IR allocation leadsan order inflation that contributes for a higher utilizationof the capacity and a greater profit for the supplier. How-ever, the supplier lacks of the accurate demand information.This is a serious problem, since the accurate demand is afundamental input for all business planning for the supplier.The retailers encounter uncertain allocations and unstableprices under IR allocations. However, there is a way to in-crease their profits by ordering more. Contrary, the truth-inducing allocations let the supplier obtain the truthful de-mand. However it may not be the profit-maximizing. Theorders, the allocations and the prices become stable for theretailers. Overall, choosing an allocation mechanism is notjust choosing one policy for allocation, but also it influencesthe market behaviors through a supply chain. We believethat our model represents typical phenomena in many sup-ply chains.

This paper integrates techniques of equilibrium analysisin Economics [16, 10] for a comprehension of market rulesand mechanism design of capacity allocation [6, 20, 11].

Our market model in the downstream market is similarto Kreps and Scheinkman’s model [16] where sellers are ina price competition in a duopoly with pre-commitment ofsupply limit for each seller. The main difference is that ourmodel has an allocation process. Once allocation is executedin our model, by treating allocated quantity as supply limit,the market behavior can be explained similarly. However,

in our model, each retailer is not able to determine alloca-tion quantity (supply limit) which is dependent on allocationmechanisms and the market behaviors in the upstream mar-ket. Meanwhile, each seller is able to determine its supplylimit independently in the Kreps and Scheinkman’s model.Therefore, our model deals with more complex business sce-nario and it is very general transactions in daily business.Since Kreps and Scheinkman [16], several works on pricecompetition with capacity constraints have been made. Afirst stream of research extends the results of the Kreps andScheinkman. Vives [29] shows price equilibria in a symmet-ric oligopoly case with common capacity constraints amongsellers. Francesco [10] extends the Kreps and Scheinkmanmodel from a duopoly to an oligopoly. A second streamof research shows the limits of the Kreps and Scheinkmanmodel by assuming asymmetric cases, including imperfectcapacity pre-commitment [4, 5] and uncertain demand [25].The difference of the two streams is caused by the symmet-ric behavior in the models. Our model is relevant to bothstreams of these literatures, since feasible allocations coverboth symmetric and asymmetric cases.

The two-connected market model is common in industrialorganization. The major works are vertical integration andmultilateral vertical contracting in [14, 21, 24]. The aimsof these papers are closely related to ours. They show howmarket rules in the downstream market affects the strate-gic choice in the upstream market and how strategic choiceinfluences the market behavior in the downstream market.However, they do not consider capacity allocation mecha-nisms in the two-connected market model.

Understanding the market behaviors based on the marketrule in the successive market is important to design alloca-tion mechanisms in supply chain management, since allo-cation mechanisms affect market behaviors in the successivemarkets and the market behaviors based on the market rulesaffect some properties of mechanisms. This point is a maindifference from some works of mechanism design for supplychain models [6, 2, 20, 30, 3] except for [11]. Therefore,our results are useful in order to design and analyze B2Bmarketplaces and supply chain management.

For future work, it is important to consider market mech-anism design in dynamic environments. Recently, some re-searches proposes approaches on mechanism design in a dy-namic environment [8, 26, 22, 13]. However, they are re-stricted to a single market. We would like to propose anddevelop an adaptive supply chain solution on e-marketplacesfor the future work.

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