TRANSSHIPMENTS IN HAZARDOUS ENVIRONMENTS: COOPERATIVE VERSUS NONCOOPERATIVE QUALITY CONTROL GAME

May 15, 2013 18:42 WSPC/0219-1989 151-IGTR 1350001

International Game Theory ReviewVol. 15, No. 1 (2013) 1350001 (27 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219198913500011

TRANSSHIPMENTS IN HAZARDOUS ENVIRONMENTS:COOPERATIVE VERSUS NONCOOPERATIVE QUALITY

CONTROL GAME

KONSTANTIN KOGAN and DANA SHERILL-ROFE∗

Department of ManagementBar-Ilan University, Israel

∗[email protected]

Received 14 November 2011Revised 9 December 2012Accepted 15 March 2013Published 8 May 2013

We address quality control of products undergoing multiple transshipment stages han-dled by different parties. Depending on the transportation modes, the stages may exposethe products to hostile environments, such as extreme temperatures, which could influ-ence quality. We incorporate the effect of quality on demand with the Neyman–Pearsonstatistical framework to study the effect of intra-competition between supply chain par-ties on inspection policies and thereby product quality. Specifically, we compare thesepolicies with that of a centralized supply chain, where a single decision maker choosesthe optimal inspection policies for all stages. We find that in terms of inspection pol-icy, a party with the higher probability of nonconformance tends to inspect less undercompetition compared to the system-wide optimal inspection policy. Conversely, theother party may inspect more than under the system-wide optimal policy. We determinewhen intra-competition impacts conformance quality so that regulations are of particularimportance for protecting consumers.

Keywords: Supply chains; quality control; game theory.

1. Introduction

In industry quality may be compromised due to imperfect or uncertain processes.The issue is all the more critical when several companies collaborate in manufac-turing and distributing a product. Uncertainty increases further due to each party’smotivations and preferences, that is, due to intra-competition in the underlying sup-ply chain. Consequently, an approach to quality and quality control which ignoresthe effects of competition, fails to capture the risk and economic consequences ofmultiple transshipments in hazardous environments.

Stiglitz (1987) provides an extensive review of the implications of the dependenceof quality on price in competitive markets. In particular, Stiglitz shows that a

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K. Kogan & D. Sherill-Rofe

competitive market equilibrium might exist in which demand does not equal supplyor that there may be, in equilibrium, a price (wage, interest rate) distribution. Thenan increase in supply (that is, a shift in the supply curve such that at every price agreater quantity of the good is supplied) leads to an equilibrium with a lower priceand in which a greater quantity is traded.

The game theoretic approach has been suggested for modeling the strategicuncertainty that arises in monopolistic supply chains due to firms working for oragainst each other. Specifically, Reyniers (1992) describes a quality-related gamebetween a supplier and customer involving a sales contract, a warranty and replace-ment costs in addition to inspection costs. Both parties choose a sampling plan as adecision variable. They begin with a noncooperative Stackleberg game under com-plete information and proceed to a game where the customer has no informationregarding supplier inspections.

Reyniers and Tapiero (1995a) consider the effect of supplier-producer contractson quality. They determine a Nash equilibrium for a two-person, nonzero sum game,where the supplier determines production quality by choosing which technologyto employ and the producer determines inspection policy. The costs of varioustechnologies that the supplier uses, as well as the producer’s inspection costs andpost-sale failure are accounted for. They also describe the effect of contracts and col-laboration between the parties. In another paper by Reyniers and Tapiero (1995b),the supplier invests in improving the quality of the process, while the customerinvests in product inspections. They study both cooperative and noncooperativesettings.

Starbird (2003) models competition in a two-echelon supply chain with a non-cooperative game between a customer and a supplier. The customer chooses lot sizewhile the producer decides on replenishment policy as well as on quality, definedas the product conformance probability. The model accounts for setup, holdingand manufacturing costs, as well as for quality related costs, including the cost ofprevention, internal failure (scrapping), inspections and external failure. Demand,however, is assumed to be deterministic and unaffected by the quality. Chao et al.(2009) describe a quality improvement effort game, similar to the inspection game,with a component supplier and a manufacturer who devise a contract to share prod-uct recall costs. They study a centralized model, as well as a Stackelberg game, andaccount for the costs of both parties for production, recall and quality improve-ment. They describe different scenarios regarding information about the basic com-ponent’s quality. Demand is assumed to be constant. Hsieh and Liu (2010) proposea noncooperative scenario with a supplier performing outbound inspections on com-ponents and a manufacturer performing both inbound components inspection andoutbound product inspection. In addition, the parties invest in quality improve-ment, which adds costs but reduces defectives. Sampling rates and investment inquality are determined for four game models with different degrees of availableinformation.

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Transshipments in Hazardous Environments

Tapiero (2001) provides a comparative and economic approach to strategic qual-ity control in a supply chain setting compatible with the Neyman–Pearson statisticalrisk framework. The effect of quality is accounted for by means of penalties forproduct experience with poor quality caused by type II risks in the Neyman–Pearsonframework. To control the negative impact of nonconforming parts reaching the end-users, either the type II error is minimized or a risk constraint is imposed. Equilibriaare then determined subject to risk constraints (Tapiero and Kogan, 2007) and tothe optimal production yield (Tapiero, 2001). In addition, the effect of the costs onthe type of equilibrium is studied to identify the case where the equilibrium inspec-tion policy is pure (sample or not sample) or the strategy is mixed (interior solu-tion). The results are compared with a centralized (system-wide optimal) solution.

Another approach to account for the effect of quality on the demand is to assumethat all defective parts will be returned by the consumers and thus the quantitysold (demand) is reduced by the amount of defective products (see, for example,El Ouardighi and Kim, 2010). Our approach to modeling the effect of quality ondemand integrates both the type II error and the fact that the consumers may returnnonconforming products. In particular, we assume that the demand is linear anddownward in the type II error function due to product returns. That is, the num-ber of returned products is proportional, but not necessarily equal to the numberof defective ones. Indeed, even a nonconforming drug (in terms of storage condi-tions) may still be efficient and thus not returned. On the other hand, some drugs,even those that are conforming, may have strong side-effects wrongly attributedto poor quality and which are subsequently returned to the retailer. This paper ismotivated by a quality control problem frequently encountered in pharmaceuticalsupply chains. In the first stage, the producer’s drugs are transported to a whole-saler. The second stage shifts the responsibility to the wholesaler, who dispatchesthe product to retailers (pharmacies). Depending on the mode of transportation(air, water, rail, or road) and storage, the two stages may take days or even weeksto complete, exposing the drugs to hostile environments, such as extreme tempera-tures that could impact the quality and thereby demand for the drugs. Therefore,cooling devices may be used during transportation and temperature monitoring isgenerally required. In addition, both producer and wholesaler may use samplingstrategies to determine the quality of individual lots. Accordingly, the demand forproducts is affected by both their quality and price.

Matsubayashi and Yamada (2008) employ a linear demand function in prod-uct quality to examine the effect of “perceived price” — a weighted combinationof price and quality level — on demand from each firm in a duopolistic market,assuming customer loyalty. They study the underlying Nash equilibrium whichdoes not account for inspection processes. Singer et al. (2003) describe a supplier-retailer game where quality affects demand. They assume that other than loss ofdemand, the costs of disposing nonconforming items are insignificant for the parties.In addition, they assume that the product can become nonconforming only during

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K. Kogan & D. Sherill-Rofe

production and that both parties perform quality control. In terms of the Neyman–Pearson statistical framework, they include only type I errors, where nonconformingitems are accepted. Singer et al. optimize a profit function that includes the cost ofachieving the total quality, namely, producing zero nonconforming items, and thecost factor that measures how rapidly the retailer’s cost of detecting an additionalpercentage of defective items grows. Unlike the above Neyman–Pearson framework-based papers, where the percentage of defective products detected depends on theprobability of inspection by the parties, the parties deterministically choose howmany products to detect. Specifically, the supplier determines the percentage ofnonconforming products manufactured while the retailer determines the percent-age of defective products detected.

In this work, we incorporate the effect of quality on demand (similar to Matsub-ayashi and Yamada, 2008) and the Neyman–Pearson statistical framework (Tapiero,2001; Tapiero and Kogan, 2007) to study an inspection game between a manufac-turer and wholesaler in a supply chain. We consider the endogenous demand in boththe product price and the product quality, but focus on the effect of product qual-ity on the demand. We formulate a noncooperative game between a manufacturerand a wholesaler where both parties independently select inspection strategies tomaximize their profits. We refer to our formulation as a transshipment inspectiongame in order to stress that we study the effect of hazardous conditions on productstransported from a manufacturer to a wholesaler and then from the wholesaler toa retailer.

This approach is then contrasted to a centralized decision mechanism where asingle decision maker selects inspection policies for the two parties to maximizethe system-wide profit (Kogan et al., 2010). We show that the overall supply chainquality is negatively affected by intra-supply chain competition. Furthermore, wefind that the party with the higher nonconformance probability is affected more bythe competition in terms of inspection policy, since it tends to inspect less comparedto the system-wide optimal inspection policy.

2. Problem Formulation

Consider a two-echelon supply chain consisting of a manufacturer i = 1, and awholesaler i = 2, who distributes the manufacturer’s products to retailers. Theproducts are produced and supplied in lots that are subject to nonperfect techno-logical and environmental conditions. As a result, a lot can become nonconformingwith a probability αi, i = 1, 2. To ensure quality, both parties employ statisticalprocess control with perfect inspections. Let xi be the inspection frequency of partyi, or equivalently, the probability that a particular lot be inspected. Accordingly,the probability of a nonconforming lot reaching the wholesaler is

β1 = α1(1 − x1). (1)

The wholesaler is not able to detect the manufacturer’s defectives, but may intro-duce new defectives when handling product lots and supplying them to the retailer.

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Transshipments in Hazardous Environments

Hence, the probability of a nonconforming lot reaching the retailers, β2, includes:(i) conforming lots 1 − α1 from the manufacturer in which defectives are intro-duced by the wholesaler α2 and which are not inspected 1 − x2 by the wholesaler,(1 − x2)(1 − α1)α2; and (ii) nonconforming lots from the manufacturer α1 (notinspected by the manufacturer, 1−x1) and thus reaching the wholesaler (1−x1)α1.These latter lots remain nonconforming even if inspected by the wholesaler andregardless of whether or not the wholesaler introduces new defectives when ship-ping the products to the retailer, (1 − x1)α1:

β2 = (1 − x2)(1 − α1)α2 + (1 − x1)α1. (2)

Note that the “probability risks”, βi, associated with sampling strategies xi arecalled type II errors in the Neyman–Pearson statistical control framework (Wether-hill, 1977; Tapiero, 1996). Accordingly, the quality is defined by the probability ofa nonconforming lot not reaching consumers, Q = 1 − β2. We assume that con-sumer demand per lot, q, is endogenous q = q(p, Q), downward sloping in lot pricep and upward sloping in lot quality. Following Matsubayashi and Yamada (2008),we assume linear demand in price and quality, q(p, Q) = a − a1p + a2Q, which, inour notation, can be presented as

q(p, Q) = a − a1p − a2β2, (3)

where a is the potential demand; a1 is the consumer sensitivity to the productprice; and a2 is the penalty (in terms of consumer demand) for nonconforming lotsreaching consumers.

We denote by m the margin that the wholesaler gains from each lot sold. Con-sequently, when the two parties maximize their profits, πi they incur the lot pro-cessing costs ci and lot inspection costs Ii. Consequently, we derive the followingtwo problems.

The manufacturer’s problem (the first firm)

maxx1

π1(x1, x2) = maxx1

(a − a1p − a2β2)(p − m − I1x1 − c1)

s.t. 0 ≤ x1 ≤ 1.(4)

The wholesaler’s problem (the second firm)

maxx2

π2(x1, x2) = maxx2

(a − a1p − a2β2)(m − I2x2 − c2)

s.t. 0 ≤ x2 ≤ 1.(5)

We assume that

a − a1p − a2(α1 + α2) ≥ 0 (6)

and that both the manufacturer and the wholesaler are sustainable

p − m − I1 − c1 ≥ 0, (7)

m − I2 − c2 ≥ 0. (8)

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This ensures that the optimization in (4) and (5) will not result in negative demandeven when β2 attains its maximum value, α2 + α1.

In terms of our motivating example, the two problems imply that themanufacturer (i = 1) attaches a logger to an outgoing lot with probability x1

and inspects this lot when handing it over to the wholesaler (i = 2). Similarly, thewholesaler attaches a logger to an incoming lot with probability x2 and inspects itupon delivery to a pharmacy. The lot’s quality is identified by the maximum timeτ that the drugs, which make up the lot, can be exposed to a temperature of overT ◦C. The probability αi that the temperature will be higher than T for more thanτ time units, making every supplied lot nonconforming, is known for the specificgeographical area and season. Specifically, let fY,Z(· , ·) be a bivariate probabilitydensity function of the random variables, temperature Y and exposure time Z, andlet Ai be the transportation time from party i. Then,

αi = P (T < Y ; τ < Z ≤ Ai) =∫ ∞

T

∫ Ai

τ

fY,Z(y, z)dzdy.

Thus, given αi, i = 1, 2 and all demand and cost parameters, each party selectsan inspection strategy xi, which affects the other party’s profit, resulting in thenoncooperative game that we study in the next section.

Note that due to the nature of the products that we consider, the noncon-formance probabilities must be low. If this is not the case, then the inspectiongame specified above must also take into account the likelihood of possible lawsuitswhen some of the delivered nonconforming drugs turn out to be life-threatening.This would imply adding high penalties to the objective functions and making fullinspection by all parties the only viable policy. Thus, in what follows, we focuson the most realistic case of low-level hazardous conditions and assume, to avoidunnecessarily complicated mathematical expressions, that αi, i = 1, 2 are small andthereby their product, α1α2, is negligible. Then Eq. (2) is simplified to

β2 = (1 − x2)α2 + (1 − x1)α1. (9)

3. Equilibrium Inspection Policies

We start off by observing that problems (4) and (5) are concave and thus theoptimal response functions of the both parties are unique. All proofs are relocatedto Appendix A.

Proposition 1. Problems (4) and (5) are concave in x1 and x2, respectively.

Since both constraints in (4) and (5) are linear, every strategy x1, x2 lies in aconvex, closed and bounded region. Then the fact proven in Proposition 1, that theplayer’s payoff function is concave in its own strategy implies the transshipmentinspection game (4)–(5) is concave, which leads to the following result (Arrow andDebreu, 1954; Rosen, 1965).

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Proposition 2. The transshipment inspection games (4) and (5) always have aNash equilibrium point.

To simplify the following exposition, we introduce two important notations. Oneis the ratio between the profit margin and the two firms’ marginal inspection costs,

Θ1 =p − m − c1

I1,

Θ2 =m − c2

I2

which we refer to as the marginal indicators of a firm’s motivation. The other

ρ =a − a1p

a2

is the ratio between the consumer demand for perfect products and consumer sen-sitivity to quality, hereafter referred to as the relative customer insensitivity toquality. We next derive the conditions under which both parties exercise a mixedstrategy, i.e., Nash equilibrium solutions xN

1 , xN2 are 0 < xN

1 < 1 and 0 < xN2 < 1.

Proposition 3. If 2α1Θ1 − α2Θ2 − 2α1 + α2 < ρ < 2α1Θ1 − α2Θ2 + α1 + α2 and2α2Θ2 − Θ1α1 + α1 − 2α2 < ρ < 2α2Θ2 − Θ1α1 + α1 + α2, then

(a) the inspection games (4) and (5) are submodular, i.e., if one party increasesinspection intensity, the other decreases it ;

(b) the Nash equilibrium of the inspection games (4) and (5) is determined by themixed strategies

xN1 =

13

(2Θ1 − α2

α1Θ2 −

(ρ − (α1 + α2)

α1

))(10)

and

xN2 =

13

(2Θ2 − α1

α2Θ1 −

(ρ − (α2 + α1)

α2

)). (11)

A number of observations follow from Proposition 3. First of all, the probabilityof inspections (see Eqs. (10) and (11)) by a party is proportional to its marginalmotivation indicator (i.e., it is proportional to the profit margin per unit inspectioncost) and inversely proportional to the other party’s marginal motivation indicatoras well as to relative customer insensitivity to quality.

Note that the equilibrium conditions derived are based on multiple require-ments. Accordingly, we are unable to conclude whether the equilibrium is uniqueunder other than the Proposition 3 conditions. To elaborate on this matter, wenext consider the difference between the marginal motivation indicators of the twofirms, weighted with their respective nonconformance probabilities, Θ2α2 − Θ1α1.Specifically, the next theorem focuses on two cases: −α1 < Θ2α2 − Θ1α1 ≤ 0 and0 ≤ Θ2α2 − Θ1α1 ≤ α2 − α1. This captures most equilibrium conditions, whilepreserving the expository simplicity. The proof of the theorem is based on technical

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Propositions A.1 and A.2 (handling the cases where one party opts for a pure strat-egy, while the other exercises a mixed inspection strategy) which are relocated toAppendix A.

Theorem 1. Let −α1 < Θ2α2 − Θ1α1 ≤ α2 − α1. Then the Nash equilibrium isunique and is determined as follows.

If ρ ≤ Θ2α2 − α2 (12),then x1 = x2 = 1.If Θ2α2 − α2 < ρ ≤ 2Θ1α1 − Θ2α2 − 2α1 + α2, (A.5)then x1 = 1 and 0 < x2 < 1 is determined by Eq. (A.4).

When Θ2α2 − Θ1α1 ≥ 0:If ρ < 2α1Θ1 − Θ2α2 + α1 + α2,

then 0 < x1 < 1 and 0 < x2 < 1 are determined by Eqs. (10) and (11), respectively.If 2α1Θ1 − Θ2α2 + α1 + α2 ≤ ρ < Θ2α2 + α1 + α2,

then x1 = 0 and 0 < x2 < 1 is determined by Eq. (A.3).If ρ ≥ Θ2α2 + α1 + α2, then x1 = x2 = 0.

When Θ2α2 − Θ1α1 < 0:If ρ < 2α2Θ2 − Θ1α1 + α1 + α2,

then 0 < x1 < 1 and 0 < x2 < 1 are determined by Eqs. (10) and (11), respectively;If 2Θ2α2 − Θ1α1 + α1 + α2 ≤ ρ < Θ1α1 + α2 + α1, then 0 < x1 < 1 is determinedby Eq. (A.1) and x1 = 0.If ρ ≥ Θ1α1 + α1 + α2, then x1 = x2 = 0.

When Θ2α2 − Θ1α1 = 0:If Θ2α2 − α2 < ρ ≤ Θ2α2 − 2α1 + α2,

then x1 = 1 and 0 < x2 < 1 is determined by Eq. (A.4).If ρ < α2Θ2 + α1 + α2,

then 0 < x1 < 1 and 0 < x2 < 1 are determined by Eqs. (10) and (11), respectively.If ρ ≥ Θ2α2 + α1 + α2, then x1 = x2 = 0.

Based on Theorem 1, we observe that for very low values of the relative customerinsensitivity to quality, ρ, both parties perform full inspection, regardless of thedifference in the loss of the marginal profit due to nonconformance, α1Θ1 − α2Θ2.Higher values of ρ induce the party with the higher loss of the marginal profitdue to nonconformance (i.e., higher Θiαi, i = 1, 2) to either inspect more orreduce inspection intensity to a rate that is slower than that of the party withlower marginal loss. As ρ further increases, i.e., consumers become less sensitiveto quality, the party with the lower value of Θiαi is the first to completely stopinspections.

Note that the first firm always starts to reduce its inspection intensity first, whilethe second firm still performs full inspections. This asymmetry at high values ofcustomer sensitivity to quality is due to our assumption that −α1 < Θ2α2−Θ1α1 ≤α2 − α1, which implies that the difference between the second and first firm’s

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Transshipments in Hazardous Environments

marginal loss is smaller than the difference in the corresponding nonconformityprobabilities. Accordingly, the first firm can afford to reduce inspections first whencustomer sensitivity is high. This result is derived in Theorem 1 from PropositionA.1 (see Appendix A, condition (iv)). Formally, it is also shown in the proof ofTheorem 1 that the symmetric condition from Proposition A.1 (condition (ii)), isunfeasible for −α1 < Θ2α2 − Θ1α1 ≤ α2 − α1. If we assume an opposite relation-ship, α2 − α1 < Θ2α2 − Θ1α1, condition (ii) of Proposition A.1 becomes feasiblethereby causing the second firm to start reducing its inspections first. Similarly,the extreme situations in which one firm performs full inspection while the otherperforms none (Proposition A.2, conditions (ii) and (iii)) are also precluded by ourassumption.

4. Analytical Results: Centralized Versus DecentralizedSupply Chains

To further elucidate the effect of intra-competition on the supply chain’s partiesand the consequent effect on quality, we now compare the Nash equilibrium fromthe previous section and the system-wide optimal solution that Kogan et al. (2010)determined for the centralized supply chain of the same setting.

Proposition 4. If ρ ≤ Θ2α2 − α2, then both parties perform full inspection andneither is affected by the competition. That is, the Nash solution is identical to thesystem-wide optimal solution.

From Proposition 4, we observe that the higher the probability of nonconfor-mance of a lot handled by the second firm, the lower the customer sensitivity toquality can drop, while both firms will still perform full inspection in centralizedand decentralized supply chains.

Since the number of possible scenarios when comparing the two types of supplychains grows exponentially, we next assume the inspection costs are symmetric,I1I2

= χ = 1, and that the parties’ profit margin is moderate, that is, 1 ≤ Θ1 ≤ 2,1 ≤ Θ2 ≤ 2. The next two propositions provide a comprehensive comparison of thecentralized and decentralized inspection policies for a wide range of supply chainparameters when α2−α1 > 0. The comparison shows that the parties are not alwaysaffected by the intra-supply chain competition and that when they are affected, theymay inspect more as well as less than the system-wide optimal solution.

Proposition 5. Let α2−α1 > 0 and α1Θ1+α1Θ2−2α1 < 2α1Θ1−α2Θ2−2α1+α2.If ρ < α1Θ1 + α1Θ2, then the wholesaler inspects less than the system-wide

optimal full inspection.

Furthermore:

• If 2α1Θ1 −α2Θ2 − 2α1 + α2 < ρ < 2α2Θ2 −α1Θ1 + 3α1Θ2 − 2α1 − 2α2, then themanufacturer inspects less than the system-wide optimal partial inspections.

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K. Kogan & D. Sherill-Rofe

• If ρ = 2α2Θ2−α1Θ1+3α1Θ2−2α1−2α2, then the manufacturer performs partialinspections and is unaffected by the competition.

• If 2α2Θ2−α1Θ1+3α1Θ2−2α1−2α2 < ρ < α1Θ1+α1Θ2, then the manufacturerinspects more than the system-wide optimal partial inspections.

Proposition 5 exemplifies the effect of intra-competition on product quality. Specif-ically, we find that the wholesaler, whose probability of nonconformance is highercompared to the manufacturer (α2 − α1 > 0), inspects less in the competitive sce-nario than in the centralized supply chain. Proposition 6 elaborates further on thisfinding.

Proposition 6. Let α2 − α1 > 0 and α2Θ1 + α2Θ2 + α1 − α2 < 2α1Θ1 − α2Θ2 +α1 + α2:

(i) If α1Θ1 +α1Θ2 ≤ ρ ≤ α2Θ1 +α2Θ2 +α1 −α2, then the manufacturer inspects(unlike the system-wide optimal solution of no inspections), while the whole-saler inspects less than the system-wide optimal full inspection policy.

(ii) If α2Θ1+α2Θ2+α1−α2 < ρ < 2α1Θ1−α2Θ2+α1+α2, then the manufacturerinspects (unlike the system-wide optimal solution of no inspections), while thewholesaler inspects less than the system-wide optimal partial inspection policy.

(iii) If 2α1Θ1 −α2Θ2 +α1 +α2 ≤ ρ < α2Θ2 +α1 +α2, then the manufacturer doesnot inspect at all and is unaffected by the competition, while the wholesalerinspects less than the system-wide optimal partial inspection policy.

(iv) If α2Θ2 + α1 + α2 ≤ ρ < α2Θ1 + α2Θ2 + α1 + α2, then the manufacturer doesnot inspect at all and is unaffected by the competition, while the wholesalerinspects less than the system-wide optimal partial inspections policy.

(v) If ρ ≥ α2Θ1 + α2Θ2 + α1 + α2, then both parties do not inspect at all andneither is affected by the competition.

Otherwise, when α2Θ1 + α2Θ2 + α1 − α2 > 2α1Θ1 − α2Θ2 + α1 + α2:

(vi) If α1Θ1 + α1Θ2 ≤ ρ < 2α1Θ1 − α2Θ2 + α1 + α2, then the manufacturerinspects (unlike the system-wide optimal solution of no inspections), while thewholesaler inspects less than the system-wide optimal full inspection policy.

(vii) If 2α1Θ1 − Θ2α2 + α1 + α2 ≤ ρ ≤ α2Θ1 + α2Θ2 + α1 − α2, then the manu-facturer does not inspect at all and is unaffected by the competition, while thewholesaler inspects less than the system-wide optimal full inspection policy.

(viii) If α2Θ1 + Θ2α2 + α1 − α2 < ρ < α2Θ2 + α1 + α2, then the manufacturerdoes not inspect at all and is unaffected by competition, while the wholesalerinspects less than the system-wide optimal partial inspection policy.

(ix) If Θ2α2 + α1 + α2 ≤ ρ < α2Θ1 + Θ2α2 + α1 + α2, then both parties do notinspect at all. The manufacturer is unaffected by the competition while thewholesaler is affected by the competition since her system-wide optimal policyis partial inspections.

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(x) If ρ ≥ α2Θ1 + Θ2α2 + α1 + α2, then both parties do not inspect at all andneither is affected by the competition.

Based on conditions (v) and (x) of Proposition 6, we observe that the higher theprobability of nonconformance of a lot handled by both firms, the higher the valueof ρ (the relative customer insensitivity to quality) from which the two firms will notperform inspections in the centralized or decentralized supply chains. Furthermore,the conditions of Propositions 5 and 6 demonstrate that when the probability ofnonconformance of a lot handled by the second firm is higher than that of the firstfirm, the second firm tends to perform fewer inspections in the decentralized supplychain compared to the system-wide optimal solution.

The subsequent two propositions treat the cases of α2−α1 < 0 (see Appendix A).The remaining cases are covered in the following proposition.

Proposition 7. (i) If max{α1Θ1 + α1 + α2,

α1Θ1 + α1Θ2 − α1 + α2

}< ρ < α1Θ1 + Θ2α1 + α1 +

α2, then the manufacturer inspects less than the system-wide optimal partialinspections, while the wholesaler does not inspect at all and is unaffected by thecompetition.

(ii) Otherwise, if ρ ≥ α1Θ1 + Θ2α1 + α1 + α2, then both parties do not inspect atall and neither is affected by the competition.

Symmetric to Propositions 5 and 6, Propositions A.3, A.4 and 7 show that whenthe probability of nonconformance of a lot handled by the first firm is higher thanthat of the second, the first firm tends to perform fewer inspections in the decen-tralized supply chain. Furthermore, Propositions 4 through 7, A.3 and A.4, implythat the Nash equilibrium inspection strategies for the two parties depend on threekey indicators, which we formally introduced in Sec. 3. The first indicator is therelative customer’s insensitivity to quality, i.e., the ratio between customer demandunder perfect product quality and customer sensitivity to quality, ρ. Our resultsshow that this indicator implies that the lower the relative insensitivity to quality,the higher the expected intensity of inspections. The second indicator is determinedfor two parties separately. It is the ratio between the profit margin of a firm andthe marginal inspection cost of the firm, Θ1 and Θ2. For this marginal motivationindicator, we find that the higher the firm’s unit cost of inspection (or the lower theprofit margin), the lower the intensity of inspections the firm exercises. Finally, thethird indicator is the marginal loss due to nonconformity or, the product of the sec-ond indicator of a firm and the probability of nonconformance for the firm. In otherwords, this indicator is the loss of profit due to nonconformance (αiΘi). Althoughthe third indicator has a more complex impact on the parties’ optimal behavior,the firm ranked lower on this indicator is always the first to stop inspections oncecustomer sensitivity to quality becomes sufficiently low. This also implies that evenwhen a party with a high marginal loss is the first to start reducing inspection inten-sity, as consumer sensitivity to quality decreases, this party will reduce inspections

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so slowly that it will stop inspecting entirely only after the other party has stoppedthe inspections completely.

5. Discussion: Numerical Examples and the Interpretation

In this section, we elaborate on the effect of intra-competition on inspection policiesand product quality as well as discuss coordination of the supply chain.

5.1. The effect of intra-competition on the manufacturer’s and

wholesaler’s inspection policies

Propositions 5 and 6 show that when the difference between the parties’ relativemargins is moderate, the gap between their nonconformance probabilities is criti-cal. Indeed, given the wholesaler’s higher probability of nonconformance comparedto the manufacturer’s, the manufacturer’s product quality is largely unaffected bythe intra-competition. Or, the effect is positive since his probability of nonconfor-mance (α1) is lower than the wholesaler’s (α2). Specifically, we observe that themanufacturer will often choose to inspect more frequently under the competitionthan under the system-wide optimal solution of the centralized supply chain. Thewholesaler, on the other hand, will almost always choose to perform fewer inspec-tions compared to the system-wide optimal policy (see Proposition 6). This is tosay, the probability of nonconformance is a decisive factor in terms of the effect ofcompetition on product quality.

To better understand the derived effect of the difference, α2 − α1 > 0, werely on two important observations. First, based on the difference between thethird indicator value for the two parties (i.e., expected marginal loss due tononconformance) employed in Theorem 1 and the assumption that χ = 1, (orI1 = I2 = I), we find that for α2Θ2 − α1Θ1 > 0 to hold when α2 − α1 > 0,inequality p−m− c1 − I > m− c2 − I must also hold. That is, the manufacturer’sprofit margin is higher than the wholesaler’s. Second, with respect to the central-ized supply chain, we observe that from I1

I2= χ = 1 and α2I1 − α1I2 > 0 we have

α2 − α1 > 0. These two observations facilitate understanding the first two scenar-ios from Theorem 1 and the corresponding theorem from Kogan et al. (2010). Forthese scenarios in particular, we find that a higher profit margin for the manufac-turer motivates him to inspect more under the intra-competition (compared to thesystem-wide optimal solution) even when the probability of nonconformance at thisstage is relatively low. In response, the wholesaler reduces inspections as is typicalfor the game of substitutes (see Proposition 3). Conversely, given α2 − α1 < 0, wedetect that the manufacturer generally tends to inspect less under intra-competitioncompared to the system-wide optimal solution, even though his probability of non-conformance is relatively high (see Proposition A.4). In addition, recalling the basicassumption of Theorem 1, α2Θ2 − α1Θ1 < α2 − α1, we obtain from α2 − α1 < 0that α2Θ2 − α1Θ1 < 0 as well. Therefore, the manufacturer’s marginal profit lossdue to nonconformance is higher than the wholesaler’s. Overall, we find that the

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party with the higher probability of nonconformance is affected more by the intra-competition in terms of inspection policy, as it tends to inspect less compared tothe system-wide optimal inspection policy.

5.2. The effect of the intra-competition

on the quality in the supply chain

The effect of intra-competition on the probability of nonconforming items leavingthe supply chain, i.e., on the probability of nonconformance β is illustrated in Figs. 1and 2 for two different cases of α2 − α1 > 0 and α2 − α1 < 0.

The first case of α2 − α1 > 0 (Fig. 1) is computed for the probability of a lotbecoming nonconforming equal to α1 = 0.01 for the first firm (stage) and α2 = 0.02for the second stage. Unit inspection costs are identical for both stages, I1 = I2 = 1monetary unit. In addition, price per lot (p) is 6 monetary units; the margin thatthe wholesaler gains from each lot sold is m = 3.2 monetary units; and the lotprocessing costs are c1 = 0.9 and c2 = 1.8 monetary units for the first and secondfirms, respectively. Therefore, Θ1 = 1.9 and Θ2 = 1.4. The evolution of β for theNash equilibrium policy (dashed line) and for the system-wide optimal inspectionpolicy (solid line) as a function of ρ (derived by changing the consumer sensitivity

Fig. 1. The effect of intra-competition on the probability of nonconforming items leaving thesupply chain (β) for α2 − α1 > 0.

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Fig. 2. The effect of intra-competition on the probability of nonconforming items leaving thesupply chain (β) for α2 − α1 < 0.

to quality a2) is shown in Fig. 1. According to Propositions 4–6 and A.3, the valuesof β are calculated from the values of xi as a function of Θi and αi. For the secondcase of α2 − α1 < 0 (see Fig. 2), we assume α1 = 0.02, α2 = 0.015, I1 = I2 = 1,p = 6, m = 3, c1 = 1.4 and c2 = 1.8. Therefore, Θ1 = 1.6 and Θ2 = 1.2. Note thatthe graphs are truncated on the left side due to constraint (6) which requires inthese cases that ρ ≥ 0.03. This implies that neither firm will choose to perform fullinspection for either scenario.

Figures 1 and 2 illustrate with respect to Propositions 4–7, A.3 and A.4, thatthe overall quality of the supply chain is affected by the intra-competition for highvalues of customer sensitivity to quality. We observe that as the probabilities ofnonconformance decrease for both parties, the threshold from which both will decideto perform no inspection will increase as well, as shown in Propositions 5 and A.3.For very low consumer sensitivity to quality levels, no firm will choose to inspect,and quality will depend exclusively on the given probabilities of nonconformance.

Figures 1 and 2 illustrate that quality deteriorates under intra-supply chaincompetition for moderate values of ρ, a fact derived from Propositions 4–7, A.3and A.4. Elaborating on Propositions 6, A.4 and 7, Figs. 1 and 2 show that forlow to moderate levels of relative customer insensitivity to quality (0.003 to 0.096),the impact of intra-competition on quality rises significantly compared to lower

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Fig. 3. The effect of competition on the probability of nonconforming items leaving the supplychain (β) for α1 − α2 > 0, expressed as the difference between the intra-competitive and system-wide optimal scenarios.

values of ρ. Specifically, the gap between the conformance quality of the two typesof supply chains increases up to 1.9% (see Fig. 3 for the case of α1 − α2 > 0).This finding is due to the fact that in the decentralized supply chain both firmsperform partial inspections under intermediate values of ρ. As ρ decreases, thefirm with lower marginal loss due to nonconformance stops inspecting first. In thecentralized supply chain, on the other hand, the decisive factor is the probabilityof nonconformance rather than the marginal loss. In particular, the firm with thehigher probability of nonconformance will perform more inspections, keeping thetotal quality of the supply chain relatively high. Moreover, Propositions 6 and 7show that in a centralized supply chain, the firms always choose to stop inspectingat a higher value of ρ (insensitivity to quality). This implies that government regu-lation of conformance quality of such perishable products as drugs is of particularimportance for the low to intermediate levels of ρ.

Coordination

Both external regulations and internal supply chain coordination are criticalfor counteracting the negative impact of intra-supply chain competition onconformance quality. Increased external regulations by national and international

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bodies (the FDA, the EU, etc.) and penalizing improper monitoring of productswill ensure conformance quality and thereby coordinate the supply chain. A sim-ilar approach can be employed internally if, for example, the supply chain adoptscoordination with linear penalty/reward functions (see, for example, Golany andRothblum, 2006). Specifically, by introducing linear functions r1x1 and r2x2, thefirms’ objective functions take the following form:

π1(x1, x2) = π1(x1, x2) + r1x1 and π2(x1, x2) = π2(x1, x2) + r2x2,

and perfect coordination can be achieved by choosing r1 = ∂π2(x1,x2)∂x1

|x1=x∗1,x2=x∗

2

and r2 = ∂π1(x1,x2)∂x2

|x1=x∗1,x2=x∗

2, where the couple x∗

1 and x∗2 is a system-wide opti-

mal solution. Note that since the firms’ decision variables are constrained, thesystem-wide optimal solution is determined differently for different supply chainparameters, unlike in unconstrained formulations considered in Golany and Roth-blum (2006). To illustrate this coordination approach, we use α1 = 0.01, α2 = 0.02,I = I1 = I2 = 1, c1 = 1.2, c2 = 2.1, m = 3.2, a = 100, a1 = 1, p = 6, anda2 = 3133.3. Thus, constraint (6) is satisfied as a− a1p− a2(α1 + α2) = 0.007 > 0.Then

0.027 = Θ1α1 + Θ2α1 < 0.03 < 2Θ1α1 − Θ2α2 + α1 + α2 = 0.04

and Condition VI of Proposition 6 holds. That is, the noncooperative firms exer-cise partial inspections x1 = 0.333 and x2 = 0.466, while the system-wide opti-mal solution in such a case is x∗

1 = 0 and x∗2 = 1. Next, introducing r1 =

∂π2(x1,x2)∂x1

|x1=0,x2=1 = 3.133 and r2 = ∂π1(x1,x2)∂x2

|x1=0,x2=1 = 100.256 we constructthe Lagrangians for the modified profits of the two firms using multipliers λ1 andλ2 for binding constraints x1 = 0 and x2 = 1.

L1 = π1(x1, x2) + λ1x1 and L2 = π2(x1, x2) − λ2(x2 − 1).

Consequently differentiating the two Lagrangians in the corresponding decision vari-ables x1 and x2 and substituting x1 = 0 and x2 = 1, we find λ1 = 9.40 > 0and λ2 = 43.86 > 0. That is, all optimality conditions are met for the two firmssimultaneously and the Nash solution of the noncooperative firms is identical tothe system-wide optimal solution of the centralized supply chain. For comparison,one can verify that a mixed strategy, which was optimal for the non-cooperativefirms maximizing the original objective functions, is no longer optimal with thepenalty/reward functions. Indeed, by setting λ1 = 0 and λ2 = 0, we find with thesame Lagrangians a new interior solution, x1 = −0.66 < 0 and x2 = 1.51 > 1,which is not feasible.

6. Conclusion

We consider a supply chain consisting of a manufacturer and wholesaler. The man-ufacturer is responsible for shipping the products to the wholesaler in the firststage. In the second stage, the wholesaler supplies the products to a retailer. Ateach stage, defects can be introduced into the products and the parties employ

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statistical process control to ensure conformance quality. Since the choice of inspec-tion strategy by one party affects the other party’s profitability, this competitionwithin the supply chain is modeled as a noncooperative game. We show that anequilibrium always exists and find that this inspection game may involve pure aswell as mixed sampling strategies. We prove that the latter implies a game of strate-gic substitutes, i.e., if one party increases inspection intensity, the other decreasesit and show that the transformation between the pure and mixed strategies iscaptured by the relative customer sensitivity to quality. Since there is a virtual“explosion” of different conditions that can be imposed on the problem parametersfor which the Nash equilibrium takes a different form, it is practically impossibleto describe all Nash solutions. Therefore, we consider only some realistic cases interms of problem parameters. For each of the considered cases, we rigorously provethat the equilibrium is unique and fully describe it.

To unveil the effect of intra-competition on inspection policies, we compare theequilibrium policies with a system-wide optimal solution of a centralized supplychain. The comparison shows that competition has an overall negative effect onthe quality of lots delivered through the supply chain. Interestingly, we find that ahigher profit margin for the manufacturer motivates him to inspect more under thecompetition compared to the system-wide optimal solution, even when the proba-bility of nonconformance at this stage is relatively low. The wholesaler, on the otherhand, reduces inspections as is typical for the game of substitutes, thereby nega-tively affecting overall conformance quality. In general, we find that the party withthe higher probability of nonconformance is affected more by the intra-competitionin terms of inspection policy, as it tends to inspect less compared to the system-wide optimal inspection policy. Consequently, the gap between the conformancequality of the centralized and decentralized supply chains rises for a broad range ofintermediate values of the customer sensitivity to quality.

The consensus is that though regulatory agencies should intensify their activi-ties, the industry must play its part as well (Editorial, 2008). Drug makers that out-source manufacturing, transportation, storage and/or other activities need to investmore in control and oversight of their products and suppliers. Both external regula-tions and internal oversight and coordination are therefore critical for counteractingthe negative impact of intra-supply chain competition on conformance quality.

Appendix A

Proof of Proposition 1. Since the constraints of problems (4) and (5) are linearto prove the proposition, it is sufficient to show that the corresponding second-orderderivatives are negative:

∂2π1(x1, x2)∂x2

1

= −2a2α1I1 < 0 as a2, α1, I1 > 0;

∂2π2(x1, x2)∂x2

2

= −2a2α2I2 < 0 as a2, α2, I2 > 0.

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Proof of Proposition 3. To prove (a) it is sufficient to show that the second-ordercross-partial derivatives are negative,

∂2π1(x1, x2)∂x1∂x2

= −a2α2I1 < 0 and∂2π2(x1, x2)

∂x2∂x1= −a2α1I2 < 0.

To prove (b) we simultaneously solve the following system of equation:

∂π1(x1, x2)∂x1

= a2α1(p − m − I1x1 − c1) − I1(a − a1p − a2((1 − x2)α2

+ (1 − x1)α1)) = 0,

∂π2(x1, x2)∂x2

= a2α2(m − I2x2 − c2) − I2(a − a1p − a2((1 − x2)α2

+ (1 − x1)α1)) = 0,

which results in

xN1 =

13

(2(p− m − c1)

I1− α2

α1

(m − c2

I2

)−

(a − a1p − a2(α1 + α2)

a2α1

))and

xN2 =

13

(2(m − c2)

I2− α1

α2

(p − m − c1

I1

)−

(a − a1p − a2(α1 + α2)

a2α2

)),

while the mixed strategy conditions are straightforwardly obtained by requiringthat 0 < xN

1 < 1 and 0 < xN2 < 1:

α2

α1

(m − c2

I2

)+

(a − a1p − a2(α1 + α2)

a2α1

)

<2(p − m − c1)

I1< 3 +

α2

α1

(m − c2

I2

)+

(a − a1p − a2(α1 + α2)

a2α1

)

and

α1

α2

(p − m − c1

I1

)+

(a − a1p − a2(α1 + α2)

a2α2

)

<2(m − c2)

I2< 3 +

α1

α2

(p − m − c1

I1

)+

(a − a1p − a2(α1 + α2)

a2α2

).

The proof is completed by substituting the notations for the marginal indicator andthe relative effect of quality on customer demand.

Proposition A.1. (i) If Θ1α1 − α1 + α2 < ρ < Θ1α1 + α1 + α2 and ρ ≥ 2Θ2α2 −Θ1α1 +α1 +α2, then the Nash equilibrium of the inspection games (4) and (5)is such that the manufacturer exercises mixed strategy

xN1 =

12

(Θ1 − ρ − (α2 + α1)

α1

)(A.1)

while the wholesaler does not inspect at all, xN2 = 0.

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(ii) If Θ1α1 − α1 < ρ < Θ1α1 + α1 and ρ ≤ 2Θ2α2 − Θ1α1 + α1 − 2α2, then themanufacturer exercises mixed strategy

xN1 =

12

(Θ1 − ρ − α1

α1

)(A.2)

and the wholesaler performs full inspection, xN2 = 1.

(iii) If ρ ≥ 2Θ1α1 − Θ2α2 + α1 + α2, and Θ2α2 + α1 − α2 < ρ < Θ2α2 + α1 + α2,

then the wholesaler exercises mixed strategy

xN2 =

12

(Θ2 − ρ − (α2 + α1)

α2

)(A.3)

and the manufacturer does not inspect at all, xN1 = 0.

(iv) If Θ2α2 − α2 < ρ < Θ2α2 + α2 and ρ ≤ 2Θ1α1 − Θ2α2 − 2α1 + α2, then thewholesaler exercises mixed strategy

xN2 =

12

(Θ2 − ρ − α2

α2

)(A.4)

and the manufacturer performs full inspection, xN1 = 1.

Proof. First, by accounting for x2 = 0 we find from the first-order optimalityequation, ∂π1(x1,x2)

∂x1= 0 that

xN1 =

12

(p − m − c1

I1− a − a1p − a2(α2 + α1)

a2α1

)

(which transforms into (A.1) when using the notations ρ, Θ1 and Θ2). The firstcondition stated in this proposition,

a − a1p − a2(α2 + α1)α1a2

<p − m − c1

I1<

a − a1p − a2(α2 − α1)α1a2

is then obtained by requiring that 0 < xN1 < 1. To specify the second condition, we

construct the Lagrangian of the wholesaler’s optimization problem to account forconstraint x2 ≥ 0, with a multiplier, λ ≥ 0:

L = (a − a1p − a2β2)(m − I2x2 − c2) + λx2.

Substituting (9) into the Lagrangian and differentiating it with respect to x2, wefind,

λ = −a2α2(m − I2x2 − c2) + I2(a − a1p − a2((1 − x2)α2 + (1 − x1)α1)).

Thus, for the case of x2 = 0, we have,

λ = −a2α2(m − c2) + I2(a − a1p − a2(α2 + (1 − x1)α1)).

Substituting (A.1) for x1 we find (in the original notations),

λ = −a2α2(m − c2) +I2

2(a − a1p − a2(α2 + α1)) +

I2a2α1

2I1(p − m − c1) ≥ 0,

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which, after some manipulation, results in the second condition:

α1

α2

(p − m − c1

2I1

)+

(a − a1p − a2(α2 + α1)

2a2α2

)≥ (m − c2)

I2,

as stated in this proposition in terms of ρ, Θ1 and Θ2.For the case of x2 = 1, we find

λ = −a2α2(m − c2) +I2

2(a − a1p − a2(α1 − 2α2)) +

I2a2α1

2I1(p − m − c1) ≥ 0,

which results in the respective condition,

α1

α2

(p − m − c1

2I1

)+

(a − a1p − a2(α1 − 2α2)

2a2α2

)≤ (m − c2)

I2.

The remaining conditions are verified similarly.

From Proposition A.1 we observe that as long as its conditions hold, the inspec-tion intensity of one party does not depend on the other party’s inspection cost. Onthe other hand, the effect of the customer sensitivity to quality remains the samefor the party which exercises a mixed strategy.

Proposition A.2. (i) If ρ ≥ max{Θ1α1 + α1 + α2, Θ2α2 + α1 + α2}, then bothfirms do not inspect, i.e., x1 = x2 = 0.

(ii) If Θ1α1 + α1 ≤ ρ ≤ Θ2α2 + α1 − α2, then the manufacturer does not inspect,x1 = 0, and the wholesaler performs full inspection, i.e., x2 = 1.

(iii) If Θ2α2 + α2 ≤ ρ ≤ Θ1α1 + α2 − α1, then the manufacturer inspects all hislots, x1 = 1 and the wholesaler does not inspect at all, x2 = 0.

(iv) If ρ ≥ max{Θ1α1 − α1, Θ2α2 − α2}, then the manufacturer and the wholesalerinspect all their lots x1 = x2 = 1.

Proof. The proof is similar to that of Proposition A.1 and is therefore omitted.

Proof of Theorem 1. Employing the theorem’s assumption, −α1 < Θ2α2 −Θ1α1 ≤ α2 − α1, we observe that inequality Θ2α2 − α2 ≤ Θ1α1 − α1 holds andthereby condition (iv) of Proposition A.1 takes the form ρ ≤ Θ2α2 − α2, as statedin Theorem 1.

Similarly, from −α1 < Θ2α2 − Θ1α1 we conclude that

2Θ1α1 − Θ2α2 − 2α1 + α2 < Θ2α2 + α2

holds and condition (iv) of Proposition A.1 simplifies into Θ2α2−α2 < ρ ≤ 2Θ1α1−Θ2α2 − 2α1 + α2, since ρ ≤ 2Θ1α1 −Θ2α2 − 2α1 + α2 ensures that ρ < Θ2α2 + α2.

Note, condition (ii) of Proposition A.1 is unfeasible for the Theorem 1 assump-tion, −α1 < Θ2α2 − Θ1α1 ≤ α2 − α1, as it requires Θ1α1 − α1 < 2Θ2α2 − Θ1α1 +α1 − 2α2.

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Let Θ2α2−Θ1α1 ≥ 0. Comparing the boundaries of the conditions of Proposition3 we find that

2Θ1α1 − Θ2α2 + α1 + α2 − 2Θ2α2 + Θ1α1 − α1 − α2 = 3Θ1α1 − 3Θ2α2 ≤ 0,

and hence, Θ2α2 − Θ1α1 ≤ α2 − α1, and

2Θ1α1 − Θ2α2 − 2α1 + α2 − 2Θ2α2 + Θ1α1 − α1 + 2α2

= 3Θ1α1 − 3Θ2α2 − 3α1 + 3α2 ≥ 0.

Thus, for both conditions of Proposition 3 to hold simultaneously, we only need toimpose the first one, 2Θ1α1 −Θ2α2 − 2α1 + α2 < ρ < 2Θ1α1 −Θ2α2 + α1 + α2, asstated in this theorem. Otherwise, if Θ2α2 − Θ1α1 < 0, the condition is inversed,i.e., 2Θ2α2 − Θ1α1 + α1 + α2 < 2Θ1α1 − Θ2α2 + α1 + α2 and therefore, we have2Θ1α1 −Θ2α2 − 2α1 + α2 < ρ < 2Θ2α2 −Θ1α1 + α1 + α2, as stated in Theorem 1.

Next, with regard to inequality Θ2α2 −Θ1α1 ≥ 0, condition (iii) of PropositionA.1 may be reduced to 2Θ2α2−Θ1α1 +α1 +α2 ≤ ρ < Θ2α2 +α1 +α2. Specifically,by comparing the lower bounds we observe that

2Θ1α1 − Θ2α2 + α1 + α2 > Θ2α2 + α1 − α2,

since Θ2α2 − Θ1α1 < α2. In addition, we verify that the set defined by 2Θ2α2 −Θ1α1 + α1 + α2 ≤ ρ < Θ2α2 + α1 + α2 is not empty:

Θ2α2 + α1 + α2 − 2Θ1α1 + Θ2α2 − α1 − α2 = 2Θ2α2 − 2Θ1α1 > 0.

Similarly, with Θ2α2−Θ1α1 ≥ 0, condition (i) of Proposition A.2 can be expressedas ρ ≥ Θ2α2 + α1 + α2, since Θ2α2 + α1 + α2 > Θ1α1 + α1 + α2. Otherwise, ifΘ2α2 −Θ1α1 < 0 and Θ2α2 + α1 + α2 < Θ1α1 + α1 + α2, this condition is ensuredby ρ ≥ Θ1α1 + α1 + α2, as stated in Theorem 1.

If Θ2α2 −Θ1α1 < 0, then condition (i) of Proposition A.1 may be expressed as2Θ2α2 − Θ1α1 + α1 + α2 ≤ ρ < Θ1α1 + α1 + α2.

Specifically, since 2Θ2α2 −Θ1α1 + α1 + α2 < Θ1α1 + α1 + α2, the upper boundis greater than both lower bounds and with respect to Theorem 1 assumption,2Θ2α2 − 2Θ1α1 ≥ −2α1, we have 2Θ2α2 − Θ1α1 + α1 + α2 ≥ Θ1α1 − α1 + α2.

Finally, conditions (ii) and (iii) of Proposition A.2 are unfeasible under theassumption of this theorem as condition (ii) is based on Θ1α1+α1 ≤ Θ2α2−α2+α1

(thereby Θ2α2−Θ1α1 > α1−α2), and condition (iii) requires Θ2α2 +α2 ≤ Θ1α1 +α2 − α1 (thereby Θ2α2 − Θ1α1 ≤ −α1).

Proof of Proposition 4. Consider condition ρ ≤ Θ2α2−α2 from Theorem 1 andcondition ρ ≤ Θ1α1 + Θ2α1

χ − α1 − α1χ from Theorem 1 by Kogan et al. (2010). If

α2χ − α1 ≥ 0 and

Θ2α2 − α2 ≤ Θ1α1 +Θ2α1

χ− α1 − α1

χ. (A.5)

Then, according to these theorems, the system-wide optimal solution as well as theNash solution is to perform full inspection (intra-competition has no effect on the

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parties). Furthermore, condition (A.5) can be presented as

Θ2α2 − Θ1α1 − (α2 − α1) ≤ α1

χ(Θ2 − 1). (A.6)

Comparing (A.6) with our assumption, −α1 < Θ2α2−Θ1α1 ≤ α2−α1, and recallingthat Θ2 > 1, we observe that (A.6) always holds.

To complete this proof, we next consider the case of α2χ−α1 < 0, by comparingthe corresponding conditions: ρ ≤ Θ2α2 − α2 from Theorem 1 and ρ ≤ Θ1α2 +Θ2α2 − α2 − α2χ from Theorem 1 by Kogan et al. (2010). Thus we obtain that

Θ2α2 − α2 ≤ Θ1α2 + Θ2α2 − α2 − α2χ, (A.7)

or, α2χ(1 − Θ2) < 0, which is identical and always holds, as Θ2 > 1. That is, bothparties perform full inspection in this case as well.

Proof of Proposition 5. This proposition compares conditions from Theorem 1for the case of Θ2α2 − Θ1α2 > 0 with the corresponding conditions from Theorem1 in Kogan et al. (2010) for the case of α2 − α1 > 0. Given condition −α1 <

Θ2α2 − Θ1α1 ≤ α2 − α1, it is evident that if Θ2α2 − Θ1α1 > 0, then α2 − α1 > 0always holds.

We start by comparing the intervals defined by conditions Θ2α2 − α2 < ρ ≤2Θ1α1−Θ2α2−2α1 +α2 and ρ < Θ1α1 +Θ2α1−2α1 from Theorem 1 presented inthis paper and by Kogan et al. (2010), respectively. It is easy to observe that theseconditions become unfeasible for 1 ≤ Θ1,2 ≤ 2, as they include cases of ρ < α1 +α2

excluded by constraint (6).Thus, we only need to compare the conditions for partial inspections of the

Nash solution with the partial inspections for the centralized solution. Specifically,we require

xN1 =

13

(2Θ1 − Θ2

α2

α1−

(ρ − (α1 + α2)

α1

))< x1 =

12

(Θ1 + Θ2 − ρ

α1

),

which after some simple manipulations results in the conditions stated in thisproposition.

Proof of Proposition 6. The proof is similar to that of Propositions 4 and 5.We compare conditions from Theorem 1 for the case of Θ2α2 − Θ1α1 > 0 withthe corresponding conditions from Theorem 1 (Kogan et al., 2010) for the case ofα2 − α1 > 0.

To prove conditions (i), (ii), (vi) and (vii), we consider 2Θ1α1−Θ2α2+α1+α2 ≤ρ < Θ2α2+α1+α2 from Theorem 1 with Θ1α1+Θ2α1 ≤ ρ ≤ Θ1α2+Θ2α2+α1−α2

from Theorem 1 (Kogan et al., 2010).By comparing the left-hand sides of both conditions, we need to show that

2Θ1α1 − Θ2α2 + α1 + α2 ≥ Θ1α1 + Θ2α1, (A.8)

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or, equivalently,

Θ1α1 − Θ2α2 + α2 ≥ α1(Θ2 − 1). (A.9)

Based on −α1 < Θ2α2 − Θ1α1 ≤ α2 − α1 from Theorem 1, the left-hand side of(A.9) leads to Θ1α1 − Θ2α2 + α2 ≥ α1. In addition, based on our assumption,1 ≤ Θ2 ≤ 2, we find for the right-hand side of (A.9) that α1(Θ2 − 1) ≤ α1, that is,(A.7) always holds.

We next verify conditions (iii) and (viii). To prove them, we compare the right-hand sides of conditions 2Θ1α1 − Θ2α2 + α1 + α2 ≤ ρ < Θ2α2 + α1 + α2 fromTheorem 1 and Θ1α1 + Θ2α1 ≤ ρ ≤ Θ1α2 + Θ2α2 + α1 − α2 from Theorem 1(Kogan et al., 2010). Namely, parts (iii) and (viii) are correct if

Θ2α2 + α1 + α2 ≥ Θ1α2 + Θ2α2 + α1 − α2, (A.10)

or, equivalently,

0 ≥ α2(Θ1 − 2)2. (A.11)

The last inequality evidently holds as 1 ≤ Θ1 ≤ 2.Conditions (iii) and (viii) of Proposition 6 apply to the cases of partial inspec-

tions of the Nash equilibrium. The fact that the Nash solution implies less inspec-tions than the system-wide optimal partial inspections is shown by subtractingx2 = 1

2 (Θ1χ + Θ2 − ρ−(α2+α1)α2

) (Theorem 1 by Kogan et al., 2010) from xN2 =

12 (Θ2 − ρ−(α2+α1)

α2) (Theorem 1 of the current paper) which results in − 1

2Θ1 < 0.To complete the proof for parts (iv), (v), (ix) and (x), we compare the right-

hand side of 2Θ1α1 −Θ2α2 + α1 + α2 ≤ ρ < Θ2α2 + α1 + α2 from Theorem 1 withthe right-hand side of Θ1α2 + Θ2α2 + α1 − α2 < ρ < Θ1α2 + Θ2α2 + α1 + α2 fromTheorem 1 by Kogan et al. (2010).

Clearly, Θ1α2 + Θ2α2 + α1 + α2 > Θ2α2 + α1 + α2, as this inequality is reducedto Θ1α2 > 0, which always holds.

Proposition A.3. Let α2 − α1 < 0.

If ρ < min{2Θ2α2 − Θ1α1 + α1 + α2, Θ1α2 + Θ2α2}, then the manufacturerinspects less than the system-wide optimal full inspection policy.

Furthermore:

• If 2Θ1α1 −Θ2α2 − 2α1 + α2 < ρ < 3Θ1α2 −Θ2α2 + 2Θ1α1 − 2α1 − 2α2, then thewholesaler inspects more than the system-wide optimal partial inspection.

• If ρ = 3Θ1α2 − Θ2α2 + 2Θ1α1 − 2α1 − 2α2, then the wholesaler performs partialinspections and is unaffected by the competition.

• If 3Θ1α2−Θ2α2+2Θ1α1−2α1−2α2 < ρ < min{2Θ2α2 − Θ1α1 + α1 + α2,

Θ1α2 + Θ2α2

}, then the

wholesaler inspects less than the system-wide optimal partial inspections policy.

Proof. This proposition compares conditions from Theorem 1 for the case ofα2Θ2−α1Θ1 < 0 with the corresponding conditions from Theorem 1 (Kogan et al.,

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2010) for the case of α2 −α1 < 0. Given condition −α1 < Θ2α2 −Θ1α1 ≤ α2 −α1,it is straightforward that if α2 − α1 < 0, then α2Θ2 − α1Θ1 < 0 always holds.

We now compare the right-hand side of 2α1Θ1−Θ2α2−2α1+α2 < ρ < 2α2Θ2−Θ1α1 + α1 + α2 from Theorem 1 with the left-hand side of α2Θ1 + α2Θ2 − 2α2 <

ρ < α2Θ1 + α2Θ2 from Theorem 1 (Kogan et al., 2010) in order to show that:

2α2Θ2 − Θ1α1 + α1 + α2 > α2Θ1 + α2Θ2 − 2α2. (A.12)

Specifically, (A.12) can be expressed as α2Θ2−Θ1α1 +α1+3α2(3 − Θ1) > 0. Then,employing the assumption α2Θ2 − Θ1α1 + α1 > 0 and accounting for the fact that3α2(3 − Θ1) > 0, as 1 < Θ1 < 2, we observe that (A.12) always holds.

For part (ii), we only need to show that

xN2 =

12

(Θ2 − ρ − α2

α2

)< x2 =

12

(Θ1χ + Θ2 − ρ + α2χ − α2

α2

),

which is evidently true as 1 < Θ1 < 2 and χ = 1.Note that similar to the proof of Proposition 7, the lower boundaries (left-hand

sides) of conditions 2α1Θ1 −Θ2α2 − 2α1 +α2 < ρ < 2α2Θ2 −Θ1α1 + α1 + α2 fromTheorem 1 and α2Θ1 + α2Θ2 − 2α2 < ρ < α2Θ1 + α2Θ2 from Theorem 1 (Koganet al., 2010), are not feasible.

Proposition A.4. Let α2 − α1 < 0.

Assume Θ1α1 + α2 + α1 < Θ2α1 + Θ2α2,

(i) If 2Θ2α2 − Θ1α1 + α2 + α1 ≤ ρ < Θ1α1 + α2 + α1, then the manufacturerinspects less than the system-wide optimal full inspection policy and the whole-saler inspects less than the system-wide optimal partial inspection policy.

(ii) If Θ1α1 + α1 + α2 ≤ ρ < Θ2α1 + Θ2α2, then the manufacturer does notinspect, unlike the system-wide optimal full inspection policy, and the whole-saler inspects less than the system-wide optimal partial inspection policy.

(iii) If Θ2α1 + Θ2α2 ≤ ρ ≤ Θ1α1 + Θ2α1 − α1 + α2, then both parties do notinspect ; the manufacturer is affected by the competition as his system-wideoptimal policy is full inspection; and the wholesaler remains unaffected by thecompetition.

When 2Θ2α2−Θ1α1 +α1 +α2 < Θ1α2 +Θ2α2 < Θ1α1 +α1 +α2 < Θ1α1 +Θ2α1−α1 + α2,

(iv) If 2Θ2α2 − Θ1α1 + α1 + α2 ≤ ρ < Θ1α2 + Θ2α2, then the manufacturerinspects less than the system-wide optimal full inspection policy and the whole-saler inspects less than the system-wide optimal partial inspection policy.

(v) If Θ1α2 + Θ2α2 ≤ ρ < Θ1α1 + α1 + α2, then the manufacturer inspects lessthan the system-wide optimal full inspection policy, while the wholesaler doesnot inspect at all and is unaffected by the competition.

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(vi) If Θ1α1 + α1 + α2 ≤ ρ ≤ Θ1α1 + Θ2α1 − α1 + α2, then both parties do notinspect at all ; the manufacturer is affected by the competition as his system-wide optimal policy is full inspection; and the wholesaler is unaffected by thecompetition.

When 2Θ2α2 − Θ1α1 + α1 + α2 < Θ1α2 + Θ2α2 and Θ1α1 + Θ2α1 − α1 + α2 <

Θ1α1 + α1 + α2

(vii) If 2Θ2α2−Θ1α1+α1+α2 ≤ ρ < Θ1α2+Θ2α2, then the manufacturer inspectsless than the system-wide optimal full inspection policy and the wholesalerinspects less than the system-wide optimal partial inspection policy.

(viii) If Θ2α1 +Θ2α2 ≤ ρ ≤ Θ1α1 +Θ2α1−α1+α2, then the manufacturer inspectsless than the system-wide optimal full inspection policy, while the wholesalerdoes not inspect at all and is unaffected by the competition.

(ix) If Θ1α1 + Θ2α1 − α1 + α2 < ρ < Θ1α1 + α1 + α2, then the manufacturerinspects less than the system-wide optimal partial inspections policy, while thewholesaler performs no inspections and is unaffected by the competition.

When 2Θ2α2 − Θ1α1 + α1 + α2 > Θ1α1 + Θ2α1 − α1 + α2,

(x) If Θ1α2 +Θ2α2 ≤ ρ ≤ Θ1α1 +Θ2α1−α1 +α2, then the manufacturer inspectsless than the system-wide optimal full inspection policy, while the wholesalerinspects, unlike the system-wide optimal policy of no inspections.

(xi) If Θ1α1 + Θ2α1 − α1 + α2 < ρ < 2Θ2α2 − Θ1α1 + α1 + α2, then the manu-facturer inspects less than the system-wide optimal partial inspections policy,while the wholesaler inspects, unlike the system-wide optimal policy of noinspections.

(xii) If 2Θ2α2 − Θ1α1 + α1 + α2 ≤ ρ < Θ1α1 + α1 + α2, then the manufacturerinspects less than the system-wide optimal partial inspections policy, while thewholesaler does not inspect at all and is unaffected by the competition.

When Θ1α2 + Θ2α2 < 2Θ2α2 − Θ1α1 + α1 + α2 < Θ1α1 + Θ2α1 − α1 + α2 <

Θ1α1 + α1 + α2,

(xiii) If Θ1α2+Θ2α2 ≤ ρ < 2Θ2α2−Θ1α1+α1+α2, then the manufacturer inspectsless than the system-wide optimal full inspection policy, while the wholesalerinspects, unlike the system-wide optimal policy of no inspections.

(xiv) If 2Θ2α2−Θ1α1+α1+α2 ≤ ρ ≤ Θ1α1+Θ2α1−α1+α2, then the manufacturerinspects less than the system-wide optimal full inspection policy, while thewholesaler does not inspect at all and is unaffected by the competition.

(xv) If Θ1α1 + Θ2α1 − α1 + α2 < ρ < Θ1α1 + α1 + α2, then the manufacturerinspects less than the system-wide optimal partial inspections policy, while thewholesaler does not inspect at all and is unaffected by the competition.

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Otherwise, when Θ1α2+Θ2α2 < 2Θ2α2−Θ1α1+α1+α2 and Θ1α1+Θ2α1−α1+α2 <

Θ1α1 + α1 + α2,

(xvi) If Θ1α2+Θ2α2 ≤ ρ < 2Θ2α2−Θ1α1+α1+α2, then the manufacturer inspectsless than the system-wide optimal full inspection policy, while the wholesalerinspects otherwise than the system-wide optimal policy of no inspections.

(xvii) If 2Θ2α2 − Θ1α1 + α1 + α2 ≤ ρ < Θ1α1 + α1 + α2, then the manufacturerinspects less than the system-wide optimal full inspection policy, while thewholesaler does not inspect at all and is unaffected by the competition.

(xviii) If Θ1α1 + α1 + α2 ≤ ρ ≤ Θ1α1 + Θ2α1 − α1 + α2, then both parties do notinspect, the manufacturer is affected by the competition as his system-wideoptimal policy is full inspection, while the wholesaler is unaffected by thecompetition.

Proof. The proof for this proposition immediately follows from the assumptionsstated. For cases (ix), (xii) and (xv) it is easy to observe that

x1 =12

(Θ1 +

Θ2

χ− ρ − (α2 + α1)

α1

)> xN

1 =12

(Θ1 − ρ − (α2 + α1)

α1

)

is equivalent to Θ2 > 0, and therefore always holds.

Proof of Proposition 7. To prove both parts of the proposition, we need to showthat Θ1α1 +α1 +α2 < Θ1α1 +Θ2α1 +α1 +α2. This can be expressed as 0 < Θ2α1

which always holds.

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