Competitive Markets When Customers Anticipate Stockouts

41 Pages·2016·0.34 MB·English

Competitive Markets When Customers
∗
Anticipate Stockouts
CaryDeckA,B AmyFarmerA JoshuaFosterC
A UniversityofArkansas
B ChapmanUniversity
C UniversityofWisconsin-Oshkosh
August7,2016
Abstract
Retailersoftenworryabouthowshopperswillreactwhentheyfindanemptyshelf,but
theanticipationofexperiencingastockoutmayalsoimpactcustomerbehavior. Thisinturn
affects a retailer’s optimal price and inventory choices. In this paper we explore how the
feasibilityofvisitingasecondsellerinadifferentiatedproductduopolyimpactsmarketout-
comesusingacombinationoftheoreticalmodeling, experiments, andsimulations. Behav-
iorally,whenshopperscanonlyvisitasinglesellertheyoverweightheprobabilityofbeing
stockedoutandavoidasellertheyexpecttobetoocrowded. Aresultofthisisthataseller
maysellmoreunitsbystockingalowerinventory. Subjectsellersappeartoanticipatethis
reaction when making price and inventory choices. However, when it is costless to visit a
secondseller,customersfollowtheirdominantstrategies. Whilesubjectsellersareobserved
toengageinexcessivepricecompetition,theydonotunderstock.
Keywords: OnShelfAvailability,MarketEntryGame,ShopperReaction,Experiments
JELClassification: D40,G31,L15,C91
∗DeckgratefullyacknowledgesfinancialsupportfromtheCenterforRetailExcellenceforthisproject.Theauthors
wishtothanktheseminarparticipantsatWashingtonStateUniversity,theattendeesofthe2014SouthernEconomic
AssociationAnnualMeetings,andBrendanJoyceforexcellentresearchassistance.
1
COMPETITIVEMARKETSWHENCUSTOMERSANTICIPATESTOCKOUTS 2
“Noonegoesthereanymore. It’stoocrowded.” -YogiBerra
Stockoutsarepotentiallycostlytoretailerswhentheyrepresentmissedsalesopportunities.
Ashoppersearchingforaspecificitemmaydecidetopurchasenothing,purchaseasubstitute
good, or shop elsewhere if faced with an empty shelf.1 Gruen et al. (2002) find that about
half of stockouts result in the purchase of a substitute product. In this case the profitability
ofthesellermaybepositivelyornegativelyaffecteddependingontherelativeprofitabilityof
thesubstituteitem(seeKamakuraandRussell1989foranempiricalstudysuggestingpeople
tendtotradeup). MahajanandvanRyzin(1999)provideageneralsurveyoftheliteratureon
the impact of substitution on inventory management. Alternatively, the shopper may simply
decidetoabandontheirshoppingcartandvisitacompetitorinstead.Thisdecisiondependson
thepriceandselectionofferedbyarivalstore.2 Matsa(2011)findsthatgrocerystoreswithmore
competition,especiallyfromWalmart,aremorelikelytoavoidshortfalls.3 Ofcourse,retailers
mayprefertorunoutofstockiftheycanofferrainchecks(seeHessandGerstner1987)oras
part of a bait and switch (see Wilke, et al. 1998a,b and Hess and Gerstner 1998). Even when
shoppersarewillingtowait,thesellermayincurbackordercompensationcoststhatnegative
impactprofits(Chenetal. 2015). Theimpactofstockoutscanextendbeyondasingleshopping
trip. Andersonetal. (2006)showthatcurrentstockoutsimpactfuturepurchases(seealsoJing
andLewis2011).
Recently, there have been several papers dealing with the optimal pricing of a good con-
ditional on inventory levels (see Chen and Simichi-Levi 2010 for a survey).4 One inventory
situationthathasbeenstudiedextensivelyismarkdownpricing,wherethesellerhasmultiple
periodsoverwhichitsellsproductsandcanmarkremaininginventorydowntoinducesales.
Forexample,CachonandSwinney(2009)considertheinventoryproblemfacedbyasellerwho
anticipatesdroppingitspriceatsomepointduringtheseason. LiuandvanRyzin(2008)con-
siderstockoutriskasawaytoinducecustomerstobuyearlierintheseason(seealsoAvivand
Pazgal 2008, Allon and Bassamboo 2011, and Liu and van Ryzin 2011). Many of these opti-
1ConcernabouthowshoppersreacttoastockoutgoesbacktoatleastWalterandGrabner(1975). Morerecently,
Honhon, etal. (2010)offersadynamicprogrammingsolutiontotheoptimalassortmentproblemwhencustomers
engageinstockout-basedsubstitution.
2BalachanderandFarquhar(1995)suggestthatinsomecircumstancesstockoutsmayhaveapositiveeffectonprice
competition.
3SeeChaouch(2001),NagarajanandRajagopalan(2008)forsimilarmethodsofminimizinginventoryshortages.
4Whilemuchoftheliteratureoninventorypricingisseparatefromtheliteratureonsubstitutabilitybetweenprod-
ucts,arecentpaperbyTranschel(2011)looksattheinterplaybetweenthetwo.
3 DECK,C.,A.FARMERANDJ.FOSTER
malpricingpapersfocusonsellerswhoareinsulatedfromcompetitionordonottakeshopper
expectationsintoaccount.5
If shopping is costly (in terms of time, gas, etc.), then customers may be captive to the
retaileronthatshoppingtrip.However,priortovisitingtheseller,customerswillchoosewhere
to shop based upon their perceived likelihood of finding the desired product in stock, which
is a function of the quantity the retailer carries and the expected behavior of other shoppers,
in addition to standard considerations such as price (travel costs, loyalty, etc.). This creates a
variation of a “market entry game” played by shoppers who want to coordinate their actions
aseachshopperpreferstonotexperienceanemptyshelf, butwantstopursuethebetterdeal
ifshewillbesuccessful. Inthetraditionalmarketentrygame,playersprivatelydecideifthey
wish to take a sure payoff or enter a pool where payoffs are decreasing in the total number
of entrants. This game is modeled on firms deciding to enter a market where a monopolist
would earn more than a duopolist who in turn would earn more than a triopolist and so on.
The tension arises because there is a threshold number of entrants below which one wants to
enterandabovewhichonedoesnot. Hence,inequilibriumonlysomefirmsshouldenterthe
market,butabsentasymmetriesthereisacoordinationproblemastowhotheentrantsshould
be. Controlled laboratory experiments have consistently found that people quickly converge
toequilibriumbehaviorinaggregatedespiteconsiderableindividualheterogeneity(Rapoport
1995,Sundali,etal. 1995,Rapoport,etal. 1998,Ochs(1999)andRapoportandSeale(2008). In
fact,thispatternissostrikingthatKahneman(1988)describesitasbeing“magic.”However,as
thecoordinationproblembecomesmoredifficultdueforexampletooverconfidence(Camerer
andLovallo1999)orambiguity(BrandtsandYao2010)excessentryisoftenobserved.
When sellers create a market entry game for shoppers, carrying a larger inventory may
induce more shoppers to visit a store if the shoppers believe that they are more likely to find
the item. Alternatively, a seller could end up with excess inventory on the shelf if customers
falselyanticipateastockoutbecausetheyoverestimatetheprobabilitythatothershopperswill
visittheseller. EveryoneavoidingastoretheybelievewillstockoutistheretailvariantofYogi
Berra’sfamousquipaboutnoonegoingthereanymorebecauseitistoocrowded. Theresultis
thathavingexcessinventorycouldbeasignthattoolittleortoomuchinventorywasordered.
Inthissetting,pricebecomesadoubleedgedswordfortheretailer. Loweringthepricemakes
the seller more attractive, which should increase the number of shoppers who visit; but, this
5NagarajanandRajagopalan(2009)lookatamodelwherefirmsconsiderstockout-basedsubstitutionandcompeti-
torinventoryinanewsvendorstyleduopoly,whileAnderhubetal.(2003)providesexperimentalevidenceforpricing
strategiesinduopolieswithheterogeneousgoodsandcapacityconstraints.
COMPETITIVEMARKETSWHENCUSTOMERSANTICIPATESTOCKOUTS 4
makesavisitriskierandmayhavetheunwantedeffectofactuallydiscouragingshoppers.
This paper examines how the potential for stockouts affects buyers’ decisions regarding
where to shop. We first construct a theoretical model with a three stage game in which two
sellers offer differentiated products with one being superior to the other (i.e. customers have
agreaterwillingnesstopayforthehighqualityseller’sproduct). Thehighqualitysellerpre-
commitstoaninventorylevel,thenbothsellerspostprices,andfinallyshoppersselectwhich
seller to visit and attempt to make purchases. If it is costless to visit the low quality seller
afterexperiencingastockoutatthehighqualityseller,thenallshoppersinitiallyvisittheseller
offeringthebestdeal. However,ifitisprohibitivelycostlyforshopperstovisitasecondstore,
thenthenumberofshopperswhovisiteachsellerdependsontheinventorylevelofthehigh
qualitysellerandthebuyersurplus(valueminusprice)ateachlocation.Themodelwedevelop
isintheveinofDeneckereandPeck(1995);however,intheirmodel,sellersofferahomogenous
product and shoppers are limited to visiting a single seller (see also Peters 1984). Despite the
relativesimplicityofthemodel,itisnotpossibletofindananalyticalsolutionforasymmetric
equilibrium.6 Therefore, we rely upon simulations and controlled laboratory experiments to
understand this situation. Behaviorally, we find that shoppers are reluctant to visit the high
quality seller as its inventory increases when stockouts are costly, a result that is consistent
withprobabilityweightingandyieldsbehaviorinlinewithYogiBerra’sstatement.
1 Theoretical Model
Supposetherearetwosellersofferingverticallydifferentiatedproducts,oneofhighquality
andoneoflowquality,tonshoppers. DefinethetwosellersasH andL,respectively. Assume
each of the n shoppers desires only one unit. Further, assume all shoppers are identical and
value the respective products at V and V , where V > V . Shoppers each independently
H L H L
make a decision regarding the product to purchase, and visit that seller initially. However,
thereisapossibilityofastockoutatthehighqualityseller. Ifthisoccurs, ashoppercanthen
choose to visit the low quality seller at some cost, which is captured by depreciating V by a
L
factorofδ. Forsimplicityweconsiderthecaseswhereδ =0,1capturingaprohibitivecostand
nocostrespectively. Further,itisassumedthelowqualitysellerwillhavesufficientquantityto
servethemarket.
6Therearealsoasymmetricequilibriasuchasexactlymshoppersoutofnfollowapurestrategytovisitaparticular
seller. Thereareotherequilibriawheresomeshoppersfollowpurestrategiesofvisitingthehighqualitysellerornot
whileothershoppersdosoprobabilistically.
5 DECK,C.,A.FARMERANDJ.FOSTER
Given this structure, the type H seller will choose the inventory capacity C ∈ [0,C] they
wouldliketocarry,whereitisassumedC < n. Highqualityinventoryhasaperunitholding
costK < V ,whichissunkoncetheproductisprocured. Thiscostcanbethoughtofeither
C H
as a direct cost associated with the high quality product such as an actual price paid to the
supplierorthecostofadjustingdisplayspaceoranopportunitycostassociatedwithforegone
salesofsomealternativeproductthatcouldhavebeencarried. Forsimplicity,bothsellersare
assumedtohavenomarginalcost.Oncethehighqualityseller’sinventorylevelisdetermined,
itbecomescommonknowledge,bothsellersprivatelyandsimultaneouslysetprices,andthen
shoppersmaketheirpurchasingdecisions.7
Giventhisgeneralsetup,thestagesofthegameareasfollows:
Stage0: Naturechoosesthedepreciationfactor,δ,andthenumberofshoppers,n.
Stage1: ThetypeH sellerchoosesitsinventorylevel,C conditionalonδandn.
Stage2: EachsellerchoosesP fori=H,LconditionalonC,δ,andn.
i
Stage3: ShoppersmakepurchasedecisionsconditionalonP ,P ,C,δ,andn.
H L
Throughout this paper the business setting is taken to be a retail market where the high
qualitysellercanbethoughtofasasmallspecialtyshopwithlimitedspaceandthelowquality
sellercanbethoughtofasageneralmerchandiserthatcarriesmanysimilarsubstituteproducts.
Inthissense,thecapacitychoicecanbetakenasthecurrentinventoryofthespecialtyretailer
or as its store size, which was a choice variable when the specialty shop opened. However,
the model applies equally well to other business settings. For example, boutique restaurants
oftenhavelimitedseatingorsuppliesoffreshqualityingredientswhilefastfoodchainshavea
seeminglylimitlesssupply.
Before analyzing the game we note that although our setup is similar to the newsvendor
problem(e.g. SchweitzerandCachon2000andNagarajanandRajagopalan2009)itisdistinct
inthatdemandisendogenouslydeterminedbythestrategicinterplayoffirmsengagedinprice
settingcompetition.8 OurmodelisalsosimilartothequantityprecommitmentmodelofKreps
andScheinkman(1983). However,inourcasecustomersonlyfacethepossibilityofastockout
at one of the sellers. The reason for this design choice is that we are particularly interested
in how the possibility of experiencing a stockout impacts shopper behavior. If stockouts are
possibleatbothsellers,thentheout-of-stockriskmaybeneutralized. Onlyallowingthehigh
7ForadiscussionofasimilarproblemwhereavailabilityisunobservedseeDana(2001).
8Althoughpricehasbeenintroducedintothenewsvendorproblem,itistypicallytreatedasashiftinastochastic
demand(seePetruzziandDada1999).
COMPETITIVEMARKETSWHENCUSTOMERSANTICIPATESTOCKOUTS 6
qualitysellertostockoutcreatesthedesiredshoppertensionbetweenasafeoptionandarisky
dealwithahigherpotentialpayoff.
Solving for the equilibrium requires backward induction; thus we begin with the shopper
behaviorinstage3. Giventhatweconsiderhomogeneousshopperswhoarenotabletocoordi-
natetheiractions,wefocusonsymmetricequilibria. Itisassumedthenrisk-neutralshoppers
willchoosetheproductthatoffersthegreaterexpectedsurplus. Assuch,thedecisionwhether
to seek the high quality product depends upon their valuation of both products less the re-
spective prices as well as the probability one can obtain the product from the type H seller if
visited.
Each consumer’s strategy space is as follows: {H,L}, {H,∅}, {L,∅} and {∅,∅} where
thefirstelementistheretailertheyvisitinitially,andthesecondentryindicateswhetherthey
wouldvisitLconditionedonhavingexperiencedastockoutatH initially. Thisstrategyspace
alsoreflectstheabilityforashoppertovisitneithersellershouldbothofferanegativesurplus.
Since pricing above value results in zero revenue to the seller, a type i seller will set P ≤ V .
i i
IntheeventofastockoutatH,itisassumedthattheC unitsarerandomlyallocatedtothem
shopperswhovisitedH.9
Letλdenotetheprobabilitythatanindividualshoppervisitsthehighqualitysellerinitially.
The surplus that a shopper obtains from a type i seller is given by V −P . If the low quality
i i
selleroffersasurplusgreaterorequaltoH,thenallshopperswillvisitL.10 Thatis,ifV −P ≥
L L
V −P thenλ*=0.
H H
If,ontheotherhand,V −P <V −P thenshoppersfaceatensionbetweenthecertain
L L H H
yetlowersurplusfromLandtryingforthehighsurplusfromH.Theirdecisiondependsonthe
chanceofnotbeingstockedoutatHandthedepreciatingfactor,δ.Letp(λ|C,Q)representthe
probabilityashoppersuccessfullymakesapurchaseatthehighqualityseller,iftheQ=n−1
otherconsumersgowithprobabilityλtosellerH whohasC units. Theexpectedsurplusfrom
thedecisiontovisitH isgivenbythefollowing.
p(λ|C,Q)(V −P )+[1−p(λ|C,Q)]max{δV −P ,0} (1)
H H L L
WhenV −P <V −P thevalueδtakesin(1)willeithermakevisitingH firstariskyor
L L H H
risklessdecisionfortheshopper. Thustherearetwocasestoconsider.
9Thiswouldbethecaseifarrivalorderwasrandomandcustomerswereservedonafirst-comefirst-servedbasis.
10Thetheoreticalresultspresentedbelowarenotsensitivetothistie-breakingassumption.
7 DECK,C.,A.FARMERANDJ.FOSTER
1.1 Case1: ConsumersFaceACostlessStockout(δ = 1)
1.1.1 Stage3: ShoppersMakePurchasingDecisions
In this case a shopper who plays {H,L} and experiences a stockout will receive the same
surplus as a shopper who plays {L,∅}. The condition to visit H becomes p(λ | C,Q)(V −
H
P )+[1−p(λ|C,Q)](V −P )>V −P ,whichholdssolongasV −P <V −P . Thus
H L L L L L L H H
inthiscasethereisadominantstrategytovisitthehighqualitysellerfirst,λ*=1.
1.1.2 Stage2: SellersSetPrices
In stage 2, H and L simultaneously choose their respective prices, P and P , given the
H L
high quality seller’s capacity, C, and anticipating the reaction of shoppers in stage 3. When
δ = 1, L’sprofitfunctionisdiscontinuous. IfV −P ≥ V −P , thennshopperswillvisit
H H L L
H first. Thisleadstoastockout,andthen−C shopperswhowereunabletoprocureaunitat
H willvisitL. However,ifLsetsapricethatyieldsV −P <V −P ,thenLcapturesalln
H H L L
shoppers. Equation(2)definesL’sprofitfunction.

(n−C)PL ifVH −PH ≥VL−PL
Π = (2)
L
nPL otherwise
Thefirstcomponentinequation(2)showsLcansetarelativelyhighprice,butensuresitwill
receiverelativelyfewshopperswhenitdoesso. Thesecondcomponentinequation(2)shows
Lcanservetheentiremarket,butmustsetarelativelylowpricetodoso.
H’s profit is also discontinuous. Should H offer a non-negative surplus and V −P ≥
H H
V −P then H will sell all C units in its inventory. However, should it be that V −P <
L L H H
max{V −P ,0}thenH sellsnothing. ThisleadstothefollowingprofitfunctionforH.
L L

C(PH −KC) ifVH −PH ≥VL−PL,0
Π = (3)
H
−CKC otherwise
From(3)itisclearthatH wantstoofferthegreatersurplusandthuspreventLfromcapturing
the entire market. Additionally, L will also want to offer the greater surplus when the profit
from doing so is greater than offering the smaller surplus and capturing the residual market,
n − C. Given P , the profit to L for offering the greater surplus is greater than that from
H
COMPETITIVEMARKETSWHENCUSTOMERSANTICIPATESTOCKOUTS 8
capturingtheresidualmarketwhenn(V −(V −P )) > (n−C)V whereV −(V −P )
L H H L L H H
isthepricethatequatesthesurplusfromeachfirm. RearrangingintermsofP wefindthatL
H
willpreferofferingagreatersurplusthanH whenP > V − CV ,whichisstrictlypositive
H H n L
giventheassumptionsthatV >V andC <n.
H L
Itisrelativelystraight-forwardtodemonstratetherewillbeamixedstrategyinpricingfor
eitherfirmgiventheseassumptions. Forinstance,ifH setsapriceofP =V −CV ,thenL’s
H H n L
bestresponseisP∗ = V ,anofferofzerosurplustotheshopper. However,H’sbestresponse
L L
to L’s zero surplus is to offer a surplus marginally greater than zero, thus raising its price to
P∗ = V − (cid:15). Yet now that P − (cid:15) > V − CV , L has an incentive to offer the greater
H H H H n L
surplustocustomers. This‘one-upping’insurplusofferingsbetweenfirmsdemonstratesthat
theequilibriumpricingstrategywillbemixed.
To define the mixed strategy over prices, let the surplus, s , offered by firm i, be drawn
i
from the distribution F where F (s ) = 0 and F (s ) = 1. It is assumed that F (·) is differen-
i i i i i i
tiableeverywhere. Theprobabilitydensityfunctionwillbedenotedbyf ,whichdescribesthe
i
probabilitywithwhichafirmwillchooseagivensurplus.
GiventhemixedstrategyofH andthatP = V −s ,theexpectedprofittoLofofferinga
i i i
surplusofsis
F (s)(V −s)n+(1−F (s))[n−C](V −s). (4)
H L H L
TogenerateindifferenceinL,weset(4)equaltothegreatestprofitLcanunilaterallyguarantee
itselfwithapurestrategy. The‘securityprofit’thatLcanunilaterallyguaranteeitselfisgiven
bysettingapriceP∗ =V andthereforesellingn−C units. Thus,Πs =(n−C)V . Setting(4)
L L L L
equaltoΠs andsolvingforF (s)yieldsthemixedstrategy
L H
n−C s
F (s)= (5)
H C V −s
L
wheres∈(cid:2)0,CV (cid:3). TheexpectedsurplusforH,E [s],canbecalculatedusingtheprobability
n L H
densityfunctionfromthisexpression,whichleadsto
(cid:90) CnVL (cid:18) n−C (cid:18) n (cid:19)(cid:19)
E [s]= sf (s)ds= 1− ln V (6)
H H C n−C L
0
Toexpressthisresultintermsofexpectedprices,wefindthat
(cid:16) (cid:16) (cid:17)(cid:17)
E [P ]=V − 1− n−C ln n V .
H H H C n−C L
9 DECK,C.,A.FARMERANDJ.FOSTER
Similarly,theexpectedprofittoH forofferingasurplusofsis
F (s)[(V −s)C−K C]+(1−F (s))[−K C]. (7)
L H C L C
ThesecurityprofitforH isdeterminedbythegreatestpriceH cansetandnotgetundercutby
L,whichwaspreviouslydeterminedtobeP = V − CV . Usingthispricetodeterminethe
H H n L
security profit for H we find Πs = (cid:0)V − CV −K (cid:1)C. Equating (7) to Πs and solving for
H H n L C H
F (s)wefind
L
V − CV
F (s)= H n L (8)
L V −s
H
wheres∈(cid:2)0,CV (cid:3). CalculatingL’sexpectedsurpluswefind
n L
E [s]=(cid:90) CnVLsf (s)ds= CV −(cid:18)V − CV (cid:19)ln(cid:32) VH (cid:33) (9)
L L n L H n L V − CV
0 H n L
Thus,theexpectedpriceofLcanbeexpressedas
EL[PL]=(cid:0)1− Cn(cid:1)VL+(cid:0)VH − CnVL(cid:1)ln(cid:16)VH−VHCnVL(cid:17).
1.1.3 Stage1: HighQualitySellerSetsInventory
Finally,inconsiderationofstage1behaviorweseektheinventoryCthatmaximizesH’sex-
pectedprofit. Giventhemixedstrategyinpricing,thisisequivalenttomaximizingitssecurity
profit, ΠsH = (cid:0)VH − CnVL−KC(cid:1)C. Therefore C∗ = min(cid:110)n2VHV−LKC,C(cid:111). Table 1 summarizes
theexpectedpricesandprofitstobothsellersasH’sstage1capacitydecision, C, changesfor
aparticularsetofparameters. Theseparametervaluesarethesameasthoseusedinthelabo-
ratory experiments described in the next section. In particular we set n = 6, C = 5, K = 3,
C
V =15,andV =9. FromTable1itisclearthatwiththeseparameters,C∗ =4.
H L
Table1: ExpectedPricesandProfitsforδ =1
C P∗ Π∗ P∗ Π∗
H H L L
1 14.2 10.5 8.9 45
2 13.3 18 8.7 36
3 12.2 22.5 8.3 27
4 10.9 24 7.6 18
5 9.2 22.5 6.7 9
COMPETITIVEMARKETSWHENCUSTOMERSANTICIPATESTOCKOUTS 10
AsummaryofCase1predictionsovershopperandsellerbehaviorisprovidedasResult1
below.
Result1.Whenitiscostlessforshopperstoexperienceastockout,shoppersfollowapurestrat-
egyandbothsellerspursueamixedstrategyoverthepricestheycharge.Further,theinventory
ofthehighqualitysellermaybelessthanorequaltotheircapacityconstraint. Formally,

1 ifVH −PH ≥VL−PL
λ∗ =
0 otherwise
n−C s
F (s)=
H C V −s (10)
L
V − CV
F (s)= H n L
L V −s
H
(cid:26) (cid:27)
nV −K
C∗ =min H C,C .
2 V
L
1.2 Case2: ConsumersFaceAProhibitivelyCostlyStockout(δ = 0)
1.2.1 Stage3: ShoppersMakePurchasingDecisions
In this case, the condition for a shopper to find it optimal to pursue {H,∅} rather than
{L,∅}isp(λ|C,Q)(V −P )>V −P . Giventhesymmetry,ifthisshopperfindsitoptimal
H H L L
tovisitH thentheQ = n−1othershopperswillalsofinditoptimalandthechancethatany
particular shopper is not stocked out is C. This implies that when V −P < C(V −P ),
n L L n H H
λ∗ =1.
Whenp(λ|C,Q)(V −P )=V −P ashopperisindifferentbetween{H,∅}and{L,∅}
H H L L
andthereisamixedstrategyequilibrium. Toidentifythemixedstrategyequilibriumonemust
expresspasafunctionofλ,C,andQasin(11).
C−1(cid:18) (cid:19) Q (cid:18) (cid:19)
(cid:88) Q (cid:88) C Q
p(λ|C,Q)= λm(1−λ)Q−m+ λm(1−λ)Q−m (11)
m m+1 m
m=0 m=C
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
Pr(m<C)⇒ Pr(m≥C)⇒
ReceivesItematH RandomlyReceivesItematH
The first summation in (11) represents the probability that H does not stock out. The second
summation in (11) represents the probability that the shopper is able to buy the item from H
whenthereisastockout. Theindexvariablemisthenumberofothershopperswhoattempt