Prominence, complexity, and pricing

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ContentslistsavailableatScienceDirect

International

Journal

of

Industrial

Organization

www.elsevier.com/locate/ijio

Prominence,

complexity,

and

pricing

R

Ioana Chioveanu

DepartmentofEconomicsandFinance,BrunelUniversityLondon, UxbridgeUB83PH,UK a r t i c l e i n f o Articlehistory: Received23May2018 Revised30October2018 Accepted29December2018 Availableonline16January2019 JELclassiﬁcation: D03 D43 L13 Keywords: Oligopolymarkets Consumerconfusion Prominence Pricecomplexity Pricedispersion a bs t r a c t

This paper analyzes prominence ina homogeneous product market where two firms simultaneously choose both prices and price complexity levels. Market-widecomplexity results in consumer confusion. Confused consumers aremorelikely tobuyfromtheprominentfirm.Inequilibrium,thereis dis-persion in both prices and price complexity. The nature of equilibrium dependson prominence. Compared to its rival, theprominentfirmmakeshigherprofit,associates asmaller pricerangewithlowestcomplexity,putslowerprobabilityon lowestcomplexity,andsetsahigheraverageprice.However, higherprominencemaybenefitconsumersand,conditionalon choosinglowestcomplexity,theprominentfirm’saverageprice islower,whichisconsistentwithconfusedconsumers’bias.

R _Ioana_Chioveanu _thanks_the _Editor,_Yongmin _Chen, _and _two_anonymous _referees_for _their _valuable

comments,whichsigniﬁcantlyimprovedthepaper.SheisgratefultoU˘gurAkgün,AlbertBanal-Estañol, John Bennett, YiquanGu, Paul Heidhues,Evagelos Paﬁlis, Alexei Parakhonyak, Pierre Régibeau,Rani Spiegler, Tobias Wenzel,Chris Wilson, JidongZhou, and audiences at the annual meetings of ASSET (2016),RES(2017),EEA(2017),CES(2017),atKing’sCollegeLondon,andatBrunelUniversityLondon forusefulcomments.Theusualdisclaimerapplies.

E-mailaddress:[email protected]

https://doi.org/10.1016/j.ijindorg.2018.12.005

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1. Introduction

Price complexity is a common feature of many markets, including those for retail financialandbankingproducts,andretailsupplyofgasandelectricity.Itstemsfromthe useofmulti-parttariffsor partitionedprices,involvedortechnicallanguage,ordifferent priceformatsorinformationdisclosuremethods.Amainconcernisthatcomplexpricing stifles competition by making it harder for consumers to understand firms’ offers and identify thebestdeal.

The2015 UKCompetitionand MarketAuthorityinvestigationof theretail banking marketfoundthat“[t]herearebarrierstoaccessingandassessinginformationonPersonal Current Accountcharges” and “overdraft charges are particularly diﬃcult to compare acrossbanks,duetoboththecomplexityanddiversityofthebanks’chargingstructures”. The 2011 report by the UK Independent Commission on Banking mentions “evidence that complexity in pricing structures makes it diﬃcult for consumers to receive good value”. The 2007 EC study of EU mortgage credit markets and Woodward and Hall’s 2012studyofUSmortgagemarketsechotheseconcerns.1

Price complexityincreases thetime (oreffort) consumers needtomake a choiceand thelevelofcognitiveabilitiesandsophisticationrequiredtofindthebestoffer.So,itmay leadtoconsumerconfusionandallowhomogeneousproductsellerstosoftenprice compe-titionandincreasetheirprofits.2Experimentalresearchindicatesthatmorefragmented multi-parttariffscancreateconfusionandleadtosuboptimalconsumerchoices(see,for instance, Kalaycı and Potters, 2011; Kalaycı,2015). These findingsare consistent with evidence from the marketing literature that partitioned (or involved) pricing makes it difficultforconsumerstocomparecompetingoffers(Greenleafetal.,2016reviewsrelated work).Evidenceofbehavioralbiases hasalsobeenfoundfor USretail financeproducts (mortgagebrokerage,loans,andcreditcardservices)byWoodwardandHall(2012)and

StangoandZinman(2009a,b).3

Insome markets whereprice complexityleads to consumerconfusion, the choicesof confused buyers are aﬀected by ﬁrm prominence, which may be due to higher brand recognition (e.g., for a pioneer or incumbent product or an intensely advertised one), to product recommendationsmade byanexpert,agent, or other consumers,to a more salientlocation(ateye-level,inadisplay,oratthetopofanonlinesearch-outcomelist), or to consumers’loyalty to an already familiar brand. See Armstrong et al. (2009) for a discussionofempiricalevidenceonprominence.Forinstance,consumerswhoshopfor a mortgage or for insurance may be biased towards considering their current-account

1_Carlin₍₂₀₀₉₎_discusses_empirical_evidence_of_price_complexity_in_ﬁnancial_markets_and_concludes_that “manyofthehouseholdswhopurchaseretailﬁnancialproductsdonotunderstandwhat theyarebuying andhowmuchtheyarepayingforthesegoods”.

2_When_facing_complex_{tariﬀs/markets,}_some_consumers_may_rationally_opt_out_of_information_processing duetoitshighcost.Or,theymaybeunabletodealwiththecomplexitybecausetheyhavepoornumeracy skillsand/ormisjudgetheinformation.

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bank. Inretail energymarketsthat werepreviouslymonopolized, consumers mayfavor the‘familiar’regionalincumbent overnewentrants.4

Thispaperexplorestherelationshipbetweenpricecomplexityasanobfuscationdevice and firm prominence and its implications in otherwise homogeneous product markets. We analyzethe impact ofprominence onfirms’ pricingand complexity choices and on marketoutcomes.Inourmodel,aprominentselleranditsrivalcompeteforaunitmassof identicalconsumerswithunitdemands.Firmssimultaneouslyandindependentlychoose both theirprices and price-complexitylevels.Thetiming reflectsthe factthat in many environments,includingbankingandfinancialmarkets,firmscanchangerelativelyeasily thepriceformatsor thetechnicallanguage employedintheirpricedisclosures.

We formalize price complexity by allowing eachfirm to select a level from a closed interval. Afirm’schoice ofcomplexityaffects consumers’abilityto understanditsprice offer. As a result, the firms’ complexity choices affect market composition: some con-sumersareexpertsandpurchasethelowest-priceproduct,whileothersareconfused.5 _A marginal increaseina firm’scomplexitylevelincreases theshareofconfusedconsumers in themarket.Confusedconsumersareunableto assessthefirms’pricesandmake ran-dom choices, but arerelatively more likely toselectthe prominentproductas it enjoys higherrecognition.6

Weshowthatthenatureoftheequilibriumdependsontherelativeprominenceofthe two firms.Bothfirms havetobalanceconflictingincentiveswhensettingtheirprices:to competeaggressivelyfortheexpertsandtoexploittheconfusedconsumers.Buttheless prominentfirmhasstrongerincentivestocompeteaggressivelyasithasasmallerbaseof confusedconsumers.Inequilibrium,thisfrictionrulesoutpurestrategypricing,so both firms randomize on prices. Moreover, theprominent firm also randomizes between the lowestandthehighestpricecomplexitylevelsand,formoderatelevelsofprominence,so doesthelessprominentseller. However, iftheprominencelevel ishighenough,theless prominent sellerchooses thelowestcomplexityforsureasitbenefits more frommarket transparency.

In equilibrium, whenever a firm randomizes on complexity, there is a positive re-lationship between prices and complexity levels.7 _When _setting _a _relatively _low _price, a firm benefits from a lower complexity level as this is associated with a higher frac-tion of experts. In contrast, when a firm sets a relatively high price, it may benefit

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Hortaçsu et al. (2017) show that inattention andincumbent brands’ advantagesare sources of con-sumerinertiaintheTexanresidentialelectricitymarket.AnalysingMexico’sprivatesocialsecuritymarket,

Hastingsetal.(2017)showthatﬁrms’advertisingandsalesspending(whichcanberelatedtoprominence) aﬀects low-incomeorprice-inelasticconsumers’choices.SeealsoGiuliettiet al.(2014)forevidencefrom Britishelectricitymarkets.

5_This_model_is_open_to_a_default-bias_{interpretation}_whereby_consumers_are_initially_assigned_to_one_firm, theprominentfirmhasaninitialadvantage(i.e.,alargerbaseofconsumers),andtheextentofconsumer inertia (i.e., theshare ofbuyers whoupholdtheir default option)is endogenously determinedby firms’ complexitychoices.

6_When_confused,_the_consumers_may_use_{intermediaries}_who_steer_them_towards_the_prominent_product, mayrelyonpersuasiveadvertisements,ormayhavestrongerdefaultbiases.

7_Armstrong _and_Chen₍₂₀₀₉₎ _and_Chioveanu₍₂₀₁₂₎_identify_positive _{relationships}_between _prices_and productqualitiesinmodelswhereﬁrmsrandomizeonbothdimensions.

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fromchoosingahighcomplexitylevel,providedthatitservesa largeenoughfraction of confusedconsumers.

Themarketoutcomesreflectthedifferencesinproductsalience.Theprominentseller makes higherprofits,chooses thehighest complexitylevel withhigher probability than therival,setsalowercut-off pricebelowwhichpricesareassociatedwiththelowest com-plexity, andchoosesthemonopolypricewith positiveprobability. Asitsellsto alarger share of confused consumers,the salient firm is more likely to choose high complexity and also, fora given complexitylevel, its incentiveto set a high priceis stronger. The lessprominentseller’s priceis alwaysbelow themonopolylevel anditsaverage priceis lowerthanthat oftherival.8

Anincrease in thelevel of prominencemay lead to lowerindustry profitand higher consumer surplus. Such an increase affects firms’ pricing directly, as it reallocates the confused consumers in favour ofthe salient firm.Moreover, ithas anindirect effect on pricing as it affects firms’ probabilitiesof choosing lowest complexity. As a result, our frameworkhighlightsanovelchannelthroughwhichprominenceaffectsmarketoutcomes, relatedtothecut-off structureoffirms’equilibriumstrategies.Anincreaseinprominence (weakly)increasestheprobability withwhichthelessprominentfirmchoosesthelowest complexity, whileit strictly decreasesthecorresponding probability of thesalient firm. This tension underlies the non-monotonicity of consumer surplus in prominence. One implication is that in an environment where less prominent firms (e.g. new entrants) increase their relative salience (for instance, through advertising investments or sales efforts), thiscouldbedetrimentaltoconsumers.

Conditionalonchoosinglowestcomplexity,theprominentfirm’saveragepriceislower. In this sense,confused consumers’ bias for the prominent seller is consistent with the ranking of theaverage prices conditional on low complexity. Inan extension, we show that a qualitatively robust cut-off mixed strategy equilibrium exists for more general confusiontechnologiesifthemarginaleffect ofafirm’spricecomplexityincreasesin the rival’scomplexitychoice.9

Inspiteoftheirprevalence,pricecomplexityandfirmprominencehaveonlyrecently received attention in the economics literature. To analyze these phenomena, a recent streamoftheoreticalresearchdevelopstheframeworkinVarian(1980),byendogenizing consumerheterogeneity. Carlin (2009) examines a homogeneous product marketwhere identical firms competein both prices andpricecomplexity levels.Strategic price com-plexity leadsto consumerconfusionandsoftenspricecompetition.Confusedconsumers makerandomchoices,soeachfirmisequallylikelytobeselected.Hisfindingsare consis-tentwithobservedpatternsin retailfinancialmarkets,suchaspricedispersion,positive mark-ups, andhigherprices in more fragmentedenvironments. Ouranalysisfocuses on

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Gurunetal.(2016)showthatlenderswhoadvertisemoresellmoreexpensivemortgagesandthatthe eﬀectisstrongerforlesssophisticatedconsumers.

9_In_the_working_paper,_we_also_verify_the_robustness_of_our_qualitative_results_in_a_mo_diﬁed_mo_del_where expertconsumersarebiasedtowardstheprominentﬁrm’sproduct(i.e.willingtopayapremiumforitso longasthepriceisbelowtheirvaluation),seeChioveanu(2017).

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theinteractionbetweenpricecomplexityandprominence,andshowsthatthelatterhas animpactontheequilibriumpattern.Speciﬁcally,weidentifyconditionswhereonlythe prominent ﬁrm randomizes in pricecomplexity levels and show that consumersurplus may benon-monotonicin prominence.

Piccione and Spiegler (2012) study a duopoly marketwhere consumers are initially assigned toonefirm(theirdefaultoption)andmakepricecomparisonswitha probabil-ity whichdepends on firms’chosen priceformats. Theyconsider a more generalframe structure and identify a necessary and sufficient condition for firms to earn max–min profits in equilibrium. Theanalyses in Carlin (2009) and Piccione and Spiegler (2012, Section IV.B) focusonthe polarcasewhereboth firms areequally prominent.Spiegler (2011, Chapter10.4) providesa treatmentfor theotherpolarcasewhereallconsumers are initiallyassigned to thesame firm,so thereis extremeprominence.By allowingfor arbitrary salience levels, our analysis fills the gap between these two polar cases, and showsthat consumersurplusisnotmonotonicinprominence.

ThesymmetricoligopolyanalysisinChioveanuandZhou(2013)showsthatthe equi-librium pattern dependson therelativeeffectiveness of framedifferentiation and frame complexity assourcesofconsumerconfusion. There,anincreasein thenumber offirms induces firmsto relymore onframecomplexityandmay harmconsumers.

Gu and Wenzel (2014) propose a sequentialmodel where a prominent seller and its rival compete in prices after committing to pricecomplexity levels. Theyshow that in equilibrium firms randomize in prices, but choose deterministic complexity levels. The salient firmchoosesthehighestcomplexity forsure,whiletherival’schoice dependson themarketconditions.Consumerprotectionpolicieswhichreducetheshareofconfused consumers may backfireby making the less prominent firmincrease its complexity. In contrast to our cut-off equilibrium model, in theirs, consumer surplus monotonically decreasesin thelevelofprominence.

Astheymodelcomplexityasalong-rundecision,GuandWenzel’sinsightsarerelevant in markets where obfuscation relates to product design rather than price disclosure.10 In our framework where both prices and complexity levels can be changed frequently, theprominentﬁrmalwaysrandomizesonpricesandpricecomplexitylevels,whereasfor relativelyhighsaliencelevels,therivalchoosesthelowestcomplexityforsure.Moreover, a reductionin theshareofconfusedalwaysimprovesconsumersurplus.

In a sequential search model where all consumers sample ﬁrst one prominent ﬁrm,

Armstrongetal.(2009)demonstratethat,withhomogeneousproducts,thesalientseller sets a lowerpricethan itsrivals,industryprofitsarehigher,and consumersurplusand welfarelower thanina marketwherefirms areequally prominent.Theyalsoshowthat prominence benefits both sellers and consumers when products are vertically differen-tiated (as the highest-quality producer has the strongest incentive to become salient).

Armstrong and Zhou (2011) explore ways in which a ﬁrm can become prominent:

10_See_also_Ellison_and_Wolitzky₍₂₀₁₂₎_,_Wilson₍₂₀₁₀₎_,_and_Taylor₍₂₀₁₇₎_for_search-cost_models_of obfus-cation.

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intermediaries may steer consumers to one firm for a fee, price advertisements may affect the order in which firms’ offers are sampled, or consumers’ default biases may be a source of prominence.Seealso Rhodes (2011) for a relatedmodel and Armstrong (2017) forarecent reviewoftheorderedsearch literature.

Inourclearinghousesetting,theorderofsearchisirrelevantbutprominenceaffectsthe behaviorofconsumerswhoareconfusedbypricecomplexity.Wefocusonenvironments wherefirmscommonlyemploycomplexprices,forexample,consumerbankingandenergy retailmarkets.Prominencemightbedrivenbydefaultbiasesfavouringtheproductunder consideration or related ones or it may be due to persuasive advertisingor marketing ployswhichmakeafirm’sproductsalientinaconsumer’smindandsomorelikelytobe considered.

By considering the interplay between complexity and prominence in a model with consumer confusion, this study contributesto anemerging literature that explores the interactionbetweenboundedlyrationalconsumersandstrategicﬁrms.SeeEllison(2006),

Spiegler (2011), Huck and Zhou (2011), Grubb (2015), and Spiegler(2016) for related discussionsandsurveysofrecentwork.Ourmodelisalsorelatedtotheliteratureonprice dispersion(seeBayeetal.,2006,forareview)andexploresanasymmetricmarketwhere ﬁrms simultaneouslychoose pricesandcomplexity,andrandomize inbothdimensions.

2. Model

Consideramarketforahomogeneousproductwithtwosellers,firms1and2.Thefirms facezeromarginalcostsofproduction.Thereisaunitmassofconsumers,eachdemanding at mostone unitof theproduct and willing to payup to v=1. Thefirms compete by simultaneouslyandindependentlychoosingprices(p1andp2)andpricecomplexitylevels

(k1 andk2).Thetimingreﬂectsthefact thatin manycasesboth complexityandprices

can be changed relatively easily. The level of complexity ki captureshow diﬃcult it is

forconsumerstoassessthepriceofﬁrmi.Theﬁrmssetpricespi ∈[0,1]andcanchoose anycomplexitylevelki ∈[k,¯k]⊂R+freeofcost.

Dependingonfirms’complexitychoices,someconsumersmayfinditdifficulttoassess the priceoffers.For given k1 and k2, a fraction μ(k1, k2)≤1 of theconsumers areable

to accurately compare the price oﬀers and select the best deal (we refer to these as the‘experts’or‘informed’),but theremaining1−μ(k1,k2)consumersareconfusedand

makerandomchoices, whichmay bebiaseddueto ﬁrmprominence.Letμ(k1,k2)∈C2.

If one ﬁrmunilaterally increases thecomplexity of its price,this lowers thefraction of expert consumers in the market (∂μ/∂ki<0, for i=1,2), but does not aﬀect the

marginal impact of the rival’s price complexity on consumers (∂2μ/∂k1∂k2=0). For

simplicity,weassumethatμ(k1,k2)=1iﬀk1=k2=k.Thatis,ifbothﬁrmschoosethe

lowest complexitylevel k,all consumers areexperts and buythecheaper product.11 _In

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Section 5,weexploretherobustnessofourresultsforalternativeconfusiontechnologies with ∂2_μ_/_∂_k

1∂k2>0.

We focus on the interaction between pricecomplexity and firm prominence. In our model,prominenceisexogenous(itmay bedue,forinstance, tohigherfirmrecognition or perceivedtrustworthiness)andhasanimpactonproductchoicewhenconsumers are confused bypricecomplexity.Italsoaffectsthechoiceofinformedconsumersifthetwo firmsofferthesameprice.Withoutlossofgenerality,firm1isa‘prominent’sellersothat the consumerswhoareunable to assessthepricesdue to complexityaremore likely to purchaseitsproduct.Thatis,afractionσ∈(1/2,1)oftheconfusedconsumersbuyfrom firm 1 andtheremaining 1−σ buyfrom firm2.Similarly, ifboth firms offerthe same price,afractionσ∈(1/2,1)oftheexpertsbuyfromfirm1andtheremaining1−σbuy from firm2.Asa result,thefirms’profitsare

πi (pi ,pj ,ki ,kj )=pi ·[qi (pi ,pj )μ(ki ,kj )+si (1−μ(ki ,kj ))] where qi(pi, pj)isgivenby qi (pi ,pj )= ⎧ ⎪ ⎨ ⎪ ⎩ 1, ifpi <min{pj ,1} si , ifpi =pj ≤1 0, ifpi >min{pj ,1} fori,j ∈{1,2} andi=j with s1=σ >1/2 ands2=1−σ.

In line with closely related work, we assume that the confused consumers do not pay morethantheirreservation price(v=1). Thismaybe becausetheyhave abudget constraint and realizeat checkout (or after purchase)if the priceis higherthan v and can decline to buy or return the product. Knowing this, firms do not have incentives to set prices above consumers’valuation.12 _Consumers’_behavior _is_affected _by market-wide complexity and prominence,but independent of how the firms’prices rank. This captures theideathatconfusion dueto complexityreduces consumers’pricesensitivity and weakenspricecompetition.

3. Preliminaryanalysis

Westartbyanalyzingfirms’priceandcomplexitychoiceswhenmarket-wide complex-ity leads to consumerconfusionand one firmisprominent.Allproofs missingfrom the text arerelegatedtotheappendix,unless specifiedotherwise.Thefollowingtworesults ruleouttheexistenceof purestrategyequilibria.

Lemma 1. Thereisnoequilibriumwhereboth ﬁrmschoose pureprice-complexity strate-gies.

12_However,_it_can_be_shown_that_our_results_are_{qualitatively}_robust_when_confused_consumers_pay_up_to 1+ε for ε < μ(k , ¯_k₎_.

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Proof. Supposeﬁrmi(j=i)choosesa deterministiccomplexitylevelki(kj).

(i) If ki =kj =k, all consumers are experts (μ(k,k)=1) and the firms make zero profits by competing à la Bertrand. But then firm i could profitably deviate to kd _i=k>kandapricepi =1whichwouldresultinanon-trivialmassofconfused consumers (i.e., 1−μ(k,k)>0) and strictly positive profits. Hence, it must be that in any candidate equilibrium at least one firm (w.l.o.g. let it be i) chooses

ki>k.

(ii) By (i) for any candidate equilibrium proﬁle of price complexities (ki, kj), some

consumers are confused, i.e., 1−μ(ki ,kj )> 0. But then for any such proﬁle (ki,kj),thereisauniquepricingequilibriumwhereﬁrmsrandomizeaccordingtoa

c.d.f.on[p0,1],withp0=σ(1−μ(ki ,kj ))/[1−(1−σ)(1−μ(ki ,kj ))]>0(see,for instance, Bayeet al.,1992), andfirmimakesprofit πi =p0[1−sj (1−μ(ki ,kj ))]. But,asitmustbethatki>k,firmicouldprofitablydeviatetopd i =p0andki d =k whichwouldresultinprofitπd _i=p0[1−sj (1−μ(k,kj ))]>p0[1−sj (1−μ(ki ,kj ))] asμ(k,kj))>μ(ki,kj).So,therecanbenoequilibriumwherebothfirmschoosepure

pricecomplexitystrategies.

This analysis focuses on σ∈(1/2, 1), but the result in Lemma 1 carries over when σ=1/2. When σ=1, there is an asymmetric pure strategy equilibrium where firm 1 chooses ¯k andfirm 2 chooses k. Inthat case, whilethere are both expertandconfused consumers,butfirm2doesnotserve latterandthedeviationin part(ii)oftheproofof

Lemma 1doesnothold assj =0 wheni=1.

Lemma 1 implies that in anycandidate equilibrium at least one firmrandomizes on complexity levels. As a result, both firms face two types of consumers, confused and experts.13 _There_is_a _conflict_between_the_incentive_to _extract_all_surplus_from_confused consumers and the incentive to reduce price and compete for experts. This intuition underliesthefollowingresult,whoseproofisstandard andthereforeomitted;seeVarian (1980) andRosenthal(1980).

Lemma2. Thereisno equilibriumwhere bothﬁrms usepure pricingstrategies.

Lemmas 1 and 2 show that in any duopoly equilibrium there must be dispersion in both pricesand complexitylevels. Firmi’sstrategyspace is[0,1]×[k,k¯].Denote by ξi≡ξi(pi,ki)ﬁrmi’smixedstrategyfori=1,2.ξiisabivariatec.d.f.andcanbewritten

asξi =Fi (pi )Hi (ki |pi ),whereFi(pi)isthemarginalc.d.f.ofﬁrmi’srandompriceand

Hi(ki|pi)istheconditionalc.d.f.ofﬁrmi’scomplexitylevel.(Ifthetworandomvariables,

piandkiareindependent,Hi (ki |pi )=Hi (ki ).)ForFi(p)andHi(ki|pi)tobewell-deﬁned 13

Wefocusonacasewhere μ(k , k )=1.However,Lemma1isrobustforμ(k,k)< 1solongas ∂ μ/∂ ki< 0, for i =1, 2.Inthatcase,evenfor k i= k j=k , ﬁrmsfacebothexpertsandconfusedandsointhecandidate

priceequilibrium, π1= p 0[1−(1−σ)(1−μ(k , k ))]= σ(1−μ(k , k )).But,ﬁrm1canproﬁtablydeviateto p d

i=1and k d1=¯k as π1d= σ(1−μ(¯k , k ))> σ(1−μ(k , k )).Asatleastoneoftheﬁrmschooseski > k,part (ii)intheproofofLemma1applies.

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c.d.f.s they should be increasing ontheir supports. Using the approach in Narasimhan (1988) andBayeetal.(1992),weshowthat bothﬁrmschoosepricesaccordingtoc.d.f.s whicharedeﬁnedonacommonintervalT =[p0,1]andarecontinuouseverywhereexcept

possibly attheupperboundp=1;seeAppendixA.1.

Supposeﬁrmi=jchoosesapricepi andcomplexitylevelki.Firmi’sexpectedproﬁt,

which depends on ﬁrmi’s choices and on therival’s mixed strategy ξj, can be written

as πi (pi ,ki ,ξj )=pi 1 p i _¯ k k μ(ki ,kj (pj ))dHj (kj |pj ) dFj (pj ) +pi si 1− 1 p 0 _¯ k k μ(ki ,kj (pj ))dHj (kj |pj ) dFj (pj ) .

The expected base of confused consumers is the termin the second square bracketsin πi(pi, ki, ξj). The remaining consumers form the expected base of experts. But, the

expertspurchasefromfirmionlywhenitoffersalowerpricethanitsrival.Theexpected numberofexperts,conditionalonfirmibeingthelowpriceseller,istheterminthefirst square brackets. Firm i serves a share si of the expected base of confused consumers.

Using Leibniz’sRule,theﬁrstderivativeofπi(pi,ki,ξj)w.r.t.ki isgivenby

pi 1 p i _¯ k k ∂μ ∂ki dHj (kj |pj ) dFj (pj )−pi si 1 p 0 _¯ k k ∂μ ∂ki dHj (kj |pj ) dFj (pj ). But ∂2_μ/∂_k

i ∂kj =0,so ∂μ(ki,kj)/∂ki isindependent ofkj, andthe ﬁrstderivative

be-comes

pi ∂μ

∂ki [(1−Fj (pi ))−si ].

Then, as∂μ(ki,kj)/∂ki<0,tomaximizeitsexpected-proﬁtﬁrmichooses

ki (pi )= ⎧ ⎪ ⎨ ⎪ ⎩ k if1−Fj (pi )>si ⇔pi <pi ¯ k if1−Fj (pi )<si ⇔pi >pi k, ∀k∈[k,k¯] ifpi =pi ,

wherethethresholdpricepi isimplicitlydefinedbyFj (pi )=1−si ,wheneverpi belongs to thesupport ofFj.Lemma 1 impliesthat at leastone ofthecut-off pricespi belongs to T,asatleastone firmmixesoncomplexitylevels.Thenextresultfollows.

Proposition1. Inequilibrium,aﬁrm’scomplexitychoicedependsonlyonitsprice.Firm i chooses its priceaccording to a c.d.f. Fi(pi)with supportT =[p0,1]. If pi <pi (pi >

pi ) ﬁrm i chooses the lowest complexity k (highest complexity k¯). If pi =pi , ﬁrm i is

indiﬀerent between any complexity level k∈[k,¯k]. If the cut-oﬀ price pi ∈T, then itis

implicitlydeﬁnedbyFj (pi )=sj .Ifpi ∈/T,ﬁrmichoosesadeterministiccomplexitylevel,

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Whena firmmixes oncomplexityin equilibrium, there isa positive relationship be-tween prices and complexity levels. If pi ∈T, at all prices below the cut-off level pi , firm i chooses the lowest complexity and at all prices above pi , it chooses the highest complexitylevel.Intuitively,whenafirmchoosesa relativelyhighprice,itsincentiveto choose highcomplexityisstrongerasitreliesmore onsellingtoconfusedconsumers.In contrast, when setting a relatively low price, a firm hasa stronger incentive to choose lowcomplexityas thisresultsin alarger baseofexperts.

Wefirstanalyzea situationwhereboth firmsrandomizeoncomplexitylevels,andso thecut-off pricesdefinedinProposition1 mustsatisfypi ∈T =(p0,1)fori=1,2. This

implies that firm i chooses complexity level k with probability Fi (pi ) and complexity level ¯k with probability 1−Fi (pi ). The threshold prices pi ∈T are implicitly defined by Fj (pi )=sj for j =1,2, j=i, where sj is firm j’s share of consumers confused by

complexity(i.e.,s1=σ >1/2and s2=1−σ).Forexpositionalsimplicity,denote:

λ1≡F1(p1)andλ2≡F2(p2).

Consistencyrequires that Fi (pi )∈(0,1) andFi (pj )=si . Thefollowing conditionholds whenbothﬁrms mixonbothprices andcomplexitylevelsin equilibrium(seeAppendix A.6).

Condition1.

0<p0<p1<p2<1.

Below we illustrate the derivation of firm 1’s expected profit. Consider a price p∈[p0,p1). By Proposition 1, firm 1 associates prices in this range with complexity

levelk.Then,itsexpected proﬁtis

π1(p,k)=p

(F2(p2)−F2(p))+(1−F2(p2))μ(k,¯k)+σ(1−F2(p2))(1−μ(k,¯k))

. (1) WithprobabilityF2(p2),ﬁrm2choosesk,sothatallconsumersareexperts,i.e.,μ(k,k)=

1. The experts purchase from firm 1 if firm 2’s price is higher, which happens with probability F2(p2)−F2(p).Withprobability 1−F2(p2),firm2 chooses ¯kandthere are

μ(k,¯k) informed and 1−μ(k,k¯) confused consumers. All the informed purchase from firm1 as itoffers a lower price(firm 2 associates ¯k with prices higherthan p2) andso

doesashareσoftheconfusedconsumers.Theﬁrsttwotermsinsquarebracketscapture theexpectednumberofexperts,whilethelastterminsquarebracketsgivestheexpected number ofconfusedconsumers.

Consider firm 1’s expected profit for a price p∈[p1,p2]. By Proposition 1, firm 1

associatespricesin thisrangewithcomplexitylevel¯k.Then, itsexpectedproﬁtis π1(p,¯k)=p (F2(p2)−F2(p))μ(k,k¯)+(1−F2(p2))μ(¯k,¯k)+σ F2(p2)(1−μ(k,k¯)) + (1−F2(p2))(1−μ(¯k,k¯)) .

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The expected number of confused consumers is the term in square brackets. Firm 1 serves a fraction σ of this group. Firm 1 also serves the expert consumers if ﬁrm 2 chooses ahigherprice.WithprobabilityF2(p2)−F2(p),thereareμ(k,¯k)expertswhile,

with probability 1−F2(p2), thereare μ(¯k,¯k);this is reﬂectedbytheﬁrst two termsin

curlybrackets.

Consider firm1’sexpected profit for p∈ (p2,1]. By Proposition 1, firm1 associates

prices inthis rangewithcomplexitylevel ¯k.Then,itsexpectedproﬁtis π1(p,k¯)=p (1−F2(p))μ(k¯,k¯)+σ F2(p2)(1−μ(k,¯k))+(1−F2(p2))(1−μ(¯k,¯k)) . Echoingpreviousreasoning,withprobabilityF2(p2)ﬁrm2choosesk,inwhichcasethere

are μ(k,¯k) informed and 1−μ(k,¯k) confused consumers. A share σ of the confused consumers purchases from firm 1, the prominent seller. The experts do not purchase from firm 1 as firm 2’s price is lower. With probability 1−F2(p2), firm 2 chooses ¯k,

so there are μ(¯k,¯k)experts and 1−μ(¯k,¯k)confused consumers.A share σof confused consumers buy fromfirm 1.Theexperts purchase fromfirm 1 ifitoffers a lowerprice, whichhappenswithprobability1−F2(p).Thefirsttermin curlybracketscapturesthe

expectednumberofexperts,whiletheterminsquarebracketsgivestheexpectednumber of confusedconsumers.

InAppendixA.2,wepresent ﬁrm1’sexpected proﬁtsat p0, pˆ1andpˆ2and alsowhen

p→p1 and p→p2. There, we also derive ﬁrm2’s expected proﬁt over thethree price

ranges, using thesame approach as above. Nextsection combines these derivations to characterize the mixed strategy equilibrium and to identify a condition on the param-eter values under which both firms randomize onboth prices and complexity levels in equilibrium. Whenthis conditiondoesnothold – whichhappenswhenfirm1’slevel of prominence is relatively high – both firms mix on prices,but only theprominent firm randomizes oncomplexitylevels.

4. Equilibriumanalysis

In equilibrium, ﬁrmi’sexpected proﬁt forany price-complexitycombination (p,ki),

whichisassignedpositivedensityinequilibrium, mustbeconstant.Then, using expres-sions (A.1)–(A.3), (A.6), and (A.7) from Appendix A.2, we can write the price ratios

p0/ p1 and p0/ p2 as functions of λ2=F2(p2) and λ1=F1(p1), ﬁrm 2’s and ﬁrm 1’s

probabilities of choosing k in equilibrium, respectively. These ratios are presented in

AppendixA.3.Wethenobtaintheequilibrium valuesofλ1 andλ2,

λ1=

(1−σ)[1−σ(2−σ)(1−μ(k,¯k))]

1−(1−σ2₎₍₁₋_μ₍_k_,¯_k₎₎ and λ2=

σ[1−(1−σ2)(1−μ(k,k¯))] 1−σ(2−σ)(1−μ(k,¯k)) . (2) It canbecheckedthatλ1∈(0, 1)andλ2>0.Furthermore,λ2<1 holdsiﬀ thefollowing

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Condition2.

(1−σ)/[σ1−σ+σ2]>1−μ(k,¯k).

As μ(k¯,k¯)=2μ(k,¯k)−1 and0≤μ(¯k,¯k)<μ(k,¯k),it followsthat 1−μ(k,k¯)≤1/2. Forrelativelylowlevelsofprominence(thatis,forσ <0.71),thisconditionalwaysholds and so firm 2 mixes between the highest and the lowest price complexity levels. More generally,foragivenμ(k,k¯),theconditionissatisfiedwhenfirm1’slevelofprominenceis nottoohigh. However,Condition2getsmorestringentasfirm1’sprominenceincreases (theLHSoftheinequalityintheconditionisdecreasinginσ).Whenfirm1isprominent enough,firm2benefitsmorefrompricetransparency,asitsshareofconfusedconsumers isrelativelysmall.

In Appendix A.3, we show that when λi∈(0, 1), the consistency requirements also

hold: Fi (p1)<Fi (p2)for i=1,2, where Fi (pi )=λi and Fi (pj )=si .Also there,we ex-plore the firms’ price c.d.f.s at the upper bound of the support. Using Lemma 4, we showthatfirm2’spricec.d.f.iscontinuouseverywhere,whilefirm1hasamasspointat theupper boundofthepricec.d.f.’s support,p=1.Then,we verifythat p0,p1, andp2

arewell definedunder Condition2. Finally,wepresenttheequilibrium cut-off pricesin expressions(A.8)and(A.9),followedbythepricingc.d.f.softhetwofirms.Using(A.1), (A.4)and(2),we obtainthe equilibrium profitof firm 1(π₁∗) andthe lowerboundofthe pricesupport(p0). π∗1 =σ(1−μ(k,k¯)) 2−σ−σσ2−2σ+3(1−μ(k,k¯)) 1−σ(2−σ)(1−μ(k,¯k)) ; (3) p0=σ(1−μ(k,¯k)) 2−σ−σσ2₋₂_σ₊₃₍₁₋_μ₍_k_,¯_k₎₎ μ(k,¯k)+σ(1−σ)(σ2−_σ₊₁₎₍₁−_μ₍_k_,_k¯₎₎2. (4)

Using(4)and(A.6),wecalculateﬁrm2’sequilibriumproﬁt,

π₂∗=σ(1−μ(k,¯k)))2−σ−σ

σ2₋₂_σ₊₃₍₁₋_μ₍_k_,¯_k₎₎

1−(1−σ2₎₍₁₋_μ₍_k_,_k¯₎₎ . (5) Notethat π₁∗/π∗₂=λ2/σ=(1−σ)/λ1.

Belowwesummarizeourﬁndings.

Proposition2. UnderCondition2,intheuniquemixedstrategyequilibriumﬁrmichooses the lowest complexity k with probability λi =Fi (pi )∈(0,1), deﬁned in (2) and highest

complexity ¯k with probability 1−λi . Both ﬁrms randomizeon pricesin [p0, 1],with p0

given in (4). Firm 2’s price c.d.f. (F2) is continuous everywhere, while ﬁrm 1’s price

c.d.f.(F1)iscontinuouson[p0,1)andhasanatom atp=1.Firmiusesk(k¯)atprices

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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0

p

c.d.f.

Fig.1. Theﬁrms’pricec.d.f.sfor σ= . 6and μ(k , ¯_k₎₌_.₆_._F₁₍_p₎_is_the_bottom_line_and_F₂₍_p₎_is_the_top_line. Thedashedlinescorrespondtopricesassociatedwith k .

When ﬁrm 1’s prominence is not too high in the sense that σ >1/2, but

Condition 2 is satisfied,both firms randomize on complexity levels and prices in equi-librium. In this case, the difference in the firms’ shares of confused consumers is not too large. Inthe limit, when σ→1/2, λ1=λ2=1/2, p1=p2, and both firms’ pricing

c.d.f.s arecontinuouseverywhereon theircommonsupport.Thisis consistent withthe resultsinCarlin(2009).ThefollowingnumericalexampleandFig.1illustratetheresult in Proposition2.

Example 1. When σ=.6 and μ(k,¯k)=.6, in equilibrium ﬁrm 1 and 2 choose k with probabilities λ1=.357 and λ2=.672, respectively. The two ﬁrms randomize onprices

according tothefollowingc.d.f.s,whichareillustratedin Fig.1,

F1(p)= ⎧ ⎪ ⎨ ⎪ ⎩ .846−.284/p forp∈[p0,p2) 1.171−.474/p forp∈[p2,p1] 2.131−1.422/p forp∈(p1,1] and F2(p)= ⎧ ⎪ ⎨ ⎪ ⎩ .948−.319/p forp∈[p0,p2) 1.313−.531/p forp∈[p2,p1] 2.593−1.593/p forp∈(p1,1) ,

wherep0=.336,p1=.582,andp2=.829.Firm1andﬁrm2makeproﬁtsπ∗1=.319and

π₂∗=.284,respectively. Firm1’satom atp=1 isφ=.108.

WhenCondition2 doesnothold,theresultsin Proposition2 nolongerapply asthe derivedλ2isweaklylargerthan1(andtobeawell-deﬁnedprobabilityandforbothﬁrms

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firm 1’s prominence advantage is large enough, firm 2 serves a relatively small share of confused consumers. Then, firm 2 relies more on expert consumers and so benefits more from market transparency than from confusion. We prove thefollowing result in

AppendixA.4.

Proposition3. WhenCondition2doesnothold,intheuniquemixedstrategy equilibrium ﬁrm 2 chooses k for sure and ﬁrm 1 chooses the lowest complexity k with probability

λh ₁ =F₁h (ph ₁)andthe highestcomplexity¯kwith probability1−λh ₁,where

λh ₁ = (1−σ)[1−σ(1−μ(k,¯k))] 1−σ(1−σ)(1−μ(k,¯k)).

Bothﬁrms randomizeonpricesin[ph ₀,1],withp₀h =σ(1−μ(k,¯k)).Firm 2’spricec.d.f.

F₂h iscontinuouseverywhere,whileﬁrm1’spricec.d.f.F₁h iscontinuouson[ph ₀,1)andhas anatom atp=1.Firm 1usesk(k¯)atpricesbelow(above)ph ₁ =(1−μ(k,¯k))∈(ph ₀,1). The equilibriumproﬁts aregivenby

π_h∗₁=σ(1−μ(k,k¯)) and π∗_h₂=σ(1−μ(k,¯k)) 1−σ(1−μ(k,¯k))

1−σ(1−σ)(1−μ(k,k¯)). (6) When prominence is largeenough, firm 2 chooses thelowest complexity forsure to minimize the number of confused buyers and reduce its disadvantage. The prominent firm, as before, associates lower prices with the lowest complexity (at those prices it benefits from more transparency) and higher prices with highest complexity (at those prices it relies more on confused consumers). Specifically, firm 1 chooses complexity k

for all prices p<ph ₁∈(ph ₀,1) and k¯ for all prices p≥ph ₁. Proposition 1 then requires that firms’pricingc.d.f.ssatisfyF₂h (ph ₁)=1−σandF₁h (1)≤σ(thatis, ph ₂ ≥1).14 _The followingexampleandFig.2illustratetheresultsforrelativelyhighprominence. Example 2. When σ=.8 and μ(k,k¯)=.6, in equilibrium firm 1 chooses k with prob-ability λh ₁ =.145, while firm 2 chooses k for sure. The two firms randomize on prices accordingto thefollowingc.d.f.s,whichareillustratedin Fig.2,

F₁h (p)= .726−.232/p forp∈[ph ₀,ph ₁) 1.113−.387/p forp∈[ph ₁,1] and F₂h (p)= 1−.32/p forp∈[ph ₀,ph ₁) 1.533−.533/p forp∈[ph 1,1) ,

where ph ₀ =.32 and ph ₁=.4.Firm 1 and ﬁrm2 make proﬁtsπ_h∗₁=.32 and π_h∗₂=.232, respectively. Firm1’satom atp=1 isφh =.274.

14_As_by_Lemma₁_Fh

1(p h2)= σ, if F 1h(1) > σthenp h2< 1andthecandidate λh2= F 2(p h2) < 1.Butthisis inconsistentwithanequilibriumwhereﬁrm2chooseskforsure.

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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0

p

c.d.f.

Fig.2. Theﬁrms’pricec.d.f.s for σ=. 8and μ(k , _k¯₎₌_.₆_._Fh

1(p )is thebottomlineand F 2h(p )isthetop line.Thedashedlinescorrespondtopricesassociatedwith k .

Theremainderof thissectionusesPropositions 2and3 to exploretheroleof promi-nenceonmarketoutcomes.

InExamples1and2whereμ(k,k¯)=0.6,anincreaseinσfrom.6to.8resultsina de-creasein industryprofitfrom.603to.552.Thisshowsthat anincreasein theprominence levelmightharmindustryprofit,inwhichcaseitbenefitstheconsumers,astotalsurplus is normalized to one.It also indicates that marketswhere an entrant competes with a prominent enough incumbent may bemore competitivethan marketswhere the differ-ences inprominencebetweensuppliersarerelativelysmaller,whichmaybeofrelevance inretailelectricitymarketswherenewentrantsfacingincumbentsupplierscouldbecome moreprominentovertime.Holdingμ(k,¯k)=0.6,Fig.3illustratesindividualand aggre-gate profitsas functions ofthelevel ofprominence.Inthis case,totalindustryprofitis lowestandconsumerssurplushighestatσ=0.754,whichisthecut-off prominencelevel forthetwotypesofequilibriapresentedin Propositions2and 3.

Example3. Supposeμ(k,¯k)=0.6.Then, Condition2 holdsiﬀσ < 0.754.

Inourframework,anincreaseinthelevelofprominenceaffectspricing,andultimately profits, intwo ways. First,prominencehasa directeffect onprices as itreallocatesthe confused consumers in favourof thesalient firm. Second,prominence affects thefirms’ probabilitiesofchoosingthelowestpricecomplexity,i.e., λ1andλ2 – whichendogenize

theexpected shareofconfused consumers– and,therefore,italso hasanindirecteﬀect onprices throughthis channel.

In settings where the total share of confused consumers is exogenous, only the di-rect eﬀect plays a role. In that case, an increase in the level of prominence boosts industryproﬁtandso harmsconsumerwelfare;this canbe easilychecked,forinstance,

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0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.3 0.4 0.5 0.6

Sigma

Profits

Fig.3. Theprofitsoffirm1(mediumsolid)andfirm2(dashed),andtotalprofit(thicksolid)for μ(k , ¯_k₎₌_.₆_.

in Narasimhan (1988). Gu and Wenzel (2014) show that this resultis qualitatively ro-bust ina sequentialset-up wherea salientfirmandits rivalfirstcommitto complexity levels and then compete in prices. In their analysis, although the share of confused is determined endogenously, firms choose deterministic complexity levels and thepricing stageis similartoNarasimhan(1988).

The comparative statics of consumers surplus with respect to the prominence level in our model is different from the corresponding result in the sequential move setting of Guand Wenzel (2014). However, ournon-monotonicityresult isnotdue to the tim-ing of the game per sebut to thecut-off structure of the equilibrium – that is, to the statistical dependence between firms’ pricing and price complexity equilibrium strate-gies.15 _Therefore,_the _indirect_effect _on_firms’_{probabilities} _of_choosing _the_lowest _price complexity identifies a novel channel through whichprominence affects industry profit andconsumersurplus.Thisnoveleffectisrelatedtotheconflictingincentivesofthetwo firms identifiedinournextresult.

Corollary 1. In the mixed strategy equilibrium, ﬁrm 2’s probability of using the lowest complexity (λ2) weakly increases in the level of prominence (σ), while ﬁrm 1’s

corre-spondingprobability(λ1)decreases inσ.

An increase in the prominence level increases the salient firm’s share of confused consumers andso it lowersitsincentiveto choose thelowestpricecomplexitylevel (k). The larger is firm 1’s share of confused consumers, the more the firm benefits from

15_For_instance, _in_a_simultaneous _move _setting_where _price_format_{diﬀerentiation}_is _the_main _source_of confusion (ratherthancomplexity),priceandpriceformatdecisions areindependent inequilibrium and consumersurplusdecreasesinthedegreeofasymmetry/prominence;seeChioveanu(2019).

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confusion.Incontrast,anincreaseinprominencedecreasesthelesssalientfirm’sshareof confused consumers andso itbooststhis firm’sincentivetochoose k. Thelowerisfirm 2’sshareofconfusedconsumers,themoreitbenefits fromtransparency.

Forrelativelyhigh levelsofprominence– i.e.,whenCondition2doesnothold– ﬁrm 2 chooses k for sure (i.e. λ2=1) and, unlike its rival, it cannot directly aﬀect market

transparency by adjusting itsprobability of choosingkin response to anincrease in σ. In this range, industryproﬁt strictly increases and consumersurplus strictly decreases in σ(seeAppendixA.5forthedetails).

For relatively low levels of prominence – i.e., when Condition 2 holds – both firms canadjusttheirprobabilitiesofchoosingkinresponsetoanincreaseinσandtheyhave conflicting incentives.Inthisrange,industryprofitmay decreaseandconsumersurplus may increasein σ. Numericalexamples suggest that thisis thecasein therangeof σ’s whereλ2<1iscloseto1andwhereanincreaseinprominencemakesthelessprominent

ﬁrma moreaggressivecompetitor.

In our simultaneous move setting, the direct and indirect effects of prominence on pricingcannotbeclearlyseparatedandgeneralcomparativestaticsanalysisisintractable. However, a combination ofnumericalsimulationsand analytical resultsprovidefurther insights.Overtherangeofσ’swhereCondition2holds,industryprofithasan inverted-U shape. Outside this range, it is strictly increasing. As a result, consumer surplus is maximizedeitheratthecut-off prominencelevelforthetwotypesofequilibriapresented inPropositions2and3(thishappensforμ(k,k¯)0.8,seeExample3foranillustration) or as σ→0.5 (which happens for μ(k,k¯)0.8). The cut-off prices of firms 1 and 2, are weakly decreasing and, respectively, increasing in σ: p1 (p2) is strictly decreasing

(increasing) inσ,whileph ₁ =1−μ(k,k¯)(ph ₂ =1)areconstant andsoindependentofσ. Thelowerboundoftheﬁrms’pricesupport(p0)hasaninverted-Ushapeovertherangeof

σ ’swhere Condition2holds anditis strictlyincreasingoutsidethis range(ph 0 =σph 1).

The likelihood that the prominent ﬁrm chooses the monopoly price strictly increases in σ.

Corollary2. Inthemixedstrategyequilibrium,(i)themoreprominentfirmmakeshigher profits thanthe rival;(ii) thepricedistributionofthe prominentfirmfirstorder stochas-tically dominates the one of the less prominent firm; (iii) the more prominent firm’s average priceishigher thanthatof the lessprominent firm,and(iv)the less prominent firm choosesthe lowest complexity(k) withhigher probabilitythanthe rival.

The prominentfirmattractsa larger shareof confusedconsumers,andso it benefits more from market-wideconfusion. For this reason, itchooses thehighest levelof com-plexity with higher probability than its rival, has lower incentives to compete for the expert consumers,andtherefore itchooses a higheraverageprice. Thecombinedeffect ofcharginghigherprices (inthefirstorderstochasticdominancesense)andattractinga highershareoftheconfusedconsumersallowstheprominentfirmtomakehigherprofits inequilibrium.Confusedconsumers’biasinfavoroftheprominentfirmappearstobe

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in-consistentwiththerankingoftheaverageprices.Ournextresultfocusesontheranking of theaverageprices, conditional onthese beingassociated with thelowest complexity (k).

Corollary3. Inthe mixedstrategy equilibrium,themore prominentﬁrmchoosesalower cut-oﬀ price – below which ituses the lowest level of pricecomplexityk – than itsrival (p1< p2 when Condition 2 holds and ph 1 < ph 2 = 1 when it does not). Furthermore,

conditional on choosing the lowest complexity, the more prominent ﬁrm oﬀers a lower average pricethan its rival: E(p1|p1<p1)< E(p2|p2<p2) when Condition 2holds,

andE(p1|p1<ph 1)<E(p2|p2<ph 2)when itdoes not.

WeprovethiscorollaryinAppendixA.5andsketchheretheintuition.Thepricec.d.f.s of the two firms, conditional on pricebeing strictly lessthan p<1,are identical. This isbecauseunconditionalpricedensitiesareproportionaleverywherebelow p=1,which can beeasilyseenin Examples1and2.Combinedwiththefact thatin equilibriumthe cut-off pricebelow whichfirm1chooses kislowerthan thecut-off priceoffirm2(that is,p1<p2,ifCondition2 holds,andph 1 <ph 2,ifitdoesnot),thisprovesthecorollary.

Hence, in ourmodel, consumers’ bias for the prominent firmis consistent with the ranking of theaverage prices conditional onthe lowest complexity. For example,if in-formation onprices associated with the lowestcomplexity gets aggregated through in-teractions betweenconfused consumers (e.g. on social media),then the rakingof these conditionalprices wouldconfirmtheconsumerbiasfortheprominentfirmex-post.

Anotherinterpretation, suggestedby areferee, couldmakeconfused consumers’bias consistent with market outcomes ex-post. Suppose that confused consumers are more likely to buy from thefirm withthe largest marketshare. Then, in marketswhere the share ofexperts is smallenough,theprominent firm’smarketsharewill be larger than therival’sandsoconfusedconsumers’biasforthisfirmwouldbeconfirmedbythemarket shares.

5. Alternativeconfusiontechnologies

The main analysis assumes that a marginal increase in ﬁrm i’s complexity reduces the fraction of experts in the market but does not alter theeﬀectiveness of the rival’s marginalincreasein pricecomplexityonconsumers,thatis,∂2_μ/∂_k

1∂k2=0.Below we

prove that there exists anequilibrium which is qualitatively consistent with theone in themainanalysiswhenever∂2_μ_/_∂_k

1∂k2>0.As∂μ/∂ki =μi< 0,thisconditionrequires

that themagnitude ofthemarginal impactof ﬁrmi’scomplexitybe decreasing in ﬁrm

j’scomplexity(∂|μi|/∂kj<0).16Morespeciﬁcally,weshowthatiftherivalusesamixed

strategy with a positive relationship between price and price complexity, it is a best

16_An_example_of_confusion_technology_which_satisﬁes_this_assumption_is_μ₍_k

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response foraﬁrmtoassociateprices belowathresholdwiththelowestcomplexityand prices aboveitwiththehighestcomplexity.

Suppose that ﬁrm j uses a mixed strategy ξj so that dkj(pj)/dpj≥0. Consider the

expected proﬁtsofﬁrmipresentedinSection 3:

πi (pi ,ki ,ξj )=pi 1 p i _¯ k k μ(ki ,kj (pj ))dHj (kj |pj ) dFj (pj ) + pi si 1− 1 p 0 _¯ k k μ(ki ,kj (pj ))dHj (kj |pj ) dFj (pj ) .

Thef.o.c.ofﬁrmi’sexpectedproﬁtmaximizationw.r.t.ki requiresthat

pi 1 p i E(μi (pj )|pj )dFj (pj )−si 1 p 0 E(μi (pj ))dFj (pj ) =0 , (7)

where∂μ(ki,kj(pj))/∂ki≡μi(ki,kj(pj))givesthemarginalimpactofkionμandE(μi (pj )|

pj )= ¯ k

k μi (ki ,kj (pj ))dHj (kj |pj ) isthe expected marginal impactof anincrease in ki

onthefractionofexpertsconditionalonﬁrmj’sprice.Forgivenξj,

1

p 0E(μi (pj ))dFj (pj )

– the overallexpectedmarginal impact ofanincreasein ki onthefraction of experts–

isaconstant.Atpi =p0,theterminbracketsbecomes(1−si ) 1

p 0E(μi (pj ))dFj (pj )<0

and when pi→1,it converges to −si 1

p 0E(μi (pj ))dFj (pj )>0. So, thereis at least one

pricepi ∈(p0,1)whichsatisﬁes(7).Moreover,pi isuniqueif

d 1 p i E(μi (pj )|pj )dFj (pj ) /dpi = 1 p i d(E(μi (pj )|pj )) dpi dFj (pj )−μe i (pi )Fj (pi )>0 , where the equality follows from Leibniz’s Rule. As −μe _i(pi )>0 and Fj (pi )>0, this condition holdsif dE(μi (pj )|pj )/dpi >0. But,as dkj(pj)/dpj>0,a suﬃcient condition

is then ∂μi (ki ,kj )/∂kj =∂2μ(ki, kj)/∂ki∂kj>0. Hence, whenever ∂2μ/∂ki∂kj>0 there

existsauniquepi ∈(p0,1)whichsatisﬁes(7)anditfollowsthatﬁrmi’scomplexitylevel

choice is ki (p)= ⎧ ⎪ ⎨ ⎪ ⎩ k ifp<pi ¯ k ifp>pi k, ∀k∈[k,¯k]ifp=pi ,

whenever pi belongsto Tj thesupport of Fj. Lemma 1 impliesthat at least one ofthe

cut-oﬀ pricespi belongstoTj.Thisshowsthatamixedstrategyequilibriumliketheone

analyzedin ourbenchmarkmodel existsfora moregeneralconfusiontechnology. 6. Conclusions

We analyzethe interplay between consumer confusion due to price complexity and ﬁrmprominenceina modelwheretwo ﬁrmscompetebysimultaneously choosingprices

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andthecomplexityoftheirpriceoffers.Oneofthefirms enjoysahigherlevelof promi-nence,whichmaybeduetohigherbrandrecognition,industrydynamics,oradvertising effort/spending. Price complexityleads to consumer confusionso that some buyers are able to identify the best offer, while others may get confused. The share of confused consumers is determined by market-wide complexity.The confused consumers shop at randomandfavorthemoreprominentfirm,inthesensethattheyaremorelikelytobuy from it.

Inequilibriumthereisdispersion inboth pricesandcomplexitylevels.Thenatureof theequilibriumdependsonthelevelofprominence.Formoderatelevelsof prominence, both firms mix onpricecomplexity levels, while forhigh levels of prominence,the less prominentfirmchoosesthelowestpricecomplexityforsure.Theprominentfirmmakes higherprofits,chooseshigherpricesonaverageandthelowestcomplexitylevelwithlower probability, andsetsthemonopolypricewithpositive probability.

Inourmodel,adecrease inprominencemay increaseindustryprofitsand harm con-sumers. Inaddition, conditionalon choosingthelowest complexity,theprominent firm sets a lower price, on average, which is consistent with confused consumers’ behavior. This suggests that it may be useful to investigate in the future an alternative model whereconfusedconsumers’beliefsaboutthepricerankingisbasedontheaverageprices conditional onlowestcomplexity andwhere, as aresult, prominenceisendogenous. Fi-nally,ouranalysis showsthat a qualitativelysimilar equilibrium existswith alternative confusion technologies if the marginal impact of an increase in one firm’s complexity increases intherival’scomplexitylevel.

AppendixA

A.1. Propertiesof thepricing distributionfunctions

The proofs of the lemmata below are standard and presented in the working paper version(Chioveanu,2017).

Lemma 3. The supports of the pricing c.d.f.s, T1 and T2 are both connected intervals

(i.e., thereare nogaps ineither ofthem).

Lemma 4. Neither firm can have amass point in the interior or at the lower boundof the other firm’s pricec.d.f. support. Moreover, firm i cannot have a mass point at the upperboundof Tj iffirm jhasamass point there.

Lemma5. In equilibrium,itmust holdthatT1=T2=[p0,ph ]forp0<ph≤1.

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A.2. Expectedproﬁts

Derivationof ﬁrm1’sexpected proﬁt

• Supposeﬁrm1 choosesa pricep∈[p0,p1).

Using(1),asF2(p2)=λ2,ﬁrm1’sexpectedproﬁtsatp=p0andwhenp→p1are

π1(p0,k)=p0 1−(1−σ)(1−λ2)(1−μ(k,k¯)) ; (A.1) lim p p 1 π1(p,k)=p1 σ−(1−σ)(1−λ2)(1−μ(k,¯k)) . (A.2)

• Supposeﬁrm1 choosesa pricep∈[p1,p2].

As 1−μ(k,k¯)=μ(k,k¯)−μ(k¯,k¯), usingF2(p2)=λ2,π1(p1,¯k)=limp p 1π1(p,k),as

given in(A.2). ByProposition 1,F2(p1)=1−σand theexpected proﬁtat p=p2is

π1(p2,k¯)=p2{(1−λ2)−(1−μ(k,¯k))[2(1−σ−λ2)+σλ2]}. (A.3) • Supposeﬁrm1 choosesa pricep∈(p2,1].

As 1−μ(k,¯k)=μ(k,¯k)−μ(k¯,k¯) and F2(p2)=λ2, ﬁrm1’sexpected proﬁt becomes

π1(p,k¯)=p{(1−F2(p))(2μ(k,k¯)−1)+σ(1−μ(k,k¯))(2−λ2)}. (A.4)

It canbe checkedthat limp p 2π1(p,k¯)=π1(p2,k¯)as presentedin(A.3).

Derivationof ﬁrm2’sexpected proﬁt

• Supposeﬁrm2 choosesa pricep∈[p0,p1).

ByProposition1,thispriceisassociatedwithcomplexityk,sofirm2’sexpectedprofit is π2(p,k)=p (F1(p1)−F1(p))μ(k,k)+(1−F1(p1))μ(k,¯k) +(1−σ)F1(p1)(1−μ(k,k))+(1−F1(p1))(1−μ(k,k¯)) . (A.5) With probability F1(p1), firm 1 chooses k, so that there are μ(k, k) informed and

1−μ(k,k) confused consumers. A share 1−σ (<σ) of the confused purchases from firm 2, the less prominent seller. The experts purchase from firm 2 if firm1’s price is higher,whichhappenswithprobabilityF1(p1)−F1(p).Withprobability1−F1(p1),firm

1chooses¯k,sothereareμ(k,¯k)informedand1−μ(k,¯k)confusedconsumers.Allexperts purchasefromfirm2asitoffersalowerprice(firm1associatesk¯withpriceshigherthan

p1)andsodoesashare1−σoftheconfusedconsumers.Theﬁrsttwotermsinthecurly

brackets capture theexpected number ofexperts, whereasthe termin square brackets givestheexpectednumberofconfusedconsumers.Usingμ(k,k)=1andF1(p1)=λ1,it

follows that,

(22)

lim p p 1

π2(p,k)=p1(1−λ1)[1−σ(1−μ(k,¯k))]. (A.6) • Supposeﬁrm2 choosesapricep∈[p1,p2].

By Proposition1,itassociatesthispricewithk.Then, ﬁrm2’sexpectedproﬁtis

π2(p,k)=p{(1−F1(p))μ(k,¯k)+(1−σ)[F1(p1)(1−μ(k,k))+(1−F1(p1))(1−μ(k,k¯))]} =p(1−F1(p))μ(k,¯k)+(1−σ)(1−λ1)(1−μ(k,k¯))

.

The logic behind the expression above is similar to the one for (A.5). But when firm 1 uses k, it attracts all the experts, as it offers a lower price. It is easy to check that π2(p1,k¯)=limp p 1π2(p,k)asgivenby(A.6), andthattheexpectedprofitatp2 is

π2(p2,k)=p2(1−σ)

1−λ1(1−μ(k,¯k))

. (A.7)

• Supposeﬁrm2 choosesapricep∈(p2,1].

By Proposition 1, itassociatesthis pricewithcomplexity level¯k. Itsexpected proﬁt is π2(p,k¯)=p (1−F1(p))μ(¯k,k¯)+(1−σ)[F1(p1)(1−μ(k,¯k))+(1−F1(p1))(1−μ(k¯,k¯))] =p(1−F1(p))(2μ(k,k¯)−1)+(1−σ)(2−λ1)(1−μ(k,¯k)) .

A.3. Equilibriumanalysis

Priceratiosusing theﬁrms’constantproﬁtconditions

Inequilibrium, ﬁrmi’s expected proﬁtfor any price-complexitycombination (p, ki),

whichisassignedpositivedensityin equilibrium,mustbe constant.

Using (A.1)–(A.3), the constant proﬁt conditions for ﬁrm 1 give the following price ratiosexpressedasfunctions ofλ2:

p0 p1 = σ−(1−σ)(1−λ2)(1−μ(k,¯k)) 1−(1−σ)(1−λ2)(1−μ(k,¯k)) and p0 p2 = 1−λ2−[2(1−σ)(1−λ2)−σλ2](1−μ(k,k¯)) 1−(1−σ)(1−λ2)(1−μ(k,k¯)) .

Using (A.6) and (A.7), theconstant proﬁt conditions of ﬁrm2 leadto thefollowing priceratiosexpressed asfunctionsofλ1

p0 p1 = (1−λ1)[1−σ(1−μ(k,¯k))] 1−σ(1−λ1)(1−μ(k,¯k)) and p0 p2 = (1−σ) 1−λ1(1−μ(k,k¯)) 1−σ(1−λ1)(1−μ(k,k¯)) . Equilibriumλvalues

Weshowbelowthat equilibriumλ1 isalwayswelldeﬁnedandthatλ2 iswelldeﬁned

whenCondition2holds. Theexpressionfortheλ’sisgivenin (2).

(i)Itiseasytosee thatλ1<σandthat λ2>1−σas1>σ(1−σ)(1−μ(k,¯k)).