Competitive intensity and its two-sided effect on the boundaries of firm performance

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1 Unversty of Toronto From the SelectedWorks of Mchael D Ryall Sprng May 19, 2017 Compettve ntensty and ts two-sded effect on the boundares of frm performance Joao Montez, London Busness School Francsco Ruz-Alseda Mchael D Ryall, Unversty of Toronto Avalable at:

2 MANAGEMENT SCIENCE Artcles n Advance, pp ISSN (prnt), ISSN (onlne) Compettve Intensty and Its Two-Sded E ect on the Boundares of Frm Performance Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. Joao Montez, a Francsco Ruz-Alseda, b Mchael D. Ryall c a Faculty of Busness and Economcs (HEC), Unversty of Lausanne, 1025 Lausanne, Swtzerland; b Escuela de Admnstracón, Pontfca Unversdad Católca de Chle, Santago, Chle; c Rotman School of Management, Unversty of Toronto, Toronto, Ontaro M5S 3E6, Canada Contact: joao.montez@unl.ch (JM); f.ruz-alseda@uc.cl, (FR-A); mke.ryall@rotman.utoronto.ca, (MDR) Receved: November 27, 2013 Accepted: October 24, 2016 Publshed Onlne n Artcles n Advance: May 19, Copyrght: 2017 INFORMS Keywords: Abstract. The new perspectve emergng from strategy s value-capture stream s that the effects of competton are twofold: competton for an agent bounds ts performance from below, whle that for ts transacton partners bounds from above. Thus, assessng the ntensty of competton on ether sde s essental to understandng frm performance. Yet, the lterature provdes no formal noton of compettve ntensty wth whch to make such assessments. Rather, some authors use added value as ther central analytc concept, others the core. Added value s smple but msses the crucal, for-an-agent sde of competton. The core s theoretcally complete but dffcult to nterpret and emprcally ntractable. Ths paper formalzes three, ncreasngly general notons of compettve ntensty, all of whch mprove on added value whle avodng the complexty of the core. We analyze markets characterzed by dsjont networks of agents (e.g., supply chans), provdng several nsghts nto competton and new tools for emprcal work. Hstory: Accepted by Bruno Cassman, busness strategy. Supplemental Materal: The onlne supplement s avalable at game theory barganng theory market structure frm strategy market performance barganng 1. Introducton Today, strategy boasts a rch and growng ven of theoretcal fndngs based on the applcaton of coaltonal game theory. 1 Ths growth s due to a number of mportant features of the coaltonal game formalsm, ncludng ts explct dstncton between the value creaton and capture aspects of market transactons, ablty to represent the sort of unstructured dealngs that are typcal n real-world markets, bult-n scalablty wth respect to market sze, and operaton at a level of abstracton that frees theorsts from the need to commt to a myrad of specal assumptons on the underlyng detals of value creaton. The fndngs n ths stream have resulted n numerous nsghts nto the central queston n strategy: What drves the economc performance of agents n compettve markets? 2 Of course, new nsghts nvarably lead to new questons. In ths paper, we tackle several such questons wth the am of provdng further contrbutons. Whle our focus s on the stream wthn strategy proper, we beleve several of our fndngs wll be of nterest to researchers outsde of the feld. Let us begn by dentfyng our motvatng research questons. Strategy s a feld domnated by emprcal work. Thus, one of the most mportant questons arsng from ths new stream s how ts theoretcal clams translate nto emprcal hypotheses. Presently, the theoretcal work publshed n the strategy lterature falls nto two essental categores: work the prmary analytcal focus of whch s on added value (e.g., begnnng wth the analyss of added-value strateges n Brandenburger and Stuart 1996) versus that wth an explct nterest n the mplcatons of the core (e.g., begnnng wth MacDonald and Ryall 2004). Both approaches provde nsghts nto the effect of competton on the dstrbuton of value wthn a market. The appeal of added value s ts great smplcty. However, ths smplcty comes at a cost: By gnorng the lon s share of compettve relatons embedded n a coaltonal game, analytcal clams based on added value tend to be coarse at best and msleadng at worst. Use of the core solves the problem of theoretcal ncompleteness but does so at the cost of nformaton requrements that ncrease exponentally wth the number of agents n the market. Thus, emprcsts (and practtoners) who wsh to apply the theory face a dlemma: opt for the easlymanageable-but-loose approach of added value or, alternatvely, for the potentally-ntractable-but-tght path of the core. 3 Ths dlemma rases the followng queston: Is t possble to construct an alternatve theoretcal tool that, on the one hand, provdes greater analytcal tracton than added value whle, on the other, mtgatng the untenable nformaton requrements of the core? There s another ssue that s closely related to the precedng one. The abstract nature of the core can 1

3 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance 2 Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. mpede one s grasp of the economcs drvng theoretcal results. For example, Aumann (1973) sets up a smple buyer/seller market and shows that, accordng to the core, the buyers are worse off formng a barganng syndcate. So egregous s ths volaton of our economc sensblty that Aumann concludes that the core s a flawed soluton concept. Later, Postlewate and Rosenthal (1974) show that Aumann s concluson was premature. As we wll see, a reasonable explanaton underpns ths paradoxcal result after all. Yet, f use of the core obscures the economcs n the smple and straghtforward settng of dscrete buyer/seller markets to a master of the theory the lkes of Aumann, no less one worres about our ablty to grasp ts mplcatons n the knds of more general settngs of nterest to strategy researchers. In an appled feld such as ours, theoretcal clarty s mportant. Thus, we ask: Can we develop new analytcal tools desgned to provde greater transparency to the economcs at work under the general compettve logc of the core? A thrd consderaton arses from nformal dscussons n the lterature regardng the ndetermnacy of competton. Typcally, even consderng all of the compettve relatons mpled by a coaltonal game under the core, the amount of value captured by each agent s at best narrowed down to a nontrval nterval. Although ntervals are less convenent than the pont estmates mpled n more famlar settngs (e.g., Cournot and Bertrand games), such ndetermnacy may nevertheless be a feature of real-world markets: Takng the premses of the model serously... competton determnes an nterval wthn whch an agent s value capture les; where, wthn that nterval, actual capture lands s due to factors other than competton...the theory ponts toward a new concepton of compettve ntensty as well as the exstence and possble mportance of persuasve resources. (Gans and Ryall 2016, p. 5) If competton s role n an agent s performance s to determne a range of performance possbltes, then the queston arses whether that range s wde or narrow: That s, to what extent s competton the determnng factor for a gven agent s performance? 4 Ths has led to the suggeston (e.g., Gans et al. 2008) that the rght way to thnk about compettve ntensty s not n terms of downward pressure on proftablty (the tradtonal understandng) but, rather, n terms of the extent to whch proftablty s determned by competton per se. 5 Movng beyond nformal speculaton, the queston s the followng: What concrete form mght a mathematcal mplementaton of ths suggeston take? The contrbuton of ths paper s that t provdes answers to these questons. After a more detaled dscusson of how the added value and core approaches have been used n strategy (next secton), we proceed to develop several varatons on a new, formal noton of compettve ntensty (Secton 4). We begn wth a formalzaton that we refer to as drect compettve ntensty. Our defnton s based on smple unlateral devatons from a focal market structure. 6 It bulds on the nsght that added value to an economcally productve group outsde the frm s own mples competton for the frm whch, n turn, drves up the lower bound on the share of value the frm must capture from the economc actvtes of ts group. 7 Ths answers all three of the precedng questons: (1) t extends the addedvalue approach n a sgnfcant way wthout approachng anywhere near the nformaton demands of the core; (2) t helps llumnate the compettve economcs at work n settngs of nterest to strategy scholars (e.g., more general than dscrete buyer/seller markets); and (3) t provdes a formal response to the nformal speculaton n strategy connectng compettve ntensty to bounds on value appropraton. Due to ts nherent smplcty, we magne ths noton wll be the one most conducve to emprcal work n strategy. What then follows are two, ncreasngly complex varatons on the ntal noton of drect compettve ntensty. In the frst of these, we show (Secton 4.2) that the drect effects of competton mply certan ndrect effects. Ths leads to a more refned defnton, ndrect compettve ntensty. The underlyng nsght s that greater drect ntensty for a rval makes t less attractve to ts transacton partners (hgher appropraton by the rval leaves less resdual value for ts partners). As a result, the relatve attractveness of the frm (to the rval s partners) tends to ncrease. Ths s also a form of competton for the frm, albet an ndrect one. We show that ndrect compettve ntensty llumnates and extends our understandng to more general settngs, for example explanng the economcs at work n the classc example of Postlewate and Rosenthal (1974). Though more nformaton s demanded by ths defnton, the mplcatons n certan settngs may be suffcently promsng to warrant ts use n some emprcal studes. Our thrd and fnal varaton, whch we call generalzed compettve ntensty, pushes nsght captured by the frst two defntons to a natural lmt (Secton 4.3). Whle ths last concept may be too demandng for emprcal work, t should be of nterest to theorsts as we use t to demonstrate condtons under whch our approach leads to bounds that concde wth those of the core (though, as wll be seen, the core s used as the prmary benchmark aganst whch our concepts are formally compared at all stages of our analyss). Our three notons of compettve ntensty provde alternatve approaches to the assessment of value capture that, to varyng degrees, stop short of the full computatonal requrements of the core a feature that should be helpful for emprcal applcatons. Lke the

4 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS 3 Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. core, each dentfes an nterval of value capture for every agent. Moreover, each s consstent wth the core n the sense that a core dstrbuton also satsfes the condtons of each for all agents. Because the core accounts for all compettve relatons n any stuaton descrbed by a cooperatve game, t s lkely to reman popular n the development of general theory n strategy. Because they are closely related, our concepts provde useful nsghts nto the economcs at work n the core. The potental beneft of enhanced explanaton s, tself, a theoretcal contrbuton. Toward the recognton of ths beneft, Secton 5 uses a dfferentated products model wth capacty constrants to explore some famlar ndustry settngs n whch compettve ntensty llumnates the essental compettve forces at work. Fnally, wth respect to emprcal analyss, we close ths ntroducton wth a word of cauton. Any theoretcal soluton concept for cooperatve games that s less nformaton-ntensve than the core runs a rsk by leavng out some subset of compettve relatons: n a gven stuaton, the relatons left out may have a substantal effect on the dstrbuton of value. Thus, whle our notons of compettve ntensty are much smpler than the core and, typcally, provde substantally stronger mplcatons than added value alone, the ntervals mpled by our concepts are usually wder than those mpled by the core. (As mentoned above, analyzng these dfferences explctly s a central feature of our analyss.) Thus, care must be taken when applyng our tools emprcally. We say more about ths n Secton Coaltonal Games n Strategy: Added Value vs. the Core By way of selectve lterature revew and an ntroducton to coaltonal game theory (CGT), we use ths secton to dscuss added value and the core the two ponts on the dmenson of theoretcal complexty that bound the mddle ground we wsh to explore later n the paper. Essentally, all applcatons of coaltonal game theory n strategy adopt one or the other of these two concepts as the foundaton of ther analyses. Both approaches have been used to provde mportant economc nsghts for strategy research. We hghlght several of these as the dscusson unfolds. We begn by defnng the objects n a coaltonal game, then quckly move on to examnng the frst concept, added value. The smplcty of added value s why t appeals as an analytcal tool. Yet, t s also why added-value analyss s ncomplete and can lead to problems. By llustratng these, we set up our examnaton of the core whch follows. As we show, whle the core solves the problem of ncompleteness, t does so by ntroducng substantal complexty and other barrers to practcal use. Throughout the paper, we adopt the followng notatonal conventons. Sets are ndcated wth captal letters, whle elements of sets, scalars, vectors, and functons are all represented by small letters. Techncal terms are talczed at the pont of defnton. Value creaton. A coaltonal game s a par (N, v) n whch N {1,...n}, n < 1, s the set of agents and v: 2 n! + s the characterstc functon. An arbtrary agent s denoted 2 N and a group of agents by G N. For any group G, v(g) s the economc value that the group G can create va some collecton of feasble, mutually agreeable transactons. 8 Let v(ú) 0 and assume v s superaddtve. 9 There s no reason for the members of G to produce less than v(g) snce, were that the case, everyone n G could be made strctly better off by producng the full amount. Therefore, under superaddtvty, v(n) s what the theory predcts s the actual economc value created. We hghlght ths feature va the specal notaton V v(n). 10 We also adopt the conventon of droppng set brackets when referrng to specfc group of agents as an argument of v; for example, gven {, j, k} N, we abuse notaton and wrte v(, j, k) rather than the more cumbersome v({, j, k}). If 2 G, then G ndcates G wth removed; f < G, then G + denotes the set G wth added. Fnally, wthout loss of generalty, we assume that v s normalzed: that s, for all 2 N, v() Value capture. A dstrbuton of value s defned as an n-tuple, ( 1,..., n )2 n +. Here, s a number ndcatng the amount of value captured (or approprated ) by agent n return for ts partcpaton n the value-creatng actvtes that contrbute to the producton of V. In the case of a frm, represents ts economc proft. The objectve n CGT applcatons s to understand how the productve alternatves elaborated by v shape. A smple set of assumptons that relates v to leads to the concept of added value. Added value. Added-value analyss reles on three assumptons: () the aggregate value captured by the agents n N s V; () the agents n N must capture at least v(n ), else they could abandon and, n the process, make themselves all strctly better off; and () agents cannot be forced to take less than ther outsde value, lest they not partcpate n the market to begn wth. Formally, V, (1) 2N j v(n ) and 0 for all 2 N. (2) j2n Agent s added value s defned as av V v(n ). If s such that (1) and (2) hold, t mmedately follows that apple av for all 2 N. Note well the analytcal smplcty: for a gven agent, only two quanttes, V and

5 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance 4 Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. v(n ), are requred to compute added value. Moreover, the economc nsght s mmedate: when partcpaton cannot be forced, as s the case n a free market, an agent cannot expect to capture more than the ncremental value assocated wth ts partcpaton. Ths nsght motvates the dea that frm strateges should am to enhance added value. 12 Unfortunately, the smplcty of added value s also ts major drawback. To see why, consder a market wth two frms, a and b, and two symmetrc buyers, 1 and 2. The frms each have one unt of capacty and produce at zero cost. Buyers each demand one unt of product: they value frm a s product at 1 and b s at 0.3. The value created s V 1.3, wth ether buyer dealng wth ether frm. If a s removed, b s left to transact wth one buyer to produce 0.3. Thus, av a 1. Therefore, a apple 1. Yet, reasonng beyond added value, we also predct a 0.7. To see why, say a s buyer demands x, wth 0.7 > x > 0. Then, even f b s buyer stands to approprate all of the value n ts deal wth b ( 2 0.3), a could offer t x, whch would make both 2 (a and buyer 1) strctly better off. Thus, a 2[0.7, 1.0], wth the fnal outcome dependent on noncompettve factors, such as hagglng ablty. Suppose that a adds a socal-networkng feature for ts product that creates a complementarty wth b s. Specfcally: each buyer values b s product at 1, provded that the other buyer has a unt of a s product. Otherwse, everythng s exactly as before. Now (usng prmes to ndcate the new game), V and av 0 a 1.7. Ths s a successful added-value strategy n the sense that a ncreased ts added value. Is greater value capture mpled for a? Consder the dstrbuton and a b 0 (buyers capture all of the value). Does competton, n the sense of the alternatves elaborated by v, rule ths out? The answer s no. Even though a 0, a has no mutually mprovng offer to make to b s buyer (.e., for buyer, v(a, ) 1, but 1 already). It can be shown that a 2[0, 1.7]: a wder range than before. Thus, f a s a bad haggler for example, t may be worse off as a result of ths move an ssue mssed by the added-value analyss. The core. In the precedng example, condtons (1) and (2) lmt attenton to fve groups. Ths leaves out 10 other groups some of whch have mportant mplcatons, as we saw. To ensure that all such mplcatons are taken nto consderaton, (2) can be generalzed to j v(g) for all G N. (3) j2g A dstrbuton s sad to be consstent wth competton when t satsfes (1) and (3). The set of all such s commonly referred to as the core. For every 2 N, the core mples a compettve nterval, denoted [ mn, max ], such that s consstent wth competton only f 2 [ mn, max ] and, for every 2[ mn, max ], there s at least one contanng t that s consstent wth competton. 13 The core has several postve features wth respect to strategy theory. Frst, t elmnates the problem of omsson assocated wth added value. Second, takng the possblty of nontrval compettve ntervals serously leads to mportant nsghts, not the least of whch are () competton bounds appropraton from below as well as from above; and () beyond competton, persuasve resources those that result n appropraton above one s compettve mnmum may also play an mportant role n frm performance. Fnally, the elegant mathematcs of the core (convex analyss) s well understood and extensvely developed. 14 In practce, however, the core suffers from ts own drawbacks. The most sgnfcant of these s that ts complexty ncreases exponentally wth the sze of N. To be clear, ths s not a theoretcal ssue. As mentoned above, whether n equals two or a quadrllon, the math s elegant and tractable. 15 Rather, the combnatoral complexty s a problem n the applcaton of the theory n practce and n the testng of ts clams emprcally. For example, a market wth just 40 partcpants mples over a trllon possble groups. 16 Thus, even when N s relatvely small, the sheer amount of data requred to test the theory seems prohbtve. Ths problem s compounded by the fact that the lon s share of the groups, because they represent alternatves to the actual actvtes that generate V, are not observed. Thus, the v( ) s for these groups (counterfactuals) must be estmated n some way (perhaps, from other prmtves). Then, even f these data ssues are surmounted, a computatonal hurdle remans: dentfyng the compettve nterval for a sngle, focal frm, requres the soluton of two lnear programmng problems (for mn and max ), each of whch requres the ncluson of all of the group-related nequaltes. Even three-letter government agences mght have trouble solvng LPs wth over a trllon constrants. Ths rases a further concern: Apparently, to satsfy (3) n the real world, the agents n N are not only requred to grasp an enormous number of dfferent, potental ways of organzng ther economc actvtes, but also to know both the values that these would produce and, smultaneously, the shares of V captured by all of the other market partcpants for the actvtes actually undertaken. In most settngs, ths seems far beyond the ken of any real human beng. 17 Actual emprcal work. One of the exctng, recent developments n strategy s the entry of a number of authors whose work focuses on emprcal analyss of the clams ssung from the theory stream. Not surprsngly gven the precedng dscusson, most of these use added value as ther focal theoretcal construct. 18 The emprcal estmaton problem assocated wth the

6 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS 5 Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. core s overwhelmng, as Chatan and Mndruta (2017) pont out. Therefore, f substantal progress s to be made, new theoretcal tools must be developed for emprcal scholars. These should provde gudance on the dentfcaton of a subset of groups that s, on the one hand, emprcally tractable and, on the other, theoretcally meanngful. It s to the development of such tools that we now turn our attenton. 3. Value Parttons We begn our analyss wth a practcal observaton. When the agents engaged n one set of economc actvtes contemplate reorganzng to engage n some other, they must consder all of the costs assocated wth that reorganzaton. Unfortunately, ths dstncton s not always explcated n a cooperatve game settng. In a model, V and v(g) are typcally sad to be the economc value that can be created by N and G, respectvely. In a real-world emprcal settng, V s data value actually created. If the agents n G transacted among themselves to contrbute to V (say, as a supply chan), then v(g) s also data. However, f the agents n G dd not transact wth one another, then v(g) s a counterfactual, the outcome of transacton possbltes that could have occurred but dd not. In the latter case, v(g) must account for all of the economc costs assocated wth ts producton (e.g., swtchng costs). Thus, the real-world analyst must dentfy those groups actvely engaged n the producton of V versus those that do not form but could have done (and ensure that the values of v reflect these dstnctons). We respond to ths need by takng the dstncton between realzed and unrealzed groups as our startng pont. A collecton of groups n N, denoted P, s called a value partton f: () P s a partton of N (.e., dvson of N nto dsjont subsets) and () V P G2P v(g). We refer to each element n a value partton as a value network and, henceforth, use to denote such a group (as dstngushed from an arbtrary group G). Agents n value networks are lnked to one another va the transactons that gve rse to some porton of the overall value created n the market, V. For a gven coaltonal game (N, v), there may be multple value parttons; that s, dfferent confguratons of dsjont groups may be capable of creatng values that aggregate to V. As dscussed at the begnnng of ths secton, emprcally, only one of these confguratons nstantates to actualze V. Let us denote ths confguraton P and refer to t as the focal value partton. The elements of P are smlarly hghlghted wth astersks ( ) and referred to as the focal value networks. The dea s that, n an emprcal context, we have some way of dentfyng the s, v( )s, and, va aggregaton, V. The focal value network contanng agent s denoted. Wth (N, v) understood from the context, the set of all value parttons s denoted P. Note that the trval value partton, {N}, s always an element of P. At least one element of P corresponds to a partton that s maxmal n terms of set cardnalty the aggregate value produced by any larger partton s less than V. Formally, a value partton P 2 P s sad to be maxmal f, for each 2 P, there do not exst dsjont S, T such that v() v(s) + v(t). Presumng that P s focal because t corresponds to emprcal data does not necessarly mply that t s maxmal (.e., t may be possble to break an observed network of transactng agents nto subgroups wthout changng the overall value produced). Therefore, t s worth mentonng that the results that follow apply to any value partton, ncludng those that are not maxmal. That sad, a partton that corresponds to the most refned (maxmal) parsng of observed transactons wll yeld the most refned conclusons. 4. Quantfyng Three Shades of Compettve Intensty We buld on the dea of a focal value partton, whch s amenable to real-world nterpretaton and useful n accountng for the relatve frctons faced by agents consderng alternatves to some status quo market structure. Our ntal noton of drect compettve ntensty s the smplest and, we conjecture, most useful dea from the standpont of emprcal applcatons. We expand ths dea to ndrect ntensty whch, whle somewhat more complex, hghlghts an mportant mplcaton of drect compettve ntensty. Extendng these mplcatons to the theoretcally maxmum number of groups leads to our most general verson of compettve ntensty. Ths sequence of defntons s ncreasng n both computatonal dffculty and refnement of the bounds on a frm s range of compettve nterval Drect Compettve Intensty Assume P s gven. Usng ths as our gude and retanng added-value constrants (1) and (2), we wsh to dentfy a further subset of core constrants (3) that provdes meanngful analytcal tracton whle yet retanng much of the smplcty of the added-value construct. Obvous frst canddates are those assocated wth the focal value networks themselves: v( ) for all 2 P. (4) 2 Invokng (4) means assumng that the members of a focal network cannot be forced to capture, n aggregate, less than the value produced by that network. The defnton of a value partton combned wth (4) mples 2 v( ) for all 2 P. (5)

7 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance 6 Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. Thus, the agents of a focal value network capture among themselves precsely the value produced va ther jont economc actvtes an uncontroversal concluson. Note also, snce apple av (N), v( ) 2 apple 2 av (N) for all 2 P. (6) When we say the added value for, we refer to the added value as defned wth respect to the market as a whole, av (N). However, the noton of added value can be extended to any group: for any G N, defne av (G) v(g) v(g ) f 2 G, and av (G) v(g + ) v(g) otherwse. Gven P, an obvous quantty of nterest s av ( ). The relevant core constrants are j2 j v( ) for all 2 N, (7) where s s focal value network wth removed. Expressons (5) and (7) mply apple av ( ) for all 2 N. (8) In other words, just as an agent s added value to the market as a whole caps ts ablty to approprate, so too does ts added value to ts own value network. Ths s consstent wth our usual ntuton about added value. At ths pont, t s worth pausng to hghlght the fact that av (N)appleav ( ). The proof of ths s smple. By the defnton of added value, av (N) V v(n ). By the defnton of the focal value partton and added value to a group, av ( ) V (v( ) + P 2P \ v( )). By superaddtvty, v(n ) (v( ) + P 2P \ v( )). The concluson follows mmedately. For ths reason, our formal defntons, whch seek to dentfy the tghtest bounds, use av (N), rather than av ( ), as the relevant constrant on the upper lmt of appropraton. That sad, (8) should be kept n mnd as t s often useful n workng through the mplcatons for a partcular confguraton of value networks. Contrary to ths ntuton s the effect of s added value to networks outsde ts own. To the extent that has postve added value to a value network besdes ts own, that added value places a lower bound on ts ablty to capture value. Ths concluson requres consderaton of the followng subset of core constrants: + j2 j v( + ) for all 2 N and 2 P \. (9) We refer to the value networks outsde of s own (P \ ) as s compettve perphery. The value networks n an agent s compettve perphery compete aganst the agent to create value wth ts transacton partners but, smultaneously, aganst ts transacton partners to create value wth t. That s, external value networks compete mplctly wth the agents who transact wth one another wthn a gven network. Thus, (9) ncludes the constrants assocated wth the groups constructed by addng each to each value network n ts compettve perphery. When these hold as well, av ( + ) for all 2 N and 2 P \, (10) whch follows from (5) and (9). When agent has added value to a network n ts perphery, then those n must mpart to a share of v( ) suffcent to prevent from leavng for a better deal n the alternate network an amount no less than that added value. As mentoned, others have emphaszed the mportance of the two aspects of competton hghlghted by the core (.e., ts two-sded effect and ts ambguous nature). 19 Fndngs (8) and (10) are consstent wth these observatons. They do, however, mpart an addtonal nsght: the effect of an agent s ablty to create value wth a gven group crtcally depends on whether the group s a focal value network contanng the agent (.e., the actve network n whch the agent partcpates) or an alternatve to that network. The added value of an agent to the focal network contanng t represents an upper bound on ts ablty to capture. For any group n whch ts membershp s only potental, ts added value represents a lower bound. Thus, the agents wth whom the frm s transactng are qualtatvely dfferent from those wth whom t could transact. Ths refnement s mssed n coarser treatments such as Porter (1979), whch, for example, lumps all of a frm s buyers together as compettors. What the precedng analyss shows s, frst, t matters whether buyers are actual or potental. When the frm s actual buyers have the potental to add value to a perpheral value network, that network competes for the buyers (and aganst the frm). A frm s buyers do not compete wth t rather, external value networks do. 20 Second, potental buyers (e.g., those who would lke to transact wth the frm but cannot, say, due to capacty constrants) create competton for the frm and, thereby, may assure t postve appropraton. Wth a bt of addtonal reflecton, we come to our central dea. As we have seen, a frm s added value to a network n ts perphery mples competton for t the greater the added value, the greater the mplct ntensty of that competton. Thus, scannng the frm s perphery and takng ts maxmum added value s one way of measurng compettve ntensty for the frm. Defnton 1. The drect compettve ntensty (DCI) for 2 N n a focal value partton P s w av (C ), where C arg max av (). (11) 2P \ That s, C 2 P s any value network n s compettve perphery to whch adds maxmum value (there may be more than one). We pont out that computng w requres assessng s added values to no more than

8 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS 7 Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. n/2 groups. 21 In actual ndustres, wth large chans of raw materal supplers, parts manufacturers, orgnal equpment manufacturers (OEMs), and dstrbutors, the number of focal value networks wll be much smaller. Note as well that need know nothng about what other agents are appropratng to compute w. Snce the logc of Defnton 1 apples symmetrcally for all of the agents n the frm s focal value network, and snce the effects across agents are cumulatve (whatever value s guaranteed to the frm s transacton partners s value prohbted to t), the aggregate of all of the other ntenstes s a measure of the ntensty of competton aganst the frm. Ths leads to the followng. Defnton 2. Gven P, the drect compettve ntensty (DCI) for s partners s j2 \ w j. Then, defne s drect compettve resdual (DCR) as w mn v( ) w j, av (N). (12) j2 \ In other words, s compettve resdual s what s left over from v( ) after payng s partners exactly ther DCI, but no more than s added value to N. One may thnk of a frm s compettve resdual as the most optmstc payoff t can expect should t succeed n lmtng ts transacton partners to appropratng no more than warranted by ther own compettve ntenstes. 22 Ths leads to the followng proposton lnkng DCI to the added-value and core approaches common n the strategy lterature. Proposton 1. Gven a focal value partton P, for all 2 N, Proof. See Appendx A. mn w v(), and (13) max apple w apple av (N). (14) Defntons 1 and 2 represent an operatonalzaton of compettve ntensty that has the desrable features of beng conceptually consstent wth core-based notons of competton emphaszed n related strategy research, whle usng a much smaller set of constrants than requred to compute the core, n a way that sheds new lght on the underlyng economcs. 23 From Proposton 1, we see that a frm s DCI provdes a conservatve lower bound, and ts DCR an optmstc upper bound, on competton s effect on ts ablty to capture value. If the maxmal partton s the trval one.e., s {N} then ths approach concdes wth conventonal added-value analyss. How complex are the drect compettve ntensty bounds versus calculatons of added value or the core nterval? Suppose there are n 2 agents partcpatng n a market wth m 2 value networks. 24 Further, assume frm s value network contans z 1 agents. To compute av requres two quanttes: V and v(n ). Calculatng s core nterval requres (2 n 1) values: V and one for every nonempty, proper subset of N. In comparson, assessng the frm s DCI requres 2(m 1) values because the added value to each of the (m 1) networks, requres two peces of nformaton: v( ) and v( ). To get s DCR requres DCI to be calculated for the other (z 1) agents n. Wth the v( )s already n hand, ths requres an addtonal (m 1) values. Addtonally, the DCR requres three more values: v( ) as well as V and v(n ) for av. Altogether, then, the ntensty nterval requres K (z + 1)(m 1) + 3 values. The maxmum K arses when the networks n s compettve perphery are all made up of pars plus a one group (a sngleton f n s odd or an empty group f the number s even, so m (n z)/2 + 1), n whch case K (z + 1)((n z)/2) + 3. Ths expresson s concave n z and maxmzed at z (n 1)/2, resultng n K max 1(n + 8 1) Ths means that, compared to K max, the number of values requred to obtan the core s, approxmately, four tmes greater when n 5, 50 tmes greater when n 10, and 1,000 tmes greater when n 15. Moreover, keep n mnd, K max s an extreme upper bound; typcally, K wll actually be much smaller. Some readers may fnd t surprsng that nether the lower nor the upper bound on an agent s appropraton need drectly depend on ts added value to the market as a whole. However, t has long been known that the correspondence between added value and appropraton does not always behave accordng to ntuton. Indeed, n some reasonable cases, ncreasng a frm s added value actually decreases both endponts of ts core nterval a result so counterntutve that some have taken t as a prma face argument aganst the use of the core as a soluton concept. One of the contrbutons of DCI s that t provdes nsght nto these objectonable paradoxes, showng that they are not paradoxes at all but, nstead, the outcome of perfectly reasonable economcs. Indeed, graspng the economcs of these specal cases s a major step toward understandng competton more generally. A concrete example wll be nstructve. Example 1. Consder a market for a homogeneous good sold by two frms, labeled a and b. Each of them has capacty of two unts, whch can be produced at zero margnal cost. There are three dentcal buyers, labeled 1, 2, and 3: each has unt demand and values the frms products dentcally at u > 0. In ths stuaton, V 3u. The added value of each player, buyer or seller, s u. It can be shown that there s a sngle dstrbuton of value n the core: a b 0 and u for each buyer

9 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance 8 Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. 2{1, 2, 3}. Intutvely, the slack capacty n the market causes the frms to compete for each other s buyers to the pont of drvng prces to zero. Ths s precsely the example of Postlewate and Rosenthal (1974). 25 Let us see how close the bounds mpled by DCI come to capture ths outcome. Among the sx nontrval value parttons n P, suppose the focal one s P {{a, 1, 2}, {b, 3}}: frm a sells to buyers 1 and 2, frm b sells to buyer 3 (all possble value parttons are the same up to agent relabelng). To see that ths s a value partton smply note that v({a, 1, 2}) 2u and v({b, 3}) u, so V v({a, 1, 2})+ v({b, 3}). Startng wth buyers 1 and 2, w 1 w 2 u snce both have added value of u wth respect to b s value network. Gven the market confguraton P, frm b has excess capacty wth whch to attract one of a s buyers were that buyer to capture an amount of value less than u. For example, f frm a demanded a prce of p from buyer 1 such that u > p > 0, then frm b could offer 1 a prce of p/2, thereby makng both t and the buyer strctly better off. Under ths confguraton, w 3 0 because frm a s sellng at full capacty. Fnally, wa w 0 because frms b add no value outsde ther own networks: the products are homogeneous and all of the demand n each network s satsfed. Proposton 1 states that u apple mn 1, mn 2. Gven v({a, 1, 2}) 2u and that the sum of the amount of value captured wthn a network must exactly equal the value t produces, we deduce 1 2 u and a 0. Movng on to frm b s value network, all we get from Proposton 1 s 0 apple b, 3 apple u (.e., frm b and buyer 3 splt the value they produce n some way). Ths s where the analyss stops based only on compettve ntensty. Stll, wth a small number of calculatons, the value captured by frm a and ts buyers s pnned down exactly. Moreover, ths mproves on the standard added-value approach, whch ndcates only that the payoff of each agent s somewhere between 0 and u. Summng up ths secton, we use the structure mpled by a market value partton to show that havng added value to a network outsde one s own mples a floor to the quantty of value one must capture. Ths dentfes network-level competton for an agent. We defne the ntensty of ths sort of competton as the maxmum of such added values. Proposton 1 shows that compettve ntensty mproves on added value n two ways. Frst, t provdes a lower bound. Ths mportant, postve feature of competton s entrely mssed by added-value analyss. Second, the upper bound s tghter than added value whenever the sum of the value added by ndvdual members of a network to alternatve networks s suffcently hgh. Moreover, whle the core nterval s typcally tghter, the nformaton requred to compute ntensty bounds s bounded above by a quadratc functon of n (rather than one that s exponental n n). Just how much tracton compettve ntensty actually provdes wll depend on the partculars of the stuaton at hand. Stll, gven ts smplcty and clear economc meanng, we conjecture that the precedng approach s one that emprcsts and practtoners wll fnd useful Indrect Compettve Intensty Once the effect of compettve ntensty s grasped, a further nsght arses. Suppose a frm and a rval operate n separate value networks. The frm adds no value to the rval s network and, hence, enjoys no beneft of compettve ntensty. Even so, what f the compettve ntensty for the rval s so strong that the rval captures the lon s share of the value produced n ts network? If the frm can create value wth some of the rval s transacton partners above and beyond the meager amount that they must receve once the DCI for the rval s taken nto account, then competton for the frm does, n fact, exst. If the frm s offered less value n ts own network than the dfference between the value t can create wth ts rval s transacton and the amount they look to capture n ther own network, then the frm can cut a deal wth them that makes them all strctly better off. Competton for the rval ndrectly creates competton for the frm. Formally, suppose agent consders jonng 2 P \ as a replacement to some subset Y. Let av Y ( ) v( ) v( \Y), the value added by Y to. By Proposton 1 agent j 2 Y cannot receve less than w. j Moreover the agents n Y cannot, n aggregate, receve less than what they can produce on ther own, v(y). Therefore, the agents n \Y cannot capture more than v( ) max j2y w j, v(y). (15) Expresson (15) presents an upper bound on the opportunty cost that the agents n \Y bear by foregong transactons wth those n Y. Thus, the agents n \Y should be wllng to replace Y wth, provded asks for no more than apple v( [ \Y) v( ) max w j, v(y) j2y av ( [ \Y) av Y ( ) max w j., v(y) (16) Equaton (16) s the value added by to the agents n \Y above and beyond the upper bound on ther opportunty cost of abandonng the agents n Y. When ths value s postve and larger than w, then the group \Y generates further competton for beyond w. Note that DCI corresponds to the partcular case n whch Y ú ( jons but replaces none of ts agents). Ths leads to a defnton that ncludes both ndrect compettve ntensty and ndrect compettve resdual. j2y

10 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS 9 Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. Defnton 3. Gven P, the ndrect compettve ntensty (ICI) for 2 N s apple s max Y, 2P \ av ( [ \Y) av Y ( ) max w j, v(y). (17) The ndrect compettve ntensty for s partners s s j, j2 and the ndrect compettve resdual (ICR) of s s mn v( ) s j, av (N). (18) j2 Once agan, an agent s external alternatves create a lower bound on the value t must capture. Symmetrcally, the compettve ntensty for ts partners places an upper bound on ts ablty to capture value. As always, an agent can never capture more than ts added value. Ths leads to bounds that are (weakly) tghter than those n Proposton 1. Proposton 2. Gven a focal value partton P, for all 2 N, Proof. See Appendx B. j2y mn s w, and (19) max apple s apple w. (20) By ncreasng the scope of competton for an agent, we can tghten our estmate of ts appropraton at the cost of ncreased nformaton requrements. Ths s the case whenever there s suffcent substtutablty between some agents n one network wth some of the agents n another (whch s lkely to be the case, for example, wth supply chans). At ths pont, we stop trackng the number of groups requred to compute the bounds of nterest. They are fewer than those requred to estmate the core nterval but more (possbly by a sgnfcant number) than those requred to compute compettve ntensty. Thus, ndrect ntensty may or may not be of practcal use n real applcatons. Whether or not the data requrements are prohbtve wll vary from stuaton to stuaton. Even so, we beleve the theoretcal nsght nto how competton for one agent ndrectly generates competton for another s qute valuable. To llustrate, we return to Example 1. Example 2. Under the focal value partton P {{a, 1, 2}, {b, 3}}, the noton of drect compettve ntensty dentfed the exact compettve ntervals for some but not all players. In partcular, w 3 0 left ambguous the splt of value between frm b and buyer 3. Gven P, buyer 3 adds no value to the network {a, 1, 2} because, under the deals mpled by that network, frm a has no capacty remanng to make a sale to buyer 3. However, frm a does compete for buyer 3 n a way that s not captured by w 3. The hgh compettve ntenstes for buyers 1 and 2 mply zero value capture by frm a. Ths leaves frm a hghly motvated (n the sense of beng wllng to take even the smallest quantty of value) to replace ether of ts buyers wth buyer 3. Indeed, frm a becomes ndfferent to such a deal only when 3 u. Thus, competton from frm a drves up buyer 3 s mnmum level of appropraton n ts transacton wth frm b. The competton for buyer 3 that s so generated by the hgh compettve ntenstes for buyers 1 and 2 s captured by our noton of ndrect compettve ntensty: we have that s u (u 3 w ) w wth 2{1, 2}. Recall that w u, and thus by Proposton 2, we have mn 3 u. However, as av 3 ({b, 3}) u, we have max apple u. Thus, 3 u and, snce the value captured wthn a value network must be shared among ts members ( b + 3 u), we also have b 0. Examples such as ths have garnered consderable nterest n economcs when contrasted wth the case n whch all the buyers are merged nto a sngle monopsonst. In our example, t can be shown that a merged buyer faces a core nterval of [u, 3u] whle, n the dsaggregated case, the core ndcates that each buyer s guaranteed to capture exactly u (or 3u n aggregate). Accordng to the core, then, the merger of buyers cannot be advantageous. As Aumann (1973, p. 1) states, It seems ntutvely obvous that n a monopolstc market, the monopolst has an advantage because he can avod competton. Indeed, so obvous does Aumann consder ths ntuton that he regards counterexamples lke ths to be strong evdence aganst the ntellgblty of the core as a soluton concept. In response, Postlewate and Rosenthal (1974, by means of an example smlar to ours) show that the core accurately captures key dfferences n the underlyng economcs competton may, ndeed, work to the dsadvantage of some syndcates. Our notons of compettve ntensty provde a transparent conceptual framework by whch to understand the economcs of these and more (cf. the opacty of the core), as well as a nce means by whch to qualfy them. Wthout resortng to the core, drect compettve ntensty alone ndcates that the dsaggregated buyers must approprate between 2u and 3u due to competton from frm b for buyers 1 and 2. Addng the ndrect compettve ntensty that arses as a result (frm a ndrectly competes for buyer 3) pns down everyone s payoffs exactly. The economcs are clear: merge the buyers and compettve ntenstes vansh. There s only one

11 Montez, Ruz-Alseda, and Ryall: Compettve Intensty and Frm Performance 10 Management Scence, Artcles n Advance, pp. 1 18, 2017 INFORMS Downloaded from nforms.org by [ ] on 15 August 2017, at 12:08. For personal use only, all rghts reserved. value chan (both frms sellng to the merged buyer) and hence zero compettve ntensty of any knd. Ths s qute reasonable. Before leavng ths example, note that t also llumnates what compettve ntensty leaves out (versus the core). Accordng to the core, the merged buyer faces appropraton n the range [u, 3u]. The mnmum s due to the surplus unt of capacty between the frms, whch creates a form of wthn-network competton (the frms jostle to fll ther respectve capactes) that assures the merged buyer s capture of at least u. Snce our compettve ntenstes are nduced by external competton (whch requres nontrval value parttons), they do not pck ths up. In the next secton, we wll allow for ths Generalzed Compettve Intensty Thus far, we have consdered competton for an agent generated by external networks (drect compettve ntensty) and by subsets of such networks affected by compettve ntenstes (ndrect compettve ntensty). We beleve that ths way of conceptualzng the effects of competton s natural, straghtforward, and enlghtenng. Yet, as we saw n the mmedately precedng example, these defntons fal to account for some forms of competton (e.g., wthn-network competton). Therefore, we now drop the requrement for a value partton and focus, nstead, on the entre set of subsets of N. Ths extenson approaches the core n ts nformaton demands and, as a result, s unlkely to be especally useful n real-world applcatons. That sad, we present t now for theorsts who may be nterested n seeng a generalzaton of the basc deas presented n Sectons 4.1 and 4.2. Readers unnterested n these techncal theoretcal detals may wsh to skp ths secton. Our defnton of generalzed compettve ntensty bulds on the premse that no group G N can receve more than the value t adds to the market as a whole, av G (N), and, moreover, no more than the aggregate of ts agents added values, P j2g av j (N). We defne the general compettve ntensty (GCI) for by the threat to jon a subgroup by payng the agents n that subgroup ther full added value (.e., to the market) the most each could hope to approprate. Ths provdes an upper bound on the opportunty cost that the agents n the group bear should they forgo transactons wth. We wsh to dentfy the set of groups for whch ths value s maxmal. Formally, apple C 2 arg max G N\ v(g [ ) mn av G (N), av j (N). j2g How does the precedng expresson generalze our earler deas? By the defnton of a value partton P, V P 2P v(). Therefore, v() av (N) for any 2 P. (21) Recallng expresson (6), for all 2 P, av (N)apple av (N). 2 The dentfcaton of C n Defnton 1 s smlar to that of C, wth the dfference beng that the maxmzaton n the defnton of the former s restrcted to 2 P. As before, no group can capture a share of value greater than ts added value to the market. Ths brngs us to the followng. Defnton 4. The generalzed compettve ntensty for 2 N s w v(c [ ) mn av C (N), j2c av j (N). (22) From (21), the natural extenson of w s the generalzed resdual of 2 N: apple w mn av G[ (N) w j. G N j2g\ Denote the set of payoffs that are consstent wth generalzed compettve ntensty by GCI 2 R n 2N V, w apple apple w, (23) where w and w are the vectors composed by the w s and w s, respectvely. Stated ths way, GCI can be seen to represent an alternatve soluton concept for coaltonal games. Hence, we now establsh ts relatonshp to the core denoted by C. Proposton 3. The core s a subset of those payoffs that are compatble wth generalzed compettve ntensty.e., C GCI. Moreover, for any focal value partton P, for all 2 N, s w max Proof. See Appendx C. mn w s. Importantly, f the core for a market s nonempty, then the set of payoffs that are compatble wth generalzed compettve ntensty for that market s also nonempty. In our settng, the extreme ponts of GCI can be descrbed by vectors that, themselves, can be used to determne whether the bounds mpled by generalzed compettve ntensty fully characterze an agent s core nterval. Quant et al. (2005) descrbe how to do ths wth a set of vectors they call the largnals. Here, we take a related approach to characterzng the set of payoffs that are consstent wth the noton of generalzed compettve ntensty. Let be an ordered profle of the agents n N and (k) 2N be the agent n poston k accordng to. Let denote the set of all possble ordered profles.