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Olson vs Coase: coaltonal worth n conflct Joan Esteban Insttut d Anàls Econòmca, CSIC and József Sákovcs Unversty of Ednburgh October, 00 Abstract We analyze a model of conflct wth endogenous choce of

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Olson vs Coase: coaltonal worth n conflct Joan Esteban Insttut d Anàls Econòmca, CSIC and József Sákovcs Unversty of Ednburgh October, 00 Abstract We analyze a model of conflct wth endogenous choce of effort, where subsets of the contenders may force the resoluton to be sequental: Frst the allance fghts t out wth the rest and n case they wn later they fght t out among themselves For three-layer games, we fnd that t wll not be n the nterest of any two of them to form an allance We obtan ths result under two dfferent scenaros: equdstant references wth varyng relatve strengths, and vcnty of references wth equal dstrbuton of ower We conclude that the commonly made assumton of suer-addtve coaltonal worth s susect Key words: coalton formaton, conflct, allance JEL Numbers: D7, D74 Introducton The ssue of coalton formaton has long been of nterest to economsts A neat and useful model was develoed for ths urose: games n coaltonal form The descrton of these games conssts of a mang from subsets of ndvduals to ayoffs Under the Coasan hyothess that wthn any grou the layers would effcently get to agreement, t s natural that t has been assumed that the ayoff to the sum of two grous s no less than what the grous would obtan searately Ths assumton has been central n the lterature on coalton formaton In contrast, the vew held by Olson (965) s that the outcome of oolng efforts s sub-addtve: the larger the grou the more neffcent t becomes n the ursut of ts ends Ths s so because the belongng to a grou or a coalton does not sto ndvduals from actng n ther own rvate nterests and hence they free-rde on the nut of ther fellow coalton members The resent aer offers a frst attemt at the resoluton of the ssue of the suer- or subaddtvty of coaltonal worth, when ndvduals cannot wrte bndng contracts Hence, we look at allance formaton between artes nvolved n conflct The rocess of coalton formaton s usually seen as a one-shot, rather than a sequental rocess Secfcally, coaltons that mght themselves break u are dsregarded on the bass that they are not credble Ths s a ersuasve argument as long as the roblem s restrcted to be one-shot Yet, there are many nstances n whch, even f the artners are fully aware that ther coalton wll eventually break u, t s n ther nterest to coordnate aganst a thrd arty Examles abound The Sovet Unon and the US were alles aganst Naz Germany n ste of beng fully aware of ther flatly oosng nterests The fall of Nazsm brought the end of the allance and the begnnng of a subsequent stage of the game: the cold war The vctory of one coalton over another changes the roblem altogether and may end the allance Yet, for both alles t mght be better to kee the allance u untl they reach the stage n whch the common enemy has dsaeared Thus, n ths aer we exlore the vablty of temorary allances that take the form of a tact agreement between a subset of agents to ostone the resoluton of ther own conflct of nterest and concert effort aganst ther common oonent Perhas the best examle for the sequencng of conflct that we have n mnd s that of a revoluton Thnk of ractcally any of them: the French, the Mexcan, the ussan Esteban and ay (00) roved that when grous artcate n a conflct game aganst each other, the wn robablty contrary to Olson s clam does ncrease wth grou sze Yet, and ths s what we are nterested n here, ndvdual exected utlty s nversely related to grou sze because of the free-rdng etc The wnnng coalton almost always has broken u and vctory has degenerated nto further conflct The second reason why we thnk that the standard aroach s not arorate s that t assumes that coalton members coordnate and ontly decde the acton to be taken by each ndvdual member In ths way (most of) the lterature smly assumes away one of the man reasons resented by Olson (965) as to why large grous mght be neffectve n achevng ther goals: the free-rder roblem A natural way of allowng to test whether the free-rdng effects are suffcent to overcome any otental returns to sze s by assumng that allances merely coordnate on the common target, but that cannot wrte a bndng agreement rescrbng the actons to be taken by each member In ths aer we shall exlctly assume that each layer ndvdually chooses the amount of effort to exend Ths assumton catures the behavor of oltcal allances, n whch the alled artes do not commt to an equal level of effort Party coaltons have the urose of ncreasng the wnnng robabltes aganst the threat of a thrd arty, wthout gvng u on ther own deologcal dentty Ths, clearly, was the logc behnd the re-world-war-ii oular fronts n Euroe The two onts ust made seem to corresond well to the motvatng examles Yet, they are concetually dstnct In ths aer we shall examne the role of each of the two dfferental asects of behavor: temorary allances and ndeendent decson makng In short, ths aer should be seen as comlementary to the aroach taken by ay and Vohra (997, 999) to model non-cooeratve coalton formaton: we are not lookng for an agreement, n fact, we are not allowng any bndng agreement; and nstead of (ust) lookng for effcency, we check for the vablty of strategc manulaton of the game of conflct At the cost of restrctng the analyss to three layers, we can ncororate nto the analyss two characterstcs that are lkely to nfluence whether coaltons form In our frst model, we analyze a contest (that s, a conflct where the layers only value ther favorte outcome) for any ossble dstrbuton of the sze/strength of the ndvduals We fnd that sequencng wll never occur Next, for equal dstrbuton of ower we show that no matter how much layers value each other s favorte otons, There are many economc examles as well Two frms may decde to on forces n ther research efforts to be n a better oston n a atent race aganst a thrd, larger(?) comettor In the event of ther wnnng the race, however, they comete aganst each other n the roduct market See, for examle, Y and Shn (000) We refer the reader to ther aers as well as to Greenberg (994) and Bloch (997) and references theren for a broader lterature revew We dscuss the recent lterature below unless they comletely agree, sequencng does not occur n equlbrum For both scenaros, we also dentfy the range of arameters for whch the sequencng of conflct s welfare mrovng (as a result of reducng the resources wasted on conflct) ecently, a number of aers have looked at coalton formaton n conflct Skaerdas (998) and Tan and Wang (999) showed, wth exogenously gven effort, that (only) when the value/synergy functon of the layers s suer-addtve, sequencng of conflct does (for layers always) arse n equlbrum As we show below, the occurrence of sequencng s not robust to endogenzng the choce of effort That s, f the amount of effort ut nto conflct s a strategc varable, the moral hazard roblem arsng between alles can we out the advantages of dseconomes of scale n effort (arsng from the breakng u of conflct together wth convex cost of effort), resultng n a sub-addtve value functon n equlbrum Nou and Tan (997) analyze a model smlar to ours, but n an exlct mltary context The two man dfferences are that ther countres can sgn non-agresson acts and that the wnnng coalton can exlot the loser s resources Ther results are almost dametrcally oosed to ours Bak and Lee (00), Bloch et al (00) and Noh (00) do endogenze the outlay decsons, however they assume that there s no further conflct wthn the allance Under these assumtons coaltons may form Ths s consstent wth our result n Esteban and Sákovcs (000), where we show that f we allow the members of the coalton to effcently share the surlus n the roorton of ther would-be shares n conflct then allances may form, at least when the thrd arty s much weaker than the two alles together These results ndcate that the effect of free-rdng can be countervaled by a decrease n the aggregate cost of conflct The closest model to ours s Garfnkel (00) She also allows for a strategc game of conflct both wthn and between the allances (addtonally, wthn the wnnng allance, she assumes that there s ont roducton as well as a fght for aroraton) Her man result s that allances only occur n suffcently large economes In artcular, they do not form when there are only three layers 4 However, she does not examne as we do the effects of ether dfferental strength or vcnty of nterest The aer s structured as follows In Secton, we resent a robablstc model of conflct wth endogenous choce of effort, based on Esteban and ay (999) In Secton, we treat the case of equdstant references and varable strengths, whle n 4 For the three-layer case our result s stronger than hers, because she assumes lnear cost of effort, so no dseconomes of scale to effort In fact, even for larger economes, t s her forward lookng noton of stablty of an allance that enhances the oortuntes for ndvduals to catalze on the benefts of the reduced outlays n the frst-erod conflct Ths effect s absent n our model 4 Secton 4 we focus on the relevance of the vcnty of nterests, wth equal strength dstrbuton The numercal calculatons were done by Mathematca and are avalable from the authors, or at the followng UL: htt://wwweconedacuk/aers/mathematcal_aendx_olson_vs_coase_coaltonal _worth_n_conflctdf A model of non-cooeratve conflct In ths secton, we lay down the foundatons of our model of the conflctual resoluton of oosng nterests wth endogenous choce of effort 5 In order to gan n analytcal deth, we shall kee the conflct model at ts smlest verson ossble We shall assume that there are three ndvduals (or grous),,, Each ndvdual refers one alternatve above the rest We shall ndex alternatves by the name of the ndvdual for whch t s the most referred choce We shall thus have alternatves,, We shall denote by u the valuaton by ndvdual of alternatve, wth u u for all π We shall assume the followng valuatons: u,,,, u u -v, 0 v, and u 0 for all the other valuatons Ths smle set-u wll allow us to look at the case of contest (v ) as well as the case where two layers nterests are less dstant than the thrd one s (v ) Indvduals see the outcome of the non-cooeratve conflct game as robablstc, wth robabltes deendng on the amount of effort contrbuted by the dfferent ndvduals We shall denote by r the effort contrbuted to conflct by ndvdual/grou To fx deas, let us start by consderng the case n whch no allance or any form of coordnaton among layers takes lace The robablty of success of the alternatve ushed by layer,, s assumed to be determned by () Â where n stands for the relatve effcency wth whch ndvdual/grou turns effort nto effectve nfluence on the robablty dstrbuton, wth Sn Thus, n can be nterreted ether as the sze of grou or as the ower of the -th ndvdual We shall take the nterretaton of the vector n as the dstrbuton of ower n r n r, 5 Our model s based on Esteban and ay (999) 5 Deendng on the effort he exects the other layers to contrbute, ndvdual wll choose the amount of effort r maxmzng her exected utlty whle deductng the utlty loss roduced by the effort exended, whch we assume to be quadratc 6 That s, () Eu Â u - r u - Â n - r where v u - u 0, π Thus, accordngly wth our assumtons on references, we shall have v 0,,,, v v v, 0 v , and v for all the other valuatons Lemma The ure game of conflct has a unque Nash equlbrum characterzed by Â v - r 0,,, Proof: To smlfy notaton we shall denote by the aggregate amount of effectve effort exended, () Ân r Usng ths notaton the wnnng robablty of alternatve can be rewrtten as (4) nr nr nr Â The frst order condton for a maxmum s n Ï (5) Â v - r Ì Â v - r 0,,, r Ó It s straghtforward to check that the second order condtons are satsfed so that (5) comletely characterzes the best resonse of an agent to the efforts contrbuted by the others Therefore, the vector r Œ + s a Nash equlbrum of the conflct game f and only f r satsfes (5) for,, Next, defne the equlbrum n terms of the vector of wnnng robabltes,, and the total effort exended, Let us defne the x matrx W wth characterstc element w matrx form as n v Observe that we can rewrte the equlbrum condtons n 6 The convexty of the cost functon serves two roles: t enhances the benefts of breakng the conflct u and t ensures an nteror soluton to the equlbrum choce of effort 6 (7) W Note that s the egenvector and the egenvalue of the matrx W From the Perron- Frobenus theorem we know that a vector n the unt smlex and a ostve number solvng (7) do exst and that they are unque Consequently, we have establshed that our conflct game has a unque Nash equlbrum, characterzed by (5) QED Note that usng (5) we can rewrte the equlbrum exected utlty of layer as (6) Eu u ÔÏ - Ì ÔÓ Â v + r Ô u Ô ÔÏ - Ì ÔÓ Â v + Â v Ô Ô u + - Â v Strategc allances n contests In ths secton we assume that the layers only attach ostve value to ther referred alternatve, whle they are ndfferent between the others Ths restrcton to contests enables us to analyze the roblem for any arbtrary dstrbuton of ower We roceed by frst characterzng the equlbrum utltes that the layers exect n a contest Next, we turn to the calculaton of the equlbrum exected utltes of a game n whch an allance s formed We do ths va backward nducton, frst calculatng the equlbrum of the conflct game that arses n case the allance comes out vctorous from the frst stage Then, usng these contnuaton values, we resolve the conflct game of the frst stage In the case of contests, the exected utlty of agent s gven by (8) u - r The frst order condton (5) can now be wrtten as (9) ( - ) n r Multlyng both sdes of (9) by n and rearrangng we have Ê n ˆ Á Ë (0) Ê n ˆ + Á Ë By (9) and (0), 7 () - ) r ( and thus the equlbrum ex-ante exected utlty, (8), becomes, () u - ( - ) ( + ) In equlbrum, condton (0) has to hold for every,, Hence, addng over all layers, and takng nto account that S, we have the (unque) equlbrum value of mlctly determned by the equaton, Ên ˆ Á () Ë Â + Ê n ˆ Á Ë Pluggng ths value of nto (0) we can obtan the equlbrum vector of robabltes, Wthout loss of generalty, we wll adot the followng arametrzaton: n - n, n nm and n n(-m) We are gong to consder the ossble formaton of an allance between layers and These two layers can alter the game by forcng the conflct game to be resolved va two sub-conflcts If the allance s formed, then n the frst round the artners wll ontly fght aganst layer That s, the frst sub-conflct wll be descrbed by a lottery, whch ether gves or the allance of and as the wnner The allance members choose ther outlay searately, as a functon of ther exectatons n the second sub-conflct Ths subgame only arses f the allance wns n the frst conflct, n whch case the artners have to fght t out between themselves The resoluton of the resdual conflct n stage Usng (0), () and () we can easly obtan that the exected second-stage utlty of members of the alled arty, (w, w ), wll be m ( + m) ( - m)( - m) (4) w and w The resoluton of the frst-stage conflct Gven the contnuaton values calculated above, together wth w, we can examne the frst-stage behavor of the layers The exected utltes as a functon of ther outlays are u (r ) - / r, u (r ) (- ) w - / r, u (r ) (- ) w - / r 8 Just as n Secton, usng the frst-order condtons we obtan that ( + ) (5) u, u ( ( + )) w, u ( ( + )) w, where and are the robabltes of wnnng n case the layers exended efforts thnkng they were sequencng the conflct but n fact t was resolved n one stage That nmr s, n( -m) r and Solvng for the 's s agan suffcent to determne the exected utltes n equlbrum Let condtons, we obtan that, (6) z + z Ê Á Ë r ˆ z Then, agan usng the frst-order, z w and z w Snce agan the s add u to one, we can solve for z, usng the fact that (7) z We obtan (8) Ê nm ˆ Ë - n z z and z Ê Ë n(- m) - n - n n m w + ( - m) w ˆ z Let us set a - n and A m ( + m) + ( - m) ( - m) Substtutng (7) and (8) n back nto (6) and usng the revously comuted contnuaton values, we then have a A m ( + m) ( - m) ( - m) (9) A + a A, A + a A and A + a A Fnally, substtutng back nto (5) we obtan the exected utlty of each layer: (0) u a u A a A( A + a A) ( A + a A ) ( A - m ( + m) ) + A m( + m) ( A + a A ) ( A - ( - m) ( - m) ) a A + A ( - m)( - m) u ( A + a A) 9 The comarson wth anarchy Followng these same stes for the benchmark anarchy case, we can calculate the equlbrum exected utltes n total conflct Unfortunately, the formulas are way too long to enable us to comare them drectly wth the utltes exected from sequenced conflct Instead, we erformed a numercal analyss and resent here the lotted values of the dfference between the exected utltes under the two modes of conflct Based on numercal analyss, we can establsh the followng results: Prooston In a socety made u of three layers/grous, ) No two layers wll ever concde n referrng to form a coalton; ) The sum of the utltes of the hyothetcal artners s hgher n anarchy; ) The layer facng the coalton always rofts from sequencng; v) When ether the ont ower share of the hyothetcal artners s small (arox less than ) or they are of smlar ower wth a not too small thrd grou, sequencng would be welfare mrovng INSET FIGUE ABOUT HEE Fgure llustrates the frst statement The dark area corresonds to the arameter values for whch layer refers sequencng through an allance wth layer over layng the orgnal one-shot, anarchc conflct game Snce the fgure for layer s the mrror mage of ths one wth resect to m 5, t can be observed drectly that there are no arameters for whch both layers would concde n ther wsh to form an allance In rncle, ths does not rule t out that ther ont utlty mght be hgher wth sequencng for some dstrbuton of ower The second statement of the rooston shows that ths s not the case, though (we omt the lot) Snce the man effect of allance formaton s that the outlays n the frst erod are reduced, we obtan that the layer that s facng the allance always rofts from such a change Ths roft dfferental s strong enough to overcome the utlty decrease of the allance for an mortant range of arameters Thus, as Fgure shows, the ont surlus of socety would ncrease wth sequencng n about half the ossble cases INSET FIGUE ABOUT HEE 4 Strategc allances wth vcnty of nterests In the revous secton, we have shown that allances do not come about based on grou szes We shall now examne whether allances may arse when layers are of equal ower, but the canddates to ally have ther nterests closer than to the thrd contendng arty To facltate the dstncton wth resect to the revous model, n ths secton we 0 shall call the three layers, and k We thus assume that n h /, h,,k ecall that u k u k -v 4 The resoluton of the resdual conflct n stage Let us start agan wth the valuaton that each of the alled layers wll attach to the contnuaton of the game nto ts second stage Observe frst that, the two grous beng symmetrc, the equlbrum wnnng robabltes wll be dentcal, that s, one half Ths n turn mles that the er cata effort contrbuted by each arty wll be equal Bearng ths n mnd, we can easly deduce from the frst order condton (5) that () r v rk, and that the equlbrum utltes of ths second stage of the game, w and w k, are 8 v () w wk - 5 w 8 Notce that () 75 w 4 The resoluton of the frst-stage conflct Let us now move on to the frst stage of the game when and k oose layer If the members of the allance wn, they obtan an (exected) utlty w gven by () and zero otherwse As for layer, the utlty of wnnng s and zero otherwse The exected utlty by layer (res k) s gven by (4) Eu w r r r r r r w r + k , k and the utlty exected by layer s gven by r r r (5) Eu ( - ) - - r + r + r k The best resonses by layers (res k) and can be obtaned from the maxmzaton of (4) and (5) ( - ) w (6) r, and r r rk By symmetry, r r k and hence We can thus rewrte (6) as (7) ( - ) w, and ( - ) From (7) we can obtan the equlbrum wnnng robablty of the coalton,, as (8) w + w Usng (6), we can wrte the equlbrum utltes of layers (res k) and as (9) + Eu Euk w 4, and (0) ( - )( - ) Eu The exected utlty Eu s ncreasng n w (and hence decreasng n v) Snce 75 w, we have that the equlbrum utlty s lowest when v and hghest when v 0, that s, () 0

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