Minmax Shapley Value and Biform Contests Subgame Perfect Nash Equilibrium

Transcription

1 1 סדרת מאמרים לדיון Workng paper seres No Mnmax Shapley Value and Bform Contests Subgame Perfect Nash Equlbrum Mordecha E. Schwarz

2 142 מערכות מידע חשבונאיות ממוחשבות

3 1 Mnmax Shapley Value and Bform Contests Subgame Perfect Nash Equlbrum Mordecha E. Schwarz 1 Abstract A bform contest s a two-stage game. In the frst stage, agents exert resources to buld up ther poston towards the second stage. The second stage s ether a barganng game or a cooperatve game, but f no cooperatve soluton s acheved t turns to a rent-seekng contest. It s shown that wth a passve regulator, the second stage s a cooperatve game and f all contestants are rsk-averse, the bform contest non-cooperatvely mplements the mnmax Shapley value, as defned n the paper, n subgame perfect equlbrum. Keywords: bform game, bform contest, rent-seekng, externaltes, mnmax approach, mnmax coaltonal worth, mnmax Shapley value. JEL classfcaton: C7, D7 1 Department of Economcs and Management, The Open Unversty of Israel. Emal: mordsch@openu.ac.l. I am truly ndebted to J. Atsu Amegashe, Shmuel Ntzan and Avad Hefetz for very helpful comments on earler versons of ths work. Of course, the responsblty for every error s exclusvely mne.

4 2 1. Introducton Cooperaton among rent-seekers s usually modeled as a two-stage game. There are at least three man categores of these models. One category whch I call wolf-pack games studed a two-stage group rent-seekng wthout a specfc sharng rule of the prze among the wnnng group members. Contestants cooperate wthn group rent-seekng aganst rval groups n the frst stage, and then the ndvdual members of the wnnng group engage n another rentseekng contest for the prze n the second stage 2. A second category, whch I call a lon-pack game, assumes that the prze s dstrbuted n the second stage among the wnnng group members accordng to a pre-known specfc sharng rule. (In ths category, I also nclude models n whch the prze s a club good for the wnnng group) 3. The thrd category contans models of compromse n the shadow of conflct, whch study the endogenous choce of the sharng rule 4. In these games, agents bargan on the allocaton of the rent n the frst stage, and only f barganng fals are they engaged n a rent-seekng conflct n the second stage 5. An evolvng subclass of ths category ncludes models n whch the sharng rules are determned endogenously and specfcally for each group n the frst stage, and n the second stage all ndvduals cooperate wthn the group rent-seekng contest and determne ther contrbuton level 6. These three categores of two-stage games of cooperaton n contests are compatble, each wth certan modfcatons, wth Brandenburger and Stuart s (2007) bform game framework. A bform (or hybrd) n -player game s a two-stage game. The frst stage s non-cooperatve and s desgned to descrbe the strategc moves of the players, but the consequences of these moves are not payoffs. Instead, each profle of strategc choces at the frst stage leads to a second stage cooperatve game. Smlarly, I defne a bform contest as a two-stage game. In The man result of wolf-pack game models s that the collectve acton problem wthn groups usually reduces total contest expendtures (Bak (1993), Katz and Tokatldu (1996), Wärneryd (1998), Inderst, Müller and Wärneryd (2007)). But ths result holds n stochastc contests n whch the wnnng probablty s a contnuous functon of contestants outlays. In determnstc contests, n whch the contestant who exerts the hghest effort wns wth certanty, ths result may be reversed (Konrad (2004)). For an analyss of a determnstc contest, see Ellngsen (1991). The man result of lon-pack game models s that the pre-known sharng rule neutralzes agents rskaverson. (Ntzan (1991), Wärneryd (1998) and Skaperdas and Gan (1995)). For studes of Compromse n the shadow of conflct, see, for example, McDonald and Solow (1981), Svejnar (1986), Alexander (1992), Skaperdas (1992) and Skaperdas and Gan (1995), Esteban and Sákovcs (2002), Bayndr-Upmann and Gerber (2003) and Skaperdas (2006). The studes mentoned n prevous footnotes assumed arbtrary partton of the contestants set nto groups. For analyses of endogenous formaton of allances, see, for example, Hart (1974), Aumann and Meyerson (1988), Katz (1987) and Lnster (1994), Huck and Konrad (2002), Lee and Cheong (2005), Bloch, Sánchez- Pagés and Soubeyran (2006), Sánchez-Pagés (2007) and Glazer (2008). See, for example, Ntzan and Ueda (2011) and references there.

5 3 the frst stage, agents exert resources to buld ther poston towards the second stage. The second stage s ether a barganng game or a cooperatve game. If the bargan fals or f no cooperatve soluton s acheved, the second stage turns to a rent-seekng contest. The resources exerted n the frst stage are amed at beng n an optmal poston n the second stage under every contngency. Although bform contests as a stylzed model of cooperaton are related to numerous real world stuatons, (lke plea-barganng, arms races and peace negotatons, strkes and wage negotatons, allocaton of cartel profts and more), ths approach was rarely appled for analyzng these stuatons, probably because compromse n the shadow of conflct s known as a very complcated stuaton for modelng, and comparatve statcs were consdered mpossble 7. In Schwarz (2011a) and (2011b), I analyzed the bform contest model wth a barganng game n ts second stage. A bform contest n whch the second stage s a cooperatve game deserves a separate analyss snce, n addton to the well-known dffcultes mentoned n the lterature, the defnton of the second stage as a cooperatve game rases another dffculty, due to externaltes whch characterze the rent-seekng compettve envronment. As explaned below, these externaltes undermne the tradtonal defnton of the coaltonal worth functon and the applcaton of any cooperatve soluton concept. To overcome the externaltes problem, I follow Tauman and Watanabe s (2007) mnmax approach and defne the mnmax value of a coalton, and the mnmax Shapley value, as explaned below. The man result of the analyss s that ths type of bform contest noncooperatvely mplements the mnmax Shapley value n subgame perfect equlbrum, provded that all contestants are rsk-averse. Ths ntroducton s followed by four sectons. Secton 2 presents the benchmark one-shot rent-seekng contest. Secton 3 sets up the bform contest model, dscusses the externaltes problem of compettve envronments, presents the mnmax Shapley value approach, defnes the bform contest cooperatve soluton concept and brefly refers to the role of socal norms and regulatory polces. Secton 0 presents the man results of ths artcle. Secton 0 7 See McDonald and Solow (1981), Alexander (1992), Anbarc, Skaperdas and Syropoulos (2002), Bayndr- Upmann and Gerber (2003) and Skaperdas (2006).

6 4 summarzes, dscusses the mplcaton of the results of the analyss and suggests some drectons for further research. 2. The Benchmark One-Shot Rent-Seekng Contest Consder a set N of n contestants who are engaged n a Tullock (1980) type stochastc rentseekng contest over a prze commonly valued by z. Denote contestant s expendtures by x and assume that agent s wnnng probablty s gven by the Tullock s (1980) Contest Success Functon 8, x p x,, jn, x x x (1) 1 j jn n The parameter measures the return to scale of the rent-seekng efforts. If 0, p 1 n, N. On the other hand, f, the rent-seekng contest becomes an all-payaucton under whch the prze s awarded to the contestant who exerts the greatest effort 9. I lmt the analyss to games wth symmetrc pure-strategy Nash equlbra, and n order to assure a unque nteror equlbrum I assume 0 nn1 10. The preferences of agent N are represented by von-neumann-morgenstern utlty functons u w satsfyng u0, u 0. From contestant s pont of vew, the contest has two contngent results: wnnng (state I) or losng (state II). Denote contestant s ntal wealth by A and hs post-contest wealth by w. If the contestant wns, w A z x. If the I contestant loses, utlty functon, w A x. Each agent N seeks to maxmze the followng expected II I II (2) Euw p uw 1 p uw x x. Solvng the contestants optmzaton problem smultaneously yelds the followng frst order condtons for an nteror soluton, For axomatzaton of ths functon, see Skaperdas (1996). See also Fullerton and McAfee (1999). Baye, Kovenock and de Vres, (1993). Pérez-Castrllo and Verder, (1992).

7 5 (3) pu Eu0, N. I II I II, u uw uw and Eu pu w 1 pu w Where p p x. Wth rskneutral contestants, (3) s reduced to pz10, N. Denote equlbrum expendtures profle wth rsk-averse and rsk-neutral contestants by be shown that, x and ˆx, respectvely. It can easly n 1 (4) xˆ z, N 2. n Unfortunately, the effect of rsk-averson on expendtures s generally ambguous, although some progress has been acheved 11. On the other hand, assumng rsk-averson complcates that analyss sgnfcantly 12. Konrad and Schlesnger (1997) show that x x x x ˆ Eu ˆ u ˆ z, and applyng a second order Taylor expanson on ths condton yelds a smplfed verson of Cornes and Hartley s (2010) result: x xˆ p, N 13. Denote by R u u the Pratt (1964) absolute rsk-averson 1 2 ndex, and t follows that f N z z and R R then ˆ n n 1 x x and ˆ 1 p p. In Fgure 1, Konrad and Schlesnger s condton states that x xˆ tan a tan a, and by 1 2 the smplfed verson of Cornes and Hartley s result ths condton s equvalent to x xˆ wˆ wˆ wˆ p. 1 I II Hllman and Katz (1984) show that for small przes, rsk-averson reduces rent-dsspaton. (See also Hllman and Samet (1987) and Ntzan (1994)). But as Konrad and Schlesnger (1997) ndcated, generalzng ths result s non-trval. (See also Cornes and Hartley, (2003)). Probably, ths s the man explanaton for the prevalence of the apparently unrealstc rsk-neutralty assumpton n the rent-seekng lterature. See Mllner and Pratt (1991). Ths s a smplfed form of Cornes and Hartley s (2010) result because t s based on the Taylor expanson technque, and hence vald only for relatvely small prze contests. Cornes and Hartley (2010) proved that the prudence condton (namely 0 ) s suffcent for ths result to hold generally. u N

8 6 Fgure 1 u u ˆ I w u u ˆ II w wˆ II a 1 a 2 z wˆ ˆ I w w 3. The Bform Contest Brandenburger and Stuart (2007) descrbed a bform n -player game (or hybrd game) as a two-stage game. The frst stage s non-cooperatve and s desgned to descrbe the strategc moves of the players. However, the consequences of these moves are not payoffs. Instead, each profle of strategc choces at the frst stage leads to a second stage cooperatve game. Ths lends the compettve envronment created by the choces that the players made n the 1 n frst stage. Formally, a bform game s a collecton, VC,, where... s a profle of strateges, V : C2 n s a cooperatve game coaltonal functon and n c c 1 C s a vector of confdence ndces for each player c 0,1 14. Roughly speakng, the confdence ndces evaluate every contngent outcome n the core accordng to player s preferences 15. Smlarly, I defne a bform contest as a two-stage game. In the frst stage, agents exert resources to optmze ther poston towards the second stage, namely, to be well-equpped and prepared n case the barganng fals, or n case where no cooperatve soluton s acheved and the rent-seekng contest s to be played. Formally, a bform contest s a collecton n x, p, u, v where x 1 x s the contestants rent-seekng expendtures vector, p s the contest success functon, u s a vector of contestants utltes and v s a TU cooperatve game characterstc functon For detaled dscusson of bform games, see also Muto, Nakayama, Potters and Tjs (1998) and Tjs (1990). For a more accurate defnton and explanaton of confdence ndex, see Brandenburger and Stuart (2007) or Stuart (2005). For studes that appled a specfc cooperatve soluton n bform games see, for nstance, Grossman and Hart (1986), who appled the Shapley value; Hart and Moore (1990), who appled the Nash Barganng Soluton and Brandenburger and Stuart (2007), who appled the core.

9 7 A coalton S set F N \ S N s a subset of agents colludng n rent-seekng aganst the complementary. Denote by N the collecton of all subsets of N and by s # S the number of elements n S, (so 1s n where n # N s the total number of contestants). Let v: N be a TU cooperatve game characterstc functon whch assgns a value, namely a real number vs for every coalton S N, satsfyng v 0. The payoff of N agent S s denoted by v S. A soluton concept s an mputaton v satsfyng certan normatve axoms lke effcency and ndvdual ratonalty. Shapley (1953) suggested fve normatve axoms for a cooperatve game soluton: farness, effcency, symmetry, addtvty and the null player axom 16, and proved that the unque soluton whch satsfes these axoms assgns each agent N a value accordng to the followng formula, (5) \ s! ns1! sh v vs vs n!. SN \ v S v S s the margnal contrbuton of agent to the value of coalton S. So the Shapley value of agent N s hs average margnal contrbutons to each coalton, assumng that every order of agents s equally lkely. Externaltes The conventonal defnton of vs mplctly assumes that the coaltonal worth s ndependent of what the players outsde of S do. Ths assumpton, however, does not hold for a compettve envronment, mplyng that the conventonal defnton of vs s napproprate for bform contests. To see ths, note that the expected worth of coalton S N s 16 Formally: Farness, vv, N ; Effcency, v vn; N Symmetry, v S v S j v v j S ; j,, Addtvty, vwvw, N; The null player axom, 0, v S v S v S.

10 8 (6) E vs psz xs. Apparently, (6) characterzes a rsk-neutral contestant. Suppose that the coaltonal effort, x S, s the sum of voluntary contrbutons of the coalton s members, x S the frst stage ndvdual equlbrum expendture. Pluggng x S nto (6) yelds S x, where x s (7) x S E v S z x, S N. S x SN S In symmetrc equlbrum, x x N, mplyng that, 1 Ev S zsx, S N, k (8) where k denotes the total number of coaltons. Namely, the expected worth of a coalton S N depends on the total number of coaltons, k, and on the coalton s sze, s. In other words, the compettve envronment creates externaltes on the worth of coaltons as the expected worth of a coalton S N F depends on the partton of the complementary set,. Therefore, the worth functon n a compettve envronment s actually, : v S N F. Apparently, these externaltes undermne the tradtonal defnton of coaltonal worth and complcate the applcaton of any cooperatve soluton concept. To overcome ths dffculty, I apply Tauman and Watanabe s (2007) mnmax approach 17. The coaltonal mnmax worth, denoted by mvs, s defned as the coaltonal expected worth under the worse scenaro from the coalton s pont of vew, whch occurs when F contans 17 An alternatve average value approach was suggested by Macho-Stadler, Pérez-Castrllo and Wettsten (2006). For further dscussons of externaltes n coalton formaton, see, for example, Chander and Tulkens (2006), Chander (2007), Chander and Wooders (2010) and Chander (2010).

11 9 n s sngleton coaltons and the total number of contestants s k ns 1. It follows from (8) that (9) 1 mv S mn max z sx k xsk, k. The applcaton of Shapley value to our compettve envronment requres some modfcaton to overcome the externaltes problem. Frstly, defne the cooperatve game mv as the mnmax correspondng game of v. The mnmax Shapley value, denoted by shmv s the Shapley value of mnmax correspondng game mv. Agent s mnmax Shapley value s denoted by sh mv. The Bform Contest Cooperatve Soluton Concept n A sharng rule β, 0,1 1 assgns each contestant N a share z of the contest prze. If β s appled n the second stage, agent s utlty s u A z x. If no sharng rule s appled n the second stage, or when no cooperatve soluton s acheved, all contestants are engaged n a rent-seekng contest and the expected utlty of agent N s II I 1 Eu w p x u w p x u w. Defne agent s gan functon as (10) g, u A zx Eu w β x. The gan functon measures agent s gan from applyng the sharng rule β, over competng n the rent-seekng contest. A sharng rule β s ratonal f β x ratonal sharng rules s defned as g, 0, N g, 0, N. The set of all β β x. It can be verfed that g β, x s bounded and concave, mplyng that s bounded, compact and convex 18. Defne the set of all feasble utlty vectors U t can be also shown that n 1 β u β 1, and by the same logc, U β s also bounded, compact and convex, mplyng that ts 18 See Schwarz (2011b).

12 effcent fronter, PU n 1 10 u β β, 1, whch s the set of all utlty vectors nduced by ratonal sharng rules, s concave. A bform contest cooperatve soluton concept s a feasble and ratonal sharng rule uβ PU. Socal Norms and Regulatory Polces Socal norms relate to far solutons and sharng rules. For example, a socalst norm postulates that effort should be compensated and that compensaton should take nto account the ndvdual s effort but also hs relatve poston n the total socetal effort. Thus, a socalst sharng rule determnes the agent s share accordng to a monotoncally ncreasng functon x. On the other hand, a compettve norm postulates that every sharng rule, β, s legtmate condtonal on beng freely and unanmously approved n a proper and far process 19. Regulatory polces express the regulator s socal phlosophy. A passve regulator beleves n lassez-fare and does not ntervene at all n the second stage. An actve regulator s concerned about procedural justce, but ndfferent to dstrbutve justce. Thus, an actve regulator determnes the rules of the game (the barganng protocol), wthout any attempt to mplement a specfc sharng rule. Under an actve regulator, any feasble sharng rule β PU s allowed. On the other hand, an actve benevolent regulator also takes far dvson or dstrbutve justce consderatons nto account and seeks to mplement a sharng rule whch maxmzes a certan socal welfare functon W u β where W : n s contnuous, monotone and quas-concave. In other words, an actve benevolent regulator W seeks to mplement arg max W βp U β u β In Schwarz (2011b), I showed that socalst norms neutralze contestants rsk-averson, mplyng that n a socalst socety, contestants expendtures n subgame perfect equlbrum are proportonal to rsk-neutral agents expendtures n the benchmark one-shot contest. On the other hand, n a compettve socety, contestants subgame perfect expendtures are unambguously hgher than n the benchmark case. The ntuton of ths result s that socalst norms reduce uncertanty, contrary to compettve norms whch do not requre any a-pror functonal relaton between expendtures and share, and consequently enhance uncertanty. W For nstance, a Benthamte regulator seeks to mplement β arg max β. Myagawa u βpu N (2002) clamed that a Rawlsan regulator seeks to mplement the Kala-Smorodnsky (1975) barganng soluton. Ths clam s naccurate. A Rawlsan regulator seeks to mplement

13 11 Equlbrum A bform contest subgame perfect Nash equlbrum s a par β, x where u β PU. Namely, β, x s a bform contest subgame perfect Nash equlbrum, f applyng β as the second stage sharng rule results n x as equlbrum expendture profle, and f x s the frst stage expendtures profle and β s appled n the second stage, no contestant can beneft from devaton. The results n Schwarz (2011a) and Schwarz (2011b) were based on the assumpton of an actve regulator who mposes n the second stage a modfed verson of Mouln s (1984) barganng protocol. In ths artcle, I assume a passve regulator, mplyng that no barganng protocol s mposed n the second stage. Thus, n the second stage, agents can ether form coaltons and play a cooperatve game, or compete n a rent-seekng contest. The followng proposton provdes a necessary and suffcent condton for a subgame perfect cooperatve soluton. Proposton: A bform contest wth a passve regulator non-cooperatvely mplements the Mnmax Shapley value n subgame perfect equlbrum, f and only f R 0, N. Proof: Denote frst stage expendtures profle by x. In the second stage, agents expendtures are sunk-costs and each contestant N s characterzed by hs expendture level, x, and hs correspondng wnnng probablty, p. Hence, (11) mn max x mv p z x k xsk, x 1 mn max z x,. z x N k xsk, x n SN S W β arg max βp U u β, whch s equvalent to mplementaton of the Kala-Smorodnsky barganng soluton f and only f N u u and z z. mn

14 12 Smlarly pluggng x S x nto (9) yelds, S (12) 1 mv S mn max z sx, S N. k xsk, k Wth k coaltons, the average sze of a coalton s s average mnmax value amv of coalton S N, n k. Insertng s nto (12) yelds the 1. k (13) amvs, k z nx Now f agent F FS jons S, the total number of coaltons s reduced to k 1, mplyng that the average margnal contrbuton of agent value of S, denoted by amc, S k, s F to the average mnmax 1 amc S k av S k av S k z nx k k (14),, 1, 1. On the other hand, f agent T leaves coalton T and jon coalton S, the total number of coaltons s unchanged, mplyng that agent s average margnal contrbuton to coalton S s non-zero f and only f by jonng (leavng) S agent decreases (ncreases) the total number of coaltons. Snce by defnton, the mnmax Shapley value of agent N s hs average margnal contrbuton to the mnmax value of all coaltons, t follows that (15) n 1 sh mv amc S k n 1, k 2 n znx zx mv, N n1 k k 1 n k 2

15 13 Insertng (15) nto (10) yelds 1 1 n 1 gsh mv, x ua zx ua zx ua x n n n (16) Takng the second order Taylor expanson of (16) and rearrangng yelds (17) g shmv, x R n z 2 2 n R where R denotes the remander of the Taylor seres (assumed neglgble). It follows mmedately from (17) that g sh mv x R N., 0 0, Notce that by (17) g n0 R 0. Insertng x ˆ from (4) nto (17) yelds g sh mv, x xˆ where Rz 2, mplyng that g 0 R 0. Namely, rsk-averse (seeker) agent s gan (loss) from cooperaton s proportonal to rsk-neutral agent s rent-seekng expendture, and s negatvely (postvely) affected by the number of contestants and the return to scale rate to rent-seekng expendtures. Summary and Dscusson A bform n-player contest s a two-stage game. In the frst stage, agents exert resources to buld up ther poston towards the second stage. The second stage s ether a barganng game or a cooperatve game. If no cooperatve soluton s acheved n the second stage, contestants are engaged n a rent-seekng contest. The nature of the second stage depends on the regulator s type. An actve regulator mposes a barganng protocol, mplyng that the second stage s a barganng game. Ths artcle was confned to the analyss of a bform contest wth a passve regulator, namely a bform contest n whch the second stage s a TU cooperatve game. To overcome the externaltes problem assocated wth the rent-seekng compettve envronment, I appled Tauman and Watanabe (2007) mnmax approach and defned the coalton s mnmax worth functon as ts worth under the worse scenaro from the coalton s pont of vew. The Mnmax Correspondng Game s a cooperatve game n whch the

16 14 coaltonal worth functon s replaced by the mnmax coaltonal worth functon. An agent s mnmax Shapley value s defned as hs Shapley value n the Mnmax Correspondng Game. The man result of the above analyss s that f all contestants are rsk-averse, the bform contest wth passve regulator non-cooperatvely mplements the mnmax Shapley value n subgame perfect equlbrum. It was also shown that the rsk-averse agent s gan from cooperaton s proportonal to the rsk-neutral agent s rent-seekng expendture, and s negatvely correlated wth the number of contestants and the return to scale rate to rentseekng expendtures. The nterrelatons between cooperatve and non-cooperatve game soluton concepts are well known snce the equvalence theorems were proven n the poneerng artcles of Debreu and Scarf (1963) and Aumann (1975), who formulated the Core Equvalence Theorem. Ths theorem states that n perfectly compettve economes, the core concdes wth the set of compettve allocatons. Later works 21 produced the Value Prncple: In perfectly compettve economes, every Shapley value allocaton s compettve, and the converse holds n the dfferentable case. 22 However, two mportant features dfferentate our results from these theorems. Frst, equvalence theorems relate to non-atomc compettve envronments, namely envronments wth nfnte number of contestants 23, whereas our results relate atomc envronments n whch the contestants number s fnte 24. Secondly and maybe more mportantly, n our settng, the compettve equlbrum s not characterzed by a prce vector or a budget set. Nevertheless, these theorems underle the ntuton behnd our results, pontng out that the nterrelatons between compettve and non-compettve soluton concepts also hold for rent-seekng compettve envronments, wth or wthout externaltes. In a semnal artcle, Stgler (1964) conjectured that although colluson may maxmze aggregate profts of all frms n an olgopolstc ndustry, cartels tend to be unstable because each member frm s ncentvzed to cheat. Ths logc mght rase some questons about the necessty for ant-trust legslaton and expensve regulatory agences, but Levensten and Suslow s (2006) survey of the emprcal lterature has abundant examples of successful cartels Shapley (1964), Shapley and Shubk (1969), Aumann and Shapley (1974), Hart (1979) to menton only a few. See Hart (2001) and also Anderson (1992). See also Aumann (1964). For a dscusson of the Core Equvalence Theorem wthn an atomc envronment context, see Anderson (1992).

17 15 that remaned stable for sgnfcant perods of tme. The authors also ndcate that: (a) Cheatng s absolutely not the man cause of the collapse of cartels but what they call barganng problems, and (b) Cartels do not rely on punshment mechansms (lke prce wars ), but on admnstratve organs lke sales agences or even an external regulatory or nspectve agency. These emprcal facts are ncompatble wth classcal cartel theory. Clearly, wth serously enforced ant-trust legslaton, sales agences or other cartel managng organzatons cannot operate n the open. But the naïve Pgouvan vew of a benevolent regulator has been challenged by economsts who recognzed that poltcans and bureaucrats have ther own nterests and preferences that do not always concde wth socal welfare maxmzaton. In a later artcle, Stgler (1971) hmself postulated that regulaton s merely a devce used by the regulated ndustry to control entry and mantan ts cartelstc arrangements. Moreover, Stgler clamed that, n many cases, olgopolstc ndustres seek a regulatory umbrella and capture the regulator to obtan exemptons from legal restrctons. Appelbaum and Katz (1987) also ndcated that t s clearly preferable for sophstcated frms not to engage n ndvdual rent-seekng and furthermore, form a coalton to face the regulator and try to affect hs polces. The formaton of such a coalton s of course subject to the usual problems facng coaltons, namely enforcement, montorng, free rders etc. The bform contest approach shed some lght on the puzzle of the grand coalton (cartel) stablty. As mentoned above, wth rsk-averse agents, the grand coalton s stable and the cooperatve soluton concdes wth the mnmax Shapley value. Ths result s supported by Stgler (1971), who ponted out that the poltcal decsons take account also of the poltcal strength of the varous frms, so small frms have a larger nfluence than they would possess n an unregulated ndustry. Thus, when quotas are gven to frms, the small frms wll almost always receve larger quotas than cost mnmzng practces would allow. Ths pattern of reducng the share of large companes and augmentng the share of small companes s compatble wth allocaton accordng to the Shapley formula. The deadweght loss of welfare caused by cartelzaton of an ndustry s a well- known fact, and appears n any mcroeconomcs textbook. It s also well-known that rent-seekng contests to acqure a monopolstc poston create addtonal welfare loss by rent-dsspaton. However, as mentoned above and proved n Schwarz (2011b), rent-dsspaton rate n bform contests depends on socal norms. Compettve norms ncrease rent-dsspaton unambguously, but the

18 16 addtonal effect of cartelzaton on welfare loss under socalst norms s ambguous. Nevertheless, Cornes and Hartley s (2010) result mples that cartelzaton under socalst norms s more lkely to ncrease rent-dsspaton and welfare loss. Our results also mply that structural reforms, where new entrants are lcensed to challenge a legal monopoly n an olgopolstc competton, do not nsure an mprovement of socal welfare. The crucal pont s whether the market structure changed from a monopoly to an olgopolstc competton or to a protected cartel. There s no reason to expect that a market shft from a legal monopoly (a rent-seekng contest wnner) to a legal cartel (a coalton of rent-seekers) bears any promse for mprovement n socal welfare. As mentoned above, a hgher rate of rent-dsspaton s a more reasonable predcton, and f consumers demand s not nfntely elastc, ther welfare s surely expected to decrease. Fnally, the smplfed model of bform contest wth a passve regulator analyzed n ths artcle suggests some further research drectons for better understandng of grand coalton stablty n rent-seekng compettve envronments. Here are 5 conjectures for coaltonal nstablty factors; each deserves deep study. a. Atttude towards rsk: Our man result s condtonal on rsk-averson of all contestants. Prncpal-agent problems may neutralze rsk-averson of organzatons lke frms or states, and undermne the stablty of the cooperatve soluton. b. Socal Norms and Stablty: As mentoned above, norms affect expendtures. Further research s requred about norms effect on stablty. For example, wth compettve norms, sharng rules do not drectly relate effort to compensaton. Thus, applyng the mnmax Shapley value may trgger deprvaton feelngs among certan members of the grand coalton. c. Heterogenety: The above analyss assumed homogenous evaluaton of the prze by all contestants. Further research s requred to check whether our results hold also wth heterogeneous evaluaton 25. d. Asymmetrc Informaton: The above analyss assumed common knowledge of the prze value and ndvduals preferences. Further research s requred to check whether our results stll hold n case of prvate or asymmetrc nformaton. 25 The results of Schwarz (2011a) and (2011b) regardng a bform contest wth an actve regulator are robust to heterogeneous evaluaton.

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