Competition and Efficiency of Coalitions in Cournot Games with Uncertainty

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1 1 Competton and Effcency of Coaltons n Cournot Games wth Uncertanty Baosen Zhang, Member, IEEE, Ramesh Johar, Member, IEEE, Ram Rajagopal, Member, IEEE, arxv: v2 [cs.gt] 14 Aug 217 Abstract We nvestgate the mpact of coalton formaton on the effcency of Cournot games where producers face uncertantes. In partcular, we study a market model where frms must determne ther output before an uncertan producton capacty s realzed. In contrast to standard Cournot models, we show that the game s not effcent when there are many small frms. Instead, producers tend to act conservatvely to hedge aganst ther rsks. We show that n the presence of uncertanty, the game becomes effcent when frms are allowed to take advantage of dversty to form groups of certan szes. We characterze the tradeoff between market power and uncertanty reducton as a functon of group sze. In partcular, we compare the welfare and output obtaned wth coaltonal competton, wth the same benchmarks when output s controlled by a sngle system operator. We show when there are frms present, competton between groups of sze Ω( results n equlbra that are socally optmal n terms of welfare and groups of sze Ω( 2/3 are socally optmal n terms of producton. We also extend our results to the case of uncertan demand by establshng an equvalency between Cournot olgopoly and Cournot Olgopsony. We demonstrate our results wth real data from electrcty markets wth sgnfcant wnd power penetraton. Index Terms Cournot Games, Decson Makng under Uncertanty, Effcency, Coaltons, Electrcty Markets I. ITRODUCTIO Cournot games are among the most extensvely studed models for olgopolstc competton among multple frms. A Cournot olgopoly s a model where partcpants compete wth each other by controllng the amount of a homogeneous good that they produce. A market prce s determned as a functon of the total output of the frms. The proft of a frm s then the product of the market prce and ther output quantty, less any costs ncurred. The producers are assumed to act strategcally and ratonally to maxmze ther ndvdual profts. Ths model was ntally studed by [1]; for surveys of such models, see, e.g., [2], [3], [4]. In ths paper, we consder Cournot competton among frms who face producton uncertanty. In the model we consder, frms frst commt to an expected level of output; subsequently, actual producton s realzed, drawn from a dstrbuton parameterzed by the frm s ex ante chosen level. Any shortfall from the precommtted level ncurs a penalty. Such a model captures producton decsons by frms n envronments where commtments must be made before all relevant factors nfluencng producton are known. B. Zhang s wth the Department of Electrcal Engneerng at the Unversty of Washngton, emal: zhangbao@uw.edu. R. Johar s wth the Department of Management & Scence and Engneerng at Stanford Unversty, emal: ramesh.johar@stanford.edu. R. Rajagopal s wth the Department of Cvl and Envronmental Engneerng at Stanford Unversty, emal: ramr@stanford.edu. Electrcty markets serve as one motvatng example of such an envronment. In electrcty markets, producers submt ther bds before the targeted tme of delvery (e.g., one day ahead. However, renewable resources such as wnd and solar have sgnfcant uncertanty (even on a day-ahead tmescale. As a result, producers face uncertantes about ther actual producton capacty at the commtment stage. Our paper focuses on a fundamental tradeoff revealed n such games. On one hand, n the classcal Cournot model, effcency obtans as the number of ndvdual frms approaches nfnty, as ths weakens each frm s market power (ablty to nfluence the market prce through ther producton choce. On the other hand, ths result does not carryover when producton uncertanty s present: frms protect themselves aganst the rsk of beng unable to meet the pror commtment by underproducng relatve to the effcent level. In consderng how to recover effcent performance, we are naturally led to thnk of coaltons of frms. Informally, f frms pool together, they can mtgate ndvdual uncertanty any one of them may perceve n future producton (a law of large numbers effect. Of course, coaltons are not wthout ther downsde: coaltons possess greater market power than ndvdual frms. Indeed, ths concern s substantal, as coaltons must be of farly substantal sze to mtgate the adverse effects of producton uncertanty. In the context of electrcty markets, regulators must grapple wth the consequences of allowng wnd generators and other renewable resources to form coaltons as they bd nto the market. As a result we are led to a fundamental queston: how many coaltons should be allowed to form, and of what sze, f the regulator s nterestng n maxmzng overall market effcency? We characterze ths tradeoff by studyng the effcency of Cournot competton when producers are allowed to form coaltons. Our man contrbutons are as follows. Frst, n the model descrbed above, we characterze equlbrum among competng coaltons, as well as the socally optmal benchmark. Second, as measures of effcency, we compare both the welfare and producton output of the frms under Cournot competton wth the socally optmal welfare and output. We characterze an optmal scalng regme for coalton structure (n the lmt of many frms under whch the effcency losses can be made arbtrarly small. That s, there exst coaltons (partton of producers that acheve essentally effcent reducton n uncertanty, but have no apprecable market power. We also characterze the rate at whch effcency loss vanshes, and establsh these results when frms may have correlated uncertanty. Fnally, by establshng an equvalence between Cournot olgopoly and Cournot olgopsony, we demonstrate

2 2 our results are applcable to settngs wth demand uncertanty as well; n partcular, we llustrate the latter results n an applcaton to urban parkng allocaton. Effcency and welfare loss n Cournot games have been studed extensvely n varous contexts. Early emprcal analyss of welfare loss was performed by [5] and [6]. Analytcally, at the lmt where many frms compete, many authors showed that a compettve lmt exsts [7], [8], [9], [1]. The quantfcaton of such a lmt was consdered by [11] where the margnal costs of the frms are assumed to be constant. The work of [12] showed that for producers wth the same cost functon competng for a resource wth a dfferentable demand curve, the effcency loss s no more than 1/(2 +1 when the producers are strategc and prce antcpatory. The paper by [13] derved a more general bound for convex demand curves and [14] studes the how the loss can be estmated n practce. The loss under asymmetrc frms was studed by [15]. Most of the precedng lterature concludes that full effcency s acheved when a large number of producers are competng aganst each other. In ths paper, we show that ths s not the case when producton uncertanty plays a role n the frms profts. Proft maxmzaton under supply (or demand uncertanty falls under the well studed newsvendor problem n the operatons lterature. However, most of prevous work on ths area s concerned wth a sngle retaler [16]. Olgopolstc competton s studed n [17] for addtve demand, and n [18] for multplcatve demand; a related model wth revenue sharng between dfferent frms s dscussed n [19]. To our knowledge, none of the prevous works n ths area consder effcent coalton formaton. Another related research s area s contract desgns (see, e.g. [2], where desgners mpose penaltes to ensure that frms operate as expected. In ths paper, the penalty s from uncertantes that are ntrnsc to the problem. One closely related work to our own s [21], where the author studes the role of ntermedares between dversfcaton and competton n a large economy. Ther results are derved under the assumpton of a common randomness affectng all consumers, whereas n our work each producer faces ts own randomness (possbly correlated wth others. Ths latter effect s what creates effcency gans by allowng coaltons to form. The remander of the paper s organzed as follows. Secton II presents our basc model of Cournot competton wth two stages: producton commtments are made n the frst stage, and actual avalable producton capacty s realzed n the second stage. The market prce s determned n the frst stage based on producton commtments. In the second stage, the producer s charged a penalty f capacty s short of the frm s commtment level. We then study the same model assumng frms act n coaltons. As the focus of the paper s on the competton between groups, we do not study the nature of revenue sharng contracts wthn a group; see, e.g., [22] and [23]. In Secton III, we begn by studyng the case where frms face..d. producton uncertanty. We frst show that as the number of frms grows, the effcency loss does not vansh (due to the adverse effects of producton uncertanty. We also show that the other extreme, a grand coalton of all producers, s neffcent (due to excessve exercse of market power. We then study coaltonal competton: n partcular, we characterze the optmal group sze and the optmal rate at whch the effcency loss approaches zero. By balancng the adverse effects of market power wth the benefts of reducton n producton uncertanty, we show that a coalton sze of producers (so coaltons compete n the market s optmal, and the effcency loss s no more thano(1/. We show that same results hold under two models of correlated producton uncertanty n Secton IV. Secton V shows how the results can be appled to demand uncertanty by dervng an equvalence between Cournot olgopoly (frms competng to supply a good and Cournot olgopsony (frms competng to consume a good. Secton VI llustrates how the results can be appled n practce wth a case study of the electrcty market n the Md-Atlantc regon of the Unted States when wnd power producers are ncluded n the market; ths secton s based on materal n [24]. Secton VII concludes the paper. II. TECHICAL PRELIMIARIES In ths secton, we defne a two-stage game where multple frms compete to satsfy the demand for a sngle resource. The man dfference between the two-stage model and classcal Cournot competton s that the bddng occurs n the frst stage, but each producer has an uncertan producton capacty at the tme of delvery (second stage. Suppose there are frms. The market operates n two stages as shown n Fg. 1. At the frst stage, frm chooses a Frm Bds Frst Stage Prce Realzed Capacty Second Stage Penalty Fg. 1: The two-stage market model. Frms commt ther producton n the frst stage, and ths determnes the prce of the good. Capactes are realzed n the second stage, and penaltes are assessed f a frm s commtment s less than ts realzed capacty. commtted producton quantty x as ts bd nto the market. Let p(y denote the market prce when y unts of aggregate output s commtted. Let X denote the capacty constrant on frm s producton. ote X s a random varable at the frst stage, and s realzed n the second stage. Throughout the paper, we wll assume X s are contnuous,.e., they follow a dstrbuton wth a contnuous probablty densty functon. To focus on the effect of producton uncertanty, we assume that each frm does not have a cost for producng the resource. 1 If the promsed amount (x s larger than the capacty (X frm s penalzed by a cost q per unt short fall. Thus the 1 Our results would reman unchanged f each frm faced the same constant margnal producton cost.

3 3 cost of shortfall for frm s q(x X +. 2 Wthout loss of generalty, we set q = 1 for the remander of the paper. The assumpton of a penalty lnear n the shortfall s relatvely common n the lterature on newsvendor problems. Ths penalty allows us to capture the rsk assocated wth a shortfall,.e., promsng more than what can actually be delvered; ndeed, the penalty serves to make a frm rsk averse n ts choce of commtment level. Our man results contnue to hold even for penaltes of the form E[f(x X + ], as long as f s convex, ncreasng, and has bounded dervatve. The structure of the penalty makes two mportant assumptons: frst, that the penalty depends only on a frm s own shortfall (.e., no nter-frm externaltes; and that any excess producton capacty cannot be resold n a secondary market. In practce, these assumptons may be volated. For example, n electrcty markets, a real-tme market s run to balance the realzed supply and demand, and the study of such markets reman an mportant future drecton for us. evertheless, we beleve our model captures the frst order effects of producton uncertanty on frm behavor, and on the role coaltons play n achevng effcent outcomes. We use the notaton x to denote the quanttes of chosen by all frms except ; that s, x = (x 1,x 2,...,x 1,x +1,...,x. The expected proft for frm s ( π (x,x = p x l x E[(x X + ]. (1 l=1 When each frm s prce antcpatory, gven x, frm chooses x > to maxmze π. A ash equlbrum of the game defned by (π 1,...,π s a vector x such that for all : π (x,x π ( x,x, for all x. (2 To analyze the ash equlbrum for ths game, we make the followng assumptons on the prce functonp; ths assumpton remans n force for the entre paper. Assumpton 1. We assume that: 1 p s strctly decreasng and p( > ; 2 p(y s concave and dfferentable on y wth p ( + < ; 3 p(y as y. Snce p s decreasng, p( >, and tends to, there s a unque zero crossng pont y max such that: p(y max =. (3 These assumptons are common n the lterature (e.g., see [12]. The frst assumpton states that the prce decreases as quantty ncreases and p( > avods trval solutons. Concavty of the demand functon s largely made for analytcal convenence and to avod long dervatons. (See Remark 1 at the end of ths secton, whch suggests that key results on scalng of optmal coalton sze contnue to hold even wth weaker assumptons on demand, e.g., logconcavty. The last 2 For a real number z, let z + denote the postve part of z,.e., z + = z for z >, and otherwse. assumpton s also made for analytcal smplcty. In practce, the prce becomes zero for large enough y. Ths s analytcally undesrable snce p may not be globally concave, so n the thrd assumpton we allow p to be negatve. Ths assumpton s essentally wthout loss of generalty snce the regme of nterest s always restrcted to aggregate producton where p s non-negatve (see Proposton 1. We make the followng assumptons on the random varable X throughout the paper: Assumpton 2. For all, X s a contnuous random varable wth fnte mean. Ths assumpton s made manly for analytc convenence. It s now straght forward to show that a unque ash equlbrum exsts for the game (π 1,...,π as gven n the followng result. Proposton 1. Suppose p satsfes Assumpton 1 and the X s satsfy Assumpton 2. Then there exsts a unque ash equlbrum x for the game defned by (π 1,...,π. Furthermore, x y max. The proof of ths proposton s gven n Appendx A. Remark 1. The concavty assumpton on the nverse-demand functon p can be relaxed to much weaker condtons. For example, suppose that p s decreasng, contnuous, and there exst a y max such that p(y max =. ote p s not necessarly concave. If all of the frms experence uncertantes wth the same margnal dstrbuton, the penalty functon E[(x X + ] s dentcal for each frm. By the result of [25] (or see Theorem 1 n [26], there exsts a symmetrc equlbrum for the game defned by (π 1,...,π. Of course wthout knowng more about p, t s hard to characterze the equlbra of the game. In ths paper, we nvestgate the case of concave p, but as a consequence of the precedng result, we expect that our man conclusons should extend to more general types of nverse demand functons, at least n the case when frms are dentcal. A. The System Operator We are nterested n the effcency of the ash equlbrum of the game (π 1,...,π. In ths secton, we outlne the benchmark welfare we consder; n partcular, we focus on the maxmum welfare acheved by a centralzed system operator. To begn, gven the prce functon p, we defne aggregate consumer surplus n the usual way as: U(y = y p(zdz. (4 The system operator ams to maxmze aggregate consumer surplus, but ncurs a cost a cost based on the expected aggregated shortfall, assumng that t can control the output of all the producers. ote that a key dfference between ths work and the exstng lterature s how the cost s accounted for: the system operator s effectvely optmzng as f t s a grand coalton of all frms, wth utlty defned by aggregate consumer surplus. Ths model s nspred by the role of the

4 4 ndependent system operator (ISO n electrcty markets; see, e.g., [27]. ( ( + maxmze U x E x X (5a subject to x,. (5b The [ reason a system operator faces the aggregate shortfall, ( E x ] + X, nstead of the sum of ndvdual shortfalls, E [(x X +], s that t s always benefcal to equalze possble shortfalls n quantty by sharng resources. In the motvatng example of electrcty markets, suppose a system operator had control of all wnd farms. Then as long as the sum of total realzed wnd, X, s larger than the sum of the total commtted wnd, x, there would be no addtonal cost. The followng lemma formalzes the beneft of aggregaton to a system operator. Lemma 1. Suppose the X s satsfy Assumpton 2. Then ( + [ E x X E (x X +]. (6 The proof of ths lemma s a straghtforward applcaton of Jensen s nequalty and s gven n Appendx B. ote that the objectve functon n (5 only depends on x. Wth a change of varables, we can rewrte (5 as: ( maxmze U(y E subject to y. y + X (7a (7b By Assumpton 1, U s dfferentable and concave, so the optmal soluton to (7 s the unque postve soluton to p(y Pr ( y X =. (8 Thus we have the followng corollary. Corollary 1. A vector x s effcent (.e., solves the optmzaton problem (5 f and only f: x = y max, (9 where y max s the unque soluton to (8. Further, y max y max. Therefore at equlbrum, f the aggregate producton of frms s y max, the equlbrum s socally optmal.3 3 It s straghtforward to show, usng the fact that p s decreasng, that n any equlbrum frms produce less than y max. We omt ths standard argument. B. Effcency Rato In ths secton we defne two closely related effcency ratos. The frst s the usual noton based on the gap between welfares of the ash equlbrum and the system operator s optmal outcome. Ths s smlar to the prce of anarchy (e.g., [28], but wth the change that the benchmark s the system operator s payoff. The second effcency rato s defned as the gap between the aggregate output at the ash equlbrum and y max. Defnton 1. Consder the Cournot game (π 1,...,π. Let (x 1,...,x be the ash equlbrum of ths game and let y max be the soluton to (8. The effcency rato r W, wth respect to the welfare, s: U r W = ( x E[(x X + ] U(y max E y max X + ]. (1 The effcency rato r O, wth respect to the output, s: r O = x. (11 y max Both effcency ratos are of nterest. The rato of welfares measures how (neffcent the ash equlbrum s, and the rato of quanttes drectly compares the total output under the ash equlbrum and the aggregate socal output. For example, n electrcty markets, the ndependent system operator (ISO s typcally nterested n ensung the maxmum amount of renewable power (e.g., wnd s njected nto the market. Therefore, r O, the rato of the total output under the ash equlbrum and the maxmum possble output y max can be a more drect measure of nterest than r W, the rato of welfares. The rest of ths paper nvestgates the behavor of r W and r O as the number of frms and the number of coaltons grow. In partcular, we characterze the asymptotc scalng rate of both effcency ratos. C. Determnstc Cournot Games Before movng on to the man result n Sectons III and IV, we consder a determnstc verson of the Cournot game,.e., one wthout producton uncertanty. Understandng of ths determnstc game provdes context for our results; further, our proofs use the determnstc settng as a buldng block. In the determnstc settng, we gnore the second stage of the game. Therefore the payoff for frm s: ( π (x,x = p x l x. (12 Compared wth (1, note that the cost for shortfall s omtted. For the rest of the paper, we use overlned varables to represent quanttes n the determnstc game. Consder the game defned by (π 1,...,π. By the same reasonng as n Proposton 1, a unque ash equlbrum exsts for ths game, denoted by (x 1,...,x. The socal welfare l=1

5 5 U(y s maxmzed at y max. The effcency ratos for ths game are defned as: ( U x r W = (13 U(y max and r O = x y max. (14 The behavor of r W and r O as ncreases s well understood; see, e.g., [12]. As noted n Proposton 2, r W approaches 1 and the game becomes effcent n the lmt of many frms. As we show n Secton III, ths s no longer true f producton uncertanty s present. Proposton 2 (Corollary 18 n [12]. D. Coaltons lm r W 1 and lm r O 1. In ths secton, we defne Cournot competton among coaltons of frms. Gven frms, let S 1,...,S be a partton of {1,...,}. Let (x 1,...,x be a vector of producton levels for each frm. The aggregate producton commtment of group S k s denoted as x(s k : x(s k = S k x. Smlarly let X(S k = S k X denote the aggregate (random realzed capacty of the group S. The payoff of the group S k s defned as: ( π k (x(s k = p x(s k x(s k E[(x(S k X(S k + ]. k (15 ote that, as for the system operator, a coalton benefts by beng able to use the excess producton of one member to offset the shortfall of another. Thus the penalty ncurred by the coalton s the shortfall between ther aggregate realzed capacty and aggregate producton commtment. ote that we do not consder the nternal proft sharng contracts of each coalton; nstead, we focus on proft maxmzaton of the coalton. Gven S 1,...,S k, we can defne a Cournot game among coaltons through the payoff functons (π 1,...,π k. In ths game the acton for group S k s the aggregate producton commtment x(s k ; as wth proft, we do not focus on how ths commtment s dvded among the ndvdual frms. The game played by coaltons s a scaled verson of the orgnal game played by ndvdual frms. The key dfference s that the penalty s not lnear n the frms. By an analogous result to Lemma 1, E[(x(S k X(S k + ] S k E[(x X + ]. (16 It s ths reducton n rsk that makes coaltons useful, as we descrbe n the subsequent sectons. III. IDEPEDET FIRMS In ths secton, we consder the effcency of the two-stage Cournot game when the producton uncertanty s..d. across frms. Snce we are nterested n the large regme, we need to specfy how the random varables (X 1,...,X scale as ncreases. Recall that X models the realzed capacty of producton. Snce we hold the prce functon constant as we ncrease, we should reasonably expect that each frm wll produce an nfntesmal amount n the lmt. If we do not adjust the producton capacty accordngly, then each frm wll effectvely face no producton uncertanty n the large lmt. Formally, we adjust the scale of the producton capacty of each frm accordng to the followng assumpton. Assumpton 3. Let X be a symmetrc contnuous random varable wth E[ X 3 ] <. Let E[X] = µ and we assume µ > y max ; let Var(X = σ 2. Suppose there are frms. The random varables X 1,...,X are drawn..d. accordng to the dstrbuton of X/. Under the above assumpton, the expected total capacty s fxed at µ and s dvded evenly among the frms. The techncal assumpton of a bounded thrd moment avods random varables wth very heavy tals, and s largely made for analytcal convenence. The assumpton µ > y max streamlnes the proofs, but s not essental. We note that Assumpton 3 can be equvalently formulated by holdng the producton capacty of each frm constant, but nstead scalng the prce functonal as p as the number of producers ncreases. Under Assumpton 3, all frms are ex ante dentcal: they have the same proft and face the same producton uncertanty. In ths secton we consder coaltonal competton where the frms are dvded evenly nto groups; snce we are nterested n large scenaros, we assume, wthout loss of generalty, s a multple of. The two extreme values of are = 1 and = : the former corresponds to a grand coalton, whle the latter corresponds to competton among ndvdual frms. The next theorem s the man result of ths secton, whch relates the effcency ratos to the group sze. Theorem 1. Suppose there are frms and X 1,...,X satsfes Assumpton 3. Let (S 1,...,S be groups where each group has / frms. Let (x(s 1,...,x(S be the soluton to the game (π 1,...,π. Then the effcency ratos scales as: ( U k=1 x(s k [ k=1 E (x(s k X(S k +] r W = U(y max E y max ( 1 = 1 O 2 and k=1 r O = x(s k y max O ( ( 1 = 1 O O X + ] (17 (. (18 We focus on the dscusson of the results here; the full proof of Theorem 1 s n Appendx C. The last two terms n (17

6 6 and (18 can be nterpreted as the effects of market power and producton uncertanty, respectvely. In (17, the neffcency due to market power scales as 1/ 2, and t decreases as grows. On the other hand, the neffcency due to producton uncertanty scales as /, whch decreases as / (the number of members n each coalton grows. Smlarly, n (18, 1/ represents the neffcency due to market power and / represents the neffcency due to uncertanty. ote from (17 and (18 that r W and r O approach 1 as long as and / both grow wthout bound. The followng corollary gves the optmal coalton sze for maxmzng the rates at whch they approach 1. Corollary 2. The optmal coalton structure to maxmze the rght hand sde of (17 s to dvde frms nto Ω( 1/3 groups, each of sze Ω( 2/3. The optmal coalton structure to maxmze the rght hand sde of (18 s to dvde frms nto Ω( groups, each of sze Ω(. Ths Corollary follows drectly from balancng the terms n Theorem 1. It s nterestng to note that r W and r O are optmzed by dfferent coalton formatons. However, both rates suggest that t s more effcent to have ntermedate regmes of coalton szes other than the two extremes of ndvdual frms and the grand coalton. The result n Theorem 1 can be extended to a more general shortfall penalty, as n the followng corollary. Corollary 3. Let f be a convex ncreasng functon wth bounded dervatve, satsfyng f(x = for all x. For a group S, let ts total proft be gven by: ( π S (x S = p x k E[f(x S X S ]. (19 x ( k S Under the same condtons as Theorem 1, the effcency ratos stll scale as ( ( 1 r W = 1 O 2 O and ( 1 r O = 1 O O (. The proof of ths corollary s gven n Appendx D. Remark 2. The scalng rates n Theorem 1 and Corollary 3 are stated as lower bounds. There are nstances that demonstrate that these rates are tght. For example, let p(x = 1 x. In ths case the neffcency due to market power can be calculated exactly and scales as r W = Ω(1/ 2 and r O = Ω(1/. Let X be a contnuous random varable that satsfes Chebyshev s nequalty wth equalty (see, e.g., [29]. Then both r W and r O scale as Ω(/. (See the proof of Theorem 1 for detals. It s worthwhle to note that the scalng rates n Theorem 1 and Corollary 3 represent the asymptotc behavor of frms. Constants of the terms n the scalng rate are determned by the partcular dstrbutons of producton uncertanty, and the prce functon. We llustrate the behavor of the effcency rato at fnte wth the followng example. Example 1. Let p(y = 1 y. Let X be normally dstrbuted as (1.1,1. ote that y max = 1 < 1.1. Let X be drawn..d. accordng to the dstrbuton of X/. Fgure 2 plots the effcency rato for two groups szes: and 2/3. Theorem 1 and Corollary 2 show that groups of sze 2/3 s optmal for welfare effcency and groups of sze s optmal for rate effcency. Fgures 2a and 2b vald these clams, although there s a swtch over pont n Fg. 2b. Effcency Rato Effcency Rato / umber of Producers (a Scalng of the welfare effcency rato. 2/ umber of Producers 1 5 (b Scalng of the quantty effcency rato. Fg. 2: Scalng of the effcency ratos for groups of sze 2/3 and (for 1/3 and total groups, respectvely. The rato of welfares, r W, approaches 1 faster when groups are of sze 2/3 than when groups are of sze. The rato of quanttes,r O, approaches 1 asymptotcally faster when groups are of sze than when groups are of sze 2/3. IV. CORRELATED FIRMS In ths secton, we consder two models where frms have correlated producton uncertanty. A. Weakly Correlated Frms The O(/ terms n (17 and (18 result from the law of large numbers. The followng corollary recovers the same

7 7 result, usng a verson of the law of large numbers for correlated random varables. Corollary 4. Let X be a random varable satsfyng Assumpton 2. Let E[X] = µ > y max. Suppose there are frms. Assume the random varables X 1,...,X each have margnal dstrbuton that s the same as X/. Let (S 1,...,S be groups where each group has / frms. Let (x(s 1,...,x(S be the soluton to the game (π 1,...,π. If E[cor(X,X j ] c (2 j=1 for some c ndependent of, then the effcency ratos scale as n (17 and (18. An example of correlated X s satsfyng the above condton s where cov(x,x j Aρ j, for some fnte A and ρ < 1. Ths type of model captures a Hotellng-lke geographc structure, where frms wth smlar ndces are more lkely to face the same producton constrants. It s partcularly relevant n electrcty markets, where wnd turbnes located near each other exhbt ths behavor. The proof of Corollary 4 s gven n Appendx E. B. Strongly Correlated Frms Earler, we consdered frms wth weakly correlated producton capacty, n the sense that the correlatons between frms decays as the number of frms grows. In ths secton, we consder the case of strongly correlated producton capactes, where the correlaton between all frms remans postve as grows. When the frms have correlated capactes, results smlar to Theorem 1 are dffcult to obtan n general snce the lmtng dstrbuton of X does not necessarly concentrate; any such result wll depend on the partcular jont dstrbuton of the X s. For ths secton, we assume that the correlaton between random varables arses from an addtve model, as descrbed n the followng assumpton. Assumpton 4. Let X be a zero mean contnuous random varable wth symmetrcal densty and satsfes E[ X 3 ] <. The random varables ˆX 1,..., ˆX are drawn..d. accordng to the same dstrbuton as X/. Let Z be a contnuous random varable wth mean µ, fnte varance, and symmetrcal densty around ts mean. The random varables Z and ˆX 1,..., ˆX are ndependent. The random varable X s gven by: X = ˆX + Z for all. (21 SnceX s are strongly correlated, X no longer concentrates around ts mean. Therefore, even for a system operator that jontly controls output of all frms, there s some resdual uncertanty n the system. However, t s stll benefcal to the system to share the producton of all frms and then face the cost of the aggregate shortfall. Wth the notatons used n Assumpton 4, the system operators s problem s max y U(y E[(y Z ˆX + ]. (22 As before, let y max be the unque soluton to (22 and y max y max. Theorem 2 states that the effcency ratos between the welfares and the quanttes have the same large asymptotc behavor as n Theorem 1. Theorem 2. Let ˆX1,..., ˆX and Z be random varables that satsfy Assumpton 4. Let (S 1,...,S be a partton of (1,..., wth sze / each. Let (x(s 1,...,x(S k be the soluton to the two-stage game (π 1,...,π. Suppose µ > y max. The effcency ratos scale as: ( U k=1 x(s k [ k=1 E (x(s k X(S k +] r W = ] + U(y max E y max X and = 1 O( 1 2 O( (23 r O = k x(s ( ( k 1 y max = 1 O O. (24 The proof of Theorem 2 proceeds n smlar steps to the proof of Theorem 1 and can be found n the Appendx F. C. Smulaton Results Here we plot the effcency rato for correlated frms and compare t to ndependent frms. Smlar to Example 1, let p(y = 1 y. Let ˆX be normally dstrbuted as (1.1,.71 and let Z be normally dstrbuted as (,.71; note that wth ths defnton, the varance of ˆX +Z s 1. Let ˆX be drawn..d. accordng to the same dstrbuton as ˆX/. Fgure 3 shows the effcency rator O for groups of sze on a semlog plot. As a baselne, we also plot the effcency rato where the random varables are drawn..d. wth normal dstrbuton (1.1,1. Effcency Rato Correlated I.I.D log(umber of Producers Fg. 3: Effcency rato for correlated frms and..d. frms as a functon of the log of the number of frms. The groups sze are for both cases. We see that n the correlated case, the effcency rato ncreases much faster than the..d. case.

8 8 From Fgure 3, we can see that the effcency rato r O approaches 1 much faster f the frms are correlated. Ths s not unexpected, snce producton uncertanty s domnated by the common random varable Z, and the ndvdual randomness can be averaged out more easly. V. COUROT OLIGOPSOY There s a natural Cournot olgopsony game correspondng to the Cournot olgopoly game dscussed n prevous sectons. Instead of supplers of a common good, frms can be thought as consumers of a common good. If there are uncertantes n the demands of the frms, a natural queston arses: does coalton ncrease the effcency of Cournot olgopsony as n the case of Cournot olgopoly? We answer ths queston by showng that under certan demand sde models, all results from prevous sectons can be drectly appled through an equvalency between Cournot olgopoly and Cournot olgopsony. Before the techncal detals, let us consder the followng motvatng example of parkng n busness dstrcts. Example 2. Many ctes around the world are experencng ncreasng vehcle traffc n downtown and busness areas. Currently, most ctes allocate a fxed number of parkng spaces to a frm. 4 For frm, suppose t s allocated x number of parkng spaces. The amount of vehcles (demand of the frm that vst the frm s a random varable denoted by X. If a vehcle successfully fnds parkng, the frm derves a certan amount of utlty. If parkng s not avalable when users try to vst a frm, t gets no utlty. Therefore frm s utlty s proportonal to mn(x,x. An ncrease of parkng spaces s correlated wth the total traffc nto downtown areas. Ths ncrease n traffc could potentally ncrease congeston, and therefore parkng s tghtly controlled by the cty. We model ths cost wth a prce functon ˆp( x and frm has cost x ˆp( x. Wth the above motvaton, we defne the expected payoff for frm to be ( T (x,x = E[mn(x,X ] x ˆp x l. (25 Each frm s prce antcpatory and frm chooses x > to maxmze T for a gven x. A ash equlbrum for ths game s defned to be the same as n (2. As the analogous of Assumpton 1, we make the followng assumpton on the prce ˆp. Assumpton 5. We assume that: 1 ˆp s strctly ncreasng and < ˆp( < 1; 2 ˆp(y s convex and dfferentable ony wth ˆp ( + > ; 3 ˆp(y as y. The ˆp( < 1 assumpton s made to avod trval solutons, snce f ˆp( > 1 no frm wll choose a postve bd. ote we assume that the utlty and the prce functon, E[mn(x,X ] and ˆp, have been approprately scaled to have the same unts. 4 Excludng prvate parkng garages l=1 It s straght forward to show that a ash equlbrum exsts for the game (T 1,...,T as gven n Proposton 3. Proposton 3. Suppose ˆp satsfes Assumpton 5 and the X s satsfy Assumpton 2. Then there exsts a unque ash equlbrum x for the game defned by (T 1,...,T. The proof of ths proposton s gven n Appendx G. A. Socal Welfare and Effcency Gven the prce functon ˆp, we defne the aggregate cost as: C(y = y ˆp(zdz. (26 Smlar to the olgopoly case, a system operator wth control of all the frms would always aggregate the supply and demand as shown by Lemma 2. Lemma 2. Suppose x 1,...,x are a set of real numbers and X 1,...,X are a set real random varables each satsfyng Assumpton 2. Then ] E mn x, X E[mn(x,X ]. (27 The proof of ths lemma s gven n Appendx 2. Therefore, from a system operator s pont of vew, the optmal allocaton s characterzed by solvng the followng problem: ( maxmze E mn x, X ] C x (28a subject to x,. (28b The objectve functon n (28 only depends on the sum of allocatons, so we can wrte t as maxmze E mn y, X ] C(y (29a subject to y. (29b Let ŷ max be the soluton to (29. As n Defnton 1, we defne the two effcency ratos, one based on the welfares and one based on the quanttes. Defnton 2. Consder the Cournot game (T 1,...,T. Let (x 1,...,x be the ash equlbrum of ths game. The effcency rato r W, wth respect to the welfares, s: E[mn(x,X ] C( r W = x E mn ŷ max, ]. (3 X C(ŷ max The effcency rato r O, wth respect to the quanttes, s: r O = x. (31 ŷ max As n Sectons III and IV, we are nterested n the behavors ofr W andr O as grows. The next secton shows that there s an exact equvalence between the game (T 1,...,T studed n ths secton and the game (π 1,...,π studed n Sectons III and IV. Consequently the man results n Theorems 1 and 2 carry over drectly.

9 9 B. Equvalence between Olgopoly and Olgopsony A certan level of symmetry between olgopoly and olgopsony s commonly expected n Cournot games, but the key dffculty s to ensure that effcent allocatons and ash equlbra reman unchanged between the two models. The followng theorem s the man result of ths secton. Theorem 3. Suppose that Assumptons 2 and 5 hold. Defne a new prce p as: p(y = 1 ˆp(y. (32 Defne an assocated utlty functon U(y = y p(zdz. Then 1 Assumpton 1 s satsfed by p. 2 For any vector x, the followng holds: ( E mn X ] C x x, ( ( = U x E x + X, (33 as well as: ( T (x ;x = E[mn(x,X ] x ˆp x l ( = x p x l E[(x X + ] l = π(x ;x. (34 3 A vector x solves the system operator s problem n (5 f and only f t solves the system operator s problem n (28. 4 A vector x s a ash equlbrum of the game defned by (T 1,...,T f and only f t s a ash equlbrum of the game defned by (π 1,...,π. In Appendx I we provde proofs of the four clams above. VI. CASE STUDY OF WID ITEGRATIO I POWER SYSTEMS Ths secton apples the result of ths paper to electrcty markets. Ths materal s based on [24]. We focus on the PJM control area n the Unted States (an area that covers most of the Md-Atlantc states of the Unted States. In recent years, there has been a dramatc ncrease of wnd generaton n PJM. Among other effects, the one most pertnent to ths paper s a sgnfcant drop n energy prces as wnd penetraton grows. Fgure 4 (reproduced from [3] shows the day-ahead electrcty prce can drop by as much as 5% wth just 1% of wnd penetraton. The electrcty market n PJM and most other regons of the Unted States operate wth a two-stage structure. The frst stage occurs roughly 24 hours before the actual tme of delvery of electrcty called the day-ahead stage where generators compete to satsfy the forecasted demand. The second stage s called the real-tme stage snce t adjusts supply and demand to meet any mbalance that may occur at the tme of electrcty delvery. Snce most conventonal generators l Fg. 4: Scatter plot of PJM day-ahead prces and wnd generaton for 212. Fgure reproduced from [3]. The horzontal axs s the percentage penetraton of wnd, and the vertcal axs s the average clearng prce n the PJM area. As the amount of wnd penetraton ncreases from to 1%, the average prce n the system drops by more than half. (e.g., coal, nuclear, hydro, etc. needs at least several hours to change producton levels, most of the market s cleared at the day-ahead stage. In ths paper, we are nterested n the wnd power producers and treat conventonal generators as non-strategc enttes. From PJM market reports ([31], t seems that conventonal generators already bd ther true cost n the current market, and therefore would not change ther bd when wnd power producers enter the market (bddng lower makes no economc sense and bddng hgher decreases the chance that they are cleared. However, nvestgatng the jont acton between renewable and conventonal generators s an mportant area of future research. We thnk of wnd power producers as the frms competng n the day-ahead market. Each frm faces uncertan supply, snce ts output s determned by the wnd condtons n the future; n addton, each frm can mpact the prce as shown n Fg. 4. Therefore the Cournot game developed n ths paper can be used to model ther behavor. For an emprcal study, we use wnd data produced by atonal Renewable Energy Lab (REL for Eastern Unted States [32]. Both wnd forecast value and forecast error are ncluded n the data set. We nterpret the forecast value as the mean of the random supply and the error as the varaton n the supply. There are 32 wnd farm locatons that are n the PJM control area. Fgure 5 shows the normalzed standard devaton of the aggregate forecast as a functon of the number of wnd farms n the aggregate. Strong correlatons can be observed between these forecasts. Applyng the analyss n Secton IV, Fg. 6 shows the optmal coaltons of the wnd farms. ote the socal optmal soluton s taken to be the soluton that maxmzes the amount of wnd njected nto the market. Interested readers can fnd a much more detaled analyss n [24].

10 1 Standard Devaton (p.u umber of Producers Fg. 5: Standard devaton as a functon of the number of producers n an aggregaton. The vertcal axs s normalzed by the total capacty of the aggregaton. Fg. 6: Total amount of wnd bd nto the day-ahead market as a functon of the number of groups for the wnd farms n the REL dataset. The maxmum occurs at 3 groups, whle the mnmum occurs at a sngle group (grand coalton. VII. COCLUSIO In ths paper we nvestgated strategc behavor of frms and coaltons n a Cournot game wth producton uncertanty. We study the effcency of Cournot competton by characterzng a fundamental tradeoff: on one hand, market power ncreases as coalton sze grows; on the other hand, the cost of producton uncertanty s mtgated as coalton sze grows. We show there s a sweet spot, n the sense that there exst groups that are large enough to acheve the uncertanty reducton of the grand coalton, but small enough such that they have no sgnfcant market power. These results have mportant mplcatons for regulators n ndustres wth producton uncertanty, such as electrcty markets. In partcular, our results suggest that wthn some lmts, coalton formaton among, e.g., generators of renewables may actually ncrease overall welfare. We have valdated these results n electrcty markets n [24], where we emprcally study the welfare benefts of coaltons of (a fnte number of wnd power generators. We conclude by notng two mportant open drectons. Frst, as prevously noted, n many real markets (ncludng electrcty markets, the penalty for a producton shortfall s not exogenous. Rather, a frm may face a spot or secondary market, nto whch t can sell excess capacty, or from whch t must buy addtonal capacty to cover a shortfall. Modelng ths two-stage market game remans an mportant challenge. Second, all our results are asymptotc, though we do characterze the rate of convergence to effcency wth optmal coalton szes. Wth fntely many (potentally heterogeneous frms, the regulator faces the potentally challengng problem of computng optmal coaltons as a benchmark. Developng approaches for ths problem remans an open ssue. APPEDIX A PROOF OF PROPOSITIO 1 Proof. We frst observe that the strategy space of each frm can be restrcted to a compact set, wthout loss of generalty. For any vectorx chosen by the other frms, f Pr(X < y max >, then frm s always strctly better off by choosng than choosng a quantty larger than y max. If Pr(X < y max = and k x k >, agan frm s always strctly better off by choosng than choosng a producton larger than y max. If Pr(X < y max = and k x k =, then frm s best strategy s to maxmze x p(x. By Assumpton 1, the unque maxmzer of x p(x s strctly larger than and strctly smaller than y max. Thus, we may restrct the strategy space of a frm to [,y max ]. Snce p satsfes Assumpton 1, p( x x s concave. By Assumpton 2, E[(x X + ] exsts and E[(x X + ] s concave n x. By addtvty of concave functons, π s concave for all x. The game defned by (π 1,...,π wth strategy spaces ([,y max ],...,[,y max ] s now a strctly concave game: each payoff π s contnuous and strctly concave n x and the strategy space of frm s a nonempty compact set. By Rosen s exstence theorem (see [33], there s a unque ash equlbrum for ths game. Furthermore, suppose x > y max. Then at least one of the x s s postve. But then frm s strctly better off f t reduces x. Therefore at equlbrum, x y max. APPEDIX B PROOF OF LEMMA 1 Proof. Ths lemma follows from Jensen s nequalty. Let W 1,...,W be random varables. Let g( = ( +. Snce g s convex, we have 1 E[g( W ] 1 E[g(W ]. (35 Snce g s homogenous, multplyng both sde of (35 by gves E[g( W ] E[g(W ]. Identfy W wth x X fnshes the proof.

11 11 APPEDIX C PROOF OF THEOREM 1 Proof. We fll frst assume (18 holds and use t to prove (17. Then we prove (18 to be true. Frst we observe that all groups are symmetrc, therefore each of them has the same expected cost: k=1 [ E (x(s k X(S k +] = E [(x(s k X(S k +] [ = E (x(s k X(S k +]. For notatonal convenence, let x = x(s k and X = X(S k. ote that E[X ] = µ. Let ˆX =X µ be the zero mean verson. Smlarly, we can wrte X = µ+ ˆX, where ˆX s zero mean. Usng these notatons, the effcency rato r W s U(x E x µ ˆX ] + r W = ( U(y max E[ y max µ ˆX + ]. (36 The followng proposton s useful for boundng r W. Proposton 4. Let V and W be ndependent contnuous random varables wth symmetrc denstes. Furthermore suppose E[V] = E[W] =. Let a be a postve real number. Then [ E (V +W a +] [ E (W a +]. (37 The proof of Prop. 4 s gven later. Applyng t, we have E x µ ˆX ] [ + ( E x µ ˆX ˆX ] +, (38 where X s ndependent of ˆX and the expectaton s now taken wth respect to both ˆX and ˆX. Therefore to lower bound r W, t suffces to lower bound ] + U =U(y max E y max µ ˆX U(x E x µ ˆX ˆX ] +. (39 Insertng an ndependent copy of ˆX nto the expectaton allows us to analyze (39 usng a Taylor expanson. Let δ = y max x, U(x = U(y max δ U(y max δu (y max δ2 + 2 U (y max, (4 where we neglect the hgher order terms n the Taylor expanson. Let g(y = E y µ ˆX ] +. To obtan a Taylor expanson of E x µ ˆX ˆX ] + around y max, we use condtonal expectaton by successvely condtonng on ˆX and ˆX: E x µ ˆX ˆX ] + [ = E ˆX [E ˆX g(y max δ ˆX ˆX ]] [ ]] E ˆX [E ˆX g(y max (δ + ˆX g (y max+ (δ + ˆX 2 g (y 2 max (a = g(y max δg (y max δ2 + 2 g (y max + E[ ˆX 2 ] g (y max 2, (41 where agan the hgher order terms are neglected and (a follows from E[ ˆX ] = and the ndependence between ˆX and ˆX. Substtutng (4 and (41 nto (39 yelds: { } U U(y max g(y max U(y max δu (y max δ2 + 2 U (y max { + g(y max δg (y max δ2 + 2 g (y max + E[ ˆX } 2 ] g (y max 2 = δ(u (y max g (y max+ δ2 2 (g (y max U (y max + E[ ˆX 2 ] g (y max 2 (a = δ2 2 (g (y max U (y max + E[ ˆX 2 ] g (y max 2, (42 where (a follows from frst order optmalty condtons and fact that y max maxmzes U(y g(y. We now derve more explct formulas for the constants n (42: g (y max = d2 dy 2E[(y µ ˆX + ] y max = d dy E[1(y µ ˆX + ] = d dy Pr( ˆX y µ = ˆf(y µ, where ˆf s the pdf of ˆX; and U (y max = p (y max. By Assumpton 2, ˆf exsts; by Assumpton 1, p (y max s negatve. Therefore both g (y max and U (y max are postve constants. The devaton δ 2 can be calculated from (18. By (18, δ scales as O( 1 +O( and δ2 scales as O ( 1 2 Snce ˆX s zero mean, +O ( 1 +O E[ ˆX ] 2 = Var(X = Var ( 2 2 / = 2 1 σ2 = σ2 = O. X (.

12 12 Combnng the above arguments, we have U(y max g(y max (U(x E[(y µ ˆX ˆX + ] ( ( ( ( = O 2 +O +O 2 +O ( ( 1 = O 2 +O. By Prop. 4, E[(y µ ˆX ˆX + ] E[(y µ ˆX + ] and we have the effcency rato scales as ( ( 1 r W = 1 O 2 O. (43 We now prove (18. The strategy of the proof proceeds n two steps. Frst we consder a determnstc game wth players, and bound the dfference between the ash equlbrum of the game (π 1,...,π and the ash equlbrum of the determnstc game. Then we bound the dfference between the latter and y max. Let (S 1,...,S be an equal szed partton of (1,...,, wth each coalton of sze /. Consder a determnstc game defned by (π 1,...,π where ( π k = p x(s m x(s k. (44 m=1 Let (x(s 1,...,x(S be a ash equlbrum of ths game. By symmetry, x(s k are of the same value. We denote ths value by x ; t satsfes: p(x +p (x x =. (45 Let (x(s 1,...,x(S be a ash equlbrum of the game (π 1,...,π. Agan, ths equlbrum s symmetrc. Denote the common producton level of each coalton by x. Smlarly, denote X(S k by X. Snce E[(x X + ] s ncreasng n x, we have x x. Therefore we can rewrte x as x, for some that solves: p((x +p ((x (x Pr(X x =. (46 Subtractng (45 from (46, we have Combnng (47 and (48, we obtan a bound on, or k x(s k k x(s k, as Pr(X x p ( Pr(X y max / p. (49 ( By assumpton, µ > y max ; applyng Chebyshev s nequalty gves Pr(X y max / = O(/. Therefore = O(/. ow we bound the gap between x and y max. Let δ = y max x. Substtutng nto (45 gves: ( p(y max δ+p ymax δ (y max δ =. (5 Snce p s decreasng and δ s postve, p(y max δ+p (y max δ ( ymax. (51 Rearrangng yelds ( p(y max δ p ymax (y max δ ( ( p ymax (y max, (52 where ( follows from the fact p s negatve and decreasng (p s decreasng and concave. Snce p s concave and decreasng, and p(y max =, p(y max δ δ p( y max. (53 See Fgure 7 for an llustraton of (53. Combnng (52 and (53 gves δ p( p (y max y max y max, (54 or δ 1 Therefore δ = O(1/. y max k p (y max ymax 2. (55 p( [p(x p(x ]+[p (x (x PSfrag replacements p (x x ] Pr(X x =. p( Snce p s concave and decreasng, p (x < p (x p(y max δ δp( <. Also sncex s postve, the term n the second set of brackets,p (x (x p y max (x x s greater y max δ than or equal to zero. Also snce s postve, Pr(X y max x Pr(X k x. Therefore: Fg. 7: Graphcal demonstraton of (52. p(x p(x Pr(X x. (47 Snce p s concave and decreasng, Combnng the two parts of the proof, ( p(x p(x ( p (. (48 1 x(s k = O +O (. (56 Dvdng both sdes by y max gves the desred result: ( ( 1 r O = 1 O O. (57

13 13 Fnally we prove Prop. 4. Let f V and f W be the pdf of V and W, respectvely. Then E V +W a +] E W a +] = µ µ v µ+x µ w µ µ µ w nfty µ+x By symmetry of f V, µ w µ µ µ w By symmetry of f W, Therefore µ+x µ nfty µ+x + (u+v µf W (wf V (vdwdv (µ wf W (wf V (vdwdv vf V (vf W (wdvdw wf W (wf V (vdwdv. vf V (vf W (wdvdw =. wf W (wf V (vdwdv µ wf W (wf V (vdwdv =. E V +W a +] E W a +] = + µ. µ v µ+x µ (u+v µf W (wf V (vdwdv µf W (wf V (vdwdv APPEDIX D PROOF OF COROLLARY 3 Proof. Let (S 1,...,S be a equal szed partton of (1,..., (each of sze /. Consder the determnstc game (π 1,...,π played by the groups. Let (x(s 1,...,x(S be a ash equlbrum of ths game. Let (x(s 1,...,x(S be a ash equlbrum of the game (π 1,...,π. Snce E[f(x X ] s ncreasng n x(s k, x x, we can rewrte x as x wth. Followng the proof of Theorem 1, we obtan (n place of (49: E[f (x X ] p. (58 ( Snce f s bounded, E[f (x X k ] BPr(x X for some B. Therefore scales as O(/. Followng the same steps as n the proof of Theorem 1, the effcency rato r O scales as 1 O ( ( 1 O. An analogous proposton to Prop. 4 can be derved for a convex and ncreasng f, and a smlar Taylor expanson argument can be used to bound r W based on r O. APPEDIX E PROOF OF COROLLARY 4 Proof. Consder a group S k. It suffces to show that Pr( S k X y max / s O(/ snce the rest of the proof proceeds exactly as that of Theorem 1. Equaton (2 mples that ( ( c Pr X y max / (µ y max 2 = O. S k (59 (See, e.g., [34], for ths standard result. APPEDIX F PROOF OF THEOREM 2 Proof. Ths proof follows a smlar path to that of Theorem 1. We assume (24 to be true and prove (23 holds. Then we prove (24. By symmetry, x(s k are equal for all k and we denote k=1 x(s as x and x = x(s 1. Let X denote X(S k. By Assumpton 4, the random varable X can be related to Z as X = Z + ˆX where ˆX has the same dstrbuton as k=1 / ˆX = k=1 Therefore ( U x(s k E [(x(s k X(S k +] = U (x E ˆX. x Z ˆX + ]. Let ˆX = ˆX. Then U (x E x Z ˆX ] + r W = + ]. U (y max E y max Z ˆX Applyng Prop. 4, E x Z ˆX ] + E x Z ˆX ˆX ] + where ˆX s ndependent of ˆX. By exactly the same argument as n the proof of Theorem 1, by assumng (24, we can show ( ( 1 r W = 1 O 2 O. We prove (24 holds by frst defnng the followng ntermedate game. Defne the payoff functon: ( π k = p x(s k x(s k E[(x(S k 1 (Z µ+ ]. (6 k