Cournot games with limited demand: from multiple equilibria to stochastic equilibrium

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1 Courno games wih limie eman: from muliple equilibria o sochasic equilibrium Io Polak an Nicolas Privaul Division of Mahemaical Sciences School of Physical an Mahemaical Sciences Nanyang Technological Universiy 21 Nanyang Link Singapore February 14, 2018 Absrac We consruc Courno games wih limie eman, resuling ino cappe sales volumes accoring o he respecive proucion shares of he players. We show ha such games ami hree isinc equilibrium regimes, incluing an inermeiae regime ha allows for a range of possible equilibria. When informaion on eman is moele by a elaye iffusion process, we also show ha his inermeiae regime collapses o a single equilibrium while he oher regimes approximae he eerminisic seing as he elay ens o zero. Moreover, as he elay approaches zero, he unique equilibrium achieve in he sochasic case provies a way o selec a naural equilibrium wihin he range observe in he no lag seing. Numerical illusraions are presene when eman is moele by an Ornsein-Uhlenbeck process an price is an affine funcion of oupu. Keywors: Game heory; muliple equilibria; equilibrium selecion; limie eman; elaye informaion; sochasic conrol. Mahemaics Subjec Classificaion 2010: 91A05, 91A10, 91A18, 91B24. 1 Inroucion Courno games wih muliple equilibria uner uncerain eman have arace significan aenion in recen years, see for example 4, 3, 2. On he oher han, he sochasic conrol of iffusions wih elaye informaion has recenly been applie o 1

2 newsvenor games, cf. 5, 6 in orer o eal wih parial informaion. Uner moel uncerainy, hese resuls have been exene in 7 o he case of elaye coefficiens in he equaion riving he sae process. In his paper, we consruc Courno games in which limie eman can inuce muliple equilibria in absence of lag. This range of equilibria collapses o a single equilibrium when he informaion on eman is elaye an moele by a ranom process. We choose o moel limie eman by sharing sales proporionally o proucion in case oal oupu excees eman. This seing iffers from he basic moel in which all proucion is sol as long as prices can ajus o heir marke clearing levels. I also allows for unsol proucion o be salvage a a fixe price. In Secion 2 we consier one-sho Courno games. In his seing, Proposiion 2.1 presens a general formula for he iniviual equilibrium proucion shares, an we erive an equaion saisfie by he oal proucion. When he marke price P q is of he familiar affine form P q = a bq for a, b > 0, we show he exisence of hree ifferen regimes of equilibria, incluing an inermeiae regime ha inclues muliple equilibria, cf. Proposiion 2.2 an Figures 1 an 2. Secon, in Secion 3 we consier a seup in which eman is moele by a sochasic iffusion process D 0,T an informaion abou eman is lagge by a ime amoun δ > 0 in such a way ha firms have o ecie on heir sraegies for ime base on he elaye value D δ of a ranom eman process. Using he maximum principle for he sochasic conrol of iffusions wih elaye filraions 5, 6, we erive necessary an sufficien coniions for opimaliy of conrols in his elaye seing. In Proposiion 3.2 we presen a naural generalizaion of Proposiion 2.1 o he case of elaye conrols, base on he erivaion of an equaion for he oal proucion. A more explici formulaion is hen obaine in Proposiion 3.3 when he sochasic eman D 0,T is moele exogenously accoring o an Ornsein-Uhlenbeck process. Finally, in he seing of affine prices we show in Corollary 3.4 ha he ranges of equilibria obaine in Proposiion 2.2 reuce o a unique equilibrium a every ime, no maer how small he lag. The lagge case can be seen as a smoohing of he no lag case, as he soluion of he eerminisic no lag problem can be obaine as he limi 2

3 of he lagge seup as δ ens o 0. In aiion, as he elay approaches zero, his also provies an equilibrium selecion mechanism in he inermeiae regime, wihin he range observe in he no lag seing. As such, a small quaniy of ranom noise can resolve he absence of uniqueness appearing in he no lag case. I urns ou ha when eman is low, he equilibrium is unique in boh he lag an no lag cases, an oal oupu excees eman overproucion. When eman is high enough, iniviual equilibrium oupus are also unique an in line wih he usual Courno equilibrium, an oal oupu falls shor of eman unerproucion in boh seups. Finally, when eman is wihin an inermeiae range, he oupu exacly meeing eman belongs o he range of possible equilibria in he no lag case, while oupu falls shor of eman in he lag case. In conras, he Sackelberg seing suie in 8 oes no give rise o muliple equilibria. 2 Courno game wih limie eman We consier wo compeiors facing a eman 0 for goos price P q 0 where q := q 1 + q 2 is he sum of he proucion oupus q 1 0, q 2 0 of Players 1 an 2. We choose o moel limie eman by assuming ha he profi f i, q 1, q 2 of player i = 1, 2 is given by f i, q 1, q 2 := P q 1 + q 2 min q i, = P q 1 + q 2 q i 1 {q 1 +q 2 } + q i c i q i 2.1 q 1 + q 2 qi q 1 + q 2 1 {<q 1 +q 2 } c i q i, where c i > 0 is firm i s per uni proucion cos, i = 1, 2. In oher wors, when he oal oupu q 1 +q 2 becomes larger han, he eman is share beween Players 1 an 2 accoring o heir respecive shares q 1 /q 1 +q 2 an q 2 /q 1 +q 2 of oal proucion. The inuiion is ha a firm ha prouces more oupu generaes more exposure, so if a buyer woul ranomly buy he prouc, he probabiliy ha firm i makes a sale is q i /q 1 + q 2, i = 1, 2. In 2.1 we assume he reail price P q 1 + q 2 o be a ecreasing funcion of oal oupu. When q 1 +q 2 >, his means ha excess proucion may force own reail 3

4 prices, which may furher increase consumer surplus. As in 5, we will assume ha unsol iems are salvage a a consan uni price S 0 which can be incorporae ino he consans c 1, c 2 an he price funcion P q wihou loss of generaliy. A Nash equilibrium for he game 2.1 is a pair q 1, q 2 such ha f 1, q 1, q 2 f 1, q 1, q 2 for all q 1 0, 2.2 an f 2, q 1, q 2 f 2, q 1, q 2 for all q 2 0, 2.3 so ha neiher player has an incenive o eviae given he conrol of he oher. In Proposiion 2.1 we presen a general characerizaion of equilibrium for he game 2.1 uner limie eman, when f i, q 1, q 2, i = 1, 2, are concave funcions of q 1, q 2. Proposiion 2.1 Any Courno equilibrium acion q 1, q 2 for he game 2.1 saisfies q i = S min, S P S c i S, 1 i j 2, min, S P S c 1 + c 2 S provie ha S := q 1 + q 2 saisfies S an solves he equaion S P S + P S = c 1 + c 2 max 1, S / + P S 1 {S >}. Proof. Consier he parial erivaives f i q, i q1, q 2 = q i P q 1 + q 2 + P q 1 + q 2 1 {q 1 +q 2 } q i P q 1 + q qj P q 1 + q 2 1 q 1 + q 2 q 1 + q 2 2 {<q 1 +q 2 } c i q i P q 1 + q 2 = + qj P q 1 + q 2 min q 1 + q 2, q 1 + q 2 q 1 + q qi P q 1 + q 2 q 1 + q 2 1 {q 1 +q 2 } c i, 1 i j 2. a When q 1 + q 2 <, he firs orer coniions become q i P q 1 + q 2 + P q 1 + q 2 c i = 0, i = 1, 2, 4

5 an he equilibrium agrees wih he usual Courno equilibrium. Namely, we have q i = ci P q 1 + q 2 P q 1 + q 2 = q 1 + q 2 q 1 + q 2 P q 1 + q 2 c i q 1 + q 2 2q 1 + q 2 P q 1 + q 2 c 1 + c 2 q 1 + q 2, i = 1, 2, an by summaion we fin q 1 + q 2 P q 1 + q 2 + 2P q 1 + q 2 = c 1 + c 2. b On he oher han, if < q 1 + q 2 we have q j q i P q 1 +q 2 + q 1 + q P 2 q1 +q 2 = ci q1 +q 2, 1 i j 2, hence by summaion we have an q 1 + q 2 P q 1 + q 2 + P q 1 + q 2 = c1 + c 2 q 1 + q 2, q i = q 1 + q 2 P q 1 + q 2 c i q 1 + q 2, 1 i j 2. 2P q 1 + q 2 c 1 + c 2 q 1 + q 2 In general, we conclue ha S P S + 2P S = c 1 + c 2 max 1, S + P S 1 {S >}, an q i = q 1 + q 2 min, q 1 + q 2 P q 1 + q 2 c i q 1 + q 2 2 min, q 1 + q 2 P q 1 + q 2 c 1 + c 2 q 1 + q 2, 1 i j 2. The seing of Proposiion 2.1 allows for negaive values of he equilibrium proucion oupus q 1 an q 2 an i oes no rea he case where proucion q 1 an q 2 mees he eman, as he erivaives of he profi funcions f i, q 1, q 2, i = 1, 2, are no coninuous a his poin, an more informaion on he funcion P q is neee. In Proposiion 2.2 below we will refine he resul of Proposiion 2.1 by consraining proucion o be nonnegaive when he marke price P q is an affine funcion of he oal oupu q. 5

6 Affine prices Unil he en of his secion we focus on he case P q 1 + q 2 = a bq 1 + q 2, 2.5 wih a > 0 an b 0. We se he equilibrium oupu q i := 0 if he proucion cos c i is larger han a > 0, henceforh we focus on he case where 0 < c i < a for i = 1, 2. Similarly, negaive prices canno lea o an equilibrium. The nex proposiion shows he exisence of hree equilibrium regimes, epening on he level of he eman. Proposiion 2.2 Assume ha P q is given by 2.5, an ha an le a 2c 1 + c 2 0 an a + c 1 2c 2 0, := a c1 c 2 2b an := 2a c1 c 2. 3b a In case >, we have he unique equilibrium a 2c q 1, q c 2 =, a + c1 2c 2 = 2 c1 c2, 2, 3b 3b b b 2.6 wih q 1 + q 2 = <. b In case <, we have he unique equilibrium ac q 1, q b = c 1 + c 2 + 2b, ac 1 + b, c 1 + c 2 + 2b 2 wih < q 1 + q 2 = a c 1 + c 2 + 2b. c In case, a poin q 1, q 2 is an equilibrium if an only if + max 2 ci, a ci b b a wih q 1 + q 2 =, 1 i j 2. The proof of Proposiion 2.2 is eferre o he appenix. c q i j + b min, 2 + cj a b 2.8, 6

7 Remarks. 1 We check ha when =, he lower boun in 2.8 reas a 2c i + c j 3b q i, which is consisen wih he equaliy When =, he upper boun in 2.8 is also consisen wih he soluion given in 2.7. c q 1, q b =, c1 + b a a 3 When < <, he coniion q 1 + q 2 =, 1 i j 2, shows ha only one of he wo conrols q 1, q 2 can be chosen inepenenly. The following Figures 1 an 2 illusrae he hree ifferen regimes a-b-c of Proposiion 2.2, incluing he exisence of a range of equilibria ha resul from 2.8 in he regime c when. 1.4 q * a=7, b=2, c 1=1, c 2= Figure 1: Equilibrium oupu q 1 as a funcion of. Figure 3 shows ha we move from overproucion o unerproucion as he value of increases, while eman is exacly me in he regime c when. 7

8 1 0.9 q * a=7, b=2, c 1=1, c 2= Figure 2: Equilibrium oupu q 2 as a funcion of q *1 +q * a=7, b=2, c 1=1, c 2= Figure 3: Toal oupu q 1 + q 2. The parameric plo of Figure 4 shows he possible equilibrium quaniies q 1, q 2 epening on he level of eman > 0. We check ha every value of 0,, correspon o a unique choice of q 1, q 2, whereas he values, yiel a coninuum of equilibria locae on he isolines q 1 + q 2 =. 18 a=20, b=1, c 1 =1, c 2 = = = q *2 q * Figure 4: Parameric plo of q 1, q 2, wih is projecion ono he plane = 0. The nex hree Figures 5, 6 an 7 presen he corresponing profis f i, q 1, q 2 of players i = 1, 2 as given from 2.1, wih a = 7, b = 2, c 1 = 1, c 2 = 2. 8

9 3.5 profi of player a=7, b=2, c 1=1, c 2= Figure 5: Profi of Player profi of player a=7, b=2, c 1=1, c 2= Figure 6: Profi of Player 2. 6 oal profi a=7, b=2, c 1=1, c 2= Figure 7: Toal profi. The seing of his secion can be mae ime-epenen wih eman 0 a ime > 0, where q 1 0, q 2 0 enoe he proucion oupus of Players 1 an 2. In his case, J i q 1, q 2 := T 0 f i, q 1, q 2, 2.9 wih q 1, q 2 := q 1, q 2 0,T, enoes he oal profi of Player i = 1, 2, an a Nash equilibrium for he game 2.9 is a pair q 1, q 2 = q 1, q 2 0,T such ha 9

10 J 1 q 1, q 2 J 1 q 1, q 2 for all q 1 = q 1 0,T, 2.10 an J 2 q 1, q 2 J 2 q 1, q 2 for all q 2 = q 2 0,T In he nex secion we will consier such a ime-epenen seing in which compeiors face uncerain sochasic eman an make opimal ecisions base on hisorical aa. 3 Courno game wih elaye informaion Le B 0,T enoe a sanar Brownian moion ha generaes he P-augmene filraion F = F 0,T on a filere probabiliy space Ω, F 0,T, P. The informaion flow available o he players is represene by he elaye filraion E δ,t efine by E := F δ, δ, T, 3.1 where δ > 0 enoes he informaion elay or lag. We le A i enoe a given se of amissible conrol processes for Player i, which are mae of real-value E δ,t -preicable processes, i = 1, 2. We enow A i wih he supremum norm an enoe A := A 1 A 2. In his secion we inrouce uncerainy in he eman, which is now moele as he soluion D 0,T of a sochasic ifferenial equaion of he form D = µ D + σ D B, D 0 = 0 > 0, 3.2 where µ, σ are given preicable processes for each R, which o no epen on he proucion oupu conrol processes q 1, q 2, an such ha 3.2 amis a unique srong soluion. Given f i : R 2 R, he profi funcion of Player i efine in 2.1, i = 1, 2, 0, T, le T J i q 1, q 2 := IE e r f i D, q 1, q 2, 3.3 δ 10

11 enoe he iscoune expece profis of Players 1 an 2, where D enoes he soluion of 3.2, provie he inegrals an expecaions exis. A Nash equilibrium for he game 3.3 is a pair q 1, q 2 A such ha J 1 q 1, q 2 J 1 q 1, q 2 for all q 1 A 1, 3.4 an J 2 q 1, q 2 J 2 q 1, q 2 for all q 2 A 2, 3.5 so ha neiher player has an incenive o eviae given he conrol of he oher. Maximum principle We rely on he framework evelope in 6, uner he following coniions on he se A = A 1 A 2 of conrol processes. A1 For all u i A i an all boune β i A i here exiss ε > 0 such ha u i + sβ i A i for all s ε, ε; i = 1, 2. A2 For all 0 δ, T an all boune E 0 -measurable ranom variables α i, he conrol process β i belongs o A i ; i = 1, 2. Define he Hamilonians H i efine by β i = 1 0,T α i ω, δ, T, : R 5 R of his game by H i, q 1, q 2, y i, z i := f i, q 1, q 2 + y i µ + z i σ, 3.6 i = 1, 2. To hese Hamilonians we associae he backwar SDEs Y i, q 2, Y i Y i T = 0 = Hi D, q 1, Z i + Z i B, δ, T, 3.7 in he ajoin processes Y i an Z i, i = 1, 2, which can be solve uner he assumpions A1-A2 above. We will rely on he necessary maximum principle for 11

12 Forwar-Backwar Sochasic Differenial Equaion FBSDE games, cf. Theorem 3.1 in 1 an Theorem 2.2 in 6. In Theorem 3.1 below we specialize Theorem 2.2 of 6 o he case of unconrolle eman D as his is he case in 3.2, an we refer o Theorem 2.2 in 6 for he general case. Theorem 3.1 Uner A1 an A2, suppose ha q 1, q 2 A an ha he soluions D, Y i, Z i of Equaion 3.7 are square-inegrable an iffereniable in he irecions of he conrol processes in A. Then he following wo saemens are equivalen: 1 For all boune β 1 A 1, β 2 A 2 we have s J 1 q 1 + sβ 1, q 2 s=0 = s J 2 q 1, q 2 + sβ 2 s=0 = 0. 2 We have H i IE u D, u 1, u 2, Y i i, Z i E Courno equilibrium u 1 =q 1,u 2 =q 2 = 0, i = 1, 2, δ, T. 3.8 In paricular, when f i is given by f i, q 1, q 2 = P q 1 + q 2 q i 1 + qi + {q1+q2 } q 1 + q 1 2 {<q 1 +q 2 } c i q i, i = 1, 2, by 3.6 we fin ha 3.8 reas H i IE r D, r i, q j i, Y i, Z i E r i =q i f i = IE r D, r 1, r 2 i r 1 =q 1,r 2 =q 2 = IE P r 1 + r 2 r i r i 1 {D r 1 +r 2 } + r 1 + r 2 D+ 1 {D<r 1 +r 2 } +P r 1 + r 2 1 {D r 1 +r 2 } + = 0, 1 i j 2. r j r 1 + r 2 2 D+ 1 {D<r 1 +r 2 } c i r 1 =q 1,r 2 =q 2 12

13 Hence from he maximum principle we mus have P q 1 + q i P q 1 IE P q 1 P q 1 + q j 1 i j 2. q qi q 1 1 {q 1 +q 2 IE D } E D 1 {D<q 1 +q 2 } E = c i, 3.9 The nex Proposiion 3.2 is he counerpar of Proposiion 2.1 in he elaye seing. In he following we focus on nonnegaive equilibrium oupus, q 1, q 2 for all δ, T, which are economically relevan, alhough A i conains funcions ha may ake negaive values. Proposiion 3.2 Any Courno equilibrium acion q 1, q 2 saisfies q i = S where S := q 1 IE P S min D, S c i S E, IE 2P S min D, S c 1 + c 2 S E i = 1, 2, δ, T, q 2 solves he equaion S P S IE mind, S E + 2P S IE mind, S E = c 1 + c 2 S + P S IE D 1 {D S } E. Proof. We apply he maximum principle Theorem 3.1 o he funcional J i q 1, q 2 given in 3.3. Namely, aking f i in 3.3 given by f i, q 1, q 2 = P q 1 + q 2 q i 1 + qi + {q1+q2 } q 1 + q 1 2 {<q 1 +q 2 } c i q i, i = 1, 2, by 3.6 we fin ha 3.9 reas H i IE r D, r i, q j i, Y i, Z i E r i =q i f i = IE r D, r i, r j i r 1 =q 1,r 2 =q 2 = IE P r 1 + r 2 r i r i 1 {D r 1 +r 2 } + r 1 + r 2 D+ 1 {D<r 1 +r 2 } +P r 1 + r 2 1 {D r 1 +r 2 } + r j r 1 + r 2 2 D+ 1 {D<r 1 +r 2 } 13 r 1 =q 1,r 2 =q 2

14 c i = 0, 1 i j 2. Hence from he maximum principle we mus have P q 1 + q i P q 1 IE P q 1 P q 1 + q j q qi q 1 1 {q 1 +q 2 IE D } E D 1 {D<q 1 +q 2 } E = c i, i j 2. By summing he above wo equaions we fin ha he sum S := q 1 + q 2 saisfies he equaion 2P q 1 + q 1 P q 1 + or equivalenly, P q 1 q 1 + P q 1 IE 1 1 {q +q 2 IE D 1 {D<q 1 +q 2 } D } E 3.12 E = c 1 + c 2. 2P S IE mind, S E c 1 + c 2 S = P S IE D 1 {D S } E S P S IE mind, S E By 3.11 an 3.12 we ge P q 1 + q j = c j, q j P q 1 IE P q 1 q 1 P q 2 1 q 1 hence q j P q 1 q 1 IE D 1 2 {D<q 1 +q 2 = c j + P q 1 IE min } 1, E 1 {q 1 +q 2 D } q j P q 1 q 1 q j P q 1 D q 1 E E IE, IE D 1 {D<q 1 +q 2 min 1, D } 3.14 E q 1 E

15 an q j = P q 1 IE c j P q 1 min D 1, q 1 +q 2 1 i j 2. Hence, using 3.13 we fin i = 1, 2. q i = S IE min D 1, q 1 +q 2 E P q1 +q 2 q 1 +q 2 2 IE E D 1 {D<q 1 +q 2 }, E 3.16 IE P S min D, S c i S E IE 2P S min D, S c 1 + c 2 S E, 3.17 Ornsein-Uhlenbeck seing In orer o erive explici soluions we henceforh assume ha µ D, q = αβ D where α, β R, an σ D, q = σ, where σ is a eerminisic funcion of R. Therefore he ynamics of D is given by D = αβ D + σb whose soluion D δ,t saisfies D 0 = 0 > 0, D = D δ e αδ + β1 e αδ + δ e αs σsb s, δ, T In he sequel we enoe by Φ : R 0, 1 he sanar Gaussian cumulaive isribuion funcion. Proposiion 3.3 In he Ornsein-Uhlenbeck seing, any Courno equilibrium acion q 1, q 2 saisfies q j = S P 2 S Φ ϕd δ S ε 2P 2 S Φ ϕd δ S ε a any δ, T, 1 i j 2, where S := q 1 S P S + S P S ϕd δ Φ ϕd δ ε + c j c i P S c j P S S P S, P S S P S c 1 + c 2 15 solves he equaion S εφ ϕd δ ε + q

16 +S 2P S + S P S ϕd δ S Φ = c 1 + c 2 S, 1 i j 2, ε wih an ε 2 := ϕ := e αδ + β1 e αδ, δ e 2αs σ 2 ss, δ, T. In conras wih Proposiion 2.1, his resul covers he case where proucion q 1 an q 2 mees he eman D, as he erivaives of he profi funcions f i D, q 1, q 2, i = 1, 2, are smoohene by he inroucion of a coniional expecaion ue elaye informaion. The proof of Proposiion 3.3 is eferre o he appenix. Affine prices In he sequel we consier he affine price moel P q = a bq, which saisfies he equaion P q qp q = a. If c i a hen P q i c i = a bq i c i 0, in which case we have q i = 0, henceforh we focus on he case where 0 < c i < a, i = 1, 2. Proposiion 3.3 yiels he following corollary. When δ ens o zero, he equaion 3.20 below converges o 2.4 in Proposiion 2.1, an i also recovers he resul of Proposiion 2.2, since lim ε 0 Φx/ε = 1 0, x, x R. Corollary 3.4 In he affine price moel he Courno equilibrium acion a any δ, T is given by P 2 S ϕd δ S q j Φ + c j c i P S ac j = S ε 2P 2 S ϕd δ S, 3.20 Φ ac ε 1 + c 2 1 i j 2, where S = q 1 a 2bS ϕd δ Φ +S 2a 3bS Φ + q 2 solves he equaion S ϕd δ S εφ ϕd δ ε ε ϕd δ S = c 1 + c 2 S. ε

17 Figure 8 presens a numerical soluion of 3.21 showing he values of q 1, q 2 as funcions of he eman. q 1 q 2 q 1 + q α = 1, β = 1, δ = 0.1, σ = 0.2 a = 7, b = 2, c1 = 1, c2= Figure 8: Graphs of q 1, q 2. As he elay δ > 0 ens o zero, he lagge equilibrium converge o a limiing equilibrium which belongs o he range of equilibria of Proposiion 2.2, as illusrae in he following Figures In he inermeiae regime, his provies a way o selec an equilibrium wihin he range of equilibria available in he no lag seing δ=0 δ=0.1 δ= a=7, b=2, c 1=1, c 2= Figure 9: Equilibrium oupu q 1 as a funcion of δ=0 δ=0.1 δ= a=7, b=2, c 1=1, c 2= Figure 10: Equilibrium oupu q 2 as a funcion of. 17

18 δ=0 δ=0.1 δ= Conclusion 0.2 a=7, b=2, c 1=1, c 2= Figure 11: Toal oupu q 1 + q 2. We have shown ha he inclusion of limie eman in our moel resuls in he exisence of hree ifferen regimes, epening on he level of he eman, incluing an inermeiae regime ha shows a range of possible equilibria. When eman is low, he marke is inefficien as here will be overproucion, whereas when eman is high, here is unfulfille eman since he usual Courno equilibrium will maerialise. When eman is a an inermeiae level, i is exacly me by he combine proucion of he firms when here is no lag, bu oupu falls shor of eman in case of lag. We have also shown ha he inroucion of a elaye sochasic eman reuces he range of equilibria o a single equilibrium in he inermeiae regime. In aiion, equilibria in oher regimes converge o he equilibrium obaine in he non-elaye seing. In his sense, he lagge case can be seen as a smoohing of he no lag case. In orer o rea he case of n-player oligopolies, we woul replace 2.1 wih f i, q 1,..., q n = P q q n q i q i 1 {q 1 + +q n } + q q 1 n {<q 1 + +q n } c i q i. In his case, he opimal q i epens on he sum n q j an his will resul ino more firs orer coniions in he proof of Proposiions 2.1 an 3.2. This requires ifferen soluion echniques an will be consiere for fuure research. j=1 j i 18

19 5 Appenix: proofs of Proposiions 2.2 an 3.3 Proof of Proposiion 2.2. We have f i q i, q1, q 2 = a 2bq i bq j 1 {q 1 +q 2 <} + 1 i j 2. a If q 1 + q 2 <, 5.1 reas aq j q 1 + q 2 2 b 1 {<q 1 +q 2 } c i, f i q i, q1, q 2 = a 2bq i bq j c i, 1 i j 2, We isinguish wo cases: 1. q i, q j > 0. In his case we mus have f i q, i q1, q 2 = a 2bq i bq j c i = 0 hence an f j q j, q1, q 2 = a 2bq j bq i c j = 0, q i = a 2ci + c j 3b an q 1 + q 2 = 2a c1 c 2 q j = a + ci 2c j, 3b 3b <. 2. q i < an q j = 0. In his case we may have an inequaliy f i q, i q1, q 2 = a 2bq i bq j c i = i.e. f j q j, q1, q 2 = a 2bq j bq i c j 0, a 2bq i c i = 0 a bq i c j 0, 19

20 or a c j /b q i = a c i /2b, which ogeher wih he coniion a + c i 2c j 0, implies a + c i 2c j = 0 an in his case, he equilibrium is alreay aken care of in poin 1 above. b On he oher han, if < q 1 + q 2, 5.1 reas f i q, i q1, q 2 aq j = q 1 + q 2 b c i, 1 i j 2, 2 There are wo kins of caniae equilibria: 1. q i, q j > 0 an we mus have: f i q, i q1, q 2 aq j = q 1 + q 2 b c i = 0 2 f j q, j q1, q 2 aq i = q 1 + q 2 b c j = 0, 2 In his case, q 1 a bq 1 + q 2 2 = c2 q1 + q 2 2, hence an q 1 = q 2 a bq 1 + q 2 2 = c1 q1 + q 2 2, ac 2 + b c 1 + c 2 + 2b 2, an q2 = < q 1 + q 2 = a c 1 + c 2 + 2b. ac 1 + b c 1 + c 2 + 2b 2, 2. q i > an q j = 0 an insea we may have an inequaliy: f i q, i q1, q 2 aq j = q 1 + q 2 b c i = 0 2 f j q, j q1, q 2 aq i = q 1 + q 2 b c j 0, 2 An herefore, f i q i, q1, q 2 = b c i = 0 f j q, j q1, q 2 = a q b i cj 0, 20

21 which is impossible. c Finally we consier wha happens when, i.e. is beween he above bounaries. The sum of erivaives f 1 q, 1 q1, q 2 2 f + q, 2 q1, q 2 = a 3bq i + q j 1 {q 1 +q <}+ 2 a q 1 + q 2 b 1 {<q 1 +q 2 } c i c j 5.2 is a ecreasing funcion of q 1 + q 2 which vanishes a q 1 + q 2 =, herefore i is sricly posiive when q 1 + q 2 <, hence a leas one of he iniviual erivaives mus be sricly posiive a such poins. Therefore, q 1 + q 2 < canno be an equilibrium. In he case q 1 + q 2 > he summaion of erivaives vanishes a q 1 + q 2 = hence i is sricly negaive when q 1 + q 2 > an a leas one of he erivaives mus be negaive. Therefore q 1 + q 2 > canno be an equilibrium eiher. Consier a poin q 1, q 2 0, 0, on he line q 1 + q 2 =. This is an equilibrium if here exiss no profiable eviaions. We nee o show ha he value funcion f i, q 1, q 2 = a bq i + q j q i q i 1 {q i +q j } + q i + q 1 j {<q i +q j } c i q i, 1 i j 2, is increasing in q i 0, q i an ecreasing in q i q i,. Given ha he parial erivaive f i q i, q1, q 2 = a 2bq i bq j 1 {q i +q j <} + aq j q i + q j 2 b 1 {<q i +q j } c i is a ecreasing funcion of q i, i is necessary an sufficien o have he lef erivaive f i q i, qi, q j = a bq i b c i = a + bq j 2b c i 0,

22 an he righ erivaive saisfies f i q, i qi +, q j = aqj b c i Therefore q 1, q 2 0, 0, on he line q 1 + q 2 = is an equilibrium if an only if a + c i + + 2b q j ci + b, 1 i j 2, 5.5 b a or equivalenly a + c j + + 2b q j cj + b, b a j = 1, 2, 5.6 which yiels 2.8. Noice ha from 2.8, if here exiss q i, q j ha saisfy he inequalies, hen, or in oher wors, if is ousie hese bouns, we o no have equilibria saisfying q i + q j =. Finally, consier q 1, q 2 on he line q 1 + q 2 = wih q i = an q j = 0. In his case, he value funcion for i f i, q 1, q 2 = a bq i q i 1 {q i } + 1 {<q i } c i q i, is clearly ecreasing in q i,, an i suffices o show ha i is increasing in q i 0,, an from 5.3 for his we nee a ci. 2b On he oher han, we check ha he righ erivaive for j, 5.4 is negaive if a c j b, hence we nee a c j a ci. b 2b However, q i, q j =, 0 canno be an equilibrium unless = a cj b = a ci, 2b as we assume ha a + c i 2c j 0. In he laer case we have a + c i 2c j = 0 an he inequaliies 2.8 rea = a + cj + 2b b q i, i.e. q i =, an 0 q j 0, i.e. q j = 0, which is compaible wih he equilibrium q i, q j =, 0. Noice ha, 0 is an equilibrium if an only if is i s monopoly quaniy. 22

23 Proof of Proposiion 3.3. From 3.9 an he fac ha q 1 is E -measurable we have P q 1 + q i P q 1 IE1 {q D} E 1 q=q + q j P q 1 q qi P q 1 q 1 +q 2 IED 1 {q>d} E 1 q=q = c i, +q i j 2. From 3.18 we wrie D = δ e αδ + β1 e αδ + X where X N 0, ε 2 an ε 2 = IE δ e αs σsb s 2 = δ e 2αs σ 2 ss = σ 2 1 e 2αδ, 2α hence by he inepenence beween E := F δ an δ eαs σsb s in 3.18 we have IE1 {q D} E 1 q=q +q 2 = Pq e αδ + β1 e αδ + X = δ, q=q 1 = = Φ e x2 /2ε 2 q 1 +q 2 ϕd δ ϕd δ q 1 q 2 ε x 2πε2. +q 2 For he secon expecaion we have IED 1 {q>d} 1 q=q +q 2 = IEe αδ + β1 e αδ + X1 {q>e αδ +β1 e +X} αδ = δ, q=q 1 +q 2 = ϕd δ IE1 q>ϕd δ +X 1 q=q +q 2 = ϕd δ PX < q ϕd δ 1 q=q + +q 2 = ϕd δ Φ q 1 ϕd δ ε + IEX1 q>ϕd δ +X 1 q=q +q 2 q 1 +q 2 ϕd δ xe x2 /2ε 2 q 1 ϕd δ εφ. ε x 2πε2 Hence 5.7 reas q i P q 1 q qj P q 1 q

24 q 1 ϕd δ εφ ϕd δ Φ ε P q 1 = c j, 1 i j 2. + q j P q 1 Φ q 1 ϕd δ q 1 q 2 ε ϕd δ ε By summing he above wo equaions we fin ha he sum S := q 1 he equaion P q 1 q 1 εφ q 1 + P q 1 ϕd δ ϕd δ Φ ε 2P q 1 + q 1 P q 1 Φ = c 1 c 2, 1 i j 2. q 1 ϕd δ ε ϕd δ q 1 q 2 ε saisfies 5.9 Combining 5.8 wih 5.9, we ge q i P q 1 + q j q 1 P q q 1 P q 1 + q 1 2 P q 1 c 1 + c 2 2P q 1 + q 1 P q 1 ϕd δ q 1 q 2 Φ ε + P q 1 = c j, 1 i j 2, or q j P S S P S + q j P q 1 Φ ϕd δ q 1 q 2 ε c 1 + c 2 2P S + S P S Φ + q j P S S P S + S 2 P S ϕd δ S Φ ε ϕd δ S ε = c j S P S + S 2 P S P S S P S + S 2 P S ϕd δ S Φ ε S P S c 1 + c 2 2P S + S P S Φ 24 ϕd δ S ε, 1 i j 2,

25 which shows ha q j = S P 2 S Φ ϕd δ S ε 2P 2 S Φ ϕd δ S ε + c j c i P S c j P S S P S. P S S P S c 1 + c 2 In he limi when is large, 5.10 yiels he sysem of equaions 2bq 1 + bq 2 + c 1 = a wih soluion an sum bq 1 + 2bq 2 + c 2 = a, q 1 = a 2c1 + c 2 3b q 2 = a + c1 2c 2, 3b q 1 = 2a c1 c 2, δ, T, 3b which recover he no lag seing of Proposiion 2.2. References 1 F. Baghery an B. Øksenal. A maximum principle for sochasic conrol wih parial informaion. Soch. Anal. Appl., 253: , M.A. Caraballo, A.M. Mármol, L. Monroy, an E.M. Buirago. Courno compeiion uner uncerainy: conservaive an opimisic equilibria. Review of Economic Design, 19: , J.N.M. Lagerlöf. Equilibrium uniqueness in a Courno moel wih eman uncerainy. Topics in Theoreical Economics, 61:1 6, J.N.M. Lagerlöf. Insising on a non-negaive price: Oligopoly, uncerainy, welfare, an muliple equilibria. Inernaional Journal of Inusrial Organizaion, 25: , B. Øksenal, L. Sanal, an J. Ubøe. Sochasic Sackelberg equilibria wih applicaions o imeepenen newsvenor moels. Journal of Economic Dynamics an Conrol, 377: , B. Øksenal an A. Sulem. Forwar-backwar sochasic ifferenial games an sochasic conrol uner moel uncerainy. Journal of Opimizaion Theory an Applicaions, 1611:22 55, O. Menoukeu Pamen. Opimal conrol for sochasic elay sysems uner moel uncerainy: A sochasic ifferenial game approach. Journal of Opimizaion Theory an Applicaions, 167: , I. Polak an N. Privaul. A sochasic newsvenor game wih ynamic reail prices. Preprin, o appear in Journal of Inusrial an Managemen Opimizaion,