Dynamic Stability of the Nash Equilibrium for a Bidding Game

Transcription

1 Dynamic Stability of the Nash Equilibrium for a Bidding Game Alberto Bressan and Hongxu Wei Deartment of Mathematics, Penn State University, University Park, Pa 16802, USA s: bressan@mathsuedu, xiaoyitangwei@gmailcom December 15, 2014 Abstract A one-sided limit order book is modeled as a noncooerative game for several layers An external buyer asks for an amount X > 0 of a given asset This amount will be bought at the lowest available rice, as long as the rice does not exceed a uer bound P One or more sellers offer various quantities of the asset at different rices, cometing to fulfill the incoming order The size X of the order and the maximum accetable rice P are not a riori known, and thus regarded as random variables In this setting, we rove that a unique Nash equilibrium exists, where each seller otimally rices his assets in order to maximize his own exected rofit Furthermore, a dynamics is introduced, assuming that each layer gradually adjusts his ricing strategy in rely to the strategies adoted by all other layers In the case of (i) infinitely many small layers or (ii) two large layers with one dominating the other, we show that the ricing strategies asymtotically converge to the Nash equilibrium Keywords: bidding game, limit order book, otimal ricing strategy, Nash equilibrium, asymtotic stability 1 Introduction A bidding game related to a continuum model of the limit order book was recently considered in [4], roving the existence and uniqueness of a Nash equilibrium and determining the otimal strategies for the various agents In the basic model, it is assumed that an external buyer asks for a random amount X > 0 of a given asset This external agent will buy the amount X at the lowest available rice, as long as this rice does not exceed an uer bound P One or more sellers offer various quantities of this same asset at different rices, cometing to fulfill the incoming order, whose size is not known a riori Having observed the rices asked by his cometitors, each layer must determine an otimal 1

2 quantity sold = X Q i _ 0 P Figure 1: The height of the various columns indicates the amount of asset offered for sale at the various rices The buyer will buy a random quantity X, at the lowest ossible rice, as long as this rice does not exceed the (random) uer bound P i strategy, maximizing his exected ayoff Because of the resence of the other sellers and of the uer bound P, asking a higher rice for an asset reduces the robability of selling it Aim of the resent aer is to advance the analysis in [4] in three main directions: (i) Consider a more realistic model where the maximum accetable rice P is a random variable, not known a riori (ii) Study the average reduction in the asked rice, resulting from the cometition among sellers (iii) Introduce a dynamics in the ricing strategies and study the asymtotic stability of the Nash equilibrium We assume that the random variable X, describing the amount of stock that the external agent wants to buy, has a distribution function for which the following holds Prob{X s} = ψ(s) (11) (A1) The ma s ψ(s) is continuously differentiable and satisfies ψ(0) = 1, ψ(+ ) = 0, ψ (s) < 0 for all s > 0, (12) (lnψ(s)) 0 for all s > 0 (13) For examle, the robability distributions determined by ψ 1 (s) = e λs λ > 0, (14) ψ 2 (s) = satisfy (13), while ψ 3 (s) = e s2 does not 1 (1+s) α α > 0, 2

3 1 1 _ ψ (s) = Prob{ X > s} h(s) = Prob{ P > s} 0 s 0 max s Figure 2: Left: a robability distribution for the random variable X, describing the size of the incoming order Right: a robability distribution for the random variable P, describing the maximum rice that the buyer is willing to ay Differently from [4, 5, 6], we here assume that the maximum rice P that the buyer is willing to ay is not known a riori We thus model P as a random variable, indeendent of X, with a distribution function h(s) = Prob{P s} (15) which satisfies the following assumtions (A2) The function s h(s) is continuous, continuously differentiable for s ] 0, max [, and satisfies h(s) = 1 for s 0, h(s) = 0 for s max, h (16) (s) < 0 for 0 < s < max, (lnh(s)) < 0 for 0 < s < max For examle, one may assume that the random variable P is uniformly distributed over the interval [ 0, max ] This leads to h(s) = 1 if s 0, ( max s)/( max 0 ) if s [ 0, max ], 0 if s max (17) In our model we assume that the i-th layer owns a total amount κ i of asset, which will be labelled by the variable β [0,κ i ] He can ut all of it on sale at a given rice, or offer different ortions at different rices His strategy will be described by a nondecreasing function φ i : [0,κ i ] IR Here φ i (β) is the rice asked for the asset β In the first art of the aer we rove the existence of a unique Nash equilibrium [9], where the ricing strategy of each layer yields the maximum exected ayoff, given the strategies adoted by all other layers An exlicit formula for these ricing strategies is rovided We also consider the limit of these Nash equilibria, as the number of layers n, while the total amount of assets ut on sale remains bounded This art of our analysis extends the earlier results in [4] to the case where the uer bound P is a random variable In Section 4 we study how the average asked rice decreases as a result of the cometition among sellers As shown in Theorem 5, the ricing strategies in the Nash equilibria satisfy: 3

4 If n cometing agents ut on sale different amounts κ 1 κ n of asset, the average rice is larger then in the case where each agent offers thesame amount (κ 1 + +κ n )/n For n cometing agents, each utting on sale the same amount κ/n of asset, the average rice decreases as either n or κ increase For a fixed number of sellers n 2, if each agent has the same amount κ/n of asset to ut on sale, the average rice aroaches 0 as κ In Section 5 we introduce a dynamics, describing how the ricing strategies may evolve in time, if they are away from a Nash equilibrium More recisely, let J i (φ(β),β) = [rofit from the sale of asset β] [robability of selling asset β] be the exected ayoff for the i-th layer, achieved by utting asset β on sale at rice φ(β) If φ J i(φ(β),β) 0, then this exected ayoff can beincreased by suitably modifyingtheasked rice φ(β) We thus consider the following systems of evolution equations, corresonding to a gradient flow: t φ i(β,t) = φ J i(φ i (β,t),β) i = 1,,n (18) Notice that, if φ(β) yields the maximum exected ayoff, then the necessary conditions yield φ J i(φ(β),β) = 0, and the right hand side of (18) vanishes In case of a Nash equilibrium, this is true for every β [0,κ i ] and every i Our main concern is the asymtotic behavior of solutions to the system (18) In the cases of (i) infinitely many small layers and (ii) two large layers, with initial strategies satisfying a secific inequality assumtion, we rove that, as t, the ricing strategies asymtotically converge to the uniquenash equilibrium On the other hand, for any number n 2 of layers, if the initial strategies have disjoint rice ranges, we show that the solution to the system (18) converges to a different limit In addition to the classical aer [9], for an introduction to non-cooerative games and Nash equilibria we refer to [3, 8, 14, 15] In the case where sellers have different beliefs about the fundamental value of the asset and on the distribution of the random order X, the equilibrium ricing strategies have been studied in [5, 6] In the literature on mathematical finance, various models of the limit order book have been recently studied, mainly from the oint of view of the agents who submit the limit orders In [11, 13, 7] rices range over a discrete set of values, while in [10, 12, 1] rices are continuous and the shae of the limit order book is described by a density function An imortant achievement of these models is that, as soon as the shae of the limit order book is given, this in turn determines a corresonding rice imact function, describing how the bid and ask rices change after the execution of a market order In the resent model, as well as in [4, 5, 6], rices are allowed to vary in a continuum of values but the shae of the limit order book is not given a riori Indeed, this shae can be endogenously determined as the unique Nash equilibrium, resulting from the otimal ricing strategies imlemented by the selling agents 4

5 2 The otimization roblem for a single layer Consider an agent offering an amount κ of assets for sale By a ricing strategy we mean any nondecreasing function φ : [0,κ] IR Here φ(β) is the rice asked for asset β [0,κ] We assume that this new seller cometes with several other sellers already resent on the market To model this situation, we consider the nondecreasing function Φ() = [total amount of assets ut on sale at rice by all other agents] (21) In this case, is the new seller adots the ricing strategy φ, his exected ayoff will be J(φ) = = κ 0 κ 0 [rofit from the sale of asset β] [robability of selling asset β] dβ [ ] [φ(β) 0 ] ψ(β +Φ(φ(β))) h(φ(β)) dβ (22) Remark 1 We regard 0 as the fundamental value of the asset To every agent, keeing the asset or selling it at rice 0 is indifferent A rofit is achieved only by selling at a higher rice Remark 2 In the case where two or more sellers ut a ositive amount of assets for sale at exactly the same rice, one needs to secify who sells it first In our model, this haens when Φ has an uward jum at, and the set {β; φ(β) = } has ositive measure By taking Φ left continuous at we model the case where the new seller has riority (ie, his assets riced at are sold before those of the other agents) By taking Φ right continuous at we model the case where the other agents have riority In the case of a Nash equilibrium, however, this situation never haens Indeed, since in our model the rices range continuously over the interval [ 0, max ], the agent which does not have riority can always imrove his exected ayoff by selling at a slightly lower rice ε In this section we derive necessary and sufficient conditions in order that the ricing strategy φ be otimal From the modeling assumtions (A2) it it obvious that an otimal strategy should satisfy 0 < φ(β) < max (23) Indeed, selling at rice 0 can only roduce a loss, while the robability of selling at rice max is zero In addition, if the function Φ() is smooth, for each β [0,κ], the otimal rice φ(β) will satisfy the necessary condition φ [(φ 0) ψ(β +Φ(φ)) h(φ)] = 0 (24) Introducing the function G β () ( ) = ψ β +Φ() ) ψ (β +Φ() ( ) 1 + h (), (25) 0 h() 5

6 we see that (24) is equivalent to Φ (φ(β)) = G β (φ(β)) (26) Remark 3 From the assumtions (A1)-(A2) it follows that the function Q() = 1 + h () 0 h() is strictly decreasing on the oen interval ] 0, max [ and there exists a unique oint ] 0, max [ such that Q( ) = 0 (27) Moreover, on the interval [ 0, ] where Q 0, the assumtion (13) imlies β Gβ () 0 (28) In the secial case where ψ(s) = e λs, the formula (25) simlifies to G() = 1 λ ( ) 1 + h () (29) 0 h() Notice that in this case the right hand side is indeendent of β We also observe that, if h is the function in (17), then = ( 0 + max )/2 The following theorem extends the necessary condition (26) to the case where Φ is a nondecreasing function Since the roof is the same as for Theorem 42 in [4], we omit details Theorem 1 (necessary conditions for otimality) Let the functions ψ, h satisfy the assumtions (A1)-(A2), and let Φ : IR + IR + be a nondecreasing ma If φ : [0,κ] [ 0, max ] is an otimal ricing strategy, then for almost every β [0,κ], setting = φ(β) one has lim su ε 0 Φ(+ε) Φ() ε G β () liminf ε 0+ Φ(+ε) Φ() ε (210) To obtain the existence and an exlicit descrition of the Nash equilibrium, the following result will be used Theorem 2 (sufficient conditions for otimality) Let the functions ψ, h satisfy the assumtions (A1)-(A2), and let Φ : IR + IR + be a nondecreasing ma with Φ( 0 ) = 0 Let φ : [0,κ] [ 0, max ] be a ricing strategy such that, for ae β [0,κ], the following holds The function Φ( ) is Lischitz continuous on [ 0,φ(β)] Moreover, its derivative satisfies Φ () G β () for ae φ(β), (211) Φ () G β () for ae > φ(β) (212) 6

7 Then φ( ) is otimal Proof 1 For any given β [0,κ], consider the ma J(,β) = ( 0 ) ψ(β +Φ()) h(), (213) describing the exected ayoff achieved by utting the asset β on sale at rice We observe that J( 0,β) = J( max,β) = 0 Moreover, since Φ is nondecreasing and can have only uward jums, while ψ is decreasing, the ma (213) can only have downward jums More recisely, for any 0 1 < 2 max, J( 2,β) J( 1,β) 2 [ ] + ψ(β +Φ())h()+( 0 )ψ (β +Φ())Φ ()h()+( 0 )ψ(β +Φ())h () d 1 (214) Notice that equality holds as long as 2 φ(β), because by assumtion Φ is Lischitz continuous on [ 0,φ(β)] 2 For ae β, by (211) the integrand in (214) is 0 for [0,φ(β)] Hence the ma J(, β) is Lischitz continuous and nondecreasing on [0, φ(β)] On the other hand, by (212) the integrand in (214) is 0 for [φ(β), max ] Hence the ma J(,β) is nonincreasing on [φ(β), max ], ossibly with downward jums We conclude that, for ae β [0, κ], the function J(, β) achieves its global maximum at = φ(β) This imlies the otimality of the ricing strategy φ( ) 3 Nash equilibria 31 Finitely many cometing sellers We now consider n sellers cometing against each other We assume that i-th agent has an amount κ i of assets to offer for sale His ricing strategy will be described by the function φ i : [0,κ i ] IR For every i {1,,n}, let Φ i () = su {β [0,κ j ]; φ j (β) < } (31) j i be the total amount of assets offered by all other agents j i at rice < Then the exected ayoff for agent i is κi [ ] J i (φ i ) = [φ i (β) 0 ] ψ(β +Φ i (φ i (β))) h(φ i (β)) dβ (32) 0 Definition An n-tule of ricing strategies (φ 1,,φ n ) is a Nash equilibrium if each φ i yields the maximum exected ayoff (32) to the i-th layer, given the function Φ i determined by the strategies of all other layers 7

8 κ i β = U () i 0 0 = φ (β) i Figure 3: The functions U i () and β φ i (β) are generalized inverses of each other Notice that the otimal strategy β φ i (β) for the i-th layer must satisfy the necessary condition φ [(φ 0) ψ(β +Φ i (φ)) h(φ)] = 0, (33) for ae β [0,κ i ] Of course, this is the same as (24), with Φ relaced by Φ i To determine these equilibrium strategies, it is convenient to introduce the functions U i () = [amount ut on sale by i-th agent at rice < ], (34) u i () = U i(), U() = n U i () i=1 Notice that the U i rovides a generalized inverse to the function φ i : [0,κ i ] IR describing the strategy of the i-th layer (see Fig 3) Indeed, u to sets of measure zero, one has U() = su{β; φ i (β) < }, φ i (β) = su{; U i () < β} Let 0 < κ 1 κ 2 κ n be the amounts of asset ut on sale by the various layers We will show that the Nash equilibrium strategies are obtained as follows STEP 1 Construct a iecewise smooth function U() on the half-oen interval [ 0, [, by solving the family of ODEs ( ) U n j +1 () = ψ U() ( ) 1 ) + n j ψ (U() h () [ j, j+1 ] (35) 0 h() with terminal condition U( ) = κ n = κ i (κ n κ n 1 ) (36) Here the oints 0 < 1 2 n are inductively determined by n =, U( j+1 ) U( j ) n j +1 i=1 8 = κ j κ j 1, j = 1,,n 1 (37)

9 For notational convenience, we here define κ 0 = 0 STEP 2 For i = 1,,n 1 the otimal strategy U i is Lischitz continuous and satisfies U () [ j, j+1 ], 1 j i, U i() n j +1 = (38) 0 / [ 1, i+1 ] Moreover, U n () = { Un 1 () < n, κ n n In other words, Player n uts an amount κ n κ n 1 of assets for sale all at the rice n =, while his remaining assets are riced in the same way as Player n 1 (39) Theorem 3 Let the assumtions (A1)-(A2) hold Then the bidding game has a unique Nash equilibrium The corresonding functions U 1,,U n in (34) are determined by (35) (39) Proof 1 We begin by roving that the function U and the oints i are uniquely determined by the equations (35) (37) For this urose, we shall use backward induction on i = n,n 1,,2,1 The first ste is to solve the backward Cauchy roblem ( ) U () = 2 ψ U() ( ) 1 ) + ψ (U() h () 0 h() (310) for < n =, with terminal condition U( n ) = κ defined at (36) Observe that the right 1 hand side of (310) is strictly ositive Moreover, since the function 0 is not integrable, we have lim U() = (311) 0 + Therefore, there exists a unique oint n 1 such that U( n ) U( n 1 ) 2 This rovides the first inductive ste = κ n 1 κ n 2 Next, assume that U has been constructed on the interval [ j+1, ] If j = 0 we are done Otherwise, the function U can be extended backwards on the additional interval [ j, j+1 ] by solving the Cauchy roblem U () = n j +1 n j ( ψ ) U() ) ψ (U() ( ) 1 + h () 0 h() (312) for < j+1, with terminal condition at = j+1 rovided by the inductive ste As before, the solution U of this ODE is strictly increasing and satisfies (311) Hence there exists a unique oint j such that U( j+1 ) U( j ) = κ j κ j 1 n j +1 9

10 This achieves the inductive ste of our construction By induction, we thus obtain a function U(), defined for [ 1, ], with 0 < 1 2 n =, U( 1 ) = 0 We then set { U() = 0 if 1, U() = κ if 2 We now show that the bidding strategies U 1,,U n in (34), determined by (35) (39) rovide a Nash equilibrium Fix any i {1,,n} and consider the function Φ i () = U() U i () (313) According to our construction, the i-th layer uts his asset β [0,κ i ] on sale at a rice φ(β) which satisfies Φ i (φ(β))+β = U(φ(β)) We claim that this rice is otimal Indeed, the sufficient conditions in Theorem 2 are satisfied To fix the ideas, assume first that 1 i < n Then φ(β) [ 1, i+1 ] Moreover, Φ i() = 0 < 1 For [ 1,φ(β)], since U i () β, by (A1) we have ( ) Φ i () = ψ Φ i ()+U i () ( ) 1 ) + ψ (Φ h () i ()+U i () 0 h() ( ) ψ Φ i ()+β ) ψ (Φ i ()+β ( ) 1 + h () 0 h() = G β () Finally, for φ(β), since U i () β we have ( ) Φ i () = ψ Φ i ()+U i () ) ψ (Φ i ()+U i () ( ) 1 + h () 0 h() ( ) ψ Φ i ()+β ) ψ (Φ i ()+β ( ) 1 + h () 0 h() = G β () By Theorem 2, φ is an otimal strategy 3 Theuniqueness of the Nash equilibrium is roved in the same way as in [4] For this reason, we only summarize the main stes of the roof Considerany Nash equilibrium, and let U 1,,U n describethestrategies of thevarious sellers, as in (34) Let U() = i U i() The same arguments used in Lemma 81 in [4] yield: 10

11 (i) Thema U() is Lischitzcontinuous on thehalf-oeninterval [ 0, [ andconstant for >, ossibly with a jum at (ii) For all excet at most one index i {1,,n}, the function U i is globally Lischitz continuous (iii) There exists a minimum asking rice A and a constant δ 0 > 0 such that U() = 0 for all A, U () δ 0 for ae [ A, ] (314) Next, by Rademacher s theorem every function U i is ae differentiable on the interval [ A, ] For any, consider the subset of indices I() = {i; U i () > 0} and call N() = #I() the cardinality of this set This function is ae well defined, and Lebesgue measurable From the necessary conditions it follows that the function U satisfies the ODE U () = N() N() 1 ψ ( ) U() ) ψ (U() ( ) 1 + h (), [ A, ] (315) 0 h() As in the roof of Theorem 82 in [4] one can show that, for each i {1,,n}, the set of rices wherethe i-th layer offers assets for sale is an interval of the form [ A, i+1 ] Moreover, A = 1 2 n = n+1 = As a consequence, the functions U i, i = 1,,n, are uniquely determined by the ODEs (35) and (38), together with the equations (36), (37), and (39) This achieves the roof of uniqueness For all details, we refer to [4] Examle 1 In the secial case where ψ,h are given by (14) and (17), for any 0 < κ 1 κ n the Nash equilibrium solution is determined by the equations U i () = 0, if 1, U i () = κ i, if i+1, U i G() () = n k, if k < < k+1, k i, (316) with G() as in (29) Moreover, the oints 1 n are inductively determined by the identities n = = 0 + max 2 The case n = 4 is illustrated in Fig 4, j+1 j G() n j d = κ j κ j 1 (j = 1,,n 1) 11

12 G() * = max Figure 4: The otimal strategies u 1,,u 4 in a Nash equilibrium, assuming ψ(s) = e λs Areas of the regions 1,2,3,4 are roortional to the amount of asset ut on sale by Players 1,2,3,4 at the given rices In addition, Player 4 uts an additional amount κ 4 κ 3 of asset for sale all at the rice Notice that, for every i, an otimality condition holds: u i () > 0 = j i u j() = G() 32 Infinitely many cometing sellers We consider here the limiting case where the number of sellers aroaches infinity while the total amount of asset on sale remains bounded More recisely, for each n 1, consider amounts 0 < κ (n) 1 κ (n) 2 κ (n) n, and assume that lim n n j=1 κ (n) j = κ, lim su κ (n) n j = 0 (317) 1 j n Let U (n) () be the total amount of asset ut on sale at rice (by all layers combined) in a Nash equilibrium If the limits (317) hold, then we will show that as n one has the uniform convergence U (n) () U () (318) The function U can be characterized as the unique Lischitz continuous function such that { U() = 0 if A, U() = κ if (319), U () = ψ ( ) U() ) ψ (U() ( ) 1 + h () 0 h() for ae [ A, ], (320) for a suitable value A [ 0, ] Notice that the above equations imly that the ma [rofit from the sale of an asset at rice ] [robability of selling the asset] [ ] (321) = [ 0 ] ψ(u ()) h() 12

13 is constant over the interval [ A, ] We can thus regard U ( ) as describing the rice distribution in a Nash equilibrium with infinitely many small layers Theorem 5 Under the assumtions (A1)-(A2), consider a sequence of Nash equilibria, where as n the limits (317) hold Then the corresonding rice distributions U (n) converge uniformly to the function U, defined as the solution to (319)-(320) Proof 1 For each n, the function U (n) is constructed according to (35) (37) Therefore U (n) () = n j=1 κ (n) j κ = U () for Moreover, U (n) can have a jum at However, by the second assumtion in (317), the size of this jum goes to zero Indeed, U (n) ( ) = n j=1 κ (n) j (κ (n) n κ (n) n 1 ) κ Comaring (35) with (320), we observe that lim n U(n) ( ) = U ( ), d d U(n) () d d U (), for every < where both U (n) and U are strictly ositive This already imlies lim suu (n) () U () [ 0, ] n 2 Given ε > 0, we can find integers m,n large enough so that Call V ε the solution to 1 m+1 n m m 1+ε, κ (n) j κ ε for all n > N (322) V () = (1+ε) ψ ( We claim that ) V() ) ψ (V() Indeed, recalling (35) (37), let j=1 ( ) 1 + h (), V( ) = κ ε (323) 0 h() V ε () U (n) () for all n N, [ 0, ] (324) (n) 1 (n) 2 (n) n = the oints determined in the construction of U (n) By the second inequality in (322), for every n > N we have V ε ( (n) n m) V ε ( ) = κ ε U (n) ( (n) n m) 13

14 Moreover, the first inequality in (322) imlies d d U(n) () d d V ε() Hence (see Fig 5, right) V ε () U (n) () for every We now observe that, as ε 0, the function max{v ε (), 0} converges to U uniformly on [ 0, ] This imlies comleting the roof lim inf n U(n) () U () [ 0, ], 4 Price reduction resulting from the cometition In this section we rove some inequalities, showing how the average rice asked for the asset decreases as a result of the cometition between sellers To fix the ideas, consider n sellers, offering the amounts κ 1 κ n of asset for sale Let β φ i (β) be the corresonding Nash equilibrium ricing strategies Calling κ = κ 1 + +κ n the total amount of asset for sale, the average asked rice is A(κ 1,,κ n ) = 1 κ n i=1 κi 0 φ i β)dβ (41) In the secial case where κ 1 = = κ n = κ/n, we write ( κ A n (κ) = A n n),, κ (42) Theorem 5 Assume that the functions ψ, h satisfy (A1)-(A2) For the Nash equilibrium strategies, the following holds (i) For any given κ 1,,κ n and κ = iκ i, one has A n (κ) A(κ 1,,κ n ) (43) (ii) For any m > n one has A m (κ) A n (κ) (44) (iii) In the case where ψ(s) = e λs, for any n 2 one has κ < κ = A n (κ ) < A n (κ), (45) lim A n(κ) = 0 (46) κ 14

15 κ κ κ ε U m U # (n) U U n U V ε * * Figure 5: Left: Comaring the rice distributions U U n U m, in the roof of (43)-(44) Right: comaring the distribution functions V ε U (n), in the roof of Theorem 5 Proof 1 Let U() bethe total amount of asset ut on sale at rice, jointly by all layers Observe that, in a Nash equilibrium, this rice alway ranges within the interval [ 0, ] Hence U( 0 ) = 0, U( ) = κ, and the second inequality in (43) is obvious The average rice is comuted by the Stieltjes integral A = 0 du() = κ 0 U()d (47) In the general case of n layers, the function U is Lischitz continuous for [ 0, [, ossibly with a jum at = Indeed, U( ) = κ (κ n κ n 1 ) (48) For <, according to (312) the function U satisfies the ODE ( ) U () = n j()+1 ψ U() ( ) 1 ) + n j() ψ (U() h () 0 h() (49) for some integer-valued function j() {1,,n 1} On the other hand, call U n () the total amount ut on sale at rice in the case of n equal layers, ie with κ 1 = = κ n = κ/n In this case, the function U n is globally Lischitz continuous and rovides a solution to the Cauchy roblem U n () = n n 1 ψ ( ) U() ) ψ (U() ( ) 1 + h (), U n ( ) = κ (410) 0 h() Comaring (410) with (48)-(49), we conclude that U() U n () for all [ 0, ], see Fig 5, left By (47) this imlies the first inequality in (43) 2 If m > n, then the corresonding solutions of the Cauchy roblem (410) satisfy U m () U n () [ 0, ] 15

16 See Fig 5, left By (47) this imlies A m (κ) A n (κ) 3 To rove (45), assume ψ(s) = e λs and let κ > κ Choose 0 < A < A < so that A n G()d = κ, n 1 A n n 1 G()d = κ with G() given at (29) Then A n (κ ) = 1 ( A ) n κ + n 1 G()d A A [ = κ κ 1 κ κ κ A A κ κ κ A + κ κ A n(κ) < A n (κ) ] n n 1 G()d + κ [ ] 1 κ n κ A n 1 G()d 4 Finally, to rove (46), fix ε > 0 and define Then for any κ > κ ε we have A n (κ) = 1 κ A = κ κ ε κ n n 1 G()d [ 1 κ κ ε κ ε = 0 +ε A 0 +ε n n 1 G()d ] n n 1 G()d + κ [ ] ε 1 κ n κ ε 0 +ε n 1 G()d κ κ ε ( 0 +ε)+ κ ε κ κ, (411) for some A < 0 +ε, deending on κ As κ, the right hand side of (411) converges to 0 +ε Therefore, Since ε > 0 was arbitrary, this roves (46) 0 limsua n (κ) 0 +ε κ 5 Dynamic stability of the Nash equilibrium In this section we assume that each agent can gradually modify his own ricing strategy, in rely to the strategies adoted by all the other layers Our main interest is in the dynamic stability of the Nash equilibrium To simlify the analysis, we shall henceforth assume that the the random buying order X has exonential distribution, so that ψ(s) = e λs 16

17 51 Infinitely many small layers We firstconsider a model with avery large numberof small layers, each with a small quantity of assets Let U() = u(x)dx (51) 0 be the total amount of assets offered for sale at rice < Then the exected ayoff achieved by offering a unit amount of asset at rice is J(,U) = ( 0 )ψ(u())h() = e λu() ( 0 )h() If the ma J(,U) is not constant, the agent ricing his asset at may increase his ayoff by varying the rice according to ṗ = d ] d J(,U()) = e λu()[ h() λ( 0 )U ()h()+( 0 )h () (52) From the above, we obtain a conservation law for the rice density u() = U (), with flux Φ = ṗ u, namely { [ ] } u t + e λu() h() λ( 0 )h()u+( 0 )h () u = 0 (53) The characteristic seed is e λu()[ ] h()+( 0 )h () 2λ( 0 )h()u Notice that (53) is a conservation law with strictly concave flux Uward jums rovide admissible shocks, while downward jums are not admissible Steady states are those where the flux vanishes identically, so that u() {0, G()} for ae, where G is the function defined at (29) Let κ = u()d be the total amount of assets offered for sale The admissibility conditions imly that a unique steady state exists, namely u () = { G() if [A, ], 0 if / [ A, ] (54) Here the oints, A [ 0, max ] are uniquely determined by the identities G( ) = 0, A G()d = κ (55) If u is an entroy-admissible solution of (53), then the integrated function U(t,) = 0 u(t,x)dx (56) rovides a viscosity solution [2] to the evolution equation [ ] U t +e λu() h() λ( 0 )h()u +( 0 )h () U = 0 (57) 17

18 κ U # () W(t,) η(t) U(t,) V(t,) ξ(t) 0 * A 0 max Figure 6: In the roof of Theorem 6, the uer solution W and the lower solution V are obtained by shifting the grah of U to the left and to the right, resectively Theorem 6 Let h satisfy the assumtions (A2) Let u(0,) = ū() be an initial data suorted inside the oen interval ] 0, max [ Then, as t +, the solution of (57) converges in L 1 to the function u defined by (54)-(55) Proof 1 Consider the integrated function U in (56) By assumtion, there exist κ,δ > 0 such that at time t = 0 the initial data U() = 0 ū(x)dx satisfy { U() = 0 if 0 +δ, U() = κ if max δ (58) We shall construct a subsolution V and a suersolution W of (57) with V(t,) U(t,) W(t,), (59) lim V(t,) = lim W(t,) = t + t + U () (510) A comarison argument will thus yield the convergence U(t, ) U as t 2 As shown in Fig 6, the lower and uer solutions V,W will have the form V(t,) = U ( ξ(t)), W(t,) = U (+η(t)), (511) for suitable functions ξ,η As in Theorem 5, here U is the unique Lischitz continuous function such that h() λ( 0 )h()u ()+( 0)h () = 0 A < <, (512) { U() = 0 if A, U() = κ if, (513) 18

19 for a suitable value A [ 0, ] We recall that (512) is equivalent to By choosing we achieve U () = d d U () = G() = 1 ( ) 1 + h () λ 0 h() ξ(0) = max 0, η(0) = 0, A < < (514) V(0,) = 0 U 0 () κ = W(0,) for all [ 0, max ] (515) 3 For any ξ > 0, using (512) (514) we obtain I(ξ) = inf 0 +δ<< max δ [ h() ( 0 )h()g(+ξ)+( 0 )h () [ ] = inf ( 0)h() G() G(+ξ) 0 +δ<< max δ > 0 ] (516) If we now choose the ma t ξ(t) satisfying ξ(t) = e λκ I(ξ(t)), (517) then the function V in (511) will be a lower solution of (57) on the domain Ω = [0, [ [ 0 +δ, max δ] (518) Observing that V U on the arabolic boundary of Ω, ie on the set {0} [ 0 +δ, max δ] [0, [ { 0 +δ} [0, [ { max δ}, (519) we conclude that V(t,x) U(t,x) for all (t,x) Ω 4 Similarly, for any η > 0, using (512) (514) we obtain J(ξ) = su 0 +δ<< max δ [ h() ( 0 )h()g( η)+( 0 )h () [ ] = su 0 +δ<< max δ ( 0 )h() G() G( η) < 0 ] (520) If we now choose the ma t η(t) satisfying η(t) = e λκ J(η(t)), (521) then the function W in (511) will be a lower solution, restricted to the domain Ω in (518) Observing that U W on the arabolic boundary (519) of Ω, we conclude that V(t,x) U(t,x) for all (t,x) Ω 5 Since ξ > 0 and η > 0 imly I(ξ) > 0 and J(η) > 0, the solutions to (517), and (521) satisfy ξ(t) 0, η(t) 0 as t 19

20 Hence V(t, ) and W(t, ) both aroach U as t Since the inequalities (59) hold for every time t 0 and every [ 0 + δ, max δ], we obtain the uniform convergence U(t, ) U 6 The L 1 convergence u(t, ) u L 1 0 (522) is now roved by means of Oleinik s estimates Indeed, recalling that the flux function in (53) is a strictly concave function of u, we have an estimate of the form u(t, 2 ) u(t, 1 ) C( 2 1 ) for all 0 δ < 1 < 2 < max +δ, t 1 (523) In articular, for t 1 the total variation of u(t, ) is uniformly bounded As a consequence, the uniform convergence U(t, ) U imlies the L 1 convergence (522) 52 Two or more large layers We consider here the case of n layers, with amounts 0 < κ 1 κ n of asset to ut on sale Let U i () = u i (x)dx (524) 0 be the total amount of asset ut on sale at rice < by Player i, and let U() = n i=1 U i () At the initial time t = 0, we assume that the suorts of u 1,,u n are all contained in a comact subset of ] 0, max [ Consider a situation where each layer gradually modifies the rices asked for his assets, in rely to the strategies adoted by all other layers This can be modeled by the system of conservation laws u i,t + e λu()[ h() λ( 0 )h() j i ] u j ()+( 0 )h () u i () = 0, (525) with i = 1,,n We think of (525) as a system of n gradient flows, in connection with the functionals J i in (32) describing the exected ayoffs of the various layers The next examle shows that, for general initial data, the solution may not converge to a Nash equilibrium Examle 2 Assume that the initial data u i (0,) = ū i (), i = 1,,n, (526) are smooth and have disjoint suorts (as in Fig 7, left) Then, as long as the suorts of the functions u i (t, ) remain disjoint, the system (525) is equivalent to { u i,t + e λu()[ ] } h()+( 0 )h () u i () = 0 (527) In this case, all densities u i satisfy the same linear transort equation Hence, for every t > 0, the solutions u i (t, ) remain smooth and with disjoint suorts We now observe that every solution to the ODE ṗ = h()+( 0 )h (), (0) ] 0, max [, (528) 20

21 u 1 u2 u 2 u 1 * 0 max 0 * max Figure 7: Left: two ricing strategies u 1,u 2 with disjoint suort For this initial data, as t + the solution to the system of conservation laws (529) will aroach two oint masses concentrated at Right: two ricing strategies satisfying the ointwise inequality u 1 u 2 For this initial data, the solution to (529) converges to the unique Nash equilibrium converges to as t + Therefore, the solutions u i to the conservation laws (527) converge to Dirac masses of sizes κ i, all located at the oint When n = 1, this limit yields the otimal strategy for the single layer However, for n 2, this limit is not a Nash equilibrium By the revious examle it follows that the Nash equilibrium for n layers is not dynamically stable wrt small erturbations, in the toology of weak convergence of measures Indeed, one can always erturb an equilibrium distribution in such a way that the densities u 1,,u n have disjoint suorts For such an initial data, the solution of (525) will not converge to the Nash equilibrium Next, we rove a ositive result, in the case of two layers and with an additional assumtion on the initial data ū 1 ū 2 When n = 2, the system (525) takes the form { u 1,t + e λu()[ ] } h() λ( 0 )u 2 ()h()+( 0 )h () u 1 () = 0, { u 2,t + e λu()[ ] } (529) h() λ( 0 )u 1 ()h()+( 0 )h () u 2 () = 0 Theorem 7 Assume that the initial data are suorted on a comact subset of ] 0, max [ and satisfy ū 1 ū 2 Then, as t +, the solution of (529) converges to the Nash equilibrium, in the toology of weak convergence of measures Proof 1 Consider the additional variables z() = u 2 () u 1 (), Z() = 0 z(x)dx Subtracting the first equation in (529) from the second, one obtains the linear conservation law { z t + e λu()[ ] } h()+( 0 )h () z = 0 (530) 21

22 Since every solution to the ODE (528) aroaches as t +, for every ε > 0 we can find a time T ε > 0 large enough such that z(t,) = u 2 (t,) u 1 (t,) = 0 for all / [ ε, +ε], t > T ε (531) For t T ε this imlies Z(t,) = { 0 if < ε, κ 2 κ 1 if > +ε (532) We recall that, by assumtion, z(0,) 0 Hence u 2 (t,) u 1 (t,) = z(t,) 0 for all t 0, [ 0, max ] (533) 2 Let U 1,U 2 be as in (524) By (533) it follows that U 1, U 2, for all t, Therefore U 1 satisfies [ ] U 1,t +e λu h() λ( 0 )U 1, h()+( 0 )h () U 1, 0 (534) Define U 1 to be the unique Lischitz function such that U () = G() if A < <, U() = 0 if A, U() = κ 1 if, for some minimum asking rice A As in ste 3 of the roof of Theorem 6, we can construct a subsolution V having the form V(t,) = U 1 ( ξ(t)), with ξ(t) decreasing to zero as t + hence lim inf U 1(t,) U t + 1 () (535) Recalling that U 1, ( ) = G( ) = 0, for every small ε > 0 we can find T ε such that U 1 (t,) κ 1 ε for all t T ε, ε (536) 3 According to (531), for all t > T ε we have U 2, (t,) = U 1, (t,) for all < ε Terefore, restricted to the domain [T ε, [ [ 0, ε], the function U 1 rovides a solution to [ ] U 1,t +e λu() h() λ( 0 )h()u 1, +( 0 )h () U 1, = 0 (537) As in ste 4 of the roof of Theorem 6, we can construct a suersolution W of (537) having the form W(t,) = U 1 (+η(t)), 22

23 where η satisfies η(t) = Letting t we thus obtain { e λ(κ 1 +κ 2 ) J(η(t)) if η(t) > ε, 0 if η(t) ε (538) lim su t Since ε > 0 was arbitrary, we conclude U 1 (t,) limsu t W(t,) = U 1 (+ε) lim suu 1 (t,) U 1 () (539) t 4 Together, (535) and (539) imly the ointwise convergence U 1 (t,) U 1 () as t (540) Finally, from the roerties (532) of Z = U 2 U 1 we deduce lim t U 2 (t,) = lim t U 1 (t,) = U 1 () if <, lim t U 2 (t,) = κ 2 if > (541) By (540)-(541) the distribution functions U 1,U 2 converge ointwise ae to the Nash equilibrium distributions This comletes the roof Acknowledgments The research of the first author was artially suorted by NSF, with grant DMS : Problems of Nonlinear Control The second author worked on this roject as art of the 2014 Mathematics Advanced Study Semesters (MASS) rogram at Penn State References [1] A Alfonsi, A Fruth and A Schied, Otimal execution strategies in limit order books with general shae functions, Quantitative Finance 10 (2010), [2] M Bardi and I Cauzzo Dolcetta, Otimal Control and Viscosity Solutions of Hamilton- Jacobi-Bellman Equations, Birkhäuser, 1997 [3] A Bressan, Noncooerative differential games Milan J of Mathematics, 79 (2011), [4] A Bressan and G Facchi, A bidding game in a continuum limit order book, SIAM J Control Otim 51 (2013), [5] A Bressan and G Facchi, Discrete bidding strategies for a random incoming order, SIAM J Financial Math 5 (2014), [6] A Bressan and D Wei, A bidding game with heterogeneous layers, J Otim Theory Al 163 (2014),

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