Optimal Pricing Strategies in a Continuum Limit Order Book

Transcription

1 Optimal Pricing Strategies in a Continuum Limit Order Book Alberto Bressan and Giancarlo Facchi Department of Mathematics, Penn State University University Park, Pa 1682, USA s: bressan@mathpsuedu, facchi@mathpsuedu October 24, 212 Abstract The paper is concerned with a continuum model of the limit order book, viewed as a noncooperative game for n players An external buyer asks for a random amount X > of a given asset This amount will be bought at the lowest available price, as long as the price does not exceed a given upper bound P One or more sellers offer various quantities of the asset at different prices, competing to fulfill the incoming order, whose size is not known a priori The first part of the paper deals with solutions to the measure-valued optimal pricing problem for a single player, proving an existence result and deriving necessary and sufficient conditions for optimality The second part is devoted to Nash equilibria For a general class of random variables X and an arbitrary number of players, the existence and uniqueness of the corresponding Nash equilibrium is proved, explicitly determining the pricing strategy of each player For a different class of random variables, it is shown that no Nash equilibrium can exist The paper also describes the asymptotic limit as the total number of players approaches infinity, and provides formulas for the price impact produced by an incoming order Keywords: measure-valued optimization, optimality conditions, optimal pricing strategy, bidding game, Nash equilibrium, limit order book, price impact 1 Introduction This paper is concerned with a continuum model of the limit order book in a stock market, viewed as a noncooperative game for n players Our main goal is to study the existence and uniqueness of a Nash equilibrium, determining the optimal bidding strategies of the various agents who submit limit orders We consider a one-sided limit order book In our basic setting, we assume that an external buyer asks for a random amount of X > of shares of a certain asset This external agent 1

2 will buy the amount X at the lowest available price, as long as this price does not exceed a given upper bound P One or more sellers offer various quantities of this asset at different prices, competing to fulfill the incoming order, whose size is not known a priori Having observed the prices asked by his competitors, each seller must determine an optimal strategy, maximizing his expected payoff Of course, when other sellers are present, asking a higher price for a stock reduces the probability of selling it In our model we assume that the i-th player owns an amount κ i of stock He can put all of it on sale at a given price, or offer different portions at different prices In general, his strategy will thus be described by a measure µ i on IR +, where µ i [, p] denotes the total amount of shares put on sale by the i-th player at a price p In practice, it is clear that prices can take only a discrete set of values However, by studying a continuum model where strategies are described by Radon measures one obtains clear-cut results on existence or non-existence of Nash equilibria, and clean, explicit solution formulas In general, it turns out that the Nash equilibrium consists of measures which are absolutely continuous wrt Lebesgue measure Several recent papers [9], [12], [5] deal with the modeling of the limit order book from the point of view of the agents who submit the limit orders These models are intrinsically discrete in the price variable: limit orders can be submitted at prices p 1,, p N } and to each price there corresponds a queue of limit orders, which are to be executed according to a first-infirst-out schedule The shape of the limit order book is determined by the prices at which the various agents decide to submit their limit orders On the other hand, in [8], [1], [1] the prices are continuous and the shape of the limit order book is described by a density An important achievement of these models is that, when the shape of the limit order book is given, this determines a corresponding price impact function The price impact function describes how the execution of a market order affects the underlying asset prices, ie it describes how the bid and ask prices change after the execution of a market order Clearly, this is a quantity of key importance in the modeling of financial markets and an understanding of the price impact function allows us to gain insight in the mechanism of price formation In [8] the limit order book has a block shape and this gives rise to a linear price impact In [1] and [1] the limit order book density has a general shape which is described by a measure These papers, given the order book shape as input, mainly consider the problem of optimal execution of trades by means of market orders In our model, prices are allowed to vary in a continuum of values but the shape of the limit order book is not given a priori Indeed, we prove that this shape can be endogenously determined as the unique Nash equilibrium, resulting from the optimal pricing strategies implemented by the selling agents The paper is organized as follows In Section 2 we consider the optimization problem for a single agent, who observes the limit orders submitted by his competitors and wishes to optimally price the sale of his own assets We also introduce a fundamental distinction between two classes of random variables: Type A and Type B These two types yield completely different results when Nash equilibria are studied Under general assumptions, the existence of an optimal pricing strategy is proved in Section 3 Necessary conditions for optimality are derived in Sections 4 and 5 For random variables of 2

3 Type B, these imply that the optimal strategy always consists in putting all the assets for sale at the same price In Section 6 we prove some sufficient conditions for optimality Sections 7 and 8 are devoted to the study of Nash equilibria We consider n players, putting on sale quantities κ 1,, κ n of the same asset We say that an n-tuple of pricing strategies µ 1,, µ n provides a Nash equilibrium if each µ i provides an optimal strategy for the i-th agent, in reply to the bidding strategies of all the other agents When the random buying order X is a random variable of type A, we prove that this noncooperative game admits a unique Nash equilibrium, which is explicitly determined On the other hand, if the random variable X is of type B, we show that no Nash equilibrium can exist In Section 9 we consider an asymptotic limit, where the total number of sellers approaches infinity, while the amount of asset put on sale by each agent approaches zero In this case, the limit order book approaches a well defined shape, determined by the probability distribution of the random variable X From this model, one can deduce the price impact of an incoming buying order of size X Some explicit examples are provided in Section 1 In addition to the classical paper [7], for an introduction to non-cooperative games and Nash equilibria we refer to [3, 6, 13, 14] 2 The optimization problem for a single player A general optimization problem for one agent can be formulated as follows non-negative random variable, with distribution function Let X be a Throughout the following we shall assume P robx s} = 1 ψs 21 A1 The map s ψs is continuously differentiable and satisfies ψ = 1, ψ+ =, ψ s < for all s > 22 We shall consider two main classes of random variables, depending on the decay properties of the function ψ Definition 1 We say that a probability distribution is of type A if ln ψs for all s > 23 of type B if ln ψs < for all s > 24 For example, the probability distributions determined by ψ 1 s = e λs λ >, 25 ψ 2 s = s α α >, 26 3

4 are of type A, while ψ 3 s = e s2 27 yields a probability distribution of type B Roughly speaking, a probability distribution is of type A if its tail decays not faster than a negative exponential Of course, one can consider more general probability distributions, where ln ψ changes sign For such random variables, the analysis will likely be more difficult Let Φ : [, P ] IR + be a non-negative, nondecreasing function For every p, we think of Φ p as the total amount of stock offered for sale at a price p by the other agents Consider an additional seller entering the market, owning an amount κ of stock Definition 2 A pricing strategy for the new player is a nondecreasing map φ : [, κ] [p, P ] Using the Lagrangian variable β [, κ] to label a particular share in possession of the new agent, by φβ we thus denote the price at which this particular share is put on sale The total amount of shares that the new agent offers for sale at price p is thus computed by µ 1 [, p] = meas β [, κ] ; φβ p} 28 This is the push-forward of the Lebesgue measure on [, κ] wrt the map φ Next, assume that the incoming order has size X The total amount of stock sold by the new agent is } βx = sup β [, κ] ; β + Φ φβ X, 29 yielding the payoff βx φβ p dβ Here p > is the value that the new player attaches to a unit amount of stock For example, it could be the mean bid-ask price The optimization problem for the new seller can thus be formulated as Maximize: [ βx ] Jφ = E φβ p dβ 21 among all pricing strategies φ : [, κ] [, P ] Here E[ ] denotes the expectation wrt the probability distribution of the random variable X Observe that, by 21 and 29, we have the equivalent representation Jφ = κ φβ p ψ β + Φ φβ dβ 211 Remark 1 If Φ has a jump at a point ξ, this means that a positive amount of stock is offered for sale by the other agents at the price ξ Two main cases can arise 4

5 CASE 1: Φ is left continuous, ie Φ ξ = Φ ξ This means that the new agent has selling priority If he also puts on sale a positive amount of stock at the same price ξ, his stock will be the first to be sold CASE 2: Φ is right continuous, ie Φ ξ = Φ ξ+ This means that the new agent does not have selling priority If he also puts on sale a positive amount of stock at the same price ξ, his stock will be the last to be sold Notice that in Case 1 the function Φ is lower semicontinuous This property will play a key role in the proof of existence of an optimal strategy 3 Existence of an optimal strategy Our first result shows the existence of an optimal strategy for the new agent, assuming that he has selling priority Theorem 31 existence Let X be a random variable satisfying the assumptions A1 Let Φ : [, P ] IR + be a left-continuous, nondecreasing function, and let κ > Then there exists an optimal pricing strategy φ : [, κ] [p, P ] for the new agent, maximizing the expected payoff 21 Proof Let φ ν ν 1 be a maximizing sequence of pricing strategies Since all functions φ ν are non-decreasing, using Helly s compactness theorem see for example [11], p 372, by extracting a subsequence and relabeling we can achieve the pointwise convergence We claim that the strategy φ is optimal φ ν β φ β for all β [, κ] 31 Indeed, since Φ is lower semicontinuous and ψ is strictly decreasing, the composite map s ψφ s is upper semicontinuous Therefore, for every β [, κ] we have lim sup ν ψ Φ φ ν β ψ Φ φ β In turn, this yields sup Jφ = φ κ κ lim Jφ ν = ν lim ν κ φ ν β p ψ β + Φ φ ν β dβ } lim sup ν φ ν β p ψ β + Φ φ ν β dβ φ β p ψ β + Φ φ β dβ = Jφ 5

6 Example 1 If the new player does not have priority, an optimal strategy may fail to exist For example, assume that the other sellers offer a total amount of stock κ, all at the same price P This situation is described by the right continuous function Φ p = if p < P, κ if p = P 32 Assume that the new player has an amount κ of stock to put on sale For each ν 1, consider the pricing strategy φ ν β P ν 1 Then φ ν ν 1 is a maximizing sequence Writing a b = mina, b}, a + = maxa, }, the expected payoffs are Jφ ν = P ν 1 p E[X κ] However, the expected payoff P p E[X κ] could be achieved only if the new agent puts all his stock for sale at the maximum price P and has selling priority over the other agents that would correspond to Φ being left continuous However, if Φ is the function in 32, the new agent does not have priority With the strategy φ β P he only achieves [ ] Jφ = P p E X κ + κ 4 Necessary conditions In this section we seek necessary conditions for the optimality of a pricing strategy φ for the new agent For this purpose given a non-negative, nondecreasing function Φ : [, P ] IR + as in 29, we introduce the functions G β p [ = ψ β + Φ p p p ψ 1 β + Φ p] 41 For a < b κ we shall also consider the integrated function G [a,b] p = b a [ ψ β + Φ p dβ p p b a ψ 1 β + Φ p dβ] 42 Remark 2 If the random variable X is of type A, then for every p the map β G β p is non-decreasing On the other hand, if X is of type B, then the maps β G β p are strictly decreasing In this section we do not make any assumption on the left or right continuity of Φ It will thus be convenient to define the left continuous function Φ p = Φ p 43 In other words, Φ is the unique left continuous function that coincides with Φ everywhere with the possible exception of countably many points of jump Call J φ the expected payoff achieved by a pricing strategy φ : [, κ] [, P ] when Φ is replaced by Φ 6

7 Lemma 41 In the above setting, for every Φ : [, P ] IR + and κ > one has sup Jφ = φ max φ J φ 44 Proof By Theorem 31, the maximum expected payoff on the right hand side of 44 is attained Namely, there exists a pricing strategy φ such that J φ = max J φ φ Consider the strategies The corresponding payoffs satisfy κ Jφ n = φ β 1 n p Therefore sup Jφ φ κ κ φ n β = φ β 1 n 45 ψ β + Φ φ β 1 dβ n φ β p ψ β + Φ φ β 1 dβ κ n n φ β p ψ β + Φ φ β dβ κ n = J φ κ n sup Jφ n n sup J φ κ } n n = J φ = sup J φ φ 46 The converse inequality is clear Indeed, Φ p Φ p for every p Hence J φ Jφ for every admissible strategy φ : [, κ] [, P ] Given a nondecreasing map φ : [, κ] [, P ] one can isolate countably many disjoint intervals = [aj, b j ] [, κ] such that φ is constant on each S j and strictly increasing elsewhere S j Namely, defining one has S = j S j, 47 β 1 / S, β 1 < β 2 = φβ 1 < φβ 2 48 In connection with the measure µ 1 introduced at 28, we observe that the atomic part of µ 1 is the measure µ a 1 concentrated on the points φa j = φb j Indeed, µ a 1 φa j } = b j a j > Theorem 42 necessary conditions for optimality Let the random variable X satisfy the assumptions A1 and let Φ : [, P ] IR + be a nondecreasing map If φ : [, κ] [p, P ] is an optimal pricing strategy, then the following holds i For almost every β [, κ] \ S, setting x = φβ one has see fig 1 lim sup ε Φ x + ε Φ x ε G β x lim inf ε + Φ x + ε Φ x ε 49 7

8 ii If β [a i, b i ] S, with φβ = x < P for all β [a i, b i ], then lim sup ε Φ x + ε Φ x ε G a i x lim sup ε G b i x lim inf ε + G [a i,b i ] x Φ x + ε Φ x ε Φ x + ε Φ x ε lim inf ε +,, Φ x + ε Φ x ε Φ p λ L x φβ φβ +ε p Figure 1: Deriving the necessary conditions for optimality The solid line has slope λ and touches the graph of Φ at the point φβ + ε Proof 1 Assume that the second inequality in 49 does not hold at some β [, κ] \ S Setting x = φβ, this clearly implies L = lim inf ε + Φ x + ε Φ x ε < 412 Hence the nondecreasing function Φ is right continuous at the point x = φβ By continuity we can thus find λ and δ > such that L < λ < G β p for all β [β, β + δ], p [x, x + δ], 413 ψζ+p p ψ ζλ > for all p [x, x +δ], ζ [β +Φ x, β +Φ x +δλ] We claim that there exists ε ], δ] such that the following conditions hold see Fig 1 Φ p Φ x + ε + λ p x ε for all p [x, x + ε], 415 β 1 = supβ ; φβ < x + ε} < β + δ 416 Indeed, by definition of lim-inf there exists ε 2 ], δ] such that Φ x + ε 2 < Φ x + λε

9 Consider the modified function Φ p = Φ p if p / ]x, x + ε 2 ], Φ p if p ]x, x + ε 2 ] By lower semicontinuity, the function η Φ x + η λη 418 attains a strictly negative minimum on the interval [, ε 2 ] If } ε argmin Φ x + η λη η [,ε 2 ] 419 is a point where this minimum is attained, then 415 holds 3 Let φ ε+ be the perturbed strategy defined by φ ε+ β = φβ if β / [β, β 1 ], x + ε if β [β, β 1 ] 42 Since ψ <, using 415 and then , one obtains J φ ε+ Jφ = β1 β β1 β [ ] x + ε p ψ β + Φ x + ε φβ p ψ β + Φ φβ dβ [ x + ε p ψ β + Φ φβ + λx + ε φβ φβ p ψ β + Φ φβ ] dβ = = β1 x +ε β φβ β1 x +ε β φβ d [ ] p p ψ β + Φ dp φβ + λp φβ dp dβ [ ψ β + Φ φβ + λp φβ +p p ψ ] β + Φ φβ + λp φβ λ dp dβ δ >, for some positive constant δ Using Lemma 41 we conclude 421 Jφ = reaching a contradiction sup Jϕ = ϕ sup J ϕ J φ ε+ Jφ + δ, ϕ 9

10 The first inequality in 49 can be proved by an entirely similar argument 4 The two statements will be deduced as consequences of the more general necessary conditions G [ξ,b i] x lim inf ε + G [ai,ξ] x lim sup ε Φ x + ε Φ x, for all ξ [a i, b i ], ε Φ x + ε Φ x, for all ξ [a i, b i ] ε 422 Indeed, the two inequalities in 41 are obtained by observing that lim ξ b i G[ξ,b i] x = G b i x, lim ξ a i + G[a i,ξ] x = G a i x Moreover, 411 follows from the two inequalities in 422, choosing ξ = b i and ξ = a i, respectively 5 It now remains to prove 422 Assume that the first inequality in 422 fails at β [a i, b i ], and call x = φβ Then by continuity we can find λ and δ > such that lim inf ε + Φ x + ε Φ x ε < λ < G [ξ,b i] p for all p [x, x + δ], which implies that there exists c > such that bi ξ bi ψσ dσ + λp p ψ σ dσ c >, for all p [x, x + δ] 423 ξ Choose ε ], δ] such that the following conditions hold Φ p Φ x + ε + λ p x ε for all p [x, x + ε], 424 β 1 = supβ ; φβ < x + ε} < bi + δε 425 where δε, as ε Let φ ξ,ε+ be the perturbed strategy defined by φ ξ,ε+ β = x + ε if β φ 1 [x, x + ε] [ξ,, φβ otherwise 426 1

11 One obtains J φ ξ,ε+ Jφ = = β1 ξ [ ] x + ε p ψ β + Φ x + ε φβ p ψ β + Φ φβ dβ β1 x +ε ξ x +ε x φβ bi + ξ d [ ] p p ψ β + Φ dp φβ + λp φβ dp dβ φ 1 p b i [ ψ β + Φ φβ + λp φβ + ] +p p ψ β + Φ λ φβ + λp φβ dp dβ c ε + εδε = c ε + oε > 427 for ε > sufficiently small Notice that the last inequality follows from 423 and 425 Using Lemma 41 we reach a contradiction Corollary 43 Assume that Φ is piecewise C 1, and let φ be an optimal strategy Then for almost every β [, κ] \ S one has d dp Φ φβ = G β φβ 428 Indeed, for ae β [, κ] \ S one has lim sup ε Φ φβ + ε Φ φβ ε Hence 428 follows from 49 = lim inf ε + Φ φβ + ε Φ φβ ε = d dp Φ φβ Example 2 Assume that the random variable X has exponential distribution, so that ψs = P robx < s} = e λs 429 Let Φ be continuous, piecewise C 1 If φ : [, κ] [, P ] is an optimal pricing strategy, then the necessary conditions imply that the range of φ should by contained in the set } p ]p, P [ ; Φ 1 p = P } λp p 5 Atomic optimal strategies Our next goal is to prove that, if the random variable X is of type B, then any optimal pricing strategy for the new agent must be a constant Namely, all stock should be offered for sale at the same price A preliminary lemma will be needed 11

12 Lemma 51 Let X be a random variable of type B Assume that φ : [, κ] [, P ] is a pricing strategy taking exactly two values, say p 1 and p 2 Then one of the two constant strategies φ 1 β p 1 or φ 2 β p 2 yields an expected payoff strictly larger than φ Proof 1 Fix p 1 < p 2 [p, P ] For θ [, κ] consider the pricing strategy φ θ β = p1 if β [, θ], if β ]θ, κ] p 2 51 The corresponding payoff is θ κ Jφ θ = p 1 ψβ + Φ p 1 dβ + p 2 ψβ + Φ p 2 dβ 52 We claim that the maximum of Jφ θ can be attained only if θ = or θ = κ 2 Assume, on the contrary, that < θ < κ, Jφ θ = max θ [,κ] θ Jφ θ 53 The optimality conditions yield d dθ Jφθ θ=θ =, d 2 dθ 2 Jφθ θ=θ 54 In turn, these imply p 1 ψθ + Φ p 1 = p 2 ψθ + Φ p 2, p 1 ψ θ + Φ p 1 p 2 ψ θ + Φ p 2 55 We now recall that X is of type B, hence From 55 we obtain ψ ψ < Therefore s 1 < s 2 = ψ s 1 ψs 1 > ψ s 2 ψs 2 56 ψ θ + Φ p 1 ψθ + Φ p 1 ψ θ + Φ p 2 ψθ + Φ p 2 57 Since p 1 < p 2, the first equality in 55 implies that ψθ + Φ p 1 > ψθ + Φ p 2, hence s 1 = θ + Φ p 1 < θ + Φ p 2 = s 2 The inequality 811 is thus in contradiction with 56 This achieves the proof 12

13 The same argument used in the proof of Lemma 51 yields Corollary 52 Let ϕ : [, κ] [, P ] be a pricing strategy taking finitely many values p 1 < p 2 < < p m For a given k 1,, m 1}, consider the two strategies ϕ k β = ϕβ if ϕβ / pk, p k+1 }, p k if ϕβ p k, p k+1 }, Then ϕ k+ β = Jϕ ϕβ if ϕβ / pk, p k+1 }, p k+1 if ϕβ p k, p k+1 }, } max Jϕ k, Jϕ k+ 58 In other words, we can always replace a strategy taking m distinct values with a new strategy taking m 1 values and achieving at least the same payoff Remark 3 Consider the continuous function κ κ F p 1, p 2, θ, q 1, q 2 = max p 1 ψβ + q 1 dβ, p 2 } ψβ + q 2 dβ θ κ p 1 ψβ + q 1 dβ p 2 ψβ + q 2 dβ θ Let κ = Φ P The proof of Lemma 51 shows that F > on the set Ω = } p 1, p 2, θ, q 1, q 2 ; p 1 < p 2 P, < θ < κ, q 1 q 2 κ Given any ε >, consider the compact subset Ω ε = p 1, p 2, θ, q 1, q 2 ; p 1 p 2 ε p 2 P, ε < θ < κ ε, q 1 q 2 κ } Since F is strictly positive on the compact set Ω ε, it attains a strictly positive minimum δε > on Ω ε In particular, this shows that given a positive ε, we can find δε > such that the following holds Assume that p 1 p 2 ε < p 2 P and θ [ε, κ ε] Then the pricing strategy φ θ in 51 satisfies Jφ θ max α [,κ] Jφ α δε 59 Theorem 53 Assume that the random variable X is of type B and satisfies the assumption A1 Then, given any nondecreasing map Φ, any optimal solution φ of the problem 21 must be constant Proof Let φ be an optimal solution Assuming that φ is not constant, we shall derive a contradiction 13

14 1 Choose ε > and points < a < a + 2ε < b < P so that meas β [, κ] ; φβ < a} > ε, meas β [, κ] ; φβ > b + ε} > ε 51 Let δε > be the corresponding constant in 59, and choose an integer n large enough so that κ < min ε, δε} 511 n Introduce the points p j = j/n and consider the approximate, piecewise constant strategy φ n β = p j if p j φβ < p j+1 This definition yields Jφ n Jφ κ n > Jφ δε By construction, φ n takes only finitely many values p,, p N Since the random variable X is of type B, by repeatedly using Corollary 52 we can replace the strategy φ n with a strategy φ taking only three distinct values, P 1, P 2, P 3 More precisely, we can find three prices P 1, P 2, P 3 p 1,, p N }, with < P 1 a < P 2 < b ε P 3 P, 513 such that the following holds Defining the piecewise constant strategy P 1 if φ n β a, φ β = P 2 if a < φ n β < b, P 3 if φ n β b, one has 514 Jφ Jφ n If now P 2 P 1 P 3 P 2, we apply once again Corollary 52 and obtain a strategy φ of the form φ Q1 if φ β = β P 1, P 2 }, Q 2 if φ β = P 3 with Q 1 P 1, P 2 }, Q 2 = P 3, Jφ Jφ On the other hand, if now P 2 P 1 > P 3 P 2, we use Corollary 52 to obtain a strategy φ of the form φ Q1 if φ β = β = P 1, Q 2 if φ β P 2, P 3 } with Q 1 = P 1, Q 2 P 2, P 3 }, Jφ Jφ In both cases we obtain a strategy φ taking exactly two values Q 1, Q 2, with Q 2 Q 1 ε Moreover meas β [, κ] ; φ β = Q 1 } ε, meas β [, κ] ; φ β = Q 2 } ε

15 4 Finally, consider the two constant strategies φ 1β = Q 1, φ 2β = Q 2 By 516 and 59, we conclude } max Jφ 1, Jφ 2 Jφ + δε Jφ n + δε Jφ κ + δε > Jφ n This contradicts the optimality of φ, proving the theorem 6 Sufficient conditions We now consider a case where all strategies φ : [, β] [p, P ] which satisfy the necessary conditions stated in Theorem 42 are in fact optimal We make the following assumption on the regularity of Φ A2 The map s Φ s is continuous on the half-open interval [, P [ Moreover, its derivative Φ p is piecewise continuous Theorem 61 sufficient conditions Let the assumptions A1-A2 hold, and let X be a random variable of type A, so that 23 holds Moreover, assume that one has Then φ is optimal G β p Φ p for all p [p, φβ], G β p Φ p for all p [φβ, P ] 61 Proof Assuming that the new agent has priority, by Theorem 31 an optimal strategy φ exists Let now φ be any admissible strategy which satisfies the conditions 61 Consider the interpolated strategy φ θ β = θφβ + 1 θφ β 62 Since φ is optimal, to prove that φ is also optimal it thus suffices to show that d dθ J θφ + 1 θφ β 63 We have d dθ Jφθ = κ [ ] φβ φ βφ θ β p ψ β + Φ φ θ β Φ φ θ β G β φ θ β dβ Indeed, the inequality follows from the fact that ψ s < for every s, and φβ φ β = φ θ β φβ = Φ φθ β G β φ θ β, φβ φ β = φ θ β φβ = Φ φθ β G β φ θ β Hence the integrand is nonnegative for every β 15

16 Corollary 62 Assume that there exists a subinterval [x 1, x 2 ] [p, P ] such that Φ x 1 λx p < if x < x 1, = if x [x 1, x 2 ], > if x > x 2 Then a pricing strategy φ : [, κ] [, P ] is optimal if and only if it takes values inside the interval [x 1, x 2 ] Indeed, in this particular case the function G β p = 1 λp p for all p [x 1, x 2 ], β [, κ] does not depend on β and the result follows directly from Theorem 61 7 Nash Equilibria We now assume that n traders compete, selling different amounts of the same asset For i = 1,, n, let κ i be the amount of stock put on sale by the i-th agent and let φ i : [, κ i ] IR + be his pricing strategy We wish to study Nash non-cooperative equilibria, where the strategy of each player is an optimal reply to the strategies adopted by all the other players In the following, we assume that all traders have the same payoff function, and they all assign the same probability distribution to the random size X of the incoming order Definition 3 Let φ i : [, κ i] [, P ] be the pricing strategy of the i-th player Define the right continuous, non-decreasing functions Φ i p = meas β [, κ j ] ; φ j β p}, i = 1,, n 71 j i Then the n-tuple of strategies φ 1,, φ n is a Nash equilibrium solution to the bidding game if each φ i provides an optimal pricing strategy for the problem κi maximize: J i φ = φβ p ψ β + Φ i φβ dβ 72 Remark 4 The above definition does not mention the possible priority of one seller over another Indeed, priority is irrelevant, because in any Nash equilibrium it is not possible that two sellers offer positive amounts of asset at the same price p If this happens, the agent that does not have priority could offer his amount at price p ε with ε > sufficiently small, and achieve a strictly larger expected payoff This motivates our choice 71 of right-continuous functions Φ i If the random variable X is of type A, in this section we shall prove that a Nash equilibrium solution always exists, and explicitly determine the strategies of the various players On the other hand, if X is of type B, we prove that no Nash equilibrium solution can exist 16

17 As a preliminary example, given a random variable X of type A we construct the Nash equilibrium in the special case when all players have the exact same amount of shares to offer for sale Lemma 71 Nash equilibrium for identical players Assume that X is of type A and satisfies the assumptions A1 Consider n players, each one putting on sale the same amount κ = κ 1 = = κ n of asset Then the pricing strategies with φβ φ 1β = φ 2β = = φ nβ = φβ, 73 1 n ψnβ n = p + [P p ], β [, κ], 74 ψnκ provide a Nash equilibrium solution to the bidding game 72 Proof 1 Since ψ <, the pricing strategies in are strictly increasing Moreover, for i = 1,, n, the functions Φ 1 = = Φ n = Φ in 71 are all equal and satisfy Φ i φβ = Φφβ = n 1β, Φ φβ = n 1 φ β 75 By 74-75, a direct computation shows that ΦP = n 1κ, Φp = for p p A = p + [P p ] ψnκ n 1 n, 76 ψ n n 1 Φp Φp >, Φ p = p p ψ n n 1 Φp for p A < p < P 77 Here the ask price p A is the minimum price at which some of the asset is offered for sale 2 In order to check the necessary condition 428, we compute G β p = ψ β + Φp φβ p ψ β + Φp 78 By 75 and 77, this yields G β φβ = ψ nβ φβ p ψ nβ = Φ φβ, 79 showing that 428 holds 3 To prove that the n-tuple of pricing strategies in provides a Nash equilibrium, we need to show that each strategy satisfies the sufficient conditions for optimality 61 Fix any value β [, κ] and call p = φβ Consider any two prices p 1, p 2 [p, P ], with p 1 < p < p 2 As observed in Remark 2, since the random variable X is of type A, the map β G β p is nondecreasing Hence G β p 2 G β 2 p 2 = Φ p 2, 71 17

18 where β 2 > β is such that p 2 = φβ 2 Next, if p 1 > φ, there exists β 1 < β such that φβ 1 = p 1 and On the other hand, if p 1 φ, then Φ p 1 = and clearly G β p 1 G β 1 p 1 = Φ p G β p 1 > Φ p The three inequalities 71, 711, 712 show that the sufficient conditions 61 are satisfied, and therefore φ 1,, φ n provides a Nash equilibrium Remark 5 In this Nash equilibrium the expected payoff of each agent is Jφ = κ φβ p ψnβ dβ = 1 n ψnκ n 1 n P p nκ ψs 1 n ds We now extend the previous result to an arbitrary number of players, putting on sale different amounts of the asset Theorem 72 existence of a Nash equilibrium Let X be a random variable of type A, satisfying the assumptions A1 Given n 2 players, putting on sale the amounts κ 1,, κ n > of the same asset, the bidding game 72 has a Nash equilibrium Proof 1 Without loss of generality, we can assume that Define < κ 1 κ 2 κ n h 1 = κ1, h j = κj κ j 1 if 2 j n, 713 and, by backward induction, p n = P, n j n j+1 p j = p + ψn j + 1h j [p j+1 p ] if j = 1,, n

19 We claim that a Nash equilibrium solution is provided by the following pricing strategies: n 1 ψnκ1 n φ 1 β = p + [p 2 p ] if β [, κ 1 ], ψnβ φ 2 β φ j β φ n β = = = φ 1 β if β [, κ 1 ], n 2 ψn 1h2 n 1 p + [p 3 p ] if β [κ 1, κ 2 ], ψn 1β κ 1 φ j 1 β if β [, κ j 1 ], n j ψn j + 1h j n j+1 p + [p j+1 p ] if β [κ j 1, κ j ], ψn j + 1β κ j 1 φ n 1 β if β [, κ j 1 ], P if β [κ n 1, κ n ] Starting from the explicit formulas 715, a direct computation shows that the corresponding functions Φ i in 71 satisfy Φ n p Φ n 1 p Φ 1 p, for all p [p, P [ 716 Moreover, for every j = 1,, n one has see Fig 2 Φ n p for all p [p, p j+1 [, Φ j p = 717 n + 1 l Φ n p for all p [p l, p l+1 [, l > j n l To determine all functions Φ j, it thus suffices to compute Φ n This is a continuous, nondecreasing, piecewise C 1 function on [, P [, which satisfies By 717 it follows Φ np = ψ Φ n p = if p p 1, n+1 j n j Φ n p p p ψ n+1 j n j Φ n p if p j < p p j+1 n+1 l Φ n l Φ n p p [p l, p l+1 ], l > j, jp = Φ np p < p j In particular, by it follows that the necessary conditions Φ i φ iβ = G β iφ i β, stated in Corollary 43, are satisfied 3 In order to apply the sufficient condition for optimality stated in Theorem 61, given any p = φ i β, we need to check that Φ i p Gβ i p if p < p, Φ i p Gβ i p if p > p, 19 72

20 Φ np Z p p p 1 p 2 p 3 p4 = P x Figure 2: The Nash equilibrium solution in the case of 4 players Player j prices his assets within the interval [p 1, p j+1 ] For any x < P, the area of each colored region within the half-plane p < x} gives the amount of asset put on sale at price x by the corresponding player In addition, Player 4 puts an amount κ 4 κ 3 for sale at price P where G β i p is defined as in 41, with Φ replaced by Φ i : [ G β i p = ψ β + Φ i p p p ψ 1 β + Φ i p] 721 We observe that, since the random variable X is of type A, from 716 it follows G β np G β n 1 p Gβ 1 p 722 To fix the ideas, assume p [p i, p i+1 ] As in the proof of Lemma 71, we shall consider various cases CASE 1: p < p 1 In this case Φ i p = and the inequality Gβ i p > Φ i p is trivial CASE 2: p 1 < p < p We can then find β [, β ] such that φ i β = p Since the random variable X is of type A, by Remark 2 this implies G β i p G β i p = Φ ip CASE 3: p < p < p i+1 Choose β ]β, κ i ] such that φ i β = p Again by Remark 2, this implies G β i p G β i p = Φ ip CASE 4: p > p i+1 In this case, we can choose β ]κ i, κ n ] such that φ n β = p This yields G β i p G κ i i p = Gβ np = Φ np Φ ip 2

21 Theorem 73 nonexistence of a Nash equilibrium Let X be a random variable of type B, satisfying the assumptions A1 Then, for any number n 2 of players offering amounts κ 1,, κ n > of the same asset for sale, a Nash equilibrium cannot exist regardless of the selling priorities established among the players Proof 1 Assume, on the contrary, that a Nash equilibrium φ 1,, φ n exists By Theorem 53, each pricing strategy φ i must be constant, say φ i β p i, i = 1,, n We claim that i j = p i p j Otherwise, since one of the two players does not have the priority over the other, he could increase his expected payoff by pricing all his asset at p i ε, for some ε small enough 2 Let ε = min i j p i p j Choose k 1,, n} such that p k < P Then the k-th player can unilaterally increase his payoff by using the strategy φ k β = p k + ε 2 This contradiction shows that no Nash equilibrium can exist 8 Uniqueness of the Nash equilibrium In this section we prove that, if the random variable X is of type A, then the Nash equilibrium constructed in Theorem 72 is unique In the following, given an n-tuple of pricing strategies φ : [, κ i ] [, P ], we denote by F i p = sup β [, κ i ] ; φ i β p } 81 the amount of asset put on sale at price p by the i-th player Moreover, we define F p = n F i p i=1 Observe that, with these definitions, the functions Φ i in 71 are expressed by Φ i p = j i F j p = F p F i p Lemma 81 Let the n-tuple φ 1,, φ n be a Nash equilibrium Then the following holds i There exists a Lipschitz constant C such that F p 2 F p 1 Cp 2 p 1 for all p < p 1 < p 2 < P 82 21

22 ii At most one of the functions F i can have an upward jump at p = P, while all the others are Lipschitz continuous on the entire interval [, P ] iii There exists a minimum ask price p A and a constant δ > such that F p = for all p p A, F p δ for ae p [p A, P ] 83 Φ i p p p A a 1 b 1 a 2 b 2 P p Figure 3: An illustration of the proof that Φ i p is Lipschitz continuous in Lemma 81 In a Nash equilibrium, no other player can sell at a price p ]a k, b k ] Proof 1 Let and p = p + ψkp p > p, 84 C = } max G β i p ; β [, κ], p [p, P ], i 1, 2,, n} We claim that for every i 1,, n}, the set } S i = p [p, P ] ; Φ i p > Φ i q + Cp q for some q < p 86 is empty Indeed, if S i, we can write S i as a union of intervals, say S i = k ]a k, b k ] Consider any other player, say the j-th player, with j i Then meas β [, κ j ] ; φ j β ]a k, b k [ } = 87 Otherwise, the j-th player could get a strictly higher expected payoff by using the strategy φ j β = ak if φ j β ]a k, b k [, φ j β otherwise, 22

23 as the following computation shows: J φ j Jφ j = a k p ψβ + Φ j a k φ j β p ψβ + Φ j φ j β β ; φ j β [a k,b k ]} β ; φ j β [a k,b k ]} ak φ j β d p p ψ β + Φ j a k Ca k p dp dβ dp φj β p p ψ β + Φ j a k Ca k p β ; φ j β [a k,b k ]} a k ψ β + Φ j a k Ca k p C dp dβ p p ψ β + Φ j a k Ca k p φj β p p ψ β + Φ j a k Ca k p β ; φ j β [a k,b k ]} a k C G β j a k dp dβ > The first inequality follows from 86 and the Fundamental Theorem of Calculus, the third inequality follows from the fact that X is of Type A, and the strict inequality follows from the definition 85 However, if 87 holds for every j i, then the strategy φ i for the i-th player is not optimal Indeed, he could achieve a strictly higher payoff by setting bk ε if φ φ i β = i β [a k, b k [, φ i β otherwise, for some ε > sufficiently small This proves that Φ j is Lipschitz on the interval [p, P ] for every j 1,, n} Since F = 1 n Φ j, n 1 j=1 we conclude that F is Lipschitz continuous on [p, P [ Let p be a point such that F p > Then at least one agent is putting some shares on sale at the price p From the necessary conditions 428 on the best reply of any of the n players, if F >, then it satisfies the inequality F p Φ jp = ψf p p ψ F, F P K = Denote by Y p the solution to the terminal value problem Y = By direct computation we see that n κ i p [p, P [ i=1 ψy p p ψ Y, Y P = K ψy p = P p p p ψk, 23

24 which implies that Y p =, where p is given by 84 Y p F p and therefore By comparison, we see that This proves the first assertion of the Lemma p A = infp : F p > } p > p 2 The second assertion is clear: if two players put a positive amount of asset for sale at the same price P, the one that does not have priority can improve his expected payoff by selling the asset at price P ε 3 Toward a proof of iii, we show that there exists δ > small enough so that, for any p < P, the following implication holds: F p δ = F p = for all p [, p ] 88 Indeed, let 1 } δ = 2 min G β i p ; β [, κ], p [p, P ], i 1, 2,, n}, and observe that δ > By i it follows that F is differentiable at ae point p [, P ] Assume F p δ and consider the non-empty set S = p < p ; F p > F p 2δ p p } If F p = F p for all p S, recalling that F is Lipschitz continuous we conclude that F p = F p = for all p p, as claimed In the opposite case, there exist p < p such that F p < F p, F p F p 2δ p p for all p [p, p ] 89 Clearly, at least one the the players is putting some assets for sale within the price interval [p, p ], say, the i-th player This leads to a contradiction, because by 89 Hence the strategy φ i β = yields a strictly higher expected payoff: J φ i Jφ i = β ; φ i β [p,p ]} β ; φ j β [p,p ]} Φ i p Φ i p 2δ p p, p if φ i β [p, p ], φ i β otherwise, p φ i β p φ j β p p ψ β + Φ i p Φ ip G β i p dp dβ p p ψ β + Φ i p 2δ G β i p dp dβ > 24

25 Φ i p p p A p p P p Figure 4: If Φ i F is small, then the i-th player can improve his expected payoff by asking the higher price p instead of a price p [p, p ] Theorem 82 In the same setting of Theorem 72, the Nash equilibrium is unique Proof 1 Let φ 1,, φ n be a Nash equilibrium By Lemma 81, the corresponding functions F i are Lipschitz continuous on [, P [, and all except at most one of them are Lipschitz continuous on the closed interval [, P ] Moreover, there exists a minimum ask price p A such that iii in Lemma 81 holds 2 By Rademacher s theorem, every function F i is differentiable ae on [, P [ For each p, consider the set of indices Ip = i ; F p > } and call Np = #Ip the cardinality of this set By Lemma 81 the function N is well defined and Lebesgue measurable Moreover, Np 2 for ae p [p A, P ] For p [p A, P [, i Ip, let β i [, κ i ] be such that φ i β i = p Recalling 41, from the necessary conditions 49 we deduce Φ ip = G β i i p = ψf p p p ψ F p i Ip Observing that Φ ip = j i F jp, one obtains Φ i Np 1 p = F p, F i Np p = F p Np for i Ip, Φ i p = F p, F i p = for i / Ip The Lipschitz function F thus satisfies the ODE F p = Np Np 1 ψf p p p ψ F p 81 25

26 at ae point p [p A, P ] γ 1 p Φ i p Φ i p γ 1 p γ 2 p γ 2 p p A p 1 p p 2 p p A p 1 p p 2 p Figure 5: A graph of the function Φ i If player i sells something at price p 2 but nothing at price p 1, then his strategy is not optimal Left: Case 1 Right: Case 2 3 We claim that, for each i 1,, n}, the set of prices where the i-th player offers assets for sale is an interval [p A, p i+1 ] Assume, on the contrary, that this is not the case To derive a contradiction, call S i = p [p, P ] ; F i > } and L i = p [p, P ] ; p is a Lebesgue point of F i } Let and assume that Let q [p A, P ] L i \ Si [q, P ] L i S i 811 q = inf[q, P ] Li S i Then, for any δ 1 >, the following two sets are non-empty: A = [q δ 1, q ] L i \ S i, B = [q, q + δ 1 ] L i S i Indeed, B is nonempty, by the definition of infimum Moreover, if q = q then q A, otherwise A is nonempty by the definition of infimum From the necessary conditions 428 we deduce Φ ip = F p = Np Np 1 ψf p p p ψ F p n n 1 ψf p p p ψ F p for p / S i, while for p S i Φ ip = ψf p p p ψ F p 26

27 Choose the intermediate slope λ = 2n 1 2n 2 ψf q q p ψ F q 812 By continuity we can choose δ < δ 1 small enough so that λ ψf p p p ψ F p >, for all p [q δ, q + δ ] Finally, let p 1 A and p 2 B be Lebesgue points of F i and consider the two lines γ 1 p = Φ i p 1 + λp p 1, γ 2 p = Φ i p 2 + λp p 2 We split the analysis into two cases Fig 5 CASE 1: γ 1 γ 2 We then consider the intermediate point p = min p > p1 ; Φ i p = γ 1 p } Observe that p > p 1, because p 1 L i \ S i and Φ i p 1 > λ Then the new pricing strategy φ i β = p1 if φ i β [p 1, p ], φ i β otherwise, yields a strictly better expected payoff: J φ i Jφ i = p 1 p ψβ + Φ i p 1 φ i β p ψβ + Φ i φ i β β ; φ i β [p 1,p ]} β ; φ i β [p 1,p ]} p1 φ i β d p p ψ β + γ 1 p dp dβ dp φi β = p p ψ β + Φ i p β ; φ i β [p 1,p ]} p 1 ψ β + γ 1 p λ dp dβ > p p ψ β + γ 1 p CASE 2: γ 1 < γ 2 We then consider the intermediate point p = maxp < p2 ; Φ i p = γ 2 p} An entirely similar argument now shows that the new pricing strategy φ i β = p if φ i β [p, p 2 ], φ i β otherwise, 27

28 yields a strictly better expected payoff In both cases we showed that φ i is not optimal, thus reaching a contradiction 4 From the previous step it follows We claim that p n = p n+1 = P p 1 p 2 p n p n+1 P Indeed, if p n < p n+1, this means that the n-th player is the only seller in the interval [p n, p n+1 ] He could achieve a better expected payoff by taking all his assets originally on sale at a price p [p n, p n+1 ] and offering them at the price p n+1 instead This shows that p n = p n+1 Finally we show that p n = P Indeed, if this were not the case, we would have F p =, for all p ]p n, P ], contradicting the third statement in Lemma 81 9 A large number of small agents In this section we study the limiting case where the number of sellers approaches infinity, but the total amount of asset offered for sale remains bounded Example 3 Consider the simple case of n players, each one selling the same amount K/n of asset By 77 in the proof of Lemma 71, the total amount Z n p = n n 1Φp of asset put on sale at price p is found by solving the ODE n 1 n Z n = ψz np p p ψ Z n, Z np = K As n, the limit distribution Zp = lim n Z n p is clearly obtained by solving Z = ψz p p ψ, ZP = K 91 Z We wish to show that the same limit holds, without assuming that all players put on sale exactly the same amount of asset Consider a sequence of bidding games, satisfying: G1 The n-th game involves n distinct players, selling the amounts κ n,1,, κ n,n of the same asset G2 The total amount of asset put on sale in the n-th game is K n lim n K n = K = n i=1 κ n,i, with G3 The largest amount of asset put on sale by any player in the n-th game approaches zero: lim n sup1 i n κ n,i = 28

29 The next result shows that, with the above assumptions, as n the limit order book approaches a well defined shape In the following, we call Z n p the amount of asset offered for sale at price < p, in the Nash equilibrium solution 715 for the n-th game Moreover, we let Zp to be the solution to the Cauchy problem 91 Observe that the right hand side of the ODE in 91 is well defined and uniformly positive as long as Z [, K] Indeed, Z p C p p for some constant C > By a comparison argument we conclude that there exists a value p A > p such that the solution of 91 satisfies Zp A =, Zp > for p A < p < P 92 We then extend the function Z to the entire interval [, P ] by setting Zp = for p [, p A ] 93 Theorem 91 Let X be a random variable of type A, satisfying the assumptions A1 Consider a sequence of games for n players, satisfying G1 G3 Then, for any ε >, the following holds lim Z np = Zp uniformly for all p [, P ], 94 n lim n Z np = Z p uniformly for all p [, p A ε] [p A + ε, P ε], 95 where Z is defined by 91, 93, and p A is determined by 92 Proof 1 For a given n 1, it is not restrictive to assume κ n,1 κ n,2 κ n,n For 1 < i n call h n,i = κn,i κ n,i 1 Moreover, set h n,1 = κn,1 In the Nash equilibrium solution for the n-th game, the total amount Z n p put on sale at price < p is characterized by the equations Z n P = K n h n,n, Z n p = for p [, p n,1 ], 96 Z np = n i + 1 n i ψz n p p p ψ Z n p Here the prices p n,i are determined by the inductive rule for p n,i < p < p n,i+1, 1 i < n 97 p n,n = P, pn,i+1 p n,i Z np n i + 1 dp = h n,i 1 for i 1 98 Recalling that ψ >, ψ <, from 97 we deduce ψz n p p p ψ Z n p Z np 2ψZ n p p p ψ Z n p, p n,1 < p < P, 99 ψz n p p p ψ Z n p Z np m + 1 m ψz n p p p ψ Z n p for p n,n m < p < P 91 29

30 2 For any fixed m 1, we claim that p n,n m P, Z n p n,n m K as n 911 Indeed, by 99 it follows that all maps Z n are increasing and uniformly Lipschitz continuous, say [ ] pa + P Z n P CP p Z n p Z n P for all p, P, for some Lipschitz constant C Since Z n P = K n h n K as n, we can find δ > such that K 2 Z np 2K for all p [P δ, P ] 913 and all n sufficiently large By 98 one has P p n,n m Z np dp m + 1 n 1 i=n m pn,i+1 p n,i Z np n dp = m + 1 n i + 1 i=n m h n,i m + 1 κ n,n 1 κ n m 1 m + 1 κ n,n as n Together, 913 and 914 imply 911 Indeed, using 91, 913 and the assumption A1, it follows that, if p n,n m < P δ, then where P p n,n m Z np dp P p n,n m m = min s [K/2, K] ψk/2 P p ψ Z n p dp m ψk/2 P p δ, 1 ψ s > By 914 we thus have p n,n m P δ for all n sufficiently large Therefore K 2 P p n,n m P p n,n m Z np dp, showing that p n,n m P as n In turn, this implies Z n p n,n m K Z n p n,n m Z n P + Z n P K CP p n,n m + K K n + κ n,n 3 By the previous step, the function Z n satisfies the differential inequalities m + 1 m ψz n p p p ψ Z n p Z np ψz n p p p ψ Z n p, p n,1 < p < p n,n m, 915 with terminal conditions at p = p n,n m satisfying 911 We now compare 915 and 911 with 91 By standard results on the continuous dependence of solutions to a Cauchy problem, for any ε > we have the convergence see Fig 6 Z n p Zp, Z np Z p, 916 3

31 K Zp Z np p A p A +ε p p 1,n P ε P Figure 6: On any subinterval [p A + ε, P ε] we have the uniform convergence Z n p Zp Since each derivative Z n is uniformly positive on the region where Z n >, this implies the convergence p n p A uniformly on the interval [p A + ε, P ε] By 99, on the region where Z n > the derivative satisfies Z np c for some constant c > and all p >, n 2 Since in 916 we can choose ε > arbitrarily small, we conclude that the value p n in 96 satisfy lim p n,1 = p A 917 n Observing that Z n p = for p [, p n,1 ], Zp = for p [, p A ], and that all functions Z, Z n are uniformly Lipschitz continuous, from 916 and 917 we deduce the convergence Examples In this section we consider in more detail the case when the probability distribution of size of incoming market order is given by 25 or 26 Example 4 Assume that the size of the incoming market order is exponentially distributed, with mean λ 1 Two competing agents put on sale the amounts κ 1 < κ 2 of shares The Nash equilibrium 715 is given by φ 2 β = P β [κ 1, κ 2 ], p + e λκ 1+λβ [P p ], β [, κ 1 ], 11 φ 1 β = p + e λκ 1+λβ [P p ], β [, κ 1 ] 31

32 The cumulative limit order book is thus given by p [p, p A [, 2 F p = λ ln p p p [p A, P [, P p κ 1 + κ 2 p = P This corresponds to a limit order book density F p = 2 λp p χ [p A,P ] p + κ 2 κ 1 δ P, where δ P denotes a unit Dirac mass located at p = P, and the ask price p A is given by p A = p + P p e λκ 1 The expected payoffs of the two agents in the Nash equilibrium configuration are given by J 1 = κ1 φ 1β p e 2λβ dβ = e λκ 1 1 e λκ 1 P p λ J 2 = J 1 + E[X κ 1 + κ 2 ] = e λκ 1 1 e λκ 2+2κ 1 P p λ We observe that an increase in the total amount put on sale by the smaller player hence by both players lowers the ask price, and also decreases the expected payoff of both competitors J 1, J 2, as κ 1 On the other hand, the larger player can increase his expected payoff by increasing the total amount of shares he puts on sale: J 2 e λκ 1 P p, as κ 2, κ 1 fixed λ Finally, using the explicit expression of the limit order book resulting from the Nash equilibrium, we can also derive an expression for the price impact function ρx, which represents the increase in the ask price in response to a market order of size X Indeed, ρx is defined by the following implicit equation Zp A + ρx = Zp A + X 12 This yields ρx = e λx 2 1P p e λκ 1 if X 2κ 1, P p A if X > 2κ 1 Example 5 Consider the asymptotic limit of a large number of small agents, putting on sale a total amount of K shares Assume that the size of the incoming market order is exponentially distributed, as in 25 In this case, the Cauchy problem 91 simplifies to Z p = 1 λp p, ZP = K 32

33 The expected payoff per unit amount of asset put on sale by any agent is given by J u = P p e λk The ask price is p A = p + P p e λk, while the price impact function is given by e λx 1P p e λk if X K, ρx = P p A if X > K Example 6 Assume that the random size X of the incoming buying order is distributed according to the power law distribution 26 Consider n players, each one putting on sale the same amount κ of shares, for a total amount of K = nκ The Nash equilibrium is thus given by 74: φ 1β = = φ nβ = φβ = p + [P p ] n nβ n α, 1 + nκ and the corresponding ask price is p n A = φ = p + [P p ] 1 + nκ 1 n n α The cumulative limit order book is thus given by Z n p = 1 + nκ 1 n α n 1 P p p p 1 n α n 1 1, p [p n A, P ] The corresponding order book density is then Z np = n αn nκ P p 1 α n n 1 p p n1 α+α nα α, p [p n A, P ] From the above expressions we can easily compute the asymptotic limit as the number of players goes to infinity, for K = nκ fixed The ask price is p A = p + [P p ] 1 + K α and the shape of the limit order book is given by Zp = 1 + K P p 1 α p p 1 α 1, p [pa, P ], Z p = 1 α 1 + K P p 1 α p p 1 α α, p [pa, P ] In this case, the price impact function is given by ρx = P p 1 + K α [1 + Xα 1], X K 33

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