Discrete Bidding Strategies for a Random Incoming Order

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1 Discrete Bidding Strategies for a Random Incoming Order Alberto Bressan and Giancarlo Facchi Department of Mathematics, Penn State University University Park, Pa 1682, USA s: bressan@mathpsuedu, facchi@mathpsuedu December 29, 213 Abstract This paper is concerned with a model of a one-sided limit order book, viewed as a noncooperative game for n players Agents offer various quantities of an asset at different prices, ranging over a finite set Ω ν = {i/νp ; i = 1,, ν}, competing to fulfill an incoming order, whose size X is not known a priori Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random variable X For a wide class of random variables, we prove that the optimal pricing strategies for each seller form a compact and convex set By a fixed point argument, this yields the existence of a Nash equilibrium for the bidding game As ν, we show that the discrete Nash equilibria converge to an equilibrium solution for a bidding game where prices range continuously over the whole interval [, P ] Keywords: measure-valued optimization, optimality conditions, optimal pricing strategy, bidding game, Nash equilibrium, limit order book 1 Introduction This paper is concerned with a bidding game for n players In the basic setting, we assume that an external buyer asks for a random amount of X > of shares of a certain asset This amount will be bought 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 pricing strategy, maximizing his expected payoff Clearly, when other sellers are present, asking a higher price for the asset increases the expected profit achieved by the sale, but reduces the probability that the asset will actually be sold This same model was recently 1

2 studied in [6], assuming that the prices asked by the bidders could range over the continuum of real numbers In such case, under natural hypotheses the authors proved that the noncooperative game admits a unique Nash equilibrium, where all players except one, at most adopt price distributions which are absolutely continuous wrt Lebesgue measure In the present paper we analyze the more realistic case where the price distributions take a discrete set of values, ranging within the finite set { k } Ω ν = p k = ν P ; k {1,, ν}, for some integer ν > 1 In our model we assume that the i-th player owns an amount κ i > of asset 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 vector µ i = µ i1,, µ iν, µ il, ν µ il = κ i, 11 where µ ik denotes the amount offered for sale by the i-th player at the price p k The paper has two main goals, namely i to study the Nash equilibrium, when prices range in a discrete set of values, and ii to show that, as ν, these discrete equilibrium solutions converge to the solution of a Nash equilibrium where prices range on a continuum interval of values [, P ] A major difference between the discrete and continuum case is how bids are prioritized When prices range over all continuum values, as proved in [6], in a Nash equilibrium no two players offer a positive amount of asset exactly at the same price Hence, if a buying order of size X arrives, the amounts sold by the various players are uniquely determined by the prices at which the sellers post their offers However, in the discrete case, it may well happen that two or more players offer positive amounts of asset for sale at exactly the same price p k In this case, one needs to specify in which order the bids of the various players will be executed As modeling assumption, we assume the following H1 Players are given a fixed ranking Hence, if they all put assets on sale at any given price p l, the buyer will start by taking assets from the first player, then from the second, and so on, until his order is fulfilled There are further differences between the models studied in [6] and in the present paper In [6], all players assigned the same probability distribution to the random variable X, and had the same payoff function Here we consider the far more general case of heterogeneous players, holding different beliefs about the fundamental value of the assets put on sale and on the random variable X describing the size of the incoming order The techniques used in the analysis are also different In [6] the existence of a Nash equilibrium was proved by solving a system of ODEs providing necessary conditions, and checking that the solution actually provided the unique Nash equilibrium to the bidding game The present paper, on the other hand, is based on the analysis of the best reply map Under suitable assumptions, we prove that this map is upper semicontinuous with compact convex values, hence by Kakutani s theorem it has a fixed point The remainder of the paper is organized as follows Section 2 collects the main definitions and assumptions of our model In particular, in we introduce three classes of l=1 2

3 random variables, which we call of Type A +, Type A, and Type B, depending on whether the function ψs = P rob{x > s} is log-convex, log-linear, or log-concave In Sections 3 and 4 we study the optimization problem faced by each single player, determining necessary conditions for optimality and studying the topological properties of the set of solutions We show that for random variables of Type A + the optimal strategy is always unique, while for random variables of Type A there is a compact, convex set of optimal strategies Finally, in the case of random variables of Type B, the set of optimal strategies may not be convex in fact, not even connected Section 5 contains our main results on the existence or non-existence of a Nash equilibria for discrete bidding strategies If each random variable X i is of type A + or A, then we prove that a Nash equilibrium exists On the other hand, if all random variables X i are of type B, then there cannot be any Nash equilibrium Remark 1 In [6] the existence result was proved by solving a system of ODEs and explicitly constructing the optimal strategies for each player That approach also yielded the uniqueness of the Nash equilibrium Here we use a more standard topological technique, applying Kakutani s fixed point theorem to a family of discrete approximation By its nature, this approach is more general but does not yield information about the uniqueness of the Nash equilibrium In Section 6 we consider the limit as ν, so that the mesh size approaches zero By possibly choosing a subsequence, we prove that a sequence of discrete Nash equilibria converges to a continuum Nash equilibrium, where strategies take values within the real line In particular, this analysis allows us to extend the existence theorem proved in [6] to the case of heterogeneous players, with different payoff functions Several recent papers have been devoted to the modeling of the limit order book In some of these models [3, 7, 13] the prices vary in a discrete set, while in other models [1, 1, 12, 14] the prices are allowed to range in a continuum set The main goal of these models is to study an optimization problem for an agent who wants to execute market orders or post limit orders, or both as in [9], maximizing a prescribed utility function The main feature of our model is that we consider the limit order book as the outcome of a noncooperative game, where several different players are trying to maximize their possibly different utility functions In this aspect, our model is similar to the model by Roşu [15] A major difference lies in the fact that in our model the utility functions of the players are determined by the probability distributions for the size of the incoming market order, and not by the expected waiting time of the order execution Nash equilibria for some models of the limit order book have been recently studied also in [4] In addition to the classical paper [11], for an introduction to non-cooperative games and Nash equilibria we refer to [5, 8, 16, 17] 2 Discrete bidding strategies Consider a bidding game for n players, competing to fulfill a incoming order whose size is not a priori known The optimization problem faced by the i-th player can be formulated in terms of the following data 3

4 i A non-negative random variable X i, whose distribution is described by ψ i s = Prob{X i s} This probability distribution reflects the beliefs of the i-th agent about the size of the incoming order ii A constant p i > describing the fundamental value of the asset, according to the i-th player This means that, for the i-th player, it is indifferent to keep the asset or to sell it at the unit price p i iii The total amount κ i > of assets put on sale the i-th player iv A nondecreasing function Φ i : Ω ν IR + Here Φ i p k describes the total amount of assets put on sale by all the other agents at prices p k, having priority over the bid of the i-th player at price p k As remarked at 11, the i-th player must optimally choose a vector µ i = µ i1,, µ iν, where µ ik denotes the amount of asset put on sale at price p k = kp /ν It will be convenient to reformulate this problem, using the variable β [, κ i ] to label a particular asset owned by the i-th agent By a pricing strategy for the i-th player we mean any function φ i : [, κ i ] Ω ν = { jp ν, } j = 1, 2,, ν 22 which is i nondecreasing, ii left continuous, and iii right continuous at β = Here φ i β is the price at which the β-asset is offered for sale Given a vector µ i1,, µ iν as in 11, since all the assets β [, κ i ] are equivalent, any function φ i : [, κ i ] Ω ν such that meas {β [, κ i ] ; φ i β = p k } = µ ik, k = 1,, ν, 23 would yield the same expected payoff By requiring that φ satisfies the properties i iii, we single out a unique such function, namely φ i β = p k if and only if k 1 µ ij < β j=1 k µ ij 24 j=1 The expected payoff for the i-th player is then computed by J i φ i, Φ i = κi φ i β p i ψ i β + Φ i φ i β dβ 25 Notice that φ i β p i is the profit achieved by selling the asset β at price φ i β, while ψ i β + Φ i φ i β is the probability that this particular asset will be actually sold We say that φ i is an optimal pricing strategy if J i φ i, Φ i J i φ i, Φ i for every other admissible strategy φ : [, κ i ] Ω ν 4

5 Next, consider n agents offering for sale quantities κ 1,, κ n of the same asset Let µ ik be the amount put on sale at price p k by the i-th agent Recalling the prioritizing rule H1, we define the functions Φ i p l = µ jl 26 k<l, j i µ jk + j<i We say that an n-tuple of strategies φ 1,, φ n is a Nash equilibrium if, defining the corresponding functions Φ i as in 26, one has J i φ i, Φ i J i φ i, Φ i for every i = 1,, n and any other pricing strategy φ i : [, κ i ] Ω ν As indicated by the analysis in [6], the existence of a Nash equilibrium strongly depends on the properties of the random variables X i Throughout the following, we assume H2 For i = 1,, n, the maps s ψ i s are twice continuously differentiable and satisfy ψ i = 1, ψ i + =, ψ is < for all s > Motivated by [6], we shall consider three main classes of random variables, depending on whether the function ψs = Prob{X s} 27 is log-convex, log-linear, or log-concave Definition We say that the random variable X in 27 is of type A + if ln ψs > for all s >, of type A if ln ψs = for all s >, 28 of type B if ln ψs < for all s > We observe that ψ is of type A if and only if ψs = e λs for some λ > On the other hand, when α > the probability distribution ψs = s α is of type A +, while ψs = e s2 yields a probability distribution of type B Roughly speaking, a probability distribution is of type A + if its density decays slower than a negative exponential, and of type B if its density decays faster than a negative exponential 5

6 3 Necessary conditions for an optimal bidding strategy Let a positive, nondecreasing function Φ i : [, P ] IR + be given, and consider the optimization problem for the i-th player, who wishes to maximize the expected payoff J i φ, Φ i in 25 over all admissible strategies φ : [, κ i ] Ω ν As remarked at 24-23, the set of admissible strategies can be identified with the set S i of ν-tuples µ i = µ i1,, µ iν, where µ ik is the amount of asset that the i-th player puts on sale at price p k In this section we derive necessary conditions for optimality In the case of two equal players, these will provide an explicit formula describing the discrete Nash equilibrium We recall that the functions φ i are assumed to be nondecreasing and left continuous, therefore at each point β the left and right limits satisfy φβ = φβ φβ+ Proposition 1 optimality conditions Given a positive, nondecreasing function Φ i let φ i : [, κ] Ω ν be a best reply for the i-th player If β ], κ i [ is a point of jump of φ, so that φ i β+ > φ i β, then φ i β+ p i ψ i β + Φi φ i β+ = φ i β p i ψ i β + Φi φ i β 39 Proof Consider the following perturbation: A direct computation yields β ε J i φ ε = + Since φ i is optimal, we conclude φ ε = { φi β+ if β [β ε, β], φ i β otherwise κ β φ i β p i ψ i β + Φ i φ i β dβ β + φ i β+ p i ψ i β + Φi φ i β+ dβ β ε d dε J iφ ε = φ i β p i ψ i β + Φi φ i β +φ i β+ p i ψ i β + Φi φ i β+ ε= Similarly, by considering the perturbation: φ ε+ = { φi β if β [β, β + ε], φ i β otherwise, we obtain the converse inequality Hence 39 holds By a similar perturbation argument, one obtains: Proposition 2 Given a positive, nondecreasing function Φ i, let µ i = µ i1,, µ iν be a best reply for the i-th player 6

7 If µ il > for some l > 1, then l 1 p l p i ψ i µ ik + Φ i p l k=1 l 1 p l 1 p i ψ i µ ik + Φ i p l 1 31 k=1 Moreover, if µ il > for some l < ν, then l l p l p i ψ i µ ik + Φ i p l p l+1 p i ψ i µ ik + Φ i p l k=1 k=1 Proof Setting µ i =, we can rewrite the optimization problem as follows: maximize : Jµ i = ν k=1 µi1 + +µ ik p k p ψ β + Φ i p k dβ, 312 µ i1 + +µ i,k 1 subject to : ν µ ik = κ i, µ ik, for all k = 1,, ν k=1 If µ il > for some l > 1, consider the perturbed strategy µ ε i = µε i1, µε i2,, µε iν defined by µ ε l 1 = µ l + ε, µ ε l = µ l ε, µ ε k = µ k if k / {l 1, l} This is clearly admissible, as long as ε µ il Since µ i is a best reply, we must have Jµ i Jµ ε i for all ε >, hence d dε Jµε i 313 ε= Replacing µ ik with µ ε ik in 312 and differentiating wrt ε, from 313 we obtain 31 The inequality 311 is proved in an entirely similar way In the particular case where the random incoming order is exponentially distributed, we obtain: Corollary 1 Assume that ψ i s = e λis, and let µ i1,, µ iν be an optimal pricing strategy for the i-th player Then µ il > = Φ i p l Φ i p l 1 1 pl p ln i, λ i p l 1 p i µ i,l 1 > = Φ i p l Φ i p l 1 1 pl p ln i λ i p l 1 p i 7

8 4 Properties of the set of best replies In this section, given a positive, nondecreasing function Φ i : Ω ν IR, we analyze the set of best replies for the i-th player These are left-continuous, nondecreasing functions φ i : [, κ i ] Ω ν = {p 1, p 2,, p ν = P } as in 22, which maximize the expected payoff 25 To simplify our notation, given < p < P, κ >, and a positive nondecreasing function Φ : Ω ν IR +, we consider the optimization problem maximize: Jφ, Φ = κ The maximum is sought among all nondecreasing functions φβ p ψ β + Φφβ dβ 414 φ : [, κ] Ω ν As in 24, setting µ l = meas {β ; φβ = p l }, the set of admissible strategies can be identified with the set of vectors S = { µ = µ 1,, µ ν ; µ l, ν l=1 } µ l = κ, Clearly S is a compact, convex subset of IR ν Setting µ =, a direct computation yields Jφ, Φ = Jµ, Φ = ν k=1 µ1 + +µ k p k p ψ β + Φp k dβ 415 µ 1 + +µ k 1 From 415 it is clear that the map µ Jµ, Φ is continuous Hence it attains a maximum on the compact set S More precisely, the set S max of vectors µ 1,, µ ν where the maximum is attained is a nonempty, compact subset of S Aim of this section is to study the geometry of this set S max Lemma 1 If the random variable X is of type A +, then the optimization problem 414 admits a unique optimal solution Proof 1 Let φ 1, φ 2 : [, κ] {p 1,, p ν } be two optimal strategies If φ 1 φ 2, then by possibly permuting the indices 1,2 we can find a maximal open interval ]a, b[ [, κ] such that φ 1 β < φ 2 β for all a < β < b 416 We claim that, for θ = a and for θ = b, the above implies θ φ 1 β p ψ β + Φφ 1 β dβ = Indeed, assume that on the contrary a φ 1 β p ψ β + Φφ 1 β dβ > θ a 8 φ 2 β p ψ β + Φφ 2 β dβ 417 φ 2 β p ψ β + Φφ 2 β dβ 418

9 Observe that the function φβ = { φ1 β if β < a, φ 2 β if β > a, is nondecreasing, because we are assuming that ]a, b[ is maximal among all open intervals satisfying 416 If 418 holds, then φ yields an expected payoff strictly greater than both φ 1 and φ 2, against the assumption The other cases are ruled out by a similar argument 2 If 416 holds, then for every θ [a, b] the interpolated strategy { φ θ φ1 β if β < θ, β = φ 2 β if β > θ, is also admissible We now define Jθ = Jφ θ, Φ We claim that Indeed, Jθ = θ φ 1 β p ψ β + Φφ 1 β dβ + J C[a, b] C 1 ]a, b[ 419 κ θ φ 2 β p ψ β + Φφ 2 β dβ, and the continuity of the map θ Jθ is clear To prove continuous differentiability, we compute d dθ Jθ = φ 1θ p ψθ + Φφ 1 θ φ 2 θ p ψθ + Φφ 2 θ 42 At points where both φ 1 and φ 2 are constant, the continuity of the right hand side of 42 is trivial Next, let θ be one of the finitely many points where φ 1 has an upward jump Since φ 1 is optimal, the necessary conditions 39 yield φ 1 θ+ p ψθ + Φφ 1 θ+ = φ 1 θ p ψθ + Φφ 1 θ The same equality holds at points where φ 2 jumps Hence the map θ Jθ is continuously differentiable 3 Next, consider an interior point θ ]a, b[ where d dθ Jθ = We claim that J attains a strict local maximum at θ Indeed, choose δ > such that φ 1, φ 2 are both constant on the interval ]θ, θ + δ] θ ]θ, θ + δ] we then have d 2 dθ 2 Jθ = d [ ] φ 1 θ p ψθ + Φφ 1 θ φ 2 θ p ψθ + Φφ 2 θ dθ For = φ 1 θ p ψ θ + Φφ 1 θ φ 2 θ p ψ θ + Φφ 2 θ [ ψ ] θ + Φφ 1 θ = φ 1 θ p ψθ + Φφ 1 θ ψθ + Φφ 1 θ ψ θ + Φφ 2 θ ψθ + Φφ 2 θ [ ] ψ θ + Φφ 2 θ + φ 1 θ p ψθ + Φφ 1 θ φ 2 θ p ψθ + Φφ 2 θ ψθ + Φφ 2 θ = Aθ + Bθ 9 421

10 We now observe that lim Aθ < 422 θ θ + because X is of type A + and hence ln ψ > On the other hand, by 42 [ ] lim Bθ = lim φ 1 θ p ψθ + Φφ 1 θ φ 2 θ p ψθ + Φφ 2 θ θ θ + θ θ + d = lim θ θ + dθ Jθ = 423 From it follows that d2 Jθ < for all θ ]θ, θ + ε], with ε > small enough dθ 2 An entirely similar argument shows that d2 Jθ < also for θ [θ ε, θ [ Hence J attains dθ 2 a strict local maximum at θ 4 As a consequence of 417, when θ = a and θ = b the maximum expected payoff is achieved: Jφ a, Φ = Jφ b, Φ = Jφ 1, Φ = Jφ 2, Φ We conclude that the function J in 419 must achieve its global minimum on [a, b] at some interior point θ ]a, b[ This implies dj dθ θ = and hence, as proved in the previous step, J must attains a strict local maximum at θ We thus reach a contradiction, proving the result Lemma 2 Assume that the random variable X is of type A, with ψs = e λs Then there exists a subset of prices Ω opt {p 1,, p ν } = Ω ν with the following property A pricing strategy φ : [, κ] Ω ν is optimal if and only if φβ Ω opt for every β Proof 1 Let φ : [, κ] Ω ν be an optimal strategy We claim that there exists a constant C such that p k pe λφp k = C 424 for all k {1,, ν} such that µ k = meas {β ; φβ = p k } > 425 Indeed, if there is only one price p k such that µ k >, then the conclusion is trivial Next, assume that φ takes values in the subset Ω = {p j1, p j2,, p jm } Ω ν, for some indices j 1 < j 2 < < j m For 1 < l m and ε small enough, consider the perturbed strategy φ ε where the amount µ ε k offered at price p k is µ k if k j l 1, k j l, µ ε k = µ k ε if k = j l 1, µ k + ε if k = j l By optimality we must have = d dε Jφε, Φ = λp jl pe λ iφp jl λp jl 1 pe λφp j l 1 ε= By induction on l = 2, 3,, m, we conclude that all quantities in 424, for µ k >, coincide 1

11 2 If φ is any other pricing strategy taking values in the same set {p j1, p j2,, p jm }, then φ is optimal as well Indeed, thanks to 424 we obtain J φ, Φ = = C ν k=1 {β ; φβ=pk } κ p k pe λβ+φp k dβ e λβ dβ = C λ 1 eλκ = Jφ, Φ 3 Let now φ, φ : [, κ] Ω ν be any two optimal pricing strategies, whose values range over the subsets Ω = {p j1, p j2,, p jm }, Ω = {pj 1, p j 2,, p j m }, respectively Since φ is optimal, by 424 there exists a constant C such that p jl pe λφp j l = C l = 1,, m Similarly, since φ is optimal, there exists a constant C such that p j l pe λφp j l = C l = 1,, m Being both optimal, φ and φ achieve the same payoff supported on Ω Ω is also optimal Hence C = C and any strategy We can now define the set Ω opt as the set of all prices that lie in the range of some optimal pricing strategy By the previous analysis, any pricing strategy φ taking values inside Ω opt is optimal Lemma 3 Let the random variable X be of type B, and let φ : [, κ] Ω ν be an optimal strategy Then φ is constant Proof By contradiction, assume that φ has a jump at β ], κ[ Then the necessary conditions 39 hold For ε sufficiently small, define { φβ φ ε β = + if β [β + ε, β ] φβ otherwise { φβ φ ε β = if β [β, β + ε] φβ otherwise for ε <, for ε > By optimality we have d dε Jφε, Φ = 426 ε= 11

12 For any ε ], ε [ sufficiently small, a similar computation as in 421 now yields d 2 dε 2 Jφε, Φ = d dε [ ] φβ p ψ β + ε + Φφβ φβ + p ψ β + ε + Φφβ + = φβ p ψ β + ε + Φφβ φβ + p ψ β + ε + Φφβ + = φβ p ψ β + ε + Φφβ ψ β + ε + Φφβ ψ β + ε + Φφβ + ψ β + ε + Φφβ ψ β + ε + Φφβ + [ ] + φβ p ψ β + ε + Φφβ φβ + p ψ β + ε + Φφβ + ψ β + ε + Φφβ + ψ β + ε + Φφβ + = Aε + Bε A similar argument as in , but using the fact that X is of type B and hence ln ψs < for all s >, we now obtain lim Aε >, lim ε + Therefore, for all ε ], ε ] sufficiently small, d 2 dε 2 Jφε, Φ > Bε = ε + Together with 426, this proves that φ cannot be an optimal strategy Assuming that the variable X is of type B, the following example shows that, for any κ >, one can construct a piecewise constant Φ with exactly one jump such that the optimization problem 414 has exactly two solutions In particular, the solution set is not convex Example 1 Assume ln ψs < for all s >, and let Φp = { if p < pk, α if p p k 427 Since by the previous Lemma any optimal pricing strategy must be constant, the only two optimal candidates are φβ p k 1, or φβ P The corresponding payoffs are Jp k 1 = p k 1 p κ ψβ dβ, 12 JP = P p α+κ α ψβ dβ

13 Consider the function fα = α+κ α ψβ dβ It is easy to see that there exists a solution α to the equation fα = p k 1 p f This follows P p from the fact that f is continuously differentiable, f <, and lim α fα = Hence, by choosing α = α in 427, we see that φβ p k 1 and φβ P are both optimal, and this shows that the set of best replies is not connected, thus not convex 5 Existence of a discrete Nash equilibrium In this section we prove the existence of a Nash equilibrium, when each probability distribution ψ i is either of type A or of type A + Theorem 1 Consider a discrete pricing game for n players, with strategies given by 22 and payoffs as in 25, 26 Assume that the selling priorities are determined by H1 and that each probability distribution ψ i is either of type A or A + Then the game admits a Nash equilibrium Proof 1 The set of of admissible strategies for the i-th player can be identified with the compact convex set ν S i = µ i = µ i1,, µ iν ; µ ij, µ ij = κ i On the cartesian product S = S 1 S n consider the multifunction R : µ 1, µ 2,, µ n R 1 R 2 R n, 528 where R i S i is the set of all best replies for player i to the strategies µ 1, µ i 1, µ i+1, µ n adopted by the other n 1 players j=1 2 By a standard argument we now show that the multifunction R in 528 has closed graph The map µ 1,, µ n J i µ 1,, µ n defined as in 25-26, describing the payoff to the i-th player, is a continuous function on S 1 S n Since each set S i is compact, the map J max i µ 1,, µ i 1, µ i+1,, µ n = max η i S i J i µ 1,, µ i 1, η i, µ i+1,, µ n is also a continuous function of its n 1 arguments Therefore, the set { GraphR = µ 1,, µ n, η 1,, η n ; } J i µ 1,, µ i 1, η i, µ i+1,, µ n = Ji max µ 1,, µ i 1, µ i+1,, µ n for all i = 1,, n 13

14 u ν v ν u i v i v ν u j p i p ν 1 p ν p j p ν 1 p ν Figure 1: Two examples of Nash equilibria for two players with the same payoff function, selling different amounts of asset Here p 1 = p 2 =, ψ 1 s = ψ 2 s = e s is closed 3 Having a closed graph, the multifunction R is upper semicontinous [2] By Lemma 2 in the previous section, if ψ i is of type A then each set R i of best replies has the form ν R i = µ i = µ i1,, µ iν ; µ ij, µ ij = κ i, µ ij = if p j / Ω opt for some subset Ω opt {p 1,, p ν } In particular, each set R i is compact and convex On the other hand, if ψ i is of type A +, then by Lemma 1 each R i reduces to a single point Applying Kakutani s fixed point theorem [2, 5], we obtain an n-tuple of admissible strategies µ 1,, µ n S 1 S n such that µ i R i µ 1, µ i 1, µ i+1, µ n for every i = 1,, n This provides a Nash equilibrium to the discrete pricing game j=1 Example 2 discrete Nash equilibrium for two players Consider a bidding game for two players with the same payoff functional More precisely, assume that in 25 one has p 1 = p 2 = p, ψ 1 s = ψ 2 s = e s Assume that Player 1 has selling priority, in case both players ask the same price For notational convenience, we write v l = µ1l, u l = µ2l Define the quantities i = sup {i ; v k = ln p k p p k 1 p, u k = ln p k+1 p p k p, } n 1 v k κ 2, j = sup j ; u k κ 1 k=i Two cases can arise, depending on the total amounts κ 1, κ 2 of assets put on sale by the two players k=j 14

15 CASE 1: κ 1 ν 1 u k = k=i u k Then a Nash equilibrium is given by: k i 1 u k i k ν 1 κ 1 ν 1 k = ν l=i u l v k = k i 1 κ 2 ν v l l=i +1 k = i v k i + 1 k ν Indeed, the following optimality conditions for Player 2 are satisfied: i Any pricing strategy φ 2 taking values within the set {p i,, p ν } yields the same payoff for Player 2; ii Any strategy φ 2 taking values on a set which is not a subset of {p i,, p ν } yields a strictly lower payoff To verify i consider any pricing strategy φ 2 : [, κ 2 ] {p 1, p 2,, p ν }, taking values inside the set {p i,, p ν } We have Jφ 2 = ν k=i φ 1 2 p k ν = p i pe v i By the definition of i it follows Jφ 2 > p i pe ln p k pe β v i ln p k p p i p dβ k=i φ 1 2 p k e β dβ = p i pe v i 1 e κ 2 p i p p i 1 p 1 e κ 2 = p i 1 p1 e κ 2, which is the payoff corresponding to the constant pricing strategy φ 2 β p i 1 Consider now another strategy, φ ε 2 β where Player 2 sells an amount ε > of shares at price p i 1 The corresponding expected payoff is ε κ2 Jφ ε 2 = p i 1 pe β dβ + p i pe v i e β dβ A simple computation shows that = p i 1 p1 e ε + p i pe v i e ε e κ 2+ε d dε Jφε 2 = e ε [p i 1 p p i pe v i 1 + e κ 2+2ε ] < p i 1 pe κ 2+ε < for all ε >, where we assume that ν is large enough so that p i 1 > p and used the fact that v i < v i We conclude that Jφ ε 2 < Jφ 2, for all ε > proving ii 15 ε

16 With similar arguments, we can verify the same optimality conditions for Player 1 If φ 1 β is any pricing strategy taking values in {p i,, p ν }, then Jφ 1 = ν k=i φ 1 1 p k p β ln k p p p k pe i 1 p dβ = p i 1 p1 e κ 1 and it is clearly not optimal to sell at lower prices, since Player 1 has the priority CASE 2: κ 1 ν 1 u k = k=i u k Then a Nash equilibrium is given by: k j 1 κ 1 ν 1 k = j u l l=j +1 u k j + 1 k ν 1 k = ν v k = k j v k j + 1 k ν 1 κ 2 ν 1 v l k = ν l=j +1 Again,we can show that any strategy φ 1 yields the same payoff to the first player if it takes values in the set {p j +1,, p ν } and that the payoff for Player 2 is the same for any pricing strategy supported on {p j,, p ν 1 } Indeed, we have Jφ 1 = ν k=j +1 uj Jφ 2 = p j p p k p e β dβ + φ 1 2 p k ν 1 k=j +1 e β u p j ln k p p j p p k p φ 1 2 p k = p j +1 pe u j 1 e κ 1, p β ln k p p e j p = p j p1 e κ 2 In Theorem 1, the assumption that every probability distribution is of type A + or A was crucial We now give another example showing that, if each probability distribution is of type B then a Nash equilibrium in general does not exist Example 3 Consider a discrete bidding game for two players, putting on sale κ 1, κ 2 > amounts of shares, and let p 1 p 2 < P be given Assume that ln ψ i s > for i = 1, 2 and all s > As usual, we assume that Player 1 has priority over Player 2 By Lemma 3, every optimal strategy φ i is constant, hence any Nash equilibrium φ 1, φ 2 has the form φ 1β p j1 = j 1P ν, φ 2β p j2 = j 2P ν, for some j 1, j 2 {1,, ν} Clearly p j1 > p 1 and p j2 > p 2 p 1, otherwise one of the payoffs would be negative If ν is large enough, then κ2 +κ 1 p ν ψ 1 βdβ < p ν 1 κ 2 We now observe that κ1 κ1 +κ 2 ψ 1 βdβ, p ν ψ 2 βdβ < p ν 1 κ 1 κ2 ψ 2 βdβ

17 if j 1 = j 2 < ν, then the strategy φ 2 p ν yields a strictly higher expected payoff for Player 2; if j 1 = j 2 = ν, then by 529 the strategy φ 2 p j2 1 yields a strictly higher expected payoff for Player 2; if j 1 < j 2, then the strategy φ 1 p j2 yields a strictly higher expected payoff for Player 1; if j 2 < j 1 < ν, then the strategy φ 1 p ν yields a strictly higher expected payoff for Player 1; if j < j 1, then the strategy φ 2 p j2 +1 yields a strictly higher expected payoff for Player 2; if j 2 = ν 1 and j 1 = ν, then by 529 the strategy φ 2 p ν 1 yields a strictly higher expected payoff for Player 1 Since the above cases exhaust all possibilities, conclude that a Nash equilibrium cannot exist 6 Convergence of discrete approximations In this section we let ν, so that the mesh size P /ν approaches zero We show that any weak limit of discrete Nash equilibria provides a Nash equilibrium for a bidding game where prices are allowed to range continuously over the reals Theorem 2 Let κ 1,, κ n > be given Assume that, for every ν 1, the n-tuple of strategies φ ν 1,, φν n provides a Nash equilibrium to the discrete bidding game in 25, 26 By selecting an infinite subset of indices I IN, one can achieve the pointwise convergence lim ν I, ν φν i β = φ i β β [, κ i ], i = 1,, n, 63 for some nondecreasing functions φ i : [, κ i ] [, P ] The n-tuple φ 1,, φ n provides a Nash equilibrium to the bidding game, with prices ranging continuously over the reals Proof 1 By possibly choosing a subsequence corresponding to an infinite subset of indices I IN, since all functions φ ν i are nondecreasing, by Helly s compactness theorem we can assume the pointwise convergence in 63, for some nondecreasing functions φ 1,, φ n 2 We claim that each limit strategy φ i is optimal for the i-th player, in reply to the left continuous function Φ i p = meas {β [, κ i ] ; φ jβ < p} 631 j i Indeed, let Φ ν i p = meas {β [, κ i ] ; φ ν j β < p} j i 17

18 Since Φ i p is nondecreasing and left continuous, hence lower semicontinuous, for every β [, κ i ] the pointwise convergence φ ν i β φ β yields Therefore lim sup ν By Fatou s lemma, we conclude Φ i φ i β lim inf ν Φν i φ ν i β ψ i β + Φ ν i φ ν i β ψ i β + Φ i φ i β J i φ i, Φ i = lim sup ν κi κi φ i β p i ψ i β + Φ i φ i β dβ φ ν i β p i ψ i β + Φ ν i φ ν i β dβ = lim sup J i φ ν i, Φ ν i ν 632 Next, let φ i : [, κ i ] [, P ] be any admissible strategy for the i-th player and let ε > be given For each ν we can construct a unique discrete-valued strategy ϕ ν i : [, κ i] Ω ν such that This strategy satisfies Indeed φ i β P ν < ϕν i β φ i β for all β [, κ i ] J i ϕ ν i, Φ i > κi J i ϕ ν i, Φ i J iφ i, Φ i κ P i ν 633 φ i β P ν p i ψ i β + Φ i ϕν i βdβ = κi φ i β p i ψ i β + Φ i ϕν i βdβ κ i P ν For all ν 1 sufficiently large we have J i φ i, Φ i κ P i ν J i φ i, Φ i J iϕ ν i, Φ i + ε J iϕ ν i, Φ ν i + 2ε J i φ ν i, Φ ν i + 2ε 634 The first inequality in 634 is an immediate consequence of 633 The second follows from the lower semicontinuity of Φ i and the pointwise relation Φ i p lim inf ν Φ ν i p The third inequality follows from the optimality of φ ν i in response to Φ ν i Together, 634 and 632 yield J i φ i, Φ i lim sup J i φ ν i, Φ ν i + 2ε J i φ i, Φ i + 2ε ν Since ε > is arbitrary, we conclude that φ i is an optimal reply to Φ i 3 Our next goal is to prove that, for each i = 1,, n, the limit function Φ i in 631 is Lipschitz continuous on [, P ] Define δ = min j ψ j κ κ n P p j >

19 We claim that, for every i {1,, n}, it is never optimal for the i-th player to put anything on sale at price p < p i + δ Indeed, by putting the same asset for sale at price P, his expected gain per unit of asset would be This proves our claim ψ j κ κ n P p i δ > p p i 4 Given an integer ν > 2P /δ, let µ ν 1,, µν n be a discrete Nash equilibrium corresponding to the mesh size P ν Fix some k < ν and assume that some agent is selling at price p k, ie µ ν ik > i=1 Let i be the player with lowest priority among those selling at price p k Then, by optimality considering the first share he is selling at price p k we obtain k 1 p k p i ψ µ ν il + k 2 µ ν ik > p k 1 p i ψ µ ν il + µ ν i,k 1 i<i i i l=1 i=1 l=1 i=1 By the mean value theorem, there exists some k 2 ζ µ ν il + µ ν i,k 1, k 1 µ ν il + µ ν i,k i i i<i l=1 i=1 l=1 i=1 [ ], κ i i=1 such that k 2 p k p i ψ l=1 i=1 µ ν il + µ ν i,k 1 + ψ ζ µ ν i,k 1 + i i i>i k 2 > p k 1 p i ψ µ ν il + µ ν i,k 1 i i l=1 i=1 i<i µ ν ik Hence µ ν i,k 1 + i>i i<i µ ν ik ψ k 2 l=1 i=1 µ ν il + ψ ζ i i µ ν ik with c = inf{ ψ ζ ; ζ [, κ i ]} >, and δ as in 635 pk p k 1 p k p i 1 c P /ν δ, 636 In order to estimate µ ν i k, we observe that somebody with higher priority than i must be selling at price p k+1, otherwise Player i would achieve a larger expected payoff by becoming the first seller at price p k+1 Call j < i the first seller at price p k+1 Then, by optimality, we have k k 1 p k+1 p j ψ µ ν il > p k p j ψ µ ν il + µ ν ik i j l=1 i=1 19 l=1 i=1

20 As before, this yields the bound i>j µ ν ik P νc δ 637 Combining 636 and 637, and recalling that j < i, we obtain i=1 µ ν ik 2P νc δ Finally, given any < a < b < P, we have i=1 meas {β ; a < φ ν i β < b} b a P /ν + 1 sup a<p k <b i=1 = a<p k <b i=1 µ ν ik µ ν ik 2 b a + P c δ ν 638 Hence, by letting ν we see that, for any j, meas {β ; φ jβ ]a, b[} 2 b a c δ This implies that every function Φ i is Lipschitz continuous on [, P [ In particular, Φ i p = Φ i p for p < P 5 We claim that, by possibly shrinking the countable set I IN ie, by taking a further subsequence, in the limit φ 1,, φ n at most one of the players puts a positive amount of assets for sale at price P For each ν 1, let µ ν 1,, µν n be a corresponding discrete Nash equilibrium Let ιν {1,, n} denote the player with lowest priority, among those who are selling something at price P : ιν = max{i ; µ ν iν > } By possibly choosing a further subsequence, we can assume that ιν = ι for all ν If ι = 1, there is nothing to prove, since only one player is selling at the highest price If ι > 1, we use the optimality conditions applied to the first share that player ι sells at price P to bound the amount of shares put on sale at P by the remaining players: P p ι ψ κ i µ ν ι ν i=1 > p ν 1 p ι ψ κ i i=1 ι i=1 µ ν iν i>ι µ ν i,ν 1 which, as before, yields ι 1 i=1 µ ν i,ν P νc δ 639 2

21 As ν, the right hand side of 639 approaches zero Combining 638 with 639, for every ε > and i ι we obtain meas {β [, κ i ] ; φ i β = P } lim sup ν meas {β [, κ i ] ; φ ν i β [P ε, P ]} < 2ε c δ Since ε > was arbitrary, for i ι one has meas {β [, κ i ] ; φ i β = P } =, proving our claim 6 According to the previous analysis, for each i ι the map p meas {β [, κ i ] ; φ i β p} is Lipschitz continuous Therefore Jφ i, Φ i = Jφ i, Φ i for all i = 1,, n Recalling step 2 and observing that Φ i p Φ i p, for every i and every admissible strategy φ i : [, κ i ] we have Jφ i, Φ i Jφ i, Φ i Jφ i, Φ i = Jφ i, Φ i proving that the n-tuple of pricing strategies φ 1,, φ n provides a Nash equilibrium As a consequence of Theorem 2, we obtain the existence of a Nash equilibrium for a pricing game with heterogeneous players, where prices can range continuously on the interval [, P ] Corollary 2 Consider a pricing game for n players, with continuum strategies φ i : [, κ i ] [, P ] and payoffs as in 25, 26 Assume that each probability distribution ψ i is either of type A or A + Then the game admits a Nash equilibrium Proof For each ν 1, consider the game where prices range over the discrete set Ω ν = { jp ν ; j = 1, 2,, ν } By Theorem 1, this game has has at least one Nash equilibrium, say φ ν 1,, φν n According to Theorem 2, we can then choose a subsequence such that 63 holds and the pointwise limit φ 1,, φ n provides a Nash equilibrium to the game with prices continuously ranging over [, P ] Remark 2 In the case where all players have the same payoff function and assign the same probability distribution to the random incoming order X, it was proved in [6] that the Nash equilibrium in the continuum model is unique In this case, the entire sequence of discrete Nash equilibria must converge to this unique limit Acknowledgment This research was partially supported by NSF, with grant DMS-11872: Problems of Nonlinear Control 21

22 References [1] A Alfonsi, A Fruth and A Schied, Optimal execution strategies in limit order books with general shape functions, Quantitative Finance 1 21, [2] J P Aubin and A Cellina, Differential inclusions Set-valued maps and viability theory Springer-Verlag, Berlin, 1984 [3] M Avellaneda and S Stoikov, High-frequency trading in a limit order book, Quantitative Finance 8 28, [4] K Bach and S Baruch, Strategic liquidity provision in limit order markets Econometrica, , [5] A Bressan, Noncooperative differential games Milan J of Mathematics, , [6] A Bressan and G Facchi, A bidding game in a continuum limit order book Preprint, 212 [7] R Cont, S Stoikov, and R Talreja, A stochastic model for order book dynamics, Operations Research 58 21, [8] E J Dockner, S Jorgensen, N V Long, and G Sorger, Differential games in economics and management science Cambridge University Press, 2 [9] F Guilbaud, and HPham, Optimal high frequency trading with limit and market orders, arxiv:11654 [1] V Henderson, and D Hobson, Optimal liquidation of derivative portfolios, Mathematical Finance , [11] J Nash, Non-cooperative games, Annals of Math , [12] A Obizhaeva, and J Wang, Optimal trading strategy and supply/demand dynamics, J Financial Markets, , 1 32 [13] T Preis, S Golke, W Paul, and J J Schneider, Multi-agent-based order book model of financial markets Europhysics Letters 75 26, [14] S Predoiu, G Shaikhet, and S Shreve, Optimal execution in a general one-sided limitorder book SIAM Journal on Financial Mathematics 2 21, [15] I Roşu, A dynamic model of the limit order book The Review of Financial Studies 22 29, [16] N Vorob ev, Foundations of game theory Noncooperative games Birkhäuser, Basel, 1994 [17] J Wang, The Theory of Games Oxford University Press,