CONVERGENCE OF PRICE PROCESSES UNDER TWO DYNAMIC DOUBLE AUCTIONS

Transcription

1 Journal of Mechanis and Institution Design ISSN: X (Print), (Online) DOI: /jid CONVERGENCE OF PRICE PROCESSES UNDER TWO DYNAMIC DOUBLE AUCTIONS Jinpeng Ma Rutgers University, USA Qiongling Li Rice University, USA ABSTRACT We study the convergence of two price processes generated by two dynaic double auctions (DA) and provide conditions under which the two price processes converge to a Walrasian equilibriu in the underlying econoy. When the conditions are not satisfied, the price processes ay result in a bubble or crash. Keywords: Double auction echaniss, increental subgradient ethods, network resource allocations. JEL Classification Nubers: D44, D INTRODUCTION A double auction (DA) echanis is a arket-clearing syste by which dispersed private inforation feeds into the syste sequentially through bilateral trading. With little concentrated inforation about total deand and supply of an asset or good available to all participants in the arketplace, it Both authors declare there are no conflicts of interest. This paper supersedes the paper Bubbles, Crashes and Efficiency with Double Auction Mechaniss (Ma & Li, 2011), which has been distributed and presented in various conferences. We thank Mark Satterthwaite for introducing us to the topic. Any errors are our own. Copyright c Jinpeng Ma, Qiongling Li / 1(1), 2016, Licensed under the Creative Coons Attribution-NonCoercial License 3.0,

2 2 Dynaic double auction is natural to ask whether the price process generated by this DA echanis converges to an equilibriu of the underlying econoy or not. Both A. Sith (1776) and Hayek (1945) raise a siilar question how a arket echanis in a laissez-faire econoy, where individual participants with little inforation about total deand and supply act solely in their self-interests, is able to integrate dispersed bits of [incoplete] inforation correctly into prices. A. Sith (1776) uses his faous invisible hand etaphor to describe its agnificence of a price echanis. Hayek (1945, p. 519) has further explored the idea: The peculiar character of the proble of a rational econoic order is deterined precisely by the fact that the knowledge of the circustances of which we ust ake use never exists in concentrated or integrated for, but solely as the dispersed bits of incoplete and frequently contradictory knowledge which all the separate individuals possess. The econoic proble of society is thus not erely a proble of how to allocate given resources [ ], it is rather a proble of the utilization of knowledge not given to anyone in its totality. He goes on by saying: This echanis would have been acclaied as one of the greatest triuphs of the huan ind if It were the result of deliberate huan design (Hayek, 1945, p. 527). It should be noted that DA echaniss eployed in real exchange arkets across the world are deliberately designed by huans. An answer to the question is iportant for understanding price deterination in an exchange arket, since DA echaniss have been widely used in equity, coodity and currency arkets, aong others. For exaple, an answer to the question is vital for understanding the efficient arkets hypothesis (Faa, 1965) and the excess volatility puzzle (Shiller, 1981). Nonetheless, it is not easy to coe up with an answer. Indeed, does a DA echanis atter for the price deterination of an asset? According to the efficient arkets hypothesis, the answer should be no since the price of an asset in an exchange arket should always follow its fundaental, with no systeatic disparity between the two that can be detected with fundaental or technical analysis. On the other hand, excess volatility suggests that the answer ay be yes, since the price of an equity can deviate fro its fundaental to a great degree and such a deviation has been realized by a DA echanis through a sequence of

3 Jinpeng Ma, Qiongling Li 3 trading between buyer and seller pairs. But, if a DA echanis really atters, how is it possible for an equity with fundaental value of 100 to be traded, say, at 300 or 50? The ain objective of this paper is to investigate if a DA echanis can generate a sequence of prices that converges to an equilibriu of the underlying econoy when individual deands and supplies are only privately known. To achieve this goal, we study a benchark odel given below: P iniize F(y) = n f i (y) + g j (y) j=1 subject to y Y, a nonepty convex subset of R d +, where f i and g j are real-valued (possibly non-differentiable) convex functions defined on the d- diensional Euclidean space R d. A large class of quasilinear econoies with sellers and n buyers can be represented by this for (see Section 2.1). For these econoies, the quantity deanded and supplied at prices y for buyer j = 1,2,,n and seller i = 1,2,, are just subsets of the subdifferentials g j (y) and f i (y), respectively, using the Fenchel duality (Ma & Nie, 2003). Thus, an equilibriu of the underlying econoy studied in this paper is an optial solution to the proble P. An Illustrative Exaple. For siplicity, consider an exchange econoy where there is a single object or asset with a finite nuber of identical copies for sale. In a dynaic double auction, a buyer subits a bid order consisting of a bid price and a bid size, and a seller subits an ask order consisting of an ask price and an ask size, with the bid price at least as high as the ask price. The bid size is the quantity the buyer is willing to buy at the bid price and the ask size is the quantity the seller is willing to sell at the ask price. The price of an object is a weighted average of the bid price and the ask price, with weight α (0,1), as in a static double auction in Chatterjee & Sauelson (1983), Myerson & Satterthwaite (1983), Wilson (1985), and Gresik (1991). Thus, given a sequence of pairs of one buyer and one seller, a sequence of prices is generated by a double auction. The next two questions are, at a given iteration, who will be the buyer and the seller pair and how are bid and ask prices are deterined? We provide two specific exaples of double auction to address the two questions. In the first double auction, we assue that the nuber of buyers equals the nuber of sellers, and buyers and sellers for two cyclic rings: a buyer ring and a seller ring (Figure 1). This syste can be realized if the buyer and seller

4 4 Dynaic double auction rings consist of two perutations of agents. A pair of a buyer and a seller is selected according to the two cyclic rings, one pair at a tie. The process starts with price X k at k. Then buyer π (1) and seller π(1) are the first pair to subit their bid and ask, respectively, based on the observed X k. After the iteration of the pair (π (1),π(1)), the next pair will be buyer π (2) and seller π(2). This iteration process ends with the pair (π (),π()) and the price X k+1. Figure 1. Iterations under a double auction, where π and π are two perutations of agents. Sellers ring π(1) π(i) π() Buyers ringπ (1) π (i) π () We need to deterine how a buyer bids and a seller asks. A buyer s bid equals the newly updated price fro the previous pair along the two rings plus a price increent that equals the product of the bid step size and the bid size. The bid step size is the price increent for one unit of the object that the buyer is willing to buy. Thus, the ore a buyer wants to buy, the higher the bid price increent. An ask price is deterined siilarly. A seller s ask price equals the newly updated price fro the previous pair along the two rings inus a price that equals the product of the ask step size and the ask size. The ask step size is now the price decreent for one unit of the object for sale. Thus, the ore a seller wants to sell, the lower the ask price. To be ore precise, let Φ i 1,k be the price at iteration k and designate the next selected pair as (π(i),π (i)). The ask price ψ π(i),k and the bid price ϕ π (i),k are deterined by, respectively, ψ π(i),k = Φ i 1,k a k S π(i) (Φ i 1,k ), ϕ π (i),k = Φ i 1,k +b k D π (i)(φ i 1,k ), (1) where S π(i) (Φ i 1,k ) and D π (i)(φ i 1,k ) are the quantity supplied (i.e. the ask size) and deanded (i.e. the bid size) at price Φ i 1,k, respectively. If S π(i) (Φ i 1,k ) and D π (i)(φ i 1,k ) are set-valued aps, equation (1) should be

5 Jinpeng Ma, Qiongling Li 5 understood with two selections fro the deand and supply. {a k } and {b k } are the ask and bid step sizes, respectively. The price Φ i,k, which is counicated to the next pair, is deterined by a weighted average of the bid and ask prices, with weight α (0,1): Φ i,k = αψ π(i),k + (1 α)ϕ π (i),k. (2) Equations (1) and (2) provide the rule on how the price at an iteration evolves fro one pair to the other along the two rings. The price process starts at Φ 0,k = X k and ends with X k+1 = Φ,k at tie k. Then this process repeats with two different perutations of agents. Thus, a sequence of prices X k, k = 0,1,2,, is generated. Note that we consider the case where is potentially a large nuber. In our second randoized double auction a pair ade up of a buyer and a seller is independently selected. Here we do not need the condition that the nuber of buyers equals the nuber of sellers because such an auction can be seen as a special case, in which the buyer ring and the seller ring in Figure 1 each consist of a single agent. Thus, equations (1) and (2) provide a sequence of prices {X k } once again. Results. Assue that the underlying econoy has a Walrasian equilibriu b and the liit li k k a k exists for two diinishing step sizes {a k } and {b k }. Suppose there is a positive scalar λ such that (see Assuption 3.2) b k n λ a k < +. (3) b Then we show that λ ust be li k k Our first ain result Theore 4.4 deonstrates that the price process {X k } ust converge to a Walrasian equilibriu price vector of the underlying econoy as long as the weight α satisfies the equality α = λ 1+λ. Beyond the existence of Walrasian equilibriu, this convergence result does not depend on privately known deands and supplies. Instead it depends on the two paraeters α and λ related to the auction for. If the weight α does not satisfy the equality α = λ 1+λ, then the price process {X k } still converges to a price but it ay be higher or lower than the equilibriu price(s). A higher than equilibriu price (i.e. bubble) is obtained when α < and a lower than equilibriu price (i.e. crash) is λ 1+λ a k. 1 1 The converse of this clai is not true. See Exaple 4.9.

6 6 Dynaic double auction λ 1+λ obtained when α > by the double auction. For exaple, α ust be right at 1 2 for λ = 1 in order for the auction to arrive at a Walrasian equilibriu. For our randoized double auction, our second ajor result Theore 4.7 shows that the above result still holds when λ is defined by λ = n li k b k a k, where n is the nuber of buyers (or agents) and is the nuber of sellers (or b objects), under Assuption 3.3. For exaple, if li k k a k = 1 and = 2n, then α ust be right at 2 3 for the auction to arrive at a Walrasian equilibriu. Once again, this conclusion does not depend on f i and g j, which are unknown to the echanis designer. The above two results are proved for a general case where there are ultiple heterogeneous objects, with each object having a finite nuber of identical copies. In the general case, {X k } is a sequence of price vectors rather than a sequence of prices. The condition on λ is also stated for the case where the liit li k b k a k ay not exist. Noises are identified as a key factor in the foration of bubbles and crashes (Shleifer, 1999). So it is of interest to see how noises ay change our results. To exaine this issue, we follow Ra et al. (2009) to introduce stochastic noises into buy and sell orders under the two DA echaniss. Interestingly, our ain results still hold for certain noises. This eans that not all noises can affect the inforational efficiency of a DA echanis. The relationship between α and λ is still the key for the convergence of the price processes under the two DA echaniss with stochastic noises. Literature. The benchark odel is the dual proble of the linear prograing relaxation in Bikhchandani & Maer (1997) that can be reforulated as a convex optiization proble in the price space without constraints by P. An optial solution to this dual P (i.e., a iniizer) is a Walrasian equilibriu price vector of the original econoy if the zero duality gap condition holds. 2 They also ention that the ascending price auction designed in Kelso & Crawford (1982) for a noted any-to-one job atching arket can be used to achieve a Walrasian equilibriu price vector for their econoy under the gross substitutes condition. A salient feature of the English auction in Kelso & Crawford (1982) is the constant increase (i.e. step size) of prices for those workers in excess deand at each iteration. Milgro (2000) studies 2 This dual approach and its related gradient ethod in search for an equilibriu in gaes can be traced back to Arrow & Hurwicz (1957). Beyond its applications in econoics and gae theory, the dual approach has any applications in other areas (see, e.g., Bertsekas, 2009; Nedić & Ozdaglar, 2009).

7 Jinpeng Ma, Qiongling Li 7 an econoy siilar to that in Bikhchandani & Maer (1997) and provides an English auction that uses a larger constant price increent in the beginning and a saller constant price increent near the end for an object in excess deand; the price process generated by such an auction can approach a Walrasian equilibriu at a faster speed when the auction begins, with approxiation errors caused by the discrete price increent being reduced near the end of the auction. Xu et al. (2015) provide a subsequent analysis of the two double auctions presented in this paper with constant step sizes and obtain several approxiation error bounds, under a ore general inforation counication structure than Figure 1, when α is at rando with unknown distributions. It is often a challenging task to design an efficient auction when the gross substitutes condition is not satisfied and agents have private reservation values over bundles. Sun & Yang (2009, 2014) develop new auction echaniss that can approach an efficient allocation when goods are substitutes and copleents. There are several other auction echaniss in the literature, see, e.g., Ausubel (2004), Gul & Stacchetti (2000), and references in both Milgro (2000) and in Sun & Yang (2014). An English auction is an ascending forat of auction in which the price of an ite is gradually increased and adjusted in each round according to reported total deand for the given total supply. When the scale of the arket is sall, this ay not cause a proble. However, when it is large, it can be difficult to know the total deand at each round or iteration. A double auction is different, since the process involves pairs of buyers and sellers at each oent in tie. A question arises as to whether such an auction process can always achieve a Walrasian equilibriu outcoe as is the case with an English auction. The question is a challenge because an equilibriu ust be defined with respect to the true deand and supply in totality. But, there is no way to know the total deand and supply in a double auction at any oent in tie. Our double auctions are largely otivated by increental subgradient ethods such as those studied in Kibardin (1980), Nedić & Bertsekas (2001), Ra et al. (2009), and Solodov & Zavriev (1998). In an increental subgradient ethod, a single sequence of step sizes is used. They are effective in a unilateral arket where one single agent updates her inforation into the price process. However, they are not as useful in a bilateral arket situation using double auctions, where there are two sequences of step sizes, one for the buyers and the other for the sellers. A coordination (or steering ) condition between the two step sizes, such as the one on λ, is required for the convergence of

8 8 Dynaic double auction the price process. If the λ condition fails to hold, our convergence results also fail, as shown in Xu et al. (2015, 2016) using nuerical siulations. Following the current study, Xu et al. (2014, 2015, 2016) also deonstrate several convergence results for the two double auctions when agents for soe Markovian chains in the iteration process. Moreover, Xu et al. (2015) study the convergence of price processes under the two double auctions when the weight α and two step sizes are independently drawn at rando, with unknown distributions. A ajor issue with double auctions is proving the existence of a Nash equilibriu; see, e.g., Satterthwaite & Willias (1989) and Jackson & Swinkels (2005). However, interdependent reservation values over bundles, assuing unit deand or supply, 3 are not a ajor concern for a Nash equilibriu. Our two dynaic double auctions are designed for an environent in which such values play an iportant role, in a odel such as the any-to-one job atching in Kelso & Crawford (1982) and the ultiple unit deand econoy in Bikhchandani & Maer (1997). These odels ay also be used for solving resource allocation probles in large scale distributional networks where agents hold dispersed private inforation; see Subsection 2.2 and e.g., Kelly et al. (1998), Nedić & Ozdaglar (2009), Ra et al. (2009). This analysis has the potential to provide solutions to real world probles such as the business to business trading in a arketplace or in ultiagent coordination systes in artificial intelligence (Xia et al., 2005). The rest of the paper is organized as follows. Section 2 introduces the odel. Section 3 presents the two DA echaniss and the ain assuptions. Section 4 discusses the ain results with DA echaniss without stochastic noises. Section 5 establishes the ain results with stochastic noises. Section 6 concludes. 2. MODEL We consider the following general proble (see also Bertsekas, 2012): P iniize F(y) f (y) + g(y) subject to y Y, 3 Kojia & Yaashita (2016) offer a latest noted exception along the line of Myerson & Satterthwaite (1983).

9 Jinpeng Ma, Qiongling Li 9 where f = n f i and g = g j. j=1 For all i = 1,2,, and j = 1,2,,n, f i : R d R and g j : R d R are convex functions and Y is a nonepty convex subset of R d +. As Bertsekas (2012) has deonstrated, such a for covers a large class of probles in the literature: a). least squares and related inference probles; b). dual optiization in separable proble; c). probles with any constraints; d). iniization of an expected value - stochastic prograing; e). Weber proble in location theory; f). distributed increental optiization-sensor networks. Here we focus on an application of the proble P to exchange econoies with indivisible assets or goods. Thus we ay assue that the price space Y is copact. Since every convex function ϕ on a copact set Y is regular Lipschitzian, the set of subgradients ϕ(y) for every y Y is a nonepty, copact, and convex set, where ϕ(y) is defined by ϕ(y) = {η ϕ(y) + η,w y ϕ(w), w}; see e.g., Clarke et al. (1988). For any two regular functions ϕ and ψ at y, the su ϕ + ψ is regular at y and We use the following notation (ϕ + ψ)(y) = ϕ(y) + ψ(y). F = inf y Y F(y), Y = {y Y F(y) = F }, dist(y,y ) = inf y Y y y where denotes the Euclidean nor Benchark Econoies with Multiple Indivisible Objects We now show how the proble P naturally captures a large class of econoies or arkets typified by the any-to-one job atching odel of Kelso & Crawford (1982) and its related econoy with indivisible objects in Bikhchandani & Maer (1997). Let M = {1,2,,} denote the set of objects and N = {1,2,,n} denote the set of agents. An agent j s reservation value function u j : 2 M R + is defined over bundles of objects in M such that u j (/0) = 0. A feasible allocation

10 10 Dynaic double auction Z is a partition (Z 0,Z 1,,Z n ) of all objects in M, in which agent j is allocated with the bundle Z j and Z 0 is the unsold bundle. Let Z denote the set of all feasible allocations. A feasible allocation Z is Pareto optial or efficient if V n u j (Z n j ) u j (Z j ), Z Z. j=1 j=1 Given a price vetor p R +, agent j s deand D j (p) consists of all bundles of M that axiize his surplus, i.e., D j (p) = {S M u j (S) i S p i u j (T ) t T p t, T M}. A pair (Z, p) of a feasible allocation Z Z and a price vector p R + is a Walrasian equilibriu if p z = 0 for all z Z 0 and Z j D j (p) for all j N. It is well-known that a Walrasian equilibriu allocation is efficient. Even though an efficient allocation always exists, a Walrasian equilibriu ay not exist, due to the nature of interdependent reservation values. Objects that are copleent often cause the proble. Two goods satisfy the gross substitutes (GS) condition if a good that is in deand and whose price is not raised will still be in deand if the price of the other good arises. The GS condition of Kelso & Crawford (1982) is a sufficient condition for existence of a Walrasian equilibriu (Bikhchandani & Maer, 1997; Gul & Stacchetti, 1999). Extensive studies of this condition can be found in Fujishige & Yang (2003), Hatfield & Kojia (2010), Hatfield & Milgro (2005) aong others. Econoies that include copleentary goods have been studied by Sun & Yang (2006, 2008, 2014). Given p R +, define π j (p) = u j (S) i S p i for S D j (p). Note that π j (p) is convex. Then the dual of the linear prograing relaxation of the integer prograing in Bikhchandani and Maer can be seen as a convex iniization proble without constraints: V = in V (p) p R + p i + n π j (p). (4) j=1 Then, we have V (p) V for all p R +. Define e : 2 M R by e i (S) = 1 for i S and e i (S) = 0 otherwise. We can write the deand correspondence ˆD j : Y R by ˆD j (p) = {e(s) S D j (p)}. Define g j (p) = π j (p) for all j = 1,2,,n and f i : R n + R by f i (p) = p i for all i = 1,2,,. Thus we obtain the general for F(p) = f (p) + g(p) subject to p Y R +. Note that the supply of an asset is the interval [0,1] at zero price. It follows fro the Fenchel duality that coŝ i (p) = f i (p) and

11 Jinpeng Ma, Qiongling Li 11 g j (p) = co ˆD j (p) for all p Y, where coc denotes the closed convex hull of the set C (Ma & Nie, 2003). A vector y is an optial solution in Y if and only if 0 n coŝ i (y) j=1 co ˆD j (y). Thus, a price vector p Y is at an equilibriu only if 0 ( f + g)(p). Because P is a dual of the linear prograing relaxation of the prial integer prograing in Bikhchandani & Maer (1997) for finding an efficient allocation, a solution y to P is a Walrasian equilibriu if and only if the duality gap is zero, i.e., V (y) = V (Bikhchandani & Maer, 1997; Ma & Nie, 2003). Note that the duality gap approaches zero for a large scale econoy (as and n go to infinite) (Bertsekas, 2009) A Congestion Control Proble with Production We introduce a data transission or congestion control proble on a given network with each link a production function (see, e.g., Kelly et al., 1998). Let N = {1,2,,n} denote the set of sources and L = {1,2,,L} the set of all undirected links. Each link l L has an increasing and convex cost function c l : [0,) [0,) such that c l (0) = 0, i.e., it costs a link l the aount c l (q) to produce capacity q 0. Let L(i) L denote the set of links used by source i N. The utility function for a source i is defined by u i : [0,) [0,), which is assued to be increasing and concave. That is, source i gains a utility u i (x i ) when it sends data at a transission rate x i. Let N(l) = {i N l L(i)} denote the set of sources that use link l. Let p R L + denote a price vector, i.e., a link charges p l per unit rate (e.g., packets per second) of data transission (Kelly et al., 1998). Define e : 2 L R L by e l (S) = 1 if l S and e l (S) = 0 otherwise. Define the supply function S l : R L + [0,) by S l (p) = {q p l q c l (q) p l z c l (z), z 0} and the deand function D i : R L + R L + by D i (p) = {e(l(i))x i u i (x i ) x i l L(i) p l u i (z) z l L(i) p l, z 0}. A triplet (p;x,q) R L + R n + R L + is a network equilibriu if a). e(l(i))x i D i (p) for all i N ; b). q l S l (p) for all l L ; c). i N(l) x i q l for all l L. Define f l (p) = p l q c l (q),q S l (p) for all l L, and g i (p) = u i (x i ) x i l L(i) p l,e(l(i))x i D i (p) for all i N. Let f (p) = l L f l (p) and g(p) = i N g i (p). We obtain the proble P, subject to p R L +. Thus, one can show that (p;x,q) is an equilibriu iff p is an optial solution to the proble P. Note that Y is nonepty. So an equilibriu always

12 12 Dynaic double auction exists with transission rates that are divisible. The proble P foralized this way is in fact the dual proble of a utility axiization proble on networks. See Kelly et al. (1998), and the exaples discussed in Nedić & Ozdaglar (2009) and Ra et al. (2009), where each link is given with a fixed capacity, no production available. A flexible capacity network is often needed in practice. Under soe ild assuptions, the first social welfare theore holds (i.e., the duality gap is zero). Thus, finding an equilibriu under the proble P is one way to solve the prial utility axiization proble. 3. TWO DOUBLE AUCTIONS We introduce two new dynaic double auctions. One double auction is designed based on the bilateral cyclic structure presented in Figure 1. The other one is based on a rando atch between the sellers and the buyers A Cyclic Double Auction (CDA) Assue that = n, which holds if the initial endowents are owned by the n agents, who act as both sellers and buyers. CDA Mechanis: Let Φ 0,k = X k. For i = 1,2,,, let ψ i,k = Φ i 1,k a k f i (Φ i 1,k ), (5) ϕ i,k = Φ i 1,k b k g i (Φ i 1,k ), (6) Φ i,k = P Y (αψ i,k + (1 α)ϕ i,k ), α (0,1), (7) where f i (Φ i 1,k ) f i (Φ i 1,k ) and g i (Φ i 1,k ) g i (Φ i 1,k ). P Y is the Euclidean projection onto Y because the weighted average of the bid and ask prices ay be out of Y in the process. Let X k+1 = Φ,k. (8) Our explanation of this auction is as follows. X k is the initial price vector for tie k. We want to obtain X k+1 with a round of iterations. To accoplish the task, the sellers and buyers for a cyclic seller ring and a cyclic buyer ring in two rando orders, i.e., two rando perutations of the agents. Then we renae these agents in the two rings with i = 1,2,,. Based on the initial price vector Φ 0,k = X k, seller 1 and buyer 1 are the first pair to subit an ask

13 Jinpeng Ma, Qiongling Li 13 order and a bid order to deterine Φ 1,k, which is then counicated to the next pair, seller 2 and buyer 2. This process continues until it reaches the last pair, seller and buyer, to obtain Φ,k. The price vector X k+1 is set to be Φ,k to end the iterations in the cycle. After obtaining X k+1, the agents are reshuffled and renaed with two new rando perutations and the process proceeds with X k+1 in the sae anner as with X k. The auction starts with a given X 0 Y R d ++, and generates a sequence of price vectors {X k }, k 0. The ask order for seller i at k consists of a vector of ask prices ψ i,k and an ask size f i (Φ i 1,k ), where f i (Φ i 1,k ) is on the supply curve f i (Φ i 1,k ). The relationship between ask prices and sizes is given by equation (14), where a k is the ask step size at round k, which is the price decreent for one unit of object for sale. The ask prices are lower than Φ i 1,k, the newly updated prices fro the previous pair. The ore the seller wants to sell an object, the lower the ask prices. Siilarly, the bid order for buyer i at k consists of a vector of bid prices ϕ i,k and a bid size g i (Φ i 1,k ), where g i (Φ i 1,k ) is on the deand curve g i (Φ i 1,k ). The relationship between bid prices and sizes is given by equation (15), where b k is the bid step size at round k, which is the price increent for one unit of object to buy. The bid prices are higher than Φ i 1,k, the newly updated prices fro the previous pair. The ore the buyer wants to buy an object, the higher the bid prices. The prices Φ i,k are a weighted average of the ask and bid prices, with a weight in (0,1), as in Chatterjee & Sauelson (1983). We provide conditions on the weight α and the two step sizes {a k } and {b k } so that the price process {X k } converges to an optial solution in Y. It ay be useful to copare CDA with the noted cyclic increental subgradient ethod (e.g., Nedić & Bertsekas, 2001). Increental Subgradient Method: Assue = n and X 0 Y. Let Φ 0,k = X k. For i = 1,2,,, let Φ i,k = P Y (Φ i 1,k a k ( f i + g i )(Φ i 1,k )), where ( f i + g i )(Φ i 1,k ) ( f i + g i )(Φ i 1,k ). Let X k+1 = Φ,k. P Y is the Euclidean projection onto the set Y. In the increental subgradient ethod, agents for a single ring in an arbitrary order (Figure 2). Prices are iterated along the ring one agent at a tie. At each iteration, only one agent i reveals his buy size in g i (Φ i 1,k )

14 14 Dynaic double auction Figure 2. An Iteration under the Increental Subgradient Method. f 1 + g 1 f i + g i f + g together with his sell size in f i (Φ i 1,k ) at prices Φ i 1,k. A key feature of their algorith is that adjustent in prices at each iteration depends on the chosen step size a k and the individual excess supply ( f i + g i )(Φ i 1,k ). They show that if the step size {a k } is diinishing, their algorith generates a sequence of prices that converges to an equilibriu in Y. CDA is different fro the increental subgradient ethod in two ways. First, the buyer and seller cyclic rings are different. We consider f and g as two different sides of the arket. Second, there are two sequences of step sizes {a k } and {b k } in CDA. This akes the convergence results for the increental subgradient ethod inapplicable to CDA because there is a new λ condition. Even if the λ condition is satisfied, the weight α and the paraeter λ ust be in a right cobination so that the process {X k } can converge to an optial solution in Y. We need to consider two step sizes because the arket using double auctions is bilateral, in contrast to the unilateral arket under the increental subgradient ethod A Randoized Double Auction (RDA) Let w k be a rando variable taking equiprobable values fro the set {1,2,,} and w k be a rando variable taking equiprobable values fro the set {1,2,,n}. Let f wk (X k ) f wk (X k ) and g w k (X k ) g w k (X k ), where if w k takes a value i, then the vector f wk (X k ) is f i (X k ), siilarly for g. Here we do not need to assue that = n. Our sequence {X k } is generated by RDA echanis as below. RDA Mechanis: Given X k, let ψ wk,k+1 = X k a k f wk (X k ), (9)

15 Jinpeng Ma, Qiongling Li 15 ϕ w k,k+1 = X k b k g w k (X k ). (10) Let X k+1 = P Y (αψ wk,k+1 + (1 α)ϕ w k,k+1), (11) where α (0,1) and P Y is the Euclidean projection onto Y. Our explanation of this auction is as follows. X k is the price vector at tie k. We want to obtain X k+1 by incorporating individual deand and supply inforation fro a pair of buyer and seller, with the seller randoly selected fro the set of sellers and the buyer randoly selected fro the set of buyers. The bid prices and bid size as well as the ask price and ask bid are deterined in the sae way as in CDA. This is equivalent to the case where a seller randoly atches with a buyer. Again, the auction starts with X 0 Y. Then (9)-(11) generate a sequence of prices X k, k = 1,2,. We are interested in the convergence of {X k } and the conditions under which {X k } converges to an optial solution in Y. Because f and g are privately known to agents, not to the echanis designer, our conditions cannot be iposed on f and g. This provides a challenge because inforation about f and g has never been revealed in totality in the two auction processes. The two auction fors are in spirit close to the decentralized arket echanis as stated in A. Sith (1776) and Hayek (1945), in contrast to a centralized arket echanis Key Assuptions We assue that the step sizes satisfy the following diinishing conditions in Assuption 3.1, which is standard for any convergence results in the literature. Assuptions 3.2 and 3.3 are the two new assuptions for two double auctions CDA and RDA. To achieve an equilibriu in Y, we ust have a right cobination of α and λ. Assuption 3.1 (Diinishing step sizes). Assue that the two sequences {a k } and {b k } of step sizes are such that (i). a k > 0 and b k > 0; (ii). a k = + and b k = +; (iii). a2 k < + and b2 k < +. Assuption 3.2. Assue that the two sequences {a k } and {b k } of step sizes

16 16 Dynaic double auction are such that there exists soe positive λ to ensure b k λa k < +. (12) Assuption 3.3. Assue that the two sequences {a k } and {b k } of step-sizes are such that there exists soe positive λ to ensure b k n λ a k < +. (13) Note the difference between Assuptions 3.2 and 3.3. Assuption 3.2 is for CDA where = n, while Assuption 3.3 is for RDA where ay be different fro n. These λ conditions (12) and (13) will be discussed in Subsection MAIN RESULTS In this section we prove our two ain results for CDA (5)-(8) and RDA (9)-(11). Lea 4.1 below is the key for both echaniss Main Result: CDA Since Y is copact, there exist scalars C 1,C 2,,C and D 1,D 2,,D such that h C i, h f i (X k ) f i (Φ i 1,k ),i = 1,2,,,k = 0,1,2, and l D i, l g i (X k ) g i (Φ i 1,k ),i = 1,2,,,k = 0,1,2,. Note that = n. Lea 4.1. Let {X k } be the sequence generated by CDA (5)-(8). Then for all y Y and k 0, we have X k+1 y 2 X k y 2 2a k α( f (X k ) f (y)) 2b k (1 α)(g(x k ) g(y)) + (αa k C + (1 α)b k D) 2,

17 Jinpeng Ma, Qiongling Li 17 where C = C i and D = D i. Reark. This lea shows that the quadratic distance of the price process {X k } to an equilibriu in Y can be bounded and that it is possible for the price process to approach an equilibriu in Y. Nonetheless, a naive choice y in Y does not work because of this ter [a k α f (X k ) + b k (1 α)g(x k )] [a k α f (y) + b k (1 α)g(y)], which is not a su for f + g as defined in P. For a choice y in Y, there is no guarantee that the ter [a k α f (X k ) + b k (1 α)g(x k )] [a k α f (y) + b k (1 α)g(y)] is always nonnegative. This is why the convergence results for the increental subgradient ethod with diinishing step sizes in Nedić & Bertsekas (2001) does not apply to CDA. Proof. Denote h i,k = f i (Φ i 1,k ) and l i,k = g i (Φ i 1,k ) for all i = 1,2,,. By the non-expansive property of projection, we have Φ i,k y 2 αψ i,k + (1 α)ϕ i,k y 2 = α(ψ i,k y) + (1 α)(ϕ i,k y) 2 = α 2 ψ i,k y 2 + (1 α) 2 ϕ i,k y 2 +2α(1 α) (ψ i,k y),(ϕ i,k y) = α 2 Φ i 1,k y a k h i,k 2 + (1 α) 2 Φ i 1,k y b k l i,k 2 +2α(1 α) (Φ i 1,k y a k h i,k ),(Φ i 1,k y b k l i,k ) = α 2 Φ i 1,k y 2 2a k α 2 h i,k,(φ i 1,k y) + α 2 a 2 k h i,k 2 +(1 α) 2 Φ i 1,k y 2 2b k (1 α) 2 l i,k,(φ i 1,k y) +(1 α) 2 b 2 k l i,k 2 + 2α(1 α) Φ i 1,k y 2 2α(1 α)a k h i,k,(φ i 1,k y) 2α(1 α)b k l i,k,(φ i 1,k y) +2α(1 α)a k b k h i,k,l i,k = Φ i 1,k y 2 2(1 α)b k l i,k,(φ i 1,k y) + αa k h i,k + (1 α)b k l i,k 2 2αa k h i,k,(φ i 1,k y) Since we have h i,k C i, l i,k D i for all k = 0,1,2,, we obtain Φ i,k y 2 Φ i 1,k y 2 + (αa k C i + (1 α)b k D i ) 2 2 (αa k h i,k + (1 α)b k l i,k ),(Φ i 1,k y)

18 18 Dynaic double auction Suing over i = 1,2,,, we get So we have Φ i,k y 2 2 Φ i 1,k y 2 + X k+1 y 2 X k y (αa k C i + (1 α)b k D i ) 2 (αa k h i,k + (1 α)b k l i,k ),(Φ i 1,k y) By the definition of subgradients h i,k and l i,k, (αa k C i + (1 α)b k D i ) 2 (αa k h i,k + (1 α)b k l i,k ),(Φ i 1,k y) h i,k,(y Φ i 1,k ) f i (y) f i (Φ i 1,k ) and Then l i,k,(y Φ i 1,k ) g i (y) g i (Φ i 1,k ). X k+1 y 2 X k y 2 2(1 α)b k (g i (Φ i 1,k ) g i (y)) So + k C i + (1 α)b k D i ) (αa 2 2αa k ( f i (Φ i 1,k ) f i (y)) X k+1 y 2 X k y 2 2αa k ( f (X k ) f (y)) 2(1 α)b k (g(x k ) g(y)) 2αa k ( f i (Φ i 1,k ) f i (X k )) + 2(1 α)b k (g i (Φ i 1,k ) g i (X k )) (αa k C i + (1 α)b k D i ) 2 Next we need to estiate f i (Φ i 1,k ) f i (X k ) and g i (Φ i 1,k ) g i (X k ). Lea Φ i 1,k X k i 1 j=1 (αa kc j + (1 α)b k D j ).

19 Jinpeng Ma, Qiongling Li 19 Proof. We show Lea by induction. Note that Φ 0,k X k = 0. Assue that it holds for i 1. Then Φ i,k X k = (αψ i,k + (1 α)ϕ i,k ) X k α ψ i,k X k + (1 α) ϕ i,k X k α Φ i 1,k a k h i,k X k + (1 α) Φ i 1,k b k l i,k X k Φ i 1,k X k + αa k C i + (1 α)b k D i by induction hypothesis i 1 j=1 This copletes the proof of Lea and So (αa k C j + (1 α)b k D j ) + αa k C i + (1 α)b k D i. f i (Φ i 1,k ) f i (X k ) g i (Φ i 1,k ) g i (X k ) Plugging into (14), we have i 1 j=1 C i (αa k C j + (1 α)b k D j ) i 1 D i (αa k C j + (1 α)b k D j ). j=1 X k+1 y 2 X k y 2 2αa k ( f (X k ) f (y)) 2(1 α)b k (g(x k ) g(y)) +2 + i 1 (αa k C i + (1 α)b k D i ) (αa k C j + (1 α)b k D j ) (αa k C i + (1 α)b k D i ) 2 = X k y 2 2αa k ( f (X k ) f (y)) 2(1 α)b k (g(x k ) g(y)) +( (αa k C i + (1 α)b k D i )) 2 = X k y 2 2αa k ( f (X k ) f (y)) 2(1 α)b k (g(x k ) g(y)) +(αa k C + (1 α)b k D) 2. This copletes the proof of Lea 4.1. Reark. To apply Lea 4.1, we ust choose a right y for CDA. To do so, we need to define the following P(α,λ): j=1 P(α,λ) iniize y Y F(y,α,λ) (α f + λ(1 α)g)(y)

20 20 Dynaic double auction where α (0,1). The paraeter λ is soe positive scalar. Then we introduce the following notation F (α,λ) = inf y Y F(y,α,λ), Y (α,λ) = {y Y F(y,α,λ) = F (α,λ)}, and dist(x,y (α,λ)), the Euclidean distance. Note that Y and Y (α,λ) are related but they are not the sae, depending on α and λ. Proposition 4.2. Let Assuptions 3.1 and 3.2 hold. Let {X k } be the price sequence generated by CDA (5)-(8). Then li inf k dist(x k,y (α,λ)) = 0. Proof. Fro Lea 4.1, we obtain for all y Y (α,λ) and k 0, X k+1 y 2 X k y 2 2(b k λa k )(1 α)(g(x k ) g(y )) 2a k [(α f + (1 α)λg)(x k ) (α f + (1 α)λg)(y )] +(αa k C + (1 α)b k D) 2. Since Y is copact, g is continuous, iage (g(y )) is bounded. That eans there exists M > 0 such that g(y) M for all y Y. Hence, for any N = 1,2,, 0 X N+1 y 2 X 0 y ( N N = I II + III + IV. N (αa k C + (1 α)b k D) 2 a k [(α f + (1 α)λg)(x k ) (α f + (1 α)λg)(y )] b k λa k ) (1 α) 2M I is a constant. When N goes to infinity, III < + since b k λa k < + and IV 2(α 2 C 2 a 2 k + (1 α)2 D 2 b 2 k ) < +.

21 Jinpeng Ma, Qiongling Li 21 Thus, II < +. We obtain li inf k [(α f + (1 α)λg)(x k) (α f + (1 α)λg)(y )] = 0. Otherwise, δ > 0 such that for any natural nuber k (α f + (1 α)λg)(x k ) (α f + (1 α)λg)(y ) > δ. And then II > δ a k = +, a contradiction. Now take a subsequence {X nk } of {X k } such that 0 (α f + (1 α)λg)(x nk ) (α f + (1 α)λg)(y ) < 1 k. By the fact that Y is copact, {X nk } has at least one accuulation point y 0, say, and since α f + (1 α)λg is continuous, we have that li (α f + (1 α)λg)(x n k ) = (α f + (1 α)λg)(y 0 ). k By the definition of {X nk }, we know that li (α f + (1 α)λg)(x n k ) = (α f + (1 α)λg)(y ), y Y (α,λ). k Hence, y 0 Y (α,λ). So This copletes the proof. li inf k dist(x k,y (α,λ)) = 0. Proposition 4.3. Let Assuptions 3.1 and 3.2 hold. Then the sequence {X k } in Proposition 4.2 converges to an optial solution y 0 Y (α,λ). Proof. Let δ k = 2a k [(α f + (1 α)λg)(x k ) (α f + (1 α)λg)(y 0 )] +2( b k λa k ) (1 α) 2M + (αa k C + (1 α)b k D) 2 > 0.

22 22 Dynaic double auction Then δ k = II + III + IV < + (see the proof of Proposition 4.2). We also have that X k+1 y 0 2 X k y (αa k C + (1 α)b k D) 2 2a k [(α f + (1 α)λg)(x k ) (α f + (1 α)λg)(y 0 )] +2( b k λa k ) (1 α) 2M X k y δ k. Then applying Proposition 1.3 in Correa & Learéchal (1993) to the result in Proposition 4.2, we have that li X k = y 0. k λ 1+λ and Theore 4.4. Let Assuptions 3.1 and 3.2 hold. Assue that α = α (0,1). Then the sequence {X k } generated by CDA (5)-(8) converges to an optial solution in Y. Proof. It follows fro Proposition 4.3 and the definition of Y (α,λ), which is the sae as Y when (1 α)λ = α holds. This copletes the proof. b Reark. Suppose the liit li k k a k exists and Assuptions 3.1 and b 3.2 hold. Then we will see in Section 4.4 that λ = li k k a k. Assue that λ = 1. Then Theore 4.4 shows that the price process {X k } converges to a Walrasian equilibriu of the underlying econoy if α = 1 2. Once λ changes, we ust also change α accordingly in order to achieve a Walrasian equilibriu. Otherwise the price process {X k } still converges but it ay not converge to a Walrasian equilibriu of the original econoy. In particular, Theore 4.4 fails if Assuption 3.2 does not hold, as shown in Xu et al. (2015, 2016) by nuerical siulations Main Result for RDA Assuption 4.5. The sequence {w k }({w k }) is a sequence of independent rando variables, each uniforly distributed over the set {1,2,,} ({1,2,,n}).

23 Jinpeng Ma, Qiongling Li 23 Furtherore, the two sequences {w k } and {w k } are independent of the sequence {X k }. Since Y is copact, f and g are regular, we have that the two sets of subgradients { f wk (X k ),k = 0,1,2, } and { g w k (X k ),k = 0,1,2, } are bounded. That is, there exist soe positive constants C 0 and D 0 such that, with probability 1, f wk (X k ) C 0 and g w k (X k ) D 0, k 0. Proposition 4.6. Let Assuptions 3.1, 3.3, and 4.5 hold. Then the sequence {X k } generated by RDA (9)-(11) converges to an optial solution in Y (α,λ) with probability 1. Proof. Since Y is copact and g is continuous, there exists M such that g(y) M for all y Y. We obtain for all k and y Y (α,λ), as in the proof of Proposition 4.2, by applying Lea 4.1 to the case with = 1: E{ X k+1 y 2 F k } X k y 2 2(1 α) b k n (g(x k) g(y)) +(αa k C 0 + (1 α)b k D 0 ) 2 2α a k ( f (X k) f (y)) X k y 2 + (αa k C 0 + (1 α)b k D 0 ) 2 2a k [(α f + λ(1 α)g)(x k) (α f + λ(1 α)g)(y)] +2 b k n λ a k g(x k) g(y) X k y 2 + 4M b k n λ a k + (αa kc 0 + (1 α)b k D 0 ) 2 2a k [(α f + λ(1 α)g)(x k) (α f + λ(1 α)g)(y)] where F k = {X 0,X 1,,X k }. Two definitions are in order. A saple path is a sequence of {X k }. For each y Y (α,λ), let Ω y denote the set containing all saple paths {X k } such that 2 a k [(α f + λ(1 α)g)(x k) (α f + λ(1 α)g)(y )] K < +, and that { X k y } converges. We need the following.

24 24 Dynaic double auction Superartingale Convergence Theore (Theore 3.1 in Nedić & Bertsekas, 2001). Let X k,z k and W k, k = 0,1,2,, be three sequences of rando variables and let F k, k = 0,1,2,, be sets of rando variables such that F k F k+1 for all k. Suppose that: (a) The rando variables X k, Z k, and W k are nonnegative, and are functions of the rando variables in F k. (b) For each k, we have E{X k+1 F k } X k Z k +W k. (c) There holds W k <. Then, we have Z k <, and the sequence X k converges to a nonnegative rando variable X, with probability 1. By the superartingale convergence theore, for each y Y (α,λ), we have that Ω y is a set of probability 1. Let {ν i } be a countable subset of the relative interior relint(y (α,λ)) that is dense in Y (α,λ). Define Ω = Ω νi. Then Ω has probability 1 since Prob( Ω νi ) i Prob( Ω νi ) = 0. For each saple path in Ω, the sequence X k ν i converges so that {X k } is bounded. By 2 we have a k [(α f + λ(1 α)g)(x k) (α f + λ(1 α)g)(y)] K < +, li [(α f + λ(1 α)g)(x k) (α f + λ(1 α)g)(y)] = 0. k Otherwise, if there exist δ > 0 such that for all k, then we have 2 (α f + λ(1 α)g)(x k ) (α f + λ(1 α)g)(y) > δ, a k [(α f + λ(1 α)g)(x k) (α f + λ(1 α)g)(y)] > δ which is ipossible. a k = +,

25 Jinpeng Ma, Qiongling Li 25 Continuity of α f + λ(1 α)g iplies that all the liit points of {X k } are belong to Y (α,λ). Since {ν i } is a dense subset of Y and X k v i converges, it follows that {X k } cannot have ore than one liit point, so it ust converge to soe vector y Y (α,λ). This copletes the proof of Proposition 4.6. Reark. In the proof above, we choose y in Y (α,λ). With such a choice, there is no guarantee that the ter 2α a k ( f (X k) f (y )) 2(1 α) b k n (g(x k) g(y )) is nonnegative for all k, as required in superartingale convergence theore. This is why we need Assuption 3.3 because we need to take the ter 4M b k n λ a k out of it. Theore 4.7. Let Assuptions 3.1, 3.3 and 4.5 hold. Assue that (1 α)λ = α and α (0,1). Then the sequence {X k } generated by RDA (9)-(11) converges to an optial solution in Y with probability 1. Proof. It follows fro Proposition 4.6 and the definition of Y (α,λ), which coincides with Y when the condition (1 α)λ = α holds. This copletes the proof. Reark. Under the assuptions in Theore 4.7, and also assue that b the liit li k k a k exists and equals 1. Moreover, assue that = 2n. Then λ = 2 because λ = n li k b k a k, by Proposition 4.10 in Section 4.4. Theore 4.7 shows that the price process {X k } converges to a Walrasian equilibriu of the underlying econoy, with probability 1, if α = 3 2. Once λ, n or has a change, we ust also change α accordingly in order to achieve a Walrasian equilibriu. Otherwise the price process {X k } still converges but it ay not converge to a Walrasian equilibriu of the underlying econoy. Once again, Assuption 3.3 is the key. If it does not hold, then Theore 4.7 fails again, as shown in Xu et al. (2015, 2016) A Nuerical Exaple The following exaple has been also studied by Xu et al. (2014, 2015, 2016). Now we use this exaple to illustrate why Theores 4.4 and 4.7 ay fail to

26 26 Dynaic double auction converge to a Walrasian equilibriu of the underlying econoy if the condition λ = 1 α α does not hold. There are three sellers, i = 1,2,3, with each seller i = 1,2,3 an initial endowent of (i + 1) units of an identical (divisible) good. There are five buyers j = 1,2,,5, each buyer j s consuer s surplus or profit function g j : R + R is obtained fro g j (y) = ax q 0 u j(q) qy, where u j : [0,) R + is j s utility function given by u j (q) = ( j + 1) + 2 ( j + 1)q. The supply curve for each seller is S i (y) = [0,i + 1] for y = 0 and S i (y) = i + 1 for y > 0, i = 1,2,3. The deand curve D j (y) = q j, where u i (q j ) = y for y >> 0, j = 1,2,,5. In this exaple, we can set f i(y) = (i+1)y for i I = {1,2,3} and g j (y) = ( j +1)+ j+1 y for j J = {1,2,,5} so that D j (y) = q j = j+1. Thus, the equilibriu price equals y 20 = y 2 9 = Note that the three sellers and five buyers have no knowledge where the equilibriu price 1.49 is. Neither do they know the total deand and supply. Each of the just subits their bid and ask based on their own private inforation. The two double auctions acting as a clearinghouse integrate individually dispersed and incoplete inforation (Hayek, 1945) into prices. While the equilibriu price equals y = 1.49, the price, to which the price process under λ(1 α) RDA converges, is α y. Thus, if α = 0.1 and λ = 1, the price process under RDA converges to the price 3y, 200% higher than the original Walrasian equilibriu price y. If α = 0.5 while λ = 9, the price process converges to 3y as well. A crash price is also possible. For exaple, with α = 0.8 and λ = 1, the price process converges to 1 2 y, 50% lower than the equilibriu price of the original econoy. Xu et al. (2015, 2016) provide siulations that are consistent with these theoretical predictions under Theore 4.7. For Theore 4.4, we ay assue that there are five sellers to the above exaple. Now the original Walrasian equilibriu becoes y = 1. With α = 0.1 and λ = 1, the CDA generates a sequence of prices that converges to 3, 200% higher than the original equilibriu price y = 1. Note that the initial distribution of endowents aong the five agents is not that iportant. b Assue that li k k a k exists and equals 1. There are soe λ such that the λ conditions in Assuptions 3.2 and 3.3 hold. Then we know that λ = b li k k a k = 1 for CDA and RDA for five sellers and five buyers econoy, the two auctions generate price sequences that converge to the equilibriu price 1

27 Jinpeng Ma, Qiongling Li 27 when α = 1 2. Any other α can result in a price that is either higher or lower than the equilibriu price 1. Consider the three sellers and five buyers econoy with RDA again. Also b assue that li k k a k exists and equals 1 and there are soe λ such that Assuption 3.3 holds. Then λ = 3 5. In order for RAD to achieve the equilibriu price 1.49, one ust have α equal 3 8. Any other α will result in a price that is either higher or lower than the equilibriu price Therefore, whether DA can achieve an equilibriu depends on a cobination of α and λ and the λ condition A Discussion about the λ Condition We have seen that λ plays a key role. In Assuption 3.2, we need λ to satisfy the condition such that b k λa k < +. Is there such a λ for any two sequences {a k } and {b k } that satisfy Assuption 3.1? Unfortunately the answer is not always affirative. The relative strength between ask and bid step sizes is quite subtle for CDA or RDA echanis. Exaple 4.8. Let a k = b k = 1 k, k is odd 1 k 2, 1 k 2, k is even k is odd 1 k, k is even Then a k λb k k [λ]+1 k is odd = k [λ]+1 k is odd = k [λ]+1 k is odd a k λb k a k λ 1 k λ b k k [λ]+1 k is odd k [λ]+1 k is odd 1 k 2 = +

28 28 Dynaic double auction for any λ. The next exaple shows that even if the liit li k b k a k exits and equals 1, there ay not exist λ that satisfies Assuption 3.2. Exaple 4.9. Let a k = 1 But k 3 4 and b k = a k (1 + a 1 3 k ). Then b k li = li (1 + 1 ) = 1. k a k k k 4 1 b k a k = a 4 3 k = 1 k = +. Note that, by Proposition 4.10 below, if there exists a λ that satisfies Assuption 3.2, it can only be 1. Hence, no λ exists and satisfies Assuption 3.2 here. The following answers what λ ust be. Proposition Let Assuption 3.1 hold. If b k λa k < + for soe λ, then the following ust hold li inf k b k b k λ lisup. a k k a k Proof. If there exist δ > 0 and k 0 such that b k a k λ > δ for all k k 0, then b k λa k δ a k = +, k=k 0 k=k 0 a contradiction. Hence, liinf k b k a k λ is one necessary condition for b k λa k < +. Siilarly, lisup k b k a k λ. This copletes the proof. Proposition Let Assuption 3.1 hold. If there exists a λ such that b k λa k < +, then it ust be unique.

29 Jinpeng Ma, Qiongling Li 29 Proof. Suppose, on the contrary, that there are two λ and λ such that Then b k λa k < + and b k λ a k < +. λ λ a k b k λa k + b k λ a k < +. But a k = +, a contradiction. This copletes the proof. Thus, there exists at ost one λ that satisfies Assuption 3.2 for any two given sequences {a k } and {b k } satisfying Assuption 3.1. But for any given λ, there are a faily of step sizes {a k } and {b k } that satisfy Assuptions 3.1 and 3.2. Let a k = 1 k and b k = 2a k + ca 2 k. Then Assuption 3.2 is satisfied with λ = 2 for any positive finite nuber c and any {a k } that satisfies Assuption DA MECHANISM WITH STOCHASTIC NOISES Let f = f i and g = g i. Assue X 0 is a rando initial vector. Let ε i,k and δ i,k denote two independent rando noise vectors. (DA) echanis with stochastic noises is defined as follows. Let Φ 0,k = X k. For i = 1,2,,, let Let ψ i,k = Φ i 1,k a k (h i,k + ε i,k ), h i,k f i (Φ i 1,k ) (14) ϕ i,k = Φ i 1,k b k (l i,k + δ i,k ), l i,k g i (Φ i 1,k ) (15) Φ i,k = P Y (αψ i,k + (1 α)ϕ i,k ), α [0,1]. (16) X k+1 = Φ,k. (17) P Y is the Euclidean projection onto Y. We define Fk i to be the σ-algebra generated by the sequence Note that F 0 k is also denoted as F k. Φ 0,0,Φ 1,0,,Φ,0,,Φ i,k.