An Incentive Compatible Double Auction for Multi-Unit Markets with Heterogeneous Goods

Transcription

1 No. 106 An Incentive Compatible Double Auction for Multi-Unit Markets with Heterogeneous Goods Markus Reinhardt December 2010 A survey conducted by the Chair of Economics and Information Systems at HHL Leipzig Graduate School of Management

2 HHL-Arbeitspapier HHL Working Paper No. 106 An Incentive Compatible Double Auction for Multi-Unit Markets with Heterogeneous Goods Markus Reinhardt ISSN (Online version) HHL Leipzig Graduate School of Management The sole responsibility for the content of the HHL Working Paper lies with the author/s. We encourage the use of the material for teaching or research purposes with reference to the source. The reproduction, copying and distribution of the paper for non-commercial purposes is permitted on condition that the source is clearly indicated. Any commercial use of the document or parts of its content requires the written consent of the author/s. For further HHL publications see

3 I. Introduction During the last decade Mechanism Design was getting more and more into the focus of economic theory as the awareness for overall social goals grew not least due to worldwide financial and economic crises. Especially, the rapid expansion of global markets and the pronounced consciousness for individual benefit pose a question for markets that facilitate overall profits instead of personal ones. Therefore, the theories of mechanism design, which offer settings, that encourage people to act according to a common social goal while at the same time maximizing their personal outcome, receive high attention. 1 On the contrary to the numerous articles that concentrate on One-Side Auctions the number of papers that work on the theory of Double Auctions is comparatively small. However, as globalized markets and electronic commerce offer varied possibilities for buying and selling worldwide goods of almost all types, in particular Double Auctions are becoming more relevant. As a consequence, a Double Auction is presented that makes use of the characteristics of large markets. Regarding the existing literature, the most famous Double Auction is beyond doubt McAffee s [5] celebrated Double Auction for single-unit environments that is individually rational and nearly efficient. In order to create the trading price and clear the market, McAffee extracts the least efficient pair of traders. Consequently, the inefficiency is limited and for large markets this loss of efficiency becomes insignificant small compared to the total surplus of the mechanism. In fact, the work of Hurwicz [1] and that of Myerson and Satterthwaite [2] demonstrated that efficient and budget balanced allocations with dominant strategies are not incentive compatible at all. Taking into account that it is impossible to provide a Double Auction that guarantees the four important properties budget balance, incentive compatibility, individual rationality and allocative efficiency at the same time, it is convenient to weaken one of them in order to ensure the others. In most cases authors decide to soften the aim of achieving allocative efficiency and concentrate on the other three properties. 2 Consequently, the main goal in this paper is to derive a Double Auction that excludes strategic behavior of the agents by sure, i.e. to provide a Double Auction that is incentive compatible at all. Thereby, the fact that the scale of future markets is likely to increase, offers options to enlarge the allocative efficiency while other important properties as budget balance, individual rationality and incentive compatibility can also be ensured. If for instance Satterthwaite and Williams [4] are considered, the likelihood of gaining from underreporting for a single agent vanishes at a rate of o(1/n) if the market grows. They showed that it is getting more complex to find a strategy that creates a higher outcome than 1 Among several other awards for researchers in the field of Mechanism Design, the Royal Swedish Academy of Science awarded the Nobel price in economics from 2007 to Hurwicz, Maskin and Myerson for their studies on Mechanism Design and the immense impact of theirs studies for economic theory. 2 In contrast to weaker formulations of for instance allocative efficiency or budget balance it is little practicable to construct approximate definitions of incentive compatibility or individual rationality. The author believes that a mechanism is either individual rational (respectively incentive compatible) or not.

4 that of revealing the truth if the number of agents increases. This property is used by many researchers to overcome the problem of strategic behavior of the agents, i.e. to limit incentive incompatibilities. One extension of McAffee s approach that uses this property was presented by Huang et al. [6]. The presented Double Auction from Huang et al. is designed for multi-unit environments with homogeneous goods, and ensures the described formulation of incentive compatibility. Nevertheless, especially for small markets Double Auctions as that from Huang et al. provide incentives for the agents to misrepresent their true values. In addition, the mechanism of Huang et al. may earn money during the auction. And consequently, it is only weakly budget balanced. Another interesting approach to circumvent the impossibility theorem of Williams and Satterthwaite was developed by Bartal, Gonen and LaMura [7]. Their idea to facilitate budget balance, individual rationality, incentive compatibility and allocative efficiency for the singleunit market is to compute pairs of agents which are given a price range instead of a definite price for trade of a single good. As the final definition of the price is left for negotiations after the auction the guarantee of efficiency or respectively incentive compatibility depends on the concrete form of the negotiation algorithm. Furthermore, the enlargement of this price range mechanism to markets with multiple and heterogeneous goods is an open question. Consequently, this paper focuses on developing a Double Auction that is incentive compatible, budget balanced and individually rational without any assumptions or relaxations also for the complex setting of a multi-unit market with heterogeneous goods. The main idea of the Double Auction is to extract a sample of buyers and sellers in order to use their individual information for the creation of the trading price and the creation of the bundles. Thereby, the incentive compatibility is ensured due to the structure of the mechanism without further assumption concerning the valuation function of the agents. As a consequence the Double Auction from this paper is furthermore named Incentive Compatible Double Auction (ICDA). Subsequent the different questions that arise considering the extraction of a group of agents are discussed. How many buyers and sellers should be removed to fix the price? Which are the criteria that define the buyers and sellers that are taken from the originals set of agents? Which rule determines the price and how is decided which goods each agent is allowed to trade? All aspects are approached in consideration of the improvement of the properties of the ICDA. Accordingly, the paper starts in the following chapter with the definition of the setting of multiunit markets with heterogeneous goods and the mathematical definition of the properties budget balance, individual rationality, incentive compatibility and allocative efficiency. Afterwards in chapter II the structure of the ICDA is introduced, before in chapter III and chapter IV the different components of the ICDA are derived. Finally, during the last chapters the properties of the ICDA and the advantages of this approach are discussed in detail. 2

5 II. Definition of the setting of a multi-unit market with heterogeneous goods Before putting the Incentive Compatible Double Auction (ICDA) into concrete terms, it is necessary to formulate some basic definitions of markets with r N multiple and heterogeneous goods to describe the intended properties of the mechanism in detail. At the beginning of each auction all goods of the environment combined in the set G = G (1),, G (n) are allocated among the n sellers s 1,, s n S. Each seller s j S owns g k (j) N units of good k = 1,, r, i.e. her personal portfolio is defined as the set G (j) = g 1 (j),, gr (j), j = 1,, n. As the environment with heterogeneous goods allows each seller to sell goods of every type at the same time, it is necessary to consider bundles of goods. As a result, instead of a separate analysis of demand and offer for each good, the valuation has to be done by valuing complete packages of different goods. These bundles consist of r quantities each for one of every heterogeneous good. Consequently, v sj : Q R +, j = 1,, n defines the valuation function of seller s j S that assigns an individual estimated value to r each bundle q Q R +. 3 The second group of agents that participate in the auction is the set of buyers b 1,, b n B, which demand bundles of the r goods from the sellers. Analogue to that of the sellers the valuation function of buyer b i B is expressed by v bi : Q R +, i = 1,, m. In addition to the individual valuation of an agent the agents utilities of a concrete bundle q Q also depend on the market price p R + r. The k-th component of the price vector p R + r corresponds to the price of the good k = 1,, r. As quasi-linearity is assumed for all agents the utility function of each buyer b i B can be described by u bi (q, p) = v bi (q) q p, i = 1,, n and that of each seller s j S can be defined by u sj (q, p) = q p v sj (q) j = 1,, m. 4 3 r The definition of the bundles q Q R + as a real valued vector allows also trades of shares of goods. 4 A natural assumption for the utility of rational individuals is that effort and earnings of one trade can be separated from each other. Therefore individual utilities are considered to be quasi-linear in this paper. 3

6 Furthermore, the participation in the Double Auction is assumed to create no additional effort for the agents, for which reason the utility for buyers and sellers who not trade equals zero. For the derivation of the efficiency property of the ICDA in chapter VI two assumptions concerning the utility functions of the agents are needed. At first, it is convenient to assume that there is at least one specific price vector p R + r and respectively one specific bundle q Q for every buyer b B and every seller s S that creates a positive outcome for them. Obviously, this is not a severe assumption as the agents as rational individuals will only participate in the Double Auction, when they could expect positive revenue. Second, it is necessary to assume that the utility functions of the agents are concave. Alike as before, this is also a natural assumption because it reflects the descending marginal utility for the agents. 5 Analyzing the valuation function makes it quite obvious which conditions have to be ensured so that the mechanism is individually rational. As rational individuals both, buyers and sellers, would not accept any trade that generates a negative outcome for them. In addition, an agent cannot be forced to trade when it is not worth to her. In this case, when no trade is carried out, the agent s utility is zero. Considering the utility function of the agents, it is clear that each buyer b B will gain from the trade of a concrete bundle q Q as long as v b (q) q p. Each seller s S on the other hand will have a non-negative outcome as long as she sells her bundle q Q with a price which is at least as high as her individual valuation, i.e. q p v s (q), and does not trade otherwise. To ensure the non-negativity of the agents utilities and thus to ensure the individual rationality of the mechanism, these conditions have to be regarded. Another important property the mechanism should provide is the budget balance, i.e. that the mechanism does not earn or lose any money. Obviously this is achieved if the sum the buyers pay is exactly the sum the sellers get for their goods. Therefore, in the ICDA the price for each sort of good is fixed and the same for all agents who trade one unit of this good. Together with the concept of the bilateral trades between one seller and one buyer it is guaranteed that the mechanism is budget balanced. Finally, the properties incentive compatibility and allocative efficiency, whose achievement is more difficult than that of the other two properties, need to be observed. A mechanism is called incentive compatible if claiming their true types is an optimal strategy for each agent, i.e. there is no other strategy than acting according to the individual information that creates a higher outcome for each of the agents. As described above the social goal this paper focuses on is the allocative efficiency, i.e. to maximize the overall usage. 6 5 To assume that the agents marginal utility decreases, is close to real life. The assumption of concave utility functions embodies that the utility of trading one additional good is getting smaller. 6 The maximization of the overall usage is the most common social choice function in mechanism design. Therefore, for the sake of simplicity allocative efficiency is often simply called efficiency. 4

7 Consequently, this property can be described as the solution of the optimization problem m i=1 n u bi q bi, p + u sj q sj, p max. q b 1,,q bm,q s1,,q sn,p j=1 Remembering the discussion from the introduction, it is obvious that it is challenging to get all agents to act truthfully while on the other hand the overall usage and not the individual usage should be maximized. The idea of the ICDA to overcome this problem is to concentrate on the achievement of incentive compatibility instead of generating a fully efficient mechanism. Therefore, the structure of the ICDA is carefully constructed to ensure that no agent will have an incentive to misrepresent her true values in order to get a better trade. Although this paper focuses on ensuring incentive compatibility it is clear that the loss of efficiency should be limited to a minimum. III. The structure of the Incentive Compatible Double Auction (ICDA) First of all, the main concept of the ICDA is to extract a subset of agents from the original sets of buyers and sellers to compute the trading price and to generate the bundles by using the information about their individual values. As the information will be used to compute the trading price vector and to define of the trading bundles the selected buyers and sellers have an influence on both. As a result they may find strategies other than revealing their true types to increase their individual profit. Consequently, the agents that are selected to define the prices must not trade in the ICDA in order to give them no incentive to act strategically, i.e. to ensure that the mechanism is incentive compatible. Afterwards the auctioneer allocates the bundles to the remaining agents according to an allocation rule. The rationing between the agents described later in chapter IV is necessary to make sure that on the one hand the agents are motivated to act truthfully and on the other hand the overall usage is maximized. Achieving a balance between individual rationality and an approximate form of allocative efficiency for the Double Auction, is the main difficulty for construction of the components price definition rule, bundles creation rule and allocation rule. 5

8 Framework of the ICDA (1) Each agent is asked to send her individual valuation to the auctioneer. (2) Then the auctioneer selects L = min(m, n) buyers b i1,, b il B and L sellers s j1,, s jl S by random in order to create the trading prices for each good. The price of good g = 1,, r is computed by using the information of the individual valuations from the selected agents. In addition, the bundles for the auction are computed out of the individual information. As the chosen agents, b i1,, b il B and respectively s j1,, s jl S, define the price and the trading bundles, they are removed from the set of the potential traders, i.e. L b il L l=1. B = l=1 and S = s jl Consequently, the set of the remaining buyers is defined as B + = B\B and that of the remaining sellers as S + = S\S. (3) Next the auctioneer allocates the goods to the traders according to allocation rule. (4) Eventually, the agents b B + and s S + will participate in the auction with a defined trading price of p g per unit for good g = 1,, r. All the remaining agents b B and s S will not trade, hence their utility equals zero. Remark: In step (2) of the ICDA the trading price vector is computed by using the individual information collected from the selected agents with an arbitrary vector function p g = p g v,, v bi b 1 il. The most important fact of framework of the ICDA is that it creates no incentive for any agent to act strategic in order to get a price that may result in a higher individual outcome. This is true as the agents, who have an influence on the trading price vector are not allowed to participate in the auction. Therefore, the concrete formulation of the price function can be used to improve the efficiency property of the mechanism. Before introducing a specific allocation rule, it is convenient to analyze which properties the framework of the ICDA offers independently of the formulation of the allocation rule. First of all, as the principle defines a common trading price p and in addition specifies the bundles which are traded, the sellers exactly get the sum paid by the buyers. As a result, no outside subsidies are needed and consequently the mechanism is budget balanced. Furthermore, the price rule ensures that the agents who trade do not have any influence on the price vector. As a consequence the agents have no incentive to signal falsely in order to get a better price. But finally, the properties individual rationality and incentive compatibility depend on the chosen allocation rule, so that the ICDA focuses on these properties while generating an appropriate allocation rule in chapter IV. As the agents b B and s S do not participate in the mechanism, it is obvious that in general the mechanism does not create a fully efficient allocation as the surplus of those 6

9 agents gets lost. Nevertheless, the idea of the ICDA is that the efficiency, lost by extracting the sample of agents, becomes negligible small if the market is sufficiently large. Thereby, it is possible to achieve an approximate efficient allocation. IV. Definition of the allocation rule As mentioned in chapter I the aim of this paper is to present a mechanism that achieves budget balance, individual rationality and incentive compatibility at the costs of a possibly lower efficiency. In the following chapters the efficiency properties of the ICDA are analyzed in detail. Afterwards it is shown that under some assumptions concerning the agents valuation functions and if the market is large enough the loss of efficiency becomes negligible small, i.e. the ICDA is asymptotically efficient. A core element of the ICDA is the introduction of an allocation rule to decide which bundles are allocated to which buyers and sellers. Alike the price definition in step (2) of the ICDA and the creation of the bundles, which are explained in the following chapter, the allocation rule plays an important role as it has to ensure the incentive compatibility and the individual rationality during the distribution process of the bundles. The complexity concerning the allocation rule arises as in addition to the properties incentive compatibility and individual rationality the efficiency of the mechanism has to be observed. Subsequently, a simple allocation rule is presented that ensures incentive compatibility and individual rationality by sure but at the costs of generally low efficiency of the mechanism. A simple allocation rule (AR1) a) Before starting the allocation a random order b i1,, b im L of the buyers from B + and a random order s j1,, s jn L of the sellers from S + are computed. b) Afterwards z = min(m L, n L) different bundles q 1,, q z of the heterogeneous goods are built by using the information of the extracted agents from B and S. c) For k = 1 to z If u bik (q k, p ) 0 and u sjk (q k, p ) 0 then b ik buys the bundle q k from s jk Else b ik and s jk will not trade End Next All agents with an index greater than z will not trade. 7

10 Proposition: The simple allocation rule (AR1) is for a given trading price vector p R + r individually rational and incentive compatible. Proof: Each agent is asked at least once if a concrete bundle is acceptable for her or not. Therefore, no agent is forced to trade if the bundle is not worth enough. Consequently, the allocation rule is individually rational. As there is only one offer by the auctioneer and the order of the agents is created randomly, the buyers and sellers have no incentive to signal falsely to get a better bundle. Obviously the optimal strategy for each agent is to accept the bundle offered if and only if the individual utility is non-negative. As a result, the allocation rule is individually rational and incentive compatible at the same time. When analyzing the simple allocation rule (AR1), some main properties can be found quite easily. As an advantage of the simple allocation rule it is unnecessary to introduce supplementary assumptions concerning the agents valuation functions to ensure the individual rationality and the incentive compatibility. Additionally, the structure of the allocation rule itself is straightforward. On the other hand, the disadvantages of the simple allocation rule are quite obvious. A trade between a buyer and a seller is executed if and only if the bundle is constructed in a way that the utilities of both agents are non-negative. Furthermore, the individual preferences of the agents are not used to increase their personal outcome. The efficiency of the simple allocation rule directly depends on the homogeneity of the agents preferences and in addition on the precision of the construction of the bundles. In general the simple allocation rule creates a very inefficient allocation. In order to improve the efficiency of the allocation rule, it is essential to install a mechanism that allows the agents to choose between different bundles so that the individual preferences are considered. The following allocation rule pays more attention to the agents preferences for a specific bundle of goods, and consequently creates a more efficient allocation. An efficient allocation rule (AR2) a) Before starting the allocation a random order b i1,, b im L of the buyers from B + and a random order s j1,, s jn L of the sellers from S + are computed. b) Afterwards z = min(m L, n L) different bundles q 1,, q z of the r different goods are built by using the information of the extracted agents from B and S. c) Each bundle q 1,, q z that creates a positive outcome considering the trading price vector p for the agents is put into their preference list. These preference lists are sorted in an ascending order according to the bundles value for the agent. 8

11 d) Before starting the allocation the following conditions have to be satisfied: d1) All bundles that are in the first position of the preference list of any buyer b ik are removed from the preference list of buyer b il with l = k + 1,, m L. This is done for all buyers b ik with k = 1,, m L 1 and in the equivalent way for all sellers s jk with k = 1,, n L 1. d2) If the number of available bundles exceeds the number of buyers or the number of sellers, then the according amount of bundles, beginning with the highest index, is removed. 7 e) At least the remaining agents are allowed to trade the bundle that is in the first position of their actual preference list. Lemma: The conditions d1) and d2) of the efficient allocation rule (AR2) does not create incentives to signal false preferences to the auctioneer. Proof: First of all, as the auctioneer allocates a bundle to the buyer and respectively to the seller with the smallest index, regarding the randomized orders from step a), the agents are indifferent of removing bundles from their preference list when they are preferred by a predecessor. This is true because, if this bundle will be traded and there are multiple solutions for the allocation, the buyer and seller with the smallest index will get this bundle. Therefore, the condition d1) is incentive compatible. Second, it is obvious that a single agent has only little influence on the number of agents involved in the mechanism. It is clear that the agent cannot gain from pretending to have an empty preference list and leaving the auction. On the other hand it is not possible for an agent to force other agents to stay in the mechanism if they cannot find bundles with an expected surplus for them. As a result, no agent has got influence in reducing or increasing the number of agents, why the condition d2) is incentive compatible, too. Lemma: If the conditions d1) and d2) are satisfied the allocation of the bundles of the allocation rule (AR2) is unique. Proof: Considering condition d2), it is guaranteed that the number of buyers and respectively the number of sellers is at least as high as the number of goods that are still available. If this is combined with condition d1), it is clear that all buyers have different bundles in first position of their final preference list. Therefore, all remaining bundles relate to exactly one first position of one buyer s and respectively one seller s preference list, and the allocation of the efficient allocation rule is unique. 7 In order to ensure that d1) and d2) are true, both steps may have to be repeated iteratively. 9

12 Proposition: The allocation rule (AR2) is individually rational and incentive compatible. Proof: First of all, the individually rationality of the efficient allocation rule is ensured as only preferences are considered for which the individual utility is positive. As a result, each buyer and each seller either gets a bundle that creates a positive outcome or will not trade. Consequently, the agents outcome of the efficient allocation rule is non-negative. Second, the incentive compatibility is guaranteed as the stability conditions d1) and d2) are incentive compatible. Due to the rule that only the first choice is allocated to each agent, it is clear that signaling the true values to the auctioneer is an optimal strategy. V. The creation of the price vector and the bundles In the previous chapters the basic framework of the ICDA and a concrete allocation rule were presented. The main focus for these components of the ICDA is to ensure the properties incentive compatibility, individual rationality and budget balance independently of the structure of the valuation functions of the buyers or sellers, and in addition to achieve a high efficiency. Finally, the questions of how to define the bundles that could be traded and how to create a price for each good are still open. Both, the definition of the price vector and the set of the trading bundles, have a direct influence on the efficiency of the ICDA. As the other intended properties incentive compatibility, individual rationality and budget balance are already ensured it is possible to design the price vector and the set of trading bundles in a way that the resulting efficiency is maximized. The ICDA is constructed to implement a mechanism that realizes the allocative efficiency as its social choice function. An allocative efficient mechanism is defined as a mechanism that maximizes the overall outcome, i.e. which maximizes the sum of the agents revenues. If it is assumed that the auctioneer has complete information about the valuation functions of the buyers and the sellers, the property allocative efficiency can be achieved by solving the maximization problem m i=1 n u bi q bi, p + u sj q sj, p max q b 1,,q bm,q s1,,q sn,p j=1 (AE). Regarding the statements about budget balance and individual rationality, it is obvious that the unrestricted formulation of the optimization problem may create an efficient allocation. But unfortunately, it prevents the mechanism to be budget balanced and individually rational. 10

13 As a consequence, the formulation from above has to be completed by the constraints u bi 0, u sj 0, i = 1,, m j = 1,, n and m n q bi = q sj i=1 j=1. 8 For the Double Auction with heterogeneous goods the result of the optimization problem provides the price vector and the trading bundles. Considering the remarks about incentive compatibility from above, it is quite obvious that it is not possible to use the information of all agents preferences because this would give them incentives to increase their personal outcome by misrepresenting the true values. Therefore, only the valuation functions of the agents that are taken from the original sets can be used for computing the optimal price vector and the trading bundles. Consequently, the price vector and the bundles for the ICDA are computed by solving the optimization problem (AE*) with the constraints L l=1 L u bil q bil, p + u sjl q sjl, p max q bi 1,,q b il,q sj,,q s 1 jl,p l=1 u bi l 0, u sjl 0, l = 1,, L l = 1,, L and b il B s jk S q bil = q sjk. 8 The constraints from (AE*) are necessary to be sure that the outcome of each agent is nonnegative and that the amount of goods that are sold is equal to that which are bought. Thus, the solution of the optimization problem (AE*) also pays attention to the properties budget balance and individual rationality. 11

14 For the ICDA the third restriction of the optimization problem has to be modified according to the requirement of bilateral trades which are used in the ICDA for the exchange of goods. Instead of the overall sum of sold and bought goods the sum of traded goods for each deal has to be equal. 9 Eventually, the price definition of the ICDA depends on the solution of the nonlinear optimization problem (AE*) for which reason the theory about solving (AE*) is important for the numerical implementation of the ICDA. First of all, the third constraint effect that in general the optimization problem (AE*) has got more than one solution. Remembering that the utility functions of the agents are assumed to be concave, some findings about the existence of a solution of (AE*) can be derived from the Kuhn-Tucker- Theory. If the so called Slater condition b B (q, p ) Q R + r with u b (q, p ) > 0 and s S (q, p ) Q R + r with u s (q, p ) > 0 (SC) is fulfilled, then there exists a solution of the (AE*). The Slater condition (SC) defines that all buyers b B and all sellers s S have a chance to find a price vector and a trading bundle with a positive outcome for them. In chapter II it was assumed that each buyer of b B and each seller s S have the ex ante opportunity to participate in the auction, i.e. that the Slater condition (SC) is true. Several numerical methods based on the Kuhn-Tucker-Theory, such as penalty algorithms or algorithms based on the steepest descent, provide adequate approaches for an effective and efficient solution of (AE*). VI. Discussion of the efficiency of the ICDA In the previous chapters the structure of the ICDA and the function of the different components of the ICDA were developed. The focus of these chapters was to install a Double Auction that ensures the three properties budget balance, incentive compatibility and individual rationality for sure. Finally, the derivation of the efficiency property is still open. Therefore, the terms and condition for an asymptotical allocative efficient ICDA are discussed in this chapter. Definition: A mechanism is called asymptotically allocative efficient, if the loss of efficiency becomes negligible small compared to the overall usage when the number of agents is sufficiently large. 9 Obviously the modification of the third constraint leads in general to a lower efficiency. In the following chapter it is shown under which assumptions the asymptotical allocative efficiency still can be achieved. 12

15 Proposition: The ICDA is asymptotically allocative efficient, if the conditions (i) (ii) (iii) the valuation functions of the agents are limited by a real value v, i.e. max b B,q Q v b (q) v and max s S,q Q v s (q) v for a v R +, r for each price vector p R + and each buyer b B with q = argmax q u b (q, p ) exists a seller s S with q = argmax q u s (q, p ) all buyers valuation functions and respectively all sellers valuation functions are taken from the same distribution, i.e. b B: v b ~V b and s S: v s ~V s are satisfied. Proof: First, it is important to show that for m, n the possibility that each agent could trade her preferred bundle converges to 1, if condition (ii) is fulfilled. Within the price definition from the ICDA the information from the sample of agents extracted in step (1) is used to approximate the valuation distributions of buyers and sellers. According to the main proposition of mathematical statistics, an empirical distribution function converges almost surely to the true distribution function, if condition (iii) is true and the size of the sample converges to infinity. Consequently, the information of the sample is a good approximation for the true distribution function of the agents valuations, if L is sufficiently large. This is ensured because L = min(m, n). Second, it is necessary to show that the loss of efficiency which is due to the extraction of some agents, converges to zero if the market is sufficiently large. Therefore, the loss of efficiency from the extraction of the sample consisting of L = min(m, n) buyers and sellers is limited by L max v b(q) + max v s (q) 2L v. b B,q Q s S,q Q As long as the usage of the traders grows faster than 2L v, the loss of efficiency caused by the extraction of the agents in step (1) of the ICDA becomes negligible small. This is true because there is a solution of the optimization problem for the creation of the bundles, for which each agent gets the chance to trade her preferred bundle. In fact, for each p R + r and each b B with q = argmax q u b exists a s S for which q = argmax q u s. Therefore, the ICDA is asymptotically allocative efficient, if the conditions of the proposition from above are satisfied. In conclusion, to complete the review of the efficiency of the ICDA, it is necessary to analyze the ICDA if conditions (i) or (ii) are not satisfied. At first the condition (i) is needed to limit the loss of efficiency that occurs from the extraction of agents from the original set of traders. Obviously the maximum valuation of each agent for every bundle has to be bounded by a v R +. Otherwise, if one agent s valuation for a certain bundle cannot be limited, i.e. becomes infinitely large, the loss of efficiency cannot be balanced in case this agent is extracted. 13

16 At least the similarity condition (ii) ensures that every buyer could find a seller whose preferred bundle is the same. If the condition (ii) is not true then the ICDA will not converge to asymptotic efficiency, even if condition (i) is satisfied. This is caused as agents may be forced to trade less preferred bundles. This characteristic is due to the definition of the ICDA that allows only bilateral trades. In worst case, if there is no overlapping between the valuation distribution of the buyers and the sellers, no trade will be carried out and consequently the efficiency equals zero. VII. Conclusion The Incentive Compatible Double Auction (ICDA) provides the important properties budget balance, individual rationality and incentive compatibility without any assumptions concerning the agents individual valuation functions even for the complex environment of a multi-unit market with heterogeneous goods. Therewith the ICDA enlarges the existing approaches of Double Auctions, which predominantly concentrated on the efficiency of the mechanism and as a result only achieve approximate formulations of the incentive compatibility. Especially for the complex environments of markets with several heterogeneous goods the ICDA strikes a new path. Due to the structure of the mechanism the achievement of the approximate allocative efficiency for multi-unit markets depends on the homogeneity of the agents valuation function. Additionally, the ICDA does not need assumptions concerning the concrete form of the valuation functions of a single agent. To achieve efficiency a sort of symmetry between the distribution of the valuation functions of buyers and sellers is needed. Ensuring the approximate formulation of allocative efficiency needs that each buyer could find a seller whose highest preference is the same. Otherwise, the ICDA will not achieve the maximum efficiency even if the market becomes large. On the contrary, for the single-unit market the approximate efficiency is ensured also without assumption (ii) concerning the agents valuations. Actually, to limit the dependence of the efficiency property on the alikeness of the buyers and the sellers preferences should be most interesting for additional work on the ICDA. The construction of an allocation rule which naturally yields a high efficiency independently of the agents preferences appears to be quite challenging. In contrast, a modification of the framework that allows also trades between groups of buyers and sellers instead of bilateral trades also seems to be promising, though the bilateral trade is a crucial part of the mechanism to ensure incentive compatibility. Consequently, the construction of an incentive compatible multilateral trade rule is recommended for further studies. 14

17 Reference List [1] Hurwicz, L. (1972). On informationally decentralized systems. In B. C. McGuire & R. Radner (Eds.), Decision and Organizations. A volume in Honor of Jakob Marschak (pp ). Amsterdam. [2] Myerson, R. B., & Satterthwaite, M. A. (1983). Efficient Mechanisms for Bilateral Trading. Journal of Economic Theory, 29( ). [3] Wilson, R. B. (1985). Incentive Efficiency Of Double Auctions. Econometrica, 53(5), [4] Satterthwaite, M. A., & Williams, S. R. (1989). The rate of convergence to efficiency in the buyer's bid double auction as the market becomes large. The Review of Economic Studies, 56(4), [5] McAffee, R. P. (1992). A Dominant Strategy Double Auction. Journal of Economic Theory, 56, Retrieved March 21, [6] Huang, P., Scheller-Wolf, A., & Sycara, K. (2002) A Strategy-Proof Multiunit Double Auction Mechanism. [7] Bartal, Y., Gonen, R., & La Mura, P. (2004). Negotiation Range Mechanisms: Exploring the Limits of Truthful Efficient Markets. Electronic Commerce,

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21 63 Wriggers, Stefan (2004) Kritische Würdigung der Means-End-Theorie im Rahmen einer Anwendung auf M-Commerce-Dienste 62 Kirchgeorg, Manfred (2003) Markenpolitik für Natur- und Umweltschutzorganisationen 61 La Mura, Pierfrancesco (2003) Correlated Equilibria of Classical Strategic Games with Quantum Signals 60 Schwetzler, Bernhard; Reimund, Carsten (2003) Conglomerate Discount and Cash Distortion: New Evidence from Germany 59 Winkler, Karsten (2003) Wettbewerbsinformationssysteme: Begriff, Anforderungen, Herausforderungen 58 Winkler, Karsten (2003) Getting Started with DIAsDEM Workbench 2.0: A Case-Based Tutorial 57 Lindstädt, Hagen (2002) Das modifizierte Hurwicz-Kriterium für untere und obere Wahrscheinlichkeiten - ein Spezialfall des Choquet-Erwartungsnutzens 56 Schwetzler, Bernhard; Piehler, Maik (2002) Unternehmensbewertung bei Wachstum, Risiko und Besteuerung Anmerkungen zum Steuerparadoxon 55 Althammer, Wilhelm; Dröge, Susanne (2002) International Trade and the Environment: The Real Conflicts 54 Kesting, Peter (2002) Ansätze zur Erklärung des Prozesses der Formulierung von Entscheidungsproblemen 53 Reimund, Carsten (2002) Internal Capital Markets, Bank Borrowing and Investment: Evidence from German Corporate Groups 52 Fischer, Thomas M.; Vielmeyer, Uwe (2002) Vom Shareholder Value zum Stakeholder Value? Möglichkeiten und Grenzen der Messung von stakeholderbezogenen Wertbeiträgen 51 Fischer, Thomas M.; Schmöller, Petra; Vielmeyer, Uwe (2002) Customer Options Möglichkeiten und Grenzen der Bewertung von kundenbezogenen Erfolgspotenzialen mit Realoptionen 50 Grobe, Eva (2003) Corporate Attractiveness : eine Analyse der Wahrnehmung von Unternehmensmarken aus der Sicht von High Potentials 49 Kirchgeorg, Manfred; Lorbeer, Alexander (2002) Anforderungen von High Potentials an Unternehmen eine Analyse auf der Grundlage einer bundesweiten Befragung von High Potentials und Personalentscheidern

22 48 Kirchgeorg, Manfred; Grobe, Eva; Lorbeer, Alexander (2003) Einstellung von Talenten gegenüber Arbeitgebern und regionalen Standorten : eine Analyse auf der Grundlage einer Befragung von Talenten aus der Region Mitteldeutschland (not published) 47 Fischer, Thomas M.; Schmöller, Petra (2001) Kunden-Controlling Management Summary einer empirischen Untersuchung in der Elektroindustrie 46 Althammer, Wilhelm; Rafflenbeul, Christian (2001) Kommunale Beschäftigungspolitik: das Beispiel des Leipziger Betriebs für Beschäftigungsförderung 45 Hutzschenreuter, Thomas (2001) Managementkapazitäten und Unternehmensentwicklung 44 Lindstädt, Hagen (2001) On the Shape of Information Processing Functions 43 Hutzschenreuter, Thomas; Wulf,Torsten (2001) Ansatzpunkte einer situativen Theorie der Unternehmensentwicklung 42 Lindstädt, Hagen (2001) Die Versteigerung der deutschen UMTS-Lizenzen eine ökonomische Analyse des Bietverhaltens 41 Lindstädt, Hagen (2001) Decisions of the Board 40 Kesting, Peter (2001) Entscheidung und Handlung 39 Kesting, Peter (2001) Was sind Handlungsmöglichkeiten? Fundierung eines ökonomischen Grundbegriffs 38 Kirchgeorg, Manfred; Kreller, Peggy (2000) Etablierung von Marken im Regionenmarketing eine vergleichende Analyse der Regionennamen "Mitteldeutschland" und "Ruhrgebiet" auf der Grundlage einer repräsentativen Studie 37 Kesting, Peter (2000) Lehren aus dem deutschen Konvergenzprozess eine Kritik des Eisernen Gesetzes der Konvergenz und seines theoretischen Fundaments 36 Hutzschenreuter, Thomas; Enders, Albrecht (2000) Möglichkeiten zur Gestaltung internet-basierter Studienangebote im Markt für Managementbildung 35 Schwetzler, Bernhard (2000) Der Einfluss von Wachstum, Risiko und Risikoauflösung auf den Unternehmenswert 34 No longer available. There will be no reissue.

23 33 Löhnig, Claudia (1999) Wirtschaftliche Integration im Ostseeraum vor dem Hintergrund der Osterweiterung der Europäischen Union: eine Potentialanalyse 32 Fischer, Thomas M. (1999) Die Anwendung von Balanced Scorecards in Handelsunternehmen 31 Schwetzler, Bernhard; Darijtschuk, Niklas (1999) Unternehmensbewertung, Finanzierungspolitiken und optimale Kapitalstruktur 30 Meffert, Heribert (1999) Marketingwissenschaft im Wandel Anmerkungen zur Paradigmendiskussion 29 Schwetzler, Bernhard (1999) Stochastische Verknüpfung und implizite bzw. maximal zulässige Risikozuschläge bei der Unternehmensbewertung 28 Fischer, Thomas M.; Decken, Tim von der (1999) Kundenprofitabilitätsrechnung in Dienstleistungsgeschäften Konzeption und Umsetzung am Beispiel des Car Rental Business 27 Fischer, Thomas M. (2000) Economic Value Added (EVA ) - Informationen aus der externen Rechnungslegung zur internen Unternehmenssteuerung? (rev. edition, July 2000) 26 Hungenberg, Harald; Wulf, Torsten (1999) The Transition Process in East Germany 25 Vilks, Arnis (1999) Knowledge of the Game, Relative Rationality, and Backwards Induction without Counterfactuals 24 Darijtschuk, Niklas (1998) Dividendenpolitik 23 Kreller, Peggy (1998) Empirische Untersuchung zur Einkaufsstättenwahl von Konsumenten am Beispiel der Stadt Leipzig 22 Löhnig, Claudia (1998) Industrial Production Structures and Convergence: Some Findings from European Integration 21 Schwetzler, Bernhard (1998) Unternehmensbewertung unter Unsicherheit Sicherheitsäquivalentoder Risikozuschlagsmethode 20 Fischer, Thomas M.; Schmitz, Jochen A. (1998) Kapitalmarktorientierte Steuerung von Projekten im Zielkostenmanagement 19 Fischer, Thomas M.; Schmitz, Jochen A. (1998) Control Measures for Kaizen Costing - Formulation and Practical Use of the Half-Life Model 18 Schwetzler, Bernhard; Ragotzky, Serge (1998) Preisfindung und Vertragsbindungen bei MBO-Privatisierungen in Sachsen