Design of optimal double auction mechanism with multi-objectives

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Abstract

This paper proposes a new approach to design optimal double auction mechanism with multi-objectives. In the optimal double auction mechanism, optimality is represented as multi-objectives to maximize the expected total revenue of sellers and buyers respectively at the same time. We give representation of allocation rules and payment rules of the optimal double auction mechanism that satisfies incentive compatibility, individual rationality, market clearing, and budget-balanced restrictions. Finally, we present a numerical example to demonstrate the function of the developed optimal double auction mechanism and its efficiency.

Highlights

► We propose a new approach to design optimal double auction mechanism with multi-objectives. ► We propose to achieve the optimality of auction by maximizing expected total revenue of sellers and buyers respectively at the same time. ► We present the allocation rules and payment rules of the optimal auction mechanism. ► We demonstrate the function of the developed optimal double auction mechanism and its efficiency with a numerical example.

Introduction

Auction is considered as one of the most effective allocation, or trade methods of resources, goods or services. In last two decades, auction theory has already become one of the most successful and active branches in economics field. The auction theory radically deals with three sorts of problem (Vijay, 2002). First, from the seller’s standpoint, why the auction mechanism not the other sale mechanisms should be used? Further, which auction format should the seller adopt to get the highest sale price or profit? The seller would face a two-lay decision-making problem: to decide the selling procedure and then to select the specific auction rule. Second, from the potential buyer’s standpoint, whether should he enter the auction? Then, under the given auction format, how does he select the optimal bidding strategies? Third, the central planner should care about the allocation efficiency of auction mechanism, i.e., whether is the auctioned object distributed to the buyer of highest value? To sum up, the whole auction theory centers round the choice and design of auction mechanisms.

The traditional auction mechanism designs for the “one to many” market structure called one-side auction, such as the work presented in Vickrey, 1961, Myerson, 1981, Laffont and Robert, 1996, Moldovanu et al., 1999, Maskin and Riley, 2000, Jehiel and Moldovanu, 2001, etc. The one-side auction only deals with a single buyer with several sellers, or a seller with several buyers. In real-life trading practice, however, each seller (or buyer) may also want to contact several buyers (or sellers) to reach the best deal. For example, in securities business, for the same kind of securities, there are always some clients who want to buy in and other clients want to sell out. By the way of double auction, the securities can be exchanged efficiently. In the transaction process, the competitions arise among buyers, as well as among sellers.

Double auction is said as a market mechanism to overcome the above the two weaknesses effectively. In addition, the double auction model possesses at least two other important features. First, rational traders in a double auction act strategically to manipulate the market-clearing price in their favor, which leads to efficient trades. Second, the double auction mimics the workings of actual interactive markets in use. With the growth of electronic commerce, designing double auction mechanisms that are applicable to emerging market structures has become an important research topic.

The proposed double auction mechanism presented in this paper is, in general, hoped to be a strategy-proof, truth-telling, dominant-strategy, incentive-compatible and individual-rational mechanism, where truthful revelation is a dominant strategy for each agent. A mechanism is individual-rational if an agent’s expected utility from participation is not less than his or her utility from nonparticipation, after the agent knows his or her own valuation of the bundle. Thus, individual rationality induces all the potential buyers and sellers to the mechanics. Also, it is well known that efficiency is one of the key criteria for a good auction mechanism. The efficiency of an auction mechanism can be measured by comparing the social welfare achieved by the mechanism with the maximum feasible social welfare with complete information. A mechanism called to be efficient if it implements an allocation that maximizes social welfare. Budget-balanced is often required in an auction. A mechanism is called to be budget-balanced if the auctioneer’s expected payoff is non-negative. Budget-balanced can hold the auctioneer in the auction.

The k-double auction is perhaps the simplest double auction model. It can be said as a family of double auction models specified parameter k. The two-player k-double auction was first introduced by Chatterjee and Samuelson (1983). Myerson and Satterthwaite (1983), using the Revelation Principle, showed that for any two probability densities of private values which are both positive over some non-empty interval, no incentive-compatible individually rational trading mechanism can be ex post efficient in the k-double auction. This mechanism has been extensively explored to the cases with many buyers and many sellers later on by Williams, 1987, Satterthwaite and Williams, 1989, and Leininger, Linhart, and Radner (1989). Rustichini, Satterthwaite, and Williams (1994) extended the Satterthwaite and Williams convergence result to general double auctions, showing that if a symmetric equilibrium exists, it must be close to truth-telling. Kadan (2007) presented sufficient conditions for the existence of an increasing equilibrium in the two-player k-double auction with affiliated private values and shows that the equilibrium is not truth-telling in general.

Keller (2006) developed a novel market-price statistics for a sealed-offer k-double auction. In the sealed-offer k-double auction with many buyers and many sellers, buyers or sellers may not simultaneously submit offers to buy and offers to sell. The bids are sealed, in the sense that buyers and sellers know only their own bids and not the bids of others. Buyers or sellers may purchase or sell multiple units by submitting multiple sealed-bids each for a single unit. Keller considered the case when k is not fixed and provides that in this situation the sealed-offer k-double auction is not only ex post classically efficient, but also ex ante Bayesian incentive efficient. However, what an auction mechanism is required to be efficient is not easy.

Hurwicz (1972) first introduced mechanism design to allocate resources efficiently under incomplete information and showed that it is impossible to implement an efficient, budget-balanced, and truth-telling mechanism, when buyers and sellers exchange single units of the same good, even in a simple exchange environment. Myerson (1981) developed the Hurwicz’s mechanism design by using optimization and extends to auction mechanism design. Myerson and Satterthwaite (1983) presented the impossibility of having an efficient, individual-rational, incentive-compatible and budget-balanced mechanism. Wilson (1985) established sufficient conditions for a double auction mechanism to be incentive-efficient. A mechanism (or a rule) is incentive-efficient if there is not another rule which would improve some agents’ expected gains from trade without reducing others’ expected gains. Cripps and Swinkels (2006) studied the efficiency of large double auctions, in which each player (seller or buyer) observes his value and then submits a bid. Trade is determined by crossing the submitted demand and supply curves. They proved that all non-trivial equilibriums of double auctions that satisfy “a little independence” are asymptotically efficient.

A continuous double auction (CDA) is an efficient market institution for real-world trading of commodities and electronic marketplaces (Ma & Leung, 2007). A continuous double auction allows buyers and sellers to continuously update their bids and asks at any time throughout the trading period and which permits trade at any time. The continuous double auction is not dominant-strategy incentive-compatible mechanism yet. Ma and Leung (2007) presented the design and analysis of a new bidding strategy for buyer and seller agents participating in agent-based continuous double auctions.

The classic results of truth-telling and individual-rational auction mechanisms were proposed by Vickrey, 1961, Clarke, 1971, and Groves (1973), under which the unique seller decides the buyers’ trading prices according to their marginal contributions to the system. An important auction mechanism jointly studied by them is called as VCG mechanism, which is dominant-strategy incentive-compatible and outcome-efficient within the private value environment. Moreover, every Bayesian incentive-compatible and efficient mechanism is payoff-equivalent to VCG mechanism from an interim perspective (Williams, 1999). The VCG mechanism, however, does not in general satisfy both ex-ante budget balance and interim individual rationality. Yoon (2001) extended and modified the Vickrey auction to double auction. Yoon provided efficient conditions under which the modified Vickrey double auction achieves full efficiency. But, budget-balanced is satisfied unless the market maker charges strictly positive fees to some participants. Yoon (2008) introduced the participatory Vickrey–Clarke–Groves mechanism that satisfies ex-ante budget balance, interim individual rationality and asymptotic efficiency. English auctions, Dutch auctions and first-price sealed-bid auctions are common auctions which all are not dominant-strategy incentive-compatible mechanism.

The literature on truth-telling budget-balanced double auction mechanisms is little. McAfee (1992) proposes a truth-telling budget-balanced double auction mechanism for a simple exchange environment, exchanging single units of the same good between buyers and sellers. Babaioff, Nisan, and Pavlov (2004) designed a truth-telling budget-balanced double auction mechanism for an exchange environment with transaction costs, in which buyers and sellers exchange single units of the same good. Babaioff and Walsh (2005) proposed a truth-telling budget-balanced double auction mechanism for a bilateral exchange environment without transaction costs, where each buyer wants to acquire a bundle of commodities. In their bilateral exchange model, each buyer wants to acquire a bundle of commodities and each seller provides a single unit of one commodity. They assumed no transaction costs and propose the known-single minded trade reduction (KSM-TR) mechanism. Chu and Shen (2006) proposed the truth-telling budget-balanced double auction mechanisms for an exchange environment with transaction costs. They designed the buyer competition (BC) mechanism and the seller competition (SC) mechanism. Unfortunately, the BC mechanism fails to be truth-telling for the sellers in the bilateral exchange environment. The sellers may have incentives to manipulate their bids and, consequently, improve their own payoffs. To address this weakness, Chu and Shen (2008) proposed two novel truth-telling mechanisms, namely, the buyer competition mechanism based on the linear relaxation of the social welfare (BC-LP mechanism) and the modified buyer competition mechanism (MBC), for generalized environment. They proved that both mechanisms are individual-rational, truth-telling and budget-balanced in the bilateral exchange environment and asymptotically efficient under some conditions. The BC-LP mechanism can be implemented by just solving two linear programs. Chu and Shen (2007) investigated two truthful double auction design approaches, namely, the Trade Reduction Approach and the Multi-Stage Design Approach, and compared the efficiency of their resulting mechanisms (including but not restricting to BC-LP mechanism and MBC mechanism) in various exchange environments.

The literature reviewed above discussed how the double auctions satisfy some of incentive compatibility, individual rationality, market clearing, budget restriction, efficiency, and optimality, but not all of these properties. In addition, the social welfare in the design of few optimal double auction mechanisms is only a form of single objective (Armstrong, 2000). As compared with previous work shown in literature, the main contribution of our proposed approach is listed as follow: (1) to propose a new design method of optimal double auction mechanism; (2) to represent optimality as multi-objectives, in sense of maximizing the expected total revenue of sellers and buyers, respectively, at the same time, to extend the case of single social welfare objective to the case of multi-objectives; (3) to provide an optimal double auction mechanism that satisfies incentive compatibility, individual rationality, market clearing, budget restriction and optimality; (4) to design the representation of allocation rule and payment rule of the optimal double auction mechanism; and (5) to show the efficiency of the optimal double auction mechanism by using the optimal double auction mechanism in an empirical study.

The remainder of this paper is organized as follows. In Section 2, we establish a multi-objectives auction model, an optimization model which includes constraint conditions to satisfy incentive compatibility, individual rationality, market clearing, and budget restriction, and multi-objectives functions to the expected total revenue of sellers and buyers. In Section 3, the representation of allocation rules and payment rules of the optimal double auction mechanism is given. In Section 4, an empirical study is presented to demonstrate the use of the developed optimal double auction mechanism and the efficiency of the mechanism. Finally, in Section 5, we present a concluding summary of this paper.

Section snippets

Double auction model with multi-objectives

In this section, we propose a model of optimal double auction mechanism with multi-objectives to satisfy some properties required. We firstly describe a general framework for the analysis and specify the restrictive assumptions used subsequently. Considering a double auction with M sellers and N buyers to participate in, we assume that each participant at most buys or sells a unit of goods and all participants are risk-neutral. For any seller i (i = 1, 2,  , M), assume that ci is the cost that he

Optimal double auction mechanism

An auction mechanism consists of a set of allocation rules and payment rules. The allocation rules, simply speaking, are to decide which sellers to sell and which buyers to buy, given a group of costs and values (c, v). If one seller is assigned to sell, his corresponding allocation function Qi(c, v) takes the value 1, or else takes the value 0. All the allocation functions of sellers compose the allocation rule for sellers, i.e., Θ = (Q1(c, v), Q2(c, v),  , QM(c, v)). Analogously, the allocation rule

Numerical example analysis and allocation efficiency

An example is used to show how our proposed double auction mechanism works. We also determine the efficiency of our auction double mechanism by the example.

Considering a double auction case with participation of 16 buyers and 16 sellers, when the auction begins, each participant submits the sealed bids to the auction organizer. Every participant only exactly knows his own value or cost, not the others’, but knows the probability distribution of the value or cost of the other participants. We

Concluding summary

Auction is an efficient approach to allocate scarce resources and to search and determine the value of scarce resources by competition (Satterthwaite & Williams, 2002). In auction, however, there is a lot of incomplete information or private information. Auction Mechanism design is an important economic topic under such incomplete information or private information to allocate resources. Hurwicz(1972) introduces mechanism design and Myerson (1981) develops it and establish the general frame of

Acknowledgements

This paper was partly supported by the City University of Hong Kong SRG Grants (Nos. 7002504 and 7002571) and the Chinese National Natural Science Foundation (Nos. 60574071 and 71071119).

References (36)

  • S.R. Williams

    Efficient performance in two agent bargaining

    Journal of Economic Theory

    (1987)
  • K. Yoon

    The modified Vickrey double auction

    Journal of Economic Theory

    (2001)
  • K. Yoon

    The participatory Vickrey–Clarke–Groves mechanism

    Journal of Mathematical Economics

    (2008)
  • M. Armstrong

    Optimal multi-object auction

    Review of Economic Studies

    (2000)
  • Babaioff, M., Nisan, N., & Pavlov, E. (2004). Mechanisms for a spatially distributed market. In Fifth ACM Conf....
  • A.K. Chatterjee et al.

    Bargaining under incomplete information

    Operations Research

    (1983)
  • L.Y. Chu et al.

    Agent competition double auction mechanism

    Management Science

    (2006)
  • L.Y. Chu et al.

    Truthful double auction mechanisms

    Operations Research

    (2008)
  • Cited by (0)

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