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On the Relationship Between the VCG Mechanism and Market Clearing On the Relationship between the VCG Mechanism and Market Clearing Takashi Tanaka1 Na Li2 Kenko Uchida3 Abstract— We consider a social cost minimization problem well. First, it may not be easy to guarantee the existence with equality and inequality constraints in which a central co- of supply function equilibria in realistic electricity markets. ordinator allocates infinitely divisible goods to self-interested N As demonstrated by [13], a pure Nash equilibrium may fail firms under information asymmetry. We consider the Vickrey- Clarke-Groves (VCG) mechanism and study its connection to an to exist even in a relatively simple supply function bidding alternative mechanism based on market clearing-price. Under model. Second, even if the supply function equilibria are the considered set up, we show that the VCG payments are known to exist, it could be a difficult task for the firms to find equal to the path integrals of the vector field of the market them efficiently [14]. Finally, unlike the mechanism design clearing prices, indicating a close relationship between the approach (discussed next), the efficiency loss is inevitable. VCG mechanism and the “clearing-price” mechanism. We then discuss its implications for the electricity market design and The mechanism design is an alternative approach to- also exploit this connection to analyze the budget balance of wards the market power mitigation. For instance, the refer- the VCG mechanism. ences [15]–[18] consider applications of the Vickrey-Clarke- Groves (VCG) mechanism [2, Ch. 23] to several market I. INTRODUCTION design problems in power system operations. The VCG Market power monitoring and mitigation are essential mechanism provides an attractive market model since it aspects of today’s electricity market operations [1]. The attains zero efficiency loss and incentive compatibility (and efficiency loss due to market manipulations by strategic firms often individual rationality [19]). It is also attractive since are not only predicted in theory [2, Ch. 12], but also doc- the existence of equilibria is guaranteed (truth-telling is an umented as real incidents [3]. While the current practice of equilibrium) and it can be trivially found by the participants. market power monitoring relies on relatively simple metrics However, a major shortcoming of the VCG mechanism is (e.g., Herfindahl–Hirschman Index [4]), further deregulation the lack of budget balance [8], which is a prominent reason and emergence of new markets (e.g., ISO-aggregator and why it is rarely used in practice. The VCG mechanism has aggregator-prosumer interfaces) necessitate advanced market other drawbacks such as vulnerability to coalitions, high power analysis [5] and appropriate market design. computational complexity, and lack of privacy [19]. The efficiency loss in oligopolistic electricity markets has For a successful electricity market design in the future, been actively studied in recent literature. Among others, a an integrated discussion on these different approaches are popular venue of research (e.g., [6]–[9]) is based on the sup- desired in the common framework, so that their pros and cons ply function bidding model by Klemperer and Meyer [10], are systematically understood. Therefore, in this paper, we in which the market clears according to the producers’ (resp. discuss different market mechanisms, namely the “clearing- consumers’) submitted supply (resp. demand) functions. In price” mechanism (the underlying mechanism for the supply [10], it is observed that if there is no demand uncertainty function bidding model) and the VCG mechanism, in a and arbitrary supply function bidding is allowed, numerous common set up. As a common set up, we consider a general multiplicity of equilibria emerges. Johari and Tsitsiklis [8] social cost minimization problem with equality and inequal- observed that “as the strategic flexibility granted to firms ity constraints. Our goal is to provide a unified view on increase, their temptation to misdeclare their cost increases as these seemingly different market mechanisms, which would well,” and considered a simple (scalar-parametrized) supply hopefully be valuable in the future studies on electricity function model with N participants and showed that the market design. price of anarchy (efficiency loss) is upper bounded by The contribution of this paper is two-fold. First, we show 1 + 1=(N − 2). In [11], it is observed that the worst case a connection between the VCG mechanism and the clearing- efficiency loss depends on the transmission network topology price mechanism by proving that the VCG payment is equal when there exist transmission constraints. The efficiency loss to the path integral of the vector fields of market clearing under transmission constraints is further studied in [12] under prices under the considered set up (Theorem 2). It also a slightly different supply function bidding model. follows from this observation that tax values calculated by While the market analysis via the supply function bidding the clearing-price mechanism and the VCG mechanism are model has been fruitful, there are several limitations as similar when individual participants’ market power is negli- 1Department of Aerospace Engineering and Engineering Mechanics, Uni- gible. This result indicates a close connection between the versity of Texas at Austin, Austin, TX, USA. [email protected]. VCG and the clearing-price mechanisms. Second, in order 2School of Engineering and Applied Sciences, Harvard University, Cam- to address the budget balance issue of the VCG mechanism, bridge, MA, USA. [email protected]. 3School of Ad- vanced Science and Engineering, Waseda University, Tokyo, Japan. we apply the above observation to analyze the worst case [email protected]. budget loss in the VCG mechanism. With some simplified ∗ assumptions, we outline how the budget loss can be analyzed The net cost for the i-th firm is Ci = fi(~xi ) + πi. An using the main result of this paper. auction mechanism is characterized by a particular choice This paper is formulated as follows. In Section II, a of a decision rule and a payment rule. general social cost minimization problem to be considered We say that an auction mechanism is (dominant strategy) ~ is formulated. The clearing-price mechanism and the VCG incentive compatible if for each firm truth-telling (fi = fi) mechanism are described in Section III. The main technical is an optimal strategy to minimize his/her net cost regardless result (path integral characterization of the VCG mechanism) of the other agents’ reporting strategy. is presented in Section IV. The budget loss of the VCG B. Cost function bidding vs. supply function bidding mechanism is studied in Section V. The cost function bidding model described above is II. PROBLEM FORMULATION closely related to the supply function bidding models [10]. Suppose that there exist N participating firms indexed by In a simple case where a production firm i is producing a i = 1; 2; :::; N and a single auctioneer. Consider a social single commodity (e.g., electricity), a supply function si(p) cost minimization problem with equality and inequality describes the desired level of production as a function of constraints: price p at which the product can be exchanged in the market. N X Mathematically, a production cost function fi(xi) and a P : min f(x) fi(xi) (1a) x , i=1 supply function si(p) are related by si(p) = argmaxxi pxi − XN f (x ): For consuming firms, there is a similar relationship s.t. g(x) , gi(xi) ≤ 0 (1b) i i i=1 between utility functions and demand functions. From this XN h(x) hi(xi) = 0: (1c) relationship (c.f., Legendre transformation), it can be seen , i=1 that a cost function bidding model is equivalent to supply For each i = 1; 2; :::; N, xi can be interpreted as production function bidding model under mild assumptions (e.g., con- by the i-th firm, or as consumption if it is negative. The 2 vexity of fi). For instance, for a > 0, fi(xi) = ax if ni i function fi : R ! R is interpreted as a production cost if 1 and only if si(p) = 2a p. Thus, assuming a linear supply the firm i is a producer, or as a negated utility function if function is equivalent to assuming a quadratic cost function. the firm is a consumer. The aggregation f(x) is referred to Alternatively, suppose that Fi is the set of non-decreasing, as the social cost function (negated social welfare function). continuously differentiable, and strongly convex functions ni k ni l Functions gi : ! and hi : ! are interpreted 0 R R R R from [0; 1) to [0; 1) such that fi(0) = 0 and f (0) = 0. as the i-th firm’s contribution to the global equality/inequality i Then fi 2 Fi if and only if si 2 Si, where Si is the set of constraints such as the power flow balance constraints, line non-decreasing continuous function from [0; 1) to [0; 1) flow limit constraints, local capacity constraints. Constraints such that si(0) = 0. For the rest of this paper, we focus on (1b) and (1c) are imposed entry-wise. the cost function bidding model only. Remark 1: We remark here that in the problem setting (1), we only consider separable objective function and con- III. CLEARING-PRICE AND VCG MECHANISMS straints function. This is usually true for electricity market, In this section, we formally introduce the clearing-price especially when DC power flow model is used where all the mechanism and the VCG mechanism for the social cost constraints are in a linear form [20]. minimization problem (1). We assume that functions gi and hi for each i = 1; 2; :::; N A. The clearing-price mechanism are publicly known. However, the function fi 2 Fi is assumed to be private, i.e., known to the i-th firm only. The The clearing-price mechanism is characterized by the family Fi of functions are assumed to be known a priori.
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