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, ∞) to [0, ∞) such that fi(0) = 0 and f (0) = 0. as the i-th firm’s contribution to the global equality/inequality i Then fi ∈ Fi if and only if si ∈ Si, where Si is the set of constraints such as the power flow balance constraints, line non-decreasing continuous function from [0, ∞) to [0, ∞) 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-PRICEAND 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 ∈ 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. An following decision and payment rules. ∗ ∗ ∗ 1 ˜ optimal solution to P will be denoted by x = (x1; ...; xN ) . Decision rule: Based on reported functions fi, i = 1, 2, ..., N, the auctioneer formulates a problem P˜, which is A. Bidding and payment model identical to P in (1) except that reported functions are used, Consider a single-round auction mechanism in which each and solve for the primal-dual optimal solution (˜x∗, µ˜∗, λ˜∗) firm is first given an opportunity to report his/her private satisfying the KKT condition:2 function f˜ ∈ F to the auctioneer. In this step, firms are i i ˜ ∗ ∗> ∗ ˜∗> ∗ ˜ ∇fi(˜x ) +µ ˜ ∇gi(˜x ) + λ ∇hi(˜x ) = 0 (2a) allowed to misreport their information and thus fi 6= fi i i i N N in general. In the next step, the auctioneer makes a social X ∗ X ∗ gi(˜xi ) ≤ 0, hi(˜xi ) = 0 (2b) decision x˜∗ = (˜x∗; ...;x ˜∗ ) based on the reported information i=1 i=1 1 N T ∗> X ∗ according to a certain decision rule. Finally, the auctioneer µ˜ gi(˜xi ) = 0 (2c) calculates payments π = (π ; ...; π ) ∈ N according to a i=1 1 N R µ˜∗ ≥ 0, (2d) certain payment rule. Throughout this paper, we consider πi as the payment from the i-th firm to the auctioneer, which N k N 2When a function f : R → R is differentiable at x ∈ R , we define can be either positive or negative. Here πi can be thought of  ∂f1 ··· ∂f1  ∂x1 ∂xN as a tax by which the i-th firm’s incentive is manipulated.  . .  the gradient matrix of f at x as ∇f(x) =  . . .  . .  1We assume x is a column vector. We write (x; y) = [x> y>]>. ∂fk ··· ∂fk ∂x1 ∂xN where (2a) is imposed for each i = 1, ..., N. A social 0 ∗ ∗ ∗ N=2 decision is determined by x˜ = (˜x1; ...;x Ñ ). −0.2 N=3 Payment rule: The auctioneer computes a vector of the N=5 N=10 “market clearing prices” 2 −0.4 N=100 +1) 1 ∗ ∗> ∗ ∗> ∗

˜ N−a p˜i =µ ˜ ∇gi(˜xi ) + λ ∇hi(˜xi ) 1 −0.6 )/(a 1 for each i = 1, ..., N. The payment (from the ﬁrm i to the −0.8 (1−2a auctioneer) is determined by 2 = N 1

C −1 clearing ∗ ∗ πi =p ˜i x˜i . (3) −1.2 The clearing-price mechanism and its variants are widely −1.4 used in practice of power system operations because of its 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 a simplicity. It is also used as an underlying market model 1 in the existing market analyses (e.g., [6]–[9]). A drawback of this mechanism is that sometimes the existence and Fig. 1. The first generator’s report a1 and the resulting cost C1. uniqueness of the solution satisfying the KKT condition are not guaranteed especially when a nonconvex problem (a1 6= 1), while other firms remain truthful (ai = 1, i = formulation is used [21]. Also, the clearing-price mechanism 2, 3, ··· ,N). The KKT condition (5) is is not incentive compatible in general, as demonstrated in the ∗ ˜∗ ∗ ˜∗ next simple example. 2a1x˜1 + λ = 0, 2˜xi + λ = 0 ∀i = 2, 3, ··· ,N N Example 1: Consider a model of power supply market X ∗ x˜i = N with N power generating firms. Let xi be the production by i=1 the i-th firm, and suppose that the total power supply must from which we have meet the total demand N. Assume that the true generation 2 2a N N cost function is given by fi(xi) = xi for every i = ˜∗ 1 ∗ λ = − , x˜1 = 1, 2, ..., N. The optimal allocation x that minimizes the social a1N − a1 + N a1N − a1 + N cost is obtained by solving a1N x˜∗ = ∀i = 2, 3, ··· ,N. i a N − a + N XN 1 1 min fi(xi) (4a) x i=1 Now, the net cost for the first firm is calculated as XN s.t. xi = N. (4b) 2 i=1 ∗ clearing N (1 − 2a1) C1 = f1(˜x1) + π1 = 2 . Clearly, the optimal solution is x∗ = (1; 1; ··· ; 1). Suppose (a1N − a1 + 1) that the function fi(xi) is publicly unknown besides the fact Figure 1 plots C1 as a function of a1 for various N. It can 2 that it belongs to the set Fi = aixi : ai > 0 . Thus, each be seen that the first generator can achieve the minimum net firm i is required to report his/her private parameter ai to 1 1 cost C1 = −1 − N 2−1 by reporting a1 = 1 + N−1 . The net the auctioneer. cost in this case is smaller than that follows from the truthful Suppose that the clearing-price mechanism is applied to report (which is −1). Thus we conclude that truthful report ˜ 2 Example 1. Based on the reported functions fi(xi) = aixi , is not a Nash equilibrium (and thus not a dominant strategy). the auctioneer solves the KKT condition It is also worth mentioning that as the number of participants ˜ ∗ ˜∗ increases (N → +∞), the optimal reporting strategy tends ∇xi fi(˜xi ) + λ = 0 ∀i = 1, 2, ··· ,N (5a) to be the truthful one (a1 → 1). N X ∗ x˜i = N (5b) If all participants are truthful in the clearing-price mech- i=1 anism, the payment for the i-th participant is for a social decision x˜∗ and the “market-clearing price” λ˜∗. clearing, truthful ∗ ∗ The auctioneer computes the payments for individual firms πi = pi xi (6) by πclearing = λ˜∗x˜∗. (Note that πclearing is negative in this i i i with p∗ = µ∗>∇g (x∗)+λ∗>∇h (x∗) where (x∗, µ∗, λ∗) is example because we follow the convention that the payment i i i i i the primal-dual optimal solution to P. is from the firm to the auctioneer by default.) Assuming all generators submit their cost functions truthfully (i.e., ˜ 2 ∗ ˜∗ B. The VCG mechanism fi(xi) = xi ), then x˜ = (1; 1; ··· ; 1) and λ = 2 satisfy the KKT condition. Hence, the reward that each generator The VCG mechanism, also known as the pivot mechanism, ˜∗ ∗ receives is calculated as −λ x˜i = 2, and the net cost for the chooses the payment rule in such a way that the i-th firm’s ∗ clearing firm i is Ci = fi(˜xi ) + πi = 1 − 2 = −1 (profit is +1). payment is equal to its externality. To this end, the mecha- To demonstrate that the clearing-price mechanism is not nism evaluates the impact of each firm’s “presence” in the incentive compatible, suppose that the first firm misreports market by introducing the following auxiliary optimization ∗ −i∗ problem for each i = 1, 2, ··· ,N: Let x˜ and x˜ (xi) be the optimal solutions to P˜ and P˜−i(x ) respectively. The net cost in this case is N i −i X ∗ VCG P (xi) : min f(xi, x−i) , fj(xj)+fi(xi) (7a) C˜i = fi(˜x ) +π ˜ x−i i i j6=i ∗ X ˜ ∗ X ˜ −i∗ N = fi(˜xi ) + fj(˜xj ) − fj(˜xj (0)). X j6=i j6=i s.t. g(xi, x−i) gj(xj)+gi(xi) ≤ 0 (7b) , Now, notice that xˆ−i∗(0) =x ˜−i∗(0), i.e., the i-th firm’s j6=i j j N reporting strategy does not alter the solution to the auxiliary X problem. Thus C˜i < Cî implies h(xi, x−i) , hj(xj)+hi(xi) = 0. (7c) j6=i ∗ X ˜ ∗ ∗ X ˜ ∗ fi(˜xi ) + fj(˜xj ) < fi(ˆxi ) + fj(ˆxj ). j6=i j6=i Decision variables in (7) are x−i = ∗ −i However, this is a contradiction to our hypothesis that xˆ is (x1; ...; xi−1; xi+1; ...; xN ). Notice that in P (xi), an an optimal solution to Pˆ. optimal decision x−i for the firms excluding i needs to be In view of Theorem 1, the VCG mechanism is said to found while xi is fixed. Denote by implement an efficient social choice function in dominant −i∗ −i∗ −i∗ −i∗ −i∗ x (xi) = x1 (xi); ...; xi−1(xi); xi+1(xi); ...; xN (xi) strategies [23]. Assuming all reports are truthful, the VCG payment becomes4 −i an optimal solution to the problem P (xi). Note that −i∗ −i∗ VCG X ∗ X −i∗ x (x ) is parametrized by x . We interpret x (0) as an πi = fj(xj ) − fj(x (0)). (9) i i j6=i j6=i j optimal social decision for the firms other than i when the i-th firm is absent from the market.3 The VCG mechanism is It is instrumental to notice the difference between the VCG characterized by the following decision and payment rules. payment (9) and that of the clearing-price mechanism (6) ˜ with truthful participants. To quantify the difference, consider Decision rule: Based on the reported functions fi, i = 1, 2, ..., N, the auctioneer formulates optimization problems an application of the VCG mechanism to the scenario in P˜ and P˜(−i)(0) ∀i = 1, ··· ,N, which are identical to P Example 1. For each i = 1, ··· ,N, the VCG payment is and P−i(0) except that reported functions are used. The computed as auctioneer determines a social decision x˜∗ as an optimal VCG X ∗ X −i∗ πi = fj(xj ) − fj(xj (0)) solution to P˜. The auctioneer also computes an optimal j6=i j6=i solution x˜−i∗(0) to P˜−i(0) for every i = 1, ··· ,N. X X N = fj(1) − fj( ) Payment rule: Payment for the firm i is calculated by j6=i j6=i N − 1 1 VCG X ˜ ∗ X ˜ −i∗ = −2 − (10) πi = fj(˜xj ) − fj(˜x (0)). (8) N − 1 j6=i j6=i j In other words, each firm receives 2 + 1 for producing The next Theorem is a straightforward application of the N−1 x = 1. Recall that the payment determined by the clearing- well-known result (e.g., [2, Ch. 23] [19, Ch.10] [23]) to the i price mechanism when all reports were truthful was 2. social cost minimization problem (1). Therefore, in this example, we have πclearing, truthful > πVCG Theorem 1: The VCG mechanism is dominant strategy i i for all i, even though in both cases the production by the incentive compatible. firms are the same. Also, notice that the difference diminishes Proof: Although this is a minor variation of the standard as N → +∞. results, we provide a proof for the sake of completeness. We first assume that there exists an index i and reporting IV. PATH INTEGRAL REPRESENTATIONS OF VCG ˜ strategies fj for j 6= i such that the firm i attains a smaller PAYMENTS ˜ net cost Ci by misreporting (i.e., fi 6= fi), and then derive In this section, we establish a mathematical relationship a contradiction. between the clearing-price and the VCG mechanisms in a ˆ ˆ−i Denote by P and P (xi) the problems (1) and (7) when general setting. We first recall some basic results from non- ˜ ˜ ˜ ˜ ∗ functions (f1, ..., fi−1, fi, fi+1, ..., fN ) are reported. Let xˆ linear programming. A feasible vector x for the problem −i∗ −i −i and xˆ (xi) be the optimal solutions to Pˆ and Pˆ (xi) −i P (xi) is said to be regular [24] if the gradient matrix respectively. The net cost for the i-th firm is ∇x−i g˜(xi, x−i) Cˆ = f (ˆx∗) +π ˆVCG i i i i ∇x−i h(xi, x−i) ∗ X ˜ ∗ X ˜ −i∗ = fi(ˆxi ) + fj(ˆxj ) − fj(ˆx (0)). j6=i j6=i j has linearly independent rows. Here, g˜(xi, x−i) is the collection of all active constraints in g(xi, x−i) ≤ 0, i.e., the ˜ ˜−i −i∗ Also, denote by P and P (xi) the problems (1) and (7) collection of indices k such that gk(xi, x−i) = 0. If x ˜ ˜ ˜ ˜ ˜ when functions (f1, ..., fi−1, fi, fi+1, ..., fN ) are reported. is regular, there exists a unique set of Lagrange multipliers

3In this paper, we interpret the i-th firm being absent from the market as 4We do not distinguish (8) and (9) since they are equal under a realistic xi = 0. See [22] for related discussions. assumption that all reports are truthful. µ−i∗ ∈ Rk, λ−i∗ ∈ Rl such that the following KKT almost everywhere in [0, 1]. We claim that the first term of conditions are satisfied [24, Proposition 3.3.1]. (14) is zero. To see this, suppose that the `-th component P −i∗ > of g (x ) + g (γ (t)) is strictly negative at some ∇ f (x−i∗) + µ−i∗ ∇g (x−i∗) j6=i j j i i xj j j j j t ∈ [0, 1]. Due to continuity, it is always possible to choose −i∗> −i∗ +λ ∇hj(xj ) = 0 ∀j 6= i (11a) an interval [t − , t + ] on which the `-th component of P −i∗ j6=i gj(xj ) + gi(γi(t)) is strictly negative. By the com- −i∗> X −i∗ µ gj(xj ) + gi(xi) = 0 (11b) plementary slackness condition (11b), the `-th component of j6=i −i∗ µ−i∗(γ (t)) is zero on this interval, and so is dµ . Thus µ−i∗ ≥ 0. i dt (11c) (14) becomes For notational simplicity, in what follows, we often suppress −i∗ −i∗ −i∗ −i∗ −i∗> X −i∗ dxj dγi the dependency of (x , µ , λ ) on xi. µ ∇g (x ) + ∇g (γ (t)) = 0 (15) j j dt i i dt For each i = 1, 2, ··· ,N, consider a smooth path γi : j6=i ni ∗ [0, 1] → R such that γi(0) = 0 and γi(1) = xi . Notice that the origin of the path corresponds to the case where almost everywhere in [0, 1]. Now, the right hand side of (12) becomes the i-th firm is absent from the market (xi = 0) and the Z 1 terminal of the path corresponds to the case where the i-th −i∗ > firm is assigned an optimal solution to (1) (x = x∗). µ (γi(t)) ∇gi(γi(t)) i i 0 For each point x ∈ γ on the path, as before, i i dγi −i∗ −i∗ > we write x (xi) to denote an optimal solution to + λ (γi(t)) ∇hi(γi(t)) dt −i −i∗ dt P (xi). Whenever x (xi) is regular, we also denote Z 1 −i∗ −i∗ X −i∗ > −i∗ by (µ (xi), λ (xi)) the corresponding unique set of = − µ (γi(t)) ∇gj(xj ) Lagrange multipliers. Notice that for each i = 1, ..., N, j6=i 0 (x−i∗(x∗), µ−i∗(x∗), λ−i∗(x )) = (x∗ , µ∗, λ∗) −i∗ i i i −i . dxj VCG + λ−i∗(γ (t))>∇h (x−i∗) dt (16) The next theorem shows that VCG payment πi is equal i j j dt to the path integral of the vector field of the market clearing Z 1 −i∗ X −i∗ dxj prices along γi. = ∇fj(xj ) dt (17) Theorem 2: Assume f, g and h are continuously differen- j6=i 0 dt ∗ X −i∗ ∗ X −i∗ tiable, and P admits an optimal solution x . Let γi : [0, 1] → = fj(x (xi )) − fj(x (0)) (18) j6=i j j6=i j ni γ (0) = 0 γ (1) = x∗ R be any smooth path such that i , i i , X ∗ X −i∗ = fj(xj ) − fj(x (0)) (19) and the following conditions are satisfied: j6=i j6=i j −i −i∗ (A) P (γi(t)) admits an optimal solution x (γi(t)) for VCG =πj . (20) every t ∈ [0, 1]. −i∗ (B) x (γi(t)) is continuously differentiable in t, and is In (16), we have used (13) and (15). In (17), the KKT condi- dx−i∗ regular almost everywhere in [0, 1]. −i∗ j tion (11a) was used. Since ∇fj(xj ) is continuous in −i∗ −i∗ dt (C) µ (γi(t)) and λ (γi(t)) are differentiable almost t, the fundamental theorem of calculus is applicable in (18). everywhere in [0, 1]. Then, the following equality holds: Remark 2: A special case of (12) can be found in [2, Ch. Z Z 1 23.C]. Theorem 2 also extends [25, Proposition 4] to general VCG ∗ ∗ dγi πi = pi (xi)dxi = pi (γi(t)) dt (12) class of problems of the form P. γi 0 dt ∗ −i∗ > −i∗ > A. Example: Simple power supply market where pi (xi) = µ (xi) ∇gi(xi) + λ (xi) ∇hi(xi). −i∗ Proof: Since x (γi(t)) is a feasible solution We consider again the power supply market in Example 1. to P−i(x ), it must satisfy the equality constraint ∗ i Recall that xi = 1 for every i is the optimal solution. So P h (x−i∗) + h (γ (t)) = 0 for every t ∈ [0, 1]. j6=i j j i i define a smooth path γi : [0, 1] → R by γi(t) = t so that Differentiating with respect to t, we obtain ∗ γi(0) = 0 and γi(1) = xi . Since hi(xi) = xi, we have −i∗ ∇x hi(xi) = 1. There is no inequality constraint. Therefore, X −i∗ dxj dγi i ∇hj(x ) + ∇hi(γi(t)) = 0. (13) (12) becomes j6=i j dt dt Z 1 Next, since the left hand side of (11b) is differentiable at VCG −i∗ almost every t ∈ [0, 1], πi = λ (xi)dxi. (21) 0 −i∗ > dµ X −i∗ Equation (21) can also be verified as follows. Note that gj(xj ) + gi(γi(t)) −i dt P (xi) is j6=i N dx−i∗ X 2 2 −i∗> X −i∗ j dγi min xj + xi + µ ∇g (x ) + ∇g (γ (t)) = 0 x j6=i j j dt i i dt −i j6=i XN s.t. xj = N − xi. (14) j6=i 휆−𝑖∗(0)

Market ∗ It is easy to verify that the optimal solution is regular and the 푥푖 clearing 푉퐶퐺 ∗ −i∗ 2(xi−N) 휋𝑖 = න 푝𝑖 (푥𝑖)푑푥𝑖 corresponding Lagrange multiplier is λ (xi) = N−1 . prices 0 Thus the right hand side of (21) is computed as (Area below the curve) 푝∗(0) Z 1 Z 1 𝑖 푐푙푒푎푟𝑖푛푔, 푡푟푢푡ℎ푓푢푙 ∗ ∗ −i∗ 2(xi − N) 1 휋𝑖 = 푝𝑖 푥𝑖 λ (xi)dxi = dxi = −2 − . 0 0 N − 1 N − 1 (Rectangle area) This result coincides with πVCG obtained in (10). i 푝∗ ∗ 𝑖 푝𝑖 (푥𝑖) B. Graphical interpretation of Theorem 2 In this subsection, we introduce a graphical interpretation of the formula (12) for the cases with ni = 1. In Figure 2, let ∗ 0 푥∗ 푥 pi (xi) (defined in Theorem 2) be the market clearing price 𝑖 𝑖 ∗ (Optimal solution to 풫) written as a function of xi. In particular, pi (0) is the market clearing price when the i-th firm’s consumption (production) Fig. 2. VCG vs clearing-price mechanisms. ∗ ∗ ∗ is zero and pi = pi (xi ) is the market clearing price when it is x∗ (i.e., the optimal solution to P). The formula (12) shows i A. Quadratic cost functions that the VCG tax (the monetary payment from the firm to the ∗ auctioneer) is equal to the area below the curve of pi (xi). Consider a special case of (1) in which each cost function 2 On the other hand, by definition (6), the tax imposed by is of the form fi(xi) = aixi with ai > 0 and the only PN the clearing-price mechanism when all reports by the firms constraint is i=1 xi = c. The next result shows that VCG clearing, truthful clearing, truthful are truthful is equal to the area of the rectangle shown in the quantity (Bloss − Bloss )/Bloss can be Figure 2. explicitly computed in terms of individual firm’s share si ∗ , VCG xi Figure 2 shows that the discrepancy between πi and . clearing, truthful c πi is closely related to the market power of the Theorem 3: For each i = 1, ..., N, assume that the cost 2 i-th firm. The next result holds for general cases with ni ≥ 1. function has the form fi(xi) = aixi with ai > 0 and the Lemma 1: Let the optimization problem P and the PN only constraint is i=1 xi = c. Then n ∗ smooth path γi : [0, 1] → R defined by γi(t) = xi t satisfy VCG clearing, truthful N 2 B − B X s the conditions in Theorem 2. Suppose, in addition, that there loss loss = i . clearing, truthful 2(1 − s ) exists a constant > 0 such that Bloss i=1 i ∗ ∗ kpi (xi) − pi k ≤ (22) Proof: From the KKT condition, for each i = 1, ..., N, −1 −i∗ P 1 we have λ (xi) = 2 (xi − c). Thus for each point xi on γi, where k · k is the Euclidean norm j6=i aj clearing, truthful on ni . Then |πVCG − π | ≤ kx∗k. R i i i ∗ −1 Z xi Proof: Due to the Cauchy-Schwarz inequality, we have VCG −i∗ X 1 1 ∗2 ∗ πi = λ (xi)dxi =2 xi −cxi . ∗ ∗ ∗ ∗ ∗ ∗ 0 j6=i aj 2 −kxi k ≤ pi (γi(t))xi − pi xi ≤ kxi k ∗ 1 2xi for all t ∈ [0, 1]. Integrating each side with respect to t, and From the KKT condition, we also have = − ∗ . ai λ VCG using the formula (12): Substituting this into the above equality, we have πi = λ∗x∗2 Z 1 Z 1 ∗ ∗ i dγi λ xi − 2(x∗−c) . Thus, πVCG = p∗(γ (t)) dt = p∗(γ (t))x∗dt, i i i i dt i i i 0 0 clearing BVCG − B N ∗ VCG clearing, truthful ∗ loss loss X VCG ∗ ∗ we obtain −kx k ≤ π − π ≤ kx k. = − πi + λ c /(−λ c) i i i i clearing i=1 Lemma 1 shows that if the price markup by firm i is Bloss XN x∗/c x∗ bounded in the sense of (22) (graphically, the function = i i ∗ VCG i=1 ∗ pi (xi) in Figure 2 is nearly constant), then πi is close 2(1 − xi /c) c clearing, truthful to π . XN s2 i = i . V. BUDGET LOSS ANALYSIS i=1 2(1 − si) Although the VCG mechanism implements efficient so- VCG cial choice functions in dominant strategies, budget balance In particular, if si = 1/N for all i, we have (Bloss − clearing, truthful clearing, truthful property is not guaranteed in general. This is a prominent Bloss )/Bloss → 0 as N → ∞. VCG reason why the VCG mechanism is rarely used in practice. Remark 3: Although Theorem 3 shows Bloss > In this section, we apply Theorem 2 to analyze the budget clearing, truthful Bloss , it should not be concluded that the budget loss VCG PN VCG loss of the VCG mechanism. Let Bloss , − i=1 πi be in the VCG mechanism is larger than that of the clearing- clearing, truthful clearing, truthful the budget loss in the VCG mechanism, and Bloss , price mechanism. This is because Bloss is evaluated PN clearing, truthful − i=1 πi be the budget loss in the clearing- with the assumption that all players are truthful, and does not price mechanism under the assumption that all players are reflect the realistic budget loss when the players are strategic truthful. in the clearing-price mechanism. B. Bounded market power [3] S. 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