1 A Structural Characterization of Market Power in Electric Power Networks

Weixuan Lin Eilyan Bitar

Abstract—We consider a market in which capacity-constrained generators compete in scalar-parameterized supply functions to serve an inelastic demand spread throughout a transmission constrained power network. The market clears according to a locational marginal pricing mechanism, in which the independent system operator (ISO) determines the generators’ production quantities to minimize the revealed cost of meeting demand, while ensuring that network transmission and generator capacity constraints are met. Under the stylizing assumption that both the ISO and generators choose their strategies simultaneously, we establish the existence of Nash equilibria for the underlying market, and derive an upper bound on the allocative efficiency loss at Nash equilibrium relative to the socially optimal level. We also characterize an upper bound on the markup of locational marginal prices at Nash equilibrium above their perfectly competitive levels. Of particular relevance to ex ante market power monitoring, these bounds reveal the role of certain market structures—specifically, the market share and residual supply index of a producer—in predicting the degree to which that producer is able to exercise market power to influence the market outcome to its advantage. Finally, restricting our attention to the simpler setting of a two-node power network, we provide a characterization of market structures under which a Braess-like paradox occurs due to the exercise of market power—that is to say, we provide a necessary and sufficient condition on market structure under which the strengthening of the network’s transmission line capacity results in the (counterintuitive) increase in the total cost of generation at Nash equilibrium.

Index Terms—Electricity markets, supply function equilibrium, market power, efficiency loss, power networks, Braess’ paradox. !

1 INTRODUCTION

We consider an electricity market design in which power (in)efficiency of supply function equilibria given various producers compete in supply functions to meet an inelastic restrictions on the parametric form of supply functions that demand distributed throughout a transmission constrained producers can bid—see, for example, [8], [9], [10], [11], power network. Such markets are vulnerable to manipula- [12], [13], [14], [15], [16], [17], [18], [19], [20]. Closest to the tion given the large leeway afforded producers in reporting market model considered in the present paper, Johari and their supply functions [1]. The potential for market ma- Tsitsiklis [6] propose a scalar-parameterized supply function nipulation is amplified by the largely inelastic nature of bidding mechanism in which N producers compete to meet electricity demand, and the presence of hard constraints on a known and inelastic demand. Under the assumption that transmission and production capacity [2], [3]. For example, each producer has sufficient capacity to serve the demand the strategic withholding of generation capacity by certain individually, they establish an upper bound on the market’s producers during the 2000-01 California electricity crisis corresponding price of anarchy given by 1+1/(N −2). More resulted in over 40 billion US dollars in added costs to recently, Xu et al. [7] provide an elegant generalization of consumers and businesses, and the bankruptcy of Pacific these results to the setting in which producers have limited Gas and Electric [4], [5]. production capacities, which are encoded in the supply In this paper, we analyze a stylized market model in functions that they bid.

arXiv:1709.09302v3 [cs.GT] 28 Jan 2019 which power producers are required to bid supply func- While the previous supply function models offer a com- tions belonging to a scalar-parameterized family defined pelling description of competition in single-node electricity in the manner of [6], [7]. Working within this setting, we markets, the characterization and analysis of supply func- elucidate the role of market structure in determining the tion equilibria becomes challenging in the presence of net- degree to which price-anticipating producers can exercise work transmission constraints [21], [22], [23], [24], [25]. For market power to influence market allocations and prices at example, it was shown in [23] that the profit maximization equilibrium. problem for each producer amounts to a mathematical pro- Related Work: The formal study of (infinite-dimensional) gram with equilibrium constraints (MPEC), which is, in gen- supply function equilibria dates back to the seminal work eral, a computationally intractable nonconvex optimization of Klemperer and Meyer [1], which revealed that essen- problem. Additionally, it is well known that, even under the tially any production profile can be supported by a supply restriction to linear or piecewise-constant supply functions, function equilibrium in the absence of demand uncertainty. supply function equilibria may fail to exist in simple two or There has subsequently emerged a large body of litera- three-node networks [26], [27]. In an effort to address such ture investigating the existence, uniqueness, and allocative difficulties in analysis, there has emerged another stream of literature that resorts to the so-called networked Cournot Weixuan Lin and Eilyan Bitar are with the School of Electrical and Com- puter Engineering, Cornell University, Ithaca, NY, 14853, USA. Emails: model to characterize the strategic interaction between pro- {wl476,eyb5}@cornell.edu ducers in constrained transmission networks. We refer the 2 reader to [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], and price markups incurred at Nash equilibria. In Section [38] for recent advances. 4, we uncover the occurrence of a transmission expansion Building on the aforementioned models of competition, paradox under a specialization of our model to the setting there are a number of papers in the literature that empiri- of a two-node power network. Section5 concludes the paper cally investigate the potential emergence of market power with a discussion on directions for future research. and assess the extent to which it exercised by producers Notation: Let R denote the set of real numbers, in actual electricity markets (e.g., by estimating price-cost and R+ the set of non-negative real numbers. Denote n > markups) [10], [13], [14], [39], [40], [41], [42], [43]. With the transpose of a vector x ∈ R by x . Let x−i = n−1 regard to market power monitoring in practice, it is not (x1, .., xi−1, xi+1, .., xn) ∈ R be the vector including all uncommon for regulating authorities to employ a variety but the ith element of x. Denote by 1 the vector of all ones. of structural indices to both assess the potential for market Denote by (·)+ the positive part function. For a univariate power ex ante, and to detect the actual exercise of market function f : R → R that is both left and right differentiable power ex post—see Twomey et al. [44] for a comprehensive at x = x, we denote by ∂−f(x)/∂x and ∂+f(x)/∂x the left overview. In particular, the residual supply index (RSI) has and right derivatives of f evaluated at x = x, respectively. empirically proven to be an effective predictor of market Finally, the operator ◦ denotes the Hadamard product be- power as measured by price-cost markup. For example, tween matrices of the same dimension. empirical analyses of hourly market data carried out by the California Independent System Operator (CAISO) reveal 2 MODELAND FORMULATION a significant negative correlation between hourly RSI and 2.1 Supply and Demand Models hourly price-cost markup [45]. Beyond empirical evidence, however, there is little theoretical justification in the litera- We consider the setting in which producers compete to ture for the observed effectiveness of the RSI as a predictor supply energy to an inelastic demand spread throughout of market power. An exception to this claim is the earlier a transmission constrained power network. The network is work of Newbery [46], which establishes an explicit rela- assumed to have a connected topology consisting of n trans- tionship between a producer’s Lerner index (at equilibrium) mission buses (or nodes) connected by m transmission lines and its RSI in the setting of a single-node Cournot oligopoly (or edges). Let V := {1, . . . , n} denote the set of all nodes. model. A basic limitation of these results, however, is their In addition, we assume that there are Ni producers located Pn inapplicability to markets with perfectly inelastic demand. at each node i ∈ V, and let N := i=1 Ni denote the total Our Contribution: In this paper, we develop a rigor- number of producers. We specify the nodal position of each n×N ous equilibrium analysis of the locational marginal pricing producer according to an incidence matrix A ∈ {0, 1} mechanism in a general network setting, where generators that is defined as ( are required to report scalar-parameterized supply func- 1, if producer j is located at node i, tions. Adopting a solution concept in which the ISO and Aij := (1) 0, otherwise. generators move simultaneously, we derive upper bounds on the worst-case efficiency loss and price markups seen Let Ni := {j | Aij = 1} be the set of producers at node i. at Nash equilibria. Of particular relevance to the design We assume that each column of the incidence matrix A has of methods for market power detection and mitigation, exactly one nonzero entry—that is, each producer is located these bounds shed light on the explicit role of (and inter- at exactly one node in the network. The formal treatment of play between) certain structural indices of market power— more general generation ownership structures—in which a specifically, the market share and residual supply index of producer’s generation capacity is allowed to span multiple a producer—in predicting the extent to which that pro- nodes in the network—is left as a direction for future work. ducer might exercise market power to influence the market The demand for energy is assumed to be perfectly inelas- n outcome to its advantage, e.g., increasing price above the tic. Accordingly, we let d ∈ R+ denote the demand profile competitive level (cf. Theorem1 and Corollary1). In Section across the network, where di represents the demand for 3.5, we empirically validate the predictive accuracy of our energy at node i. We let xj be the production quantity of theoretical upper bound on price markups using historical producer j, and denote by Cj(xj) the corresponding cost spot price data from the 1999-2000 Great Britain Electricity incurred by producer j for producing xj units of energy. We N Pool. In Section4, we specialize our equilibrium analysis denote by x := (x1, . . . , xN ) ∈ R the production profile. to the setting of a two-node power network, and provide a We make the following standard assumption regarding the characterization of market structures under which a Braess- producers’ cost functions. like paradox occurs due to the exercise of market power. That Assumption 1 (Convex Production Costs). The production is to say, we characterize a range of scenarios in which the cost C (x ) of each producer j ∈ {1,...,N} is a convex strengthening of the network’s transmission line capacity j j function that satisfies C (x ) = 0 for x ≤ 0, and C (x ) > 0 results in the (counterintuitive) increase in the total cost of j j j j j for x > 0. generation at a Nash equilibrium. j Organization: In Section2, we introduce the scalar- We also assume that each producer j ∈ {1,...,N} has a parameterized supply function bidding mechanism, and maximum production capacity Xj ≥ 0. It is important to note formulate the networked supply function game. Section3 that Assumption1 implicitly ignores producers’ “start-up establishes the existence of Nash equilibria and the unique- costs”, as the incorporation of nonzero startup costs will, in ness of the production profile that they induce; along with general, result in the discontinuity of producers’ production providing an upper bound on the worst-case efficiency loss cost functions at the origin. 3

2.2 The Economic Dispatch Problem et al. [7], and consider a capacitated version of the single- Ultimately, the objective of the independent system operator parameter supply function first proposed in [6]. Specifically, j θ ∈ (ISO) is to choose a production profile that minimizes the each producer reports a scalar parameter j R+ that true cost of serving the demand, while respecting the ca- defines a supply function of the form pacity constraints on transmission and generation facilities. θj S (p; θ ) = X − , (3) Doing so amounts to solving the so called economic dispatch j j j p (ED) problem, which is formally defined as where Sj(p; θj) denotes the maximum quantity that pro- N j p > 0 X X ducer is willing to supply at any price . Here, j minimize Cj(xj) j N is the true production capacity of producer . Note that, x∈R j=1 (2) implicit in this choice of supply function parameterization, subject to Ax − d ∈ P, is the requirement that each producer offer its full capacity

0 ≤ xj ≤ Xj, j = 1,...,N. into the power market. That is, we do not allow producers to bid their capacities strategically, as the “physical with- n Here, P ⊆ R represents the feasible set of nodal power holding” of capacity is carefully monitored and prohibited injections over the network. Adopting the assumptions on by the majority of ISOs in operation today [51], [52]. We which the so called DC power flow model [47] is based, one denote the strategy profile of all producers by the vector of N can represent the set P as a polytope bids θ := (θ1, . . . , θN ) ∈ R+ . Given the producers’ reported bids θ, the ISO’s objective n n > o P = y ∈ R | 1 y = 0, Hy ≤ c , is to choose a production allocation that minimizes the reported aggregate production cost, subject to the network 2m×n where H ∈ R denotes the shift-factor matrix, and transmission and production capacity constraints. The re- 2m c ∈ R the corresponding vector of transmission line ported cost function of producer j is defined as the integral of > capacities. We will refer to the constraint 1 y = 0 as its inverse supply function, which is given by Hy ≤ c the power balance constraint, and the constraint as x   Z θj Xj the transmission capacity constraint. Any production profile Cbj(x; θj) := dz = θj log . (4) ∗ ∗ ∗ N X − z X − x x = (x1, . . . , xN ) ∈ R that solves (2) is called efficient, and 0 j j PN ∗ the corresponding aggregate production cost j=1 Cj(xj ) With these reported costs in hand, the ISO solves the is referred to as the efficient cost. following economic dispatch (ED) problem:

N X 2.3 Scalar-parameterized Supply Function Bidding minimize Cbj(xj; θj) x∈ N R j=1 In practice, the ISO does not have access to the producers’ (5) Ax − d ∈ P, true cost information. Instead, the producers are asked to subject to report their private information to the ISO in the form of xj ≤ Xj, j = 1,...,N. supply functions, which specify the maximum quantity a Naturally, the misrepresentation of private cost informa- producer is willing to supply as a function of price. In tion by producers has the potential to induce market the majority of US electricity markets in operation today, allocations—as determined by the solution of problem (5)— it is customary for the ISO to require that each producer that are highly suboptimal (inefficient) for the original ED report a supply function in the form of a non-decreasing step problem (2). In this paper, we will attempt to understand function that is parameterized according to a finite number the role played by different market and network structures of price-quantity pairs [48]. The characterization of market in determining the extent of such inefficiency at equilibrium. equilibria that might emerge under this class of supply functions is analytically intractable, in general—even in the absence of network transmission constraints. The difficulty 2.4 Attributes of the Supply Function Parameterization in analysis derives in large part from the discontinuity of We make several remarks regarding the supply function each producer’s residual demand function [49], [50]. As a parameterization considered in this paper. First, notice that result, there is a need to resort to stylized supply function each producer’s reported cost function (4) resembles a log- equilibrium models, which appropriately restrict the class arithmic barrier function, which encodes each producer’s of supply functions from which a producer is allowed to capacity constraint in a continuously differentiable fashion. choose its bid. In principle, restrictions on the class of supply As a result, one can omit the production capacity constraint functions should be chosen in such a manner as to facilitate associated with any producer j whose bid satisfies θj > 0 mathematical analyses, while preserving the main structural from the ED problem (5) without changing its optimal solu- determinants of market power and the primary mechanisms tion. This substantially simplifies the mathematical analysis by which market power is exercised, e.g., through the “eco- of the resulting supply function game. Second, the paramet- nomic withholding” of capacity. ric family of supply functions considered in this paper is With this motivation in mind, we investigate the setting expressive enough to capture market outcomes in which in which producers are allowed to bid scalar-parameterized producers can exercise market power via the “economic supply functions, and analyze the existence and efficiency of withholding” of capacity. Qualitatively, the inverse supply market equilibria that result under price-anticipating behav- function under this choice of parameterization resembles the ior in this setting. In particular, we adopt the approach of Xu so-called “hockey stick” offer strategy in which a producer 4

offers its last few units of supply at prices that are well in problem (7) can be expressed in closed-form as an explicit excess of its true marginal cost [53]. In this context, θj/p function of the nodal supply profile q. It is not difficult to P θ > 0 may be interpreted as the amount of capacity withheld by show that if j∈Ni j , then the optimal solution to (7) producer j when the price is p. is unique and is given by We also note that, in contrast to the supply functions    considered in this paper, the more widely studied family θj X xj (qi, θ) = Xj − P ·  Xk − qi (8) of affine supply functions [9], [10], [11], [17] cannot cap- k∈N θk i k∈Ni ture the economic withholding of capacity, as producers’ for each j ∈ N . If, on the other hand, P θ = 0, then inverse supply functions are necessarily affine under this i j∈Ni j parameterization. Moreover, the analysis of affine supply any feasible solution to the local ED problem (7) is optimal. function equilibria is known to be analytically intractable in We specify one optimal solution to (7) according to the presence of production capacity constraints [11]. Xj xj (qi, θ) = P · qi (9) Xk Remark 1 (Possibility of Negative Supply). A practical k∈Ni drawback of the class of supply functions that we consider for each j ∈ Ni. With Eqs. (8)–(9) in hand, one can express is that they allow for the possibility of market allocations the production cost at node i in closed-form as Gi(qi; θ) = (i.e., solutions of the ED problem (5)) in which a producer P C (x (q , θ); θ ). j∈Ni bj j i j has a negative supply allocation. We will, however, show that such outcomes are not possible at equilibrium. Namely, Remark 2 (Nodal Decomposition of Strategy Profile). We x (q , θ) we show in Proposition1 (in Section3) and Proposition2 note that j i depends on the global strategy profile θ {θ |k ∈ N } (in AppendixA) that the production quantity of a producer only through the local strategy profile k i . This is guaranteed to be non-negative at Nash equilibria and reveals an important insight. Namely, given a fixed nodal q competitive equilibria, respectively. It is also worth noting supply profile , the explicit interaction between producers that the results of this paper continue to hold under a decouples across the different nodes in the network. Such insight will play an important role in our game theoretic modified class of supply functions given by Sj(p; θj) = n θj o analysis in Section3. max Xj − p , − , where  > 0 is an arbitrary positive constant. We forgo this treatment for ease of exposition. 2.6 Networked Supply Function Game We now present a game theoretic model of supply function 2.5 Nodal Decomposition of Economic Dispatch competition in a constrained power network. We define the In what follows, we develop a primal decomposition of set of players as N := {0, 1,...,N}, where 0 denotes the the ED problem (5), which reveals an explicit relationship ISO, and j ∈ {1,...,N} denotes the jth power producer. between an individual producer’s production quantity and In practice, the producers and ISO engage in a sequential- the aggregate production quantity at his node. We do so move game in which the producers simultaneously report n by introducing an auxiliary variable q := Ax ∈ R , which their bids, in anticipation of the ISO’s determination of q = P x production quantities and nodal prices according to the we refer to as the nodal supply profile. Here, i j∈Ni j represents the aggregate production quantity at node i, and solution of the ED problem (5). Formally, this amounts to serves as the coupling variable between the (network-wide) a multi-leader, single-follower Stackelberg game. However, ED problem and the nodal ED problems defined in terms given the generality of the setting considered in this paper, a of the local variables {xj|j ∈ Ni} at each node i ∈ V. general equilibrium analysis of a sequential-move formula- More formally, the ED problem (5) admits an equivalent tion is seemingly out of reach. We, therefore, adopt a simpli- reformulation as fying assumption, and consider a model of competition that n assumes that the producers and ISO move simultaneously.1 X minimize Gi(qi; θ) q∈ n R i=1 1. The model of simultaneous movement adopted in this paper—in which the ISO’s strategic variables are the nodal supply quantities—is subject to q − d ∈ P, (6) known to manifest in market equilibria that underpredict the intensity of competition in power networks with large transmission capacities, as qi = 0, if Ni = 0, X compared to the more plausible sequential-move formulation [33], [36]. qi ≤ Xj, if Ni > 0, i = 1, . . . , n, We refer the reader to [54, Sec. IV], which provides a detailed numerical comparison of market equilibria that result under both the sequential j∈N i and simultaneous-move formulations in a two-node network. Both where Gi(qi; θ) denotes the optimal value of the local ED models are shown to provide identical predictions if the transmission i line is congested under the simultaneous-move formulation. However, problem at node . It is defined as if the line capacity is sufficiently large such that the transmission line  is guaranteed to never congest, then the simultaneous-move formula-  X X tion is shown to predict higher price markups and a greater loss of Gi(qi; θ) := min Cbj(xj; θj) xj = qi, allocative efficiency at equilibrium than is predicted by the sequential- j∈N j∈N i i (7) move formulation. In contrast, an alternative model of simultaneous ) movement—in which the ISO’s strategic variables are the nodal price differences—will result in market equilibria that coincide with those xj ≤ Xj, ∀j ∈ Ni . predicted by the sequential-move formulation in networks with suffi- ciently large transmission capacities, but will overpredict the intensity Given a fixed nodal supply profile q, the network-wide of competition in networks with limited transmission capacities. We refer the reader to Yao et al. [36] for a comprehensive discussion on the ED problem (6) can be separated across nodes as n local ED relative advantages and disadvantages of these competing models of problems (7). Moreover, the optimal solution to the local ED simultaneous movement. 5

Specifically, we assume that the producers choose their sup- the price equal to zero. It follows that the price at each node ply function bids θ concurrent with the ISO’s determination i ∈ V is given by of the nodal supply profile q. As a result, the networked  P j∈N θj supply function game decouples across the nodes in the  i , P θ > 0   if j∈Ni j p (q, θ) = P X − q network, where the producers at each node, taking the ISO’s i j∈Ni j i (10) nodal supply quantity as given, compete only amongst  P 0, if j∈N θj = 0, themselves in determining their supply function bids. Such i a requirement of simultaneous movement can be interpreted given a nodal supply profile q and bid profile θ. as an assumption of “bounded rationality” in which the Remark 3 (Locational Marginal Pricing). When the nodal n producers only partially anticipate the impact of their bids supply profile q ∈ R is chosen to solve the ED problem on the congestion charges (i.e., the nodal prices differences) (6), the nodal pricing mechanism (10) corresponds to the that result at equilibrium—see, for example, Metzler et so called locational marginal pricing (LMP) mechanism used al. [32] and Yao et al. [36]. We note that, for reasons of in many electricity markets that are in operation today. In computational and analytical tractability, the assumption AppendixA, we show that the LMP mechanism ensures of simultaneous movement is commonly employed in the the existence of an efficient competitive equilibrium in the related literature investigating the use of Cournot models presence of transmission capacity constraints. to describe competition in transmission constrained power markets [28], [32], [33], [35], [36]. With the previous specification of nodal production quantities and prices in hand, we are now in a position to We proceed with a formal description of the market formally define the payoff of each producer j ∈ N as participants, their strategy sets, and payoff functions in the i networked supply function game. πj (q, θ) := pi (q, θ) xj (q, θ) − Cj (xj (q, θ)) . (11)

Producer j’s feasible strategy set is given by Xj := R+. 2.6.1 Independent System Operator (ISO) 2.6.3 Solution Concept The ISO chooses the production quantities of the individual X := QN X producers to minimize the reported aggregate cost, while re- Let j=0 j denote the feasible strategy set for all π := (π , π , . . . , π ) specting transmission and production capacity constraints. players, and 0 1 N denote their collection (N , X , π) Given the nodal decomposition of the ED problem devel- of payoff functions. It follows that the triple oped in Section 2.5, such choice can be reduced to the defines a normal-form game, which we shall refer to as determination of the nodal supply profile q ∈ n—which the networked supply function game for the remainder of this R (N , X , π) we define to be the strategy of the ISO. Accordingly, we paper. We describe stable outcomes of the game define the payoff of the ISO as according to the Nash equilibrium solution concept. We restrict our attention to pure strategy Nash equilibria in this n X paper, as it is straightforward to show that the networked π0 (q, θ) := − Gi(qi; θ), supply function game does not admit any non-degenerate i=1 mixed strategy Nash equilibria under Assumption2 (which where his feasible strategy set is defined as ensures the strict concavity of producers’ payoff functions).  Definition 1 (Nash Equilibrium). The pair (q, θ) ∈ X is a

 n X pure strategy Nash equilibrium (NE) of the game (N , X , π) if X0 := q ∈ R q − d ∈ P, qi ≤ Xj, if Ni > 0, the payoff of the ISO satisfies  j∈Ni ) π0 (q, θ) ≥ π0 (q, θ) for all q ∈ X0, qi = 0, if Ni = 0 . and the payoff of each producer j ∈ {1,...,N} satisfies
πj (q, θj, θ−j) ≥ πj q, θj, θ−j for all θj ∈ Xj.

2.6.2 Producers and Nodal Pricing Mechanism We let XNE ⊆ X denote the set of all pure strategy Nash (N , X , π) Each producer j ∈ Ni at node i ∈ V must choose a bid equilibria associated with the game . parameter θj ≥ 0, which specifies his supply function. In Proposition1, we show that the networked supply With a slight abuse of notation, we denote the production function game is guaranteed to admit at least one pure quantity of producer j by xj(q, θ) := xj(qi, θ), where the strategy Nash equilibrium if Assumptions1-2 are satisfied. right-hand side is specified according to Eqs. (8)–(9). Naturally, the production profile at a Nash equilibrium may In this paper, we consider a nodal pricing mechanism, i.e., differ from the efficient production profile. We, therefore, a mechanism in which prices are allowed to vary across use the price of anarchy as a measure of the allocative efficiency the different nodes in the network. In particular, if the bids loss at a Nash equilibrium [55]. i P θ > 0 submitted at node are such that j∈Ni j , then the corresponding price at node i is chosen to clear the market Definition 2 (Price of Anarchy). The price of anarchy (PoA) at that node. That is to say, the price at node i is set as the associated with the game (N , X , π) is defined according to P S (p; θ ) = q ( PN ) unique solution to the equation j∈Ni j j i. If, on Cj (xj (q, θ)) PoA := sup j=1 (q, θ) ∈ X , the other hand, the bids submitted at node i are such that PN ∗ NE P θ = 0 S (p; θ ) = X j=1 Cj(xj ) j∈Ni j , then it follows that j j j for all ∗ producers j ∈ Ni, whatever the price p. In this case, we set where x is the efficient production profile. 6

3 NASH EQUILIBRIUM residual supply index of producer j measures the extent In this section, we characterize the set of Nash equilibria of to which the remaining aggregate production capacity in the networked supply function game. Specifically, we char- the market (excluding producer j) is capable of meeting acterize the production profile and the nodal supply profile demand. More precisely, we have the following definition. at a Nash equilibrium as the unique optimal solutions to two Definition 4 (Residual Supply Index). For each node i ∈ V, different convex programs, and provide upper and lower the residual supply index (RSI) of each producer j ∈ Ni is bounds on the nodal prices at a Nash equilibrium in Section defined according to 3.2. In Sections 3.3–3.4, we use this characterization to derive P X − X upper bounds on the worst-case allocative efficiency loss k∈Ni k j RSIj := max . and nodal price markups at a Nash equilibrium. The upper qi bounds are explanatory in nature, as they reveal an explicit RSI ∈ [0, ∞) relationship between a producer’s market power and clas- The residual supply index takes values j . j sical structural indices of market power—specifically, the According to this definition, producer is said to be pivotal RSI < 1 j producer’s market share and its residual supply index. Finally, if j . That is, the removal of producer from node i in Section 3.5, we empirically evaluate the predictive quality precludes the remaining producers (at that node) from i of our theoretical upper bound on price markup using meeting the maximum nodal supply at node . Clearly, there j historical spot price data from the 1999-2000 Great Britain is potential for producer to exercise considerable market RSI  1 Electricity Pool. power if it is pivotal. If, on the other hand, j , then producer j is far from being pivotal, and will likely have little market power to influence the market clearing 3.1 Structural Market Power Indices price. In practice, the residual supply index has proved to In what follows, we provide formal definitions of a pro- be effective in predicting the exercise of market power in ducer’s market share (MS) and residual supply index (RSI). electricity markets [45], [58], [59], [60], [61]. For example, In order to define these market power indices, we first it was shown in [57] that market clearing prices are close introduce a (worst-case) measure of peak demand at a to being competitive on average if the RSI of the largest node, which we refer to as the maximum nodal supply. More producer is no less than 120%. precisely, the maximum nodal supply at node i ∈ V is Remark 4 (Network Structure). We note that our definition defined as of residual supply index reflects the potential impact that max  n qi := sup qi q ∈ R+, q − d ∈ P . (12) ‘network structure’ will have on a producer’s market power in a worst-case sense. First, each producer’s RSI is measured Clearly, the maximum nodal supply at each node depends with respect to a surrogate for its ‘nodal demand’ that on both the network’s transmission capacity and the de- corresponds to the maximum power injection that can be max > mand profile. And it holds that di ≤ qi ≤ 1 d for each feasibly supported by the network at the producer’s node. node i ∈ V. Using this (conservative) measure of peak Second, the ‘residual production capacity’ at each node is demand at a node, we define the market share of each calculated with respect to the aggregate production capacity producer as follows. at that node alone, and disregards the potential contribution Definition 3 (Market Share). The market share (MS) of a of production capacity from other nodes. Although they might appear overly conservative at first glance, the com- producer j ∈ Ni at node i ∈ V is defined as bination of these two approximations in characterizing a max min{Xj, qi } producer’s residual supply index will play a central role in MSj := max . qi our derivation of upper bounds on the worst-case allocative efficiency loss and price markups seen at Nash equilibria in Note that MS ∈ [0, 1] for all producers, and that pro- j the presence of network constraints. ducers with large (relative) market shares are more likely to possess the ability to exercise market power. Despite its prevelant use among market monitors, the market share 3.2 Characterizing Nash Equilibrium index has been criticized as an inadequate measure of mar- In Proposition1, we establish the existence of Nash equilibria ket power in unconcentrated markets in which the largest for the networked supply function game. Additionally, we producer has a small market share, but is close to being characterize the nodal supply profile and the production pivotal [45], [56]. For instance, Sheffrin [57] argues that Cali- profile that result at a Nash equilibrium as the unique fornia’s deregulated electricity markets were far from being optimal solutions to two different convex programs, and competitive during the 2000–2001 crisis, in spite of the fact provide upper and lower bounds on the nodal prices seen that no single producer had a market share exceeding 20%. at a Nash equilibrium. This characterization will play an Sheffrin goes on to claim that, on many occasions, producers integral role in our subsequent derivation of upper bounds with a market share less than 10% were able to influence the on the worst-case allocative efficiency loss and nodal price market clearing price to an unwarranted degree. markups seen at Nash equilibria. We first require the fol- In part, such critiques of the market share index have lowing assumption, which limits the ‘local market power’ served to motivate the design of alternative screening tools of each producer. for market power. One commonly used screen—originally developed by the California Independent System Operator Assumption 2 (No Pivotal Supplier). The residual supply (CAISO) [45]—is the residual supply index. Essentially, the index of each producer j ∈ {1,...,N} satisfies RSIj > 1. 7

Essentially, Assumption2 amounts to requiring that no (iii) The price pi(q, θ) at each node i ∈ V at a Nash producer be pivotal.2 The requirement of no pivotal supplier equilibrium (q, θ) satisfies is enforced in electricity markets via the performance of " − + # ∂ Cej ∂ Cej the so-called ‘pivotal supplier screen’. In the United States, pi(q, θ) ∈ , if xj(q, θ) ∈ [0,Xj), (16) for instance, a producer is said to pass the pivotal supplier ∂xj ∂xj screen if the annual peak demand can be met in the absence " − ! ∂ Cej of this producer [63, p. 18]. Additionally, the violation of pi(q, θ) ∈ , ∞ if xj(q, θ) = Xj, (17) ∂x this assumption (i.e., the presence of pivotal suppliers) is j known to manifest in large price markups and allocative for each producer j ∈ Ni, where efficiency loss at equilibrium—see [45], [64], [65], [66] for − − ∂ Cej ∂ Cej(xj(q, θ); qi) several related theoretical and empirical analyses. Moreover, := , in AppendixB, we provide an example of a two-node ∂xj ∂xj network, which reveals that, in the absence of Assumption + + ∂ Cej ∂ Cej(xj(q, θ); qi) := . 2, the allocative efficiency loss at a Nash equilibrium can be ∂x ∂x arbitrarily large for the game considered in this paper. j j n With Assumption2 in hand, we state the following (iv) The nodal supply profile q ∈ R at a Nash equilibrium result, which establishes the existence of Nash equilibria, (q, θ) is the unique optimal solution to the following provides upper and lower bounds on nodal prices at a convex program: Nash equilibrium, and shows that the production profile n X and the nodal supply profile at a Nash equilibrium are minimize Gei(qi) subject to q ∈ X0, (18) q∈ n uniquely determined as the solutions of two explicit convex R+ i=1 programs. n where the modified nodal cost functions {Gei(qi)}i=1 are Proposition 1 (Existence and Characterization of NE). Let defined as

Assumptions1-2 hold. Z qi (i) The networked supply function game (N , X , π) ad-  gi(z)dz, if Ni > 0 Gei(qi) := 0 (19) mits at least one pure strategy Nash equilibrium. 0, if Ni = 0. (ii) The production profile x (q, θ) ∈ N at a Nash equi- R The function gi(z) is defined according to Eq. (14). librium (q, θ) is the unique optimal solution to the following convex program: Several remarks are in order. First, Proposition1 im- n plies that, although there may exist a multiplicity of Nash X X minimize Cej(xj; qi) equilibria, the production profile that results at a Nash x∈ N R i=1 j∈N equilibrium is unique. Furthermore, Proposition1 provides i (15) subject to Ax − d ∈ P, an approach to the tractable calculation of the unique pro- 0 ≤ x ≤ X , j = 1,...,N, duction profile at a Nash equilibrium via the solution of j j two finite-dimensional convex programs. That is, one first N where the modified cost functions {Cej(xj; qi)}j=1 are solves problem (18) for the unique nodal supply profile defined according to Eq. (13). q at a Nash equilibrium; and then solves problem (15) to determine the unique production quantity of each producer. 2. An important limitation of Assumption2 is that it implicitly re- It is also worth noting that the modified nodal cost functions quires that there is either no producer or at least two producers at each n {Gei(·)} admit closed-form expressions for a large family node in the network. One possible way of ensuring the satisfaction of i=1 Assumption2, in practice, is to formulate the network model according of production cost functions Cj(·), e.g., piecewise-quadratic to a “reduction” of the actual power network, where a node in the functions. reduced network corresponds to a connected subnetwork of buses that We also note that Proposition1 builds upon and gen- are connected by uncongested transmission lines. One approach to con- structing such reduced network models is to leverage on common prior eralizes existing results from the literature, [6, Thm. 1] and knowledge of transmission lines that are “systematically congested” in [7, Thm 4.1], to accommodate the more general setting in practice, as this would yield a reasonable approximation of the network which there are transmission capacity constraints between according to uncongested subnetworks connected by transmission lines producers and consumers. We conclude this subsection with that are “normally” binding. We refer the reader to Yao et al. [62] for a more detailed discussion on the treatment of “systematically a brief discussion of the key ideas used in proving Propo- congested” transmission lines in constructing such network reductions. sition1. The crux of the derivation relies on the suitable

! ! xj xj 1 Z Cej(xj; qi) := 1 + Cj(xj) − Cj(z)dz (13) P X − q P X − q k∈Ni,k6=j k i k∈Ni,k6=j k i 0

 ( ) ∂−C (x ; z)  ej ej X X gi(z) := max xe ∈ argmin Cek(xk; z) xk = z and xk ≤ Xk ∀ k ∈ Ni (14) j∈Ni ∂xj x∈ N  R k∈Ni k∈Ni  8

design of cost functions to ensure equivalence between the the network. This, in turn, may increase the efficiency loss stationarity conditions for problems (15) and (18) and the at Nash equilibrium, as the PoA bound is non-decreasing best response conditions of all producers and the ISO at in the maximum nodal supply of each node. In Section 4.3, Nash equilibrium. This enables the characterization of the we examine a two-node network, and establish a necessary set of Nash equilibria according to the optimal solution and sufficient condition (cf. Lemma2) under which the sets of the convex programs (15) and (18)—a technique strengthening of the network’s transmission line capacity that is closely related to the use of potential functions in results in this seemingly paradoxical behavior. We conclude characterizing Nash equilibria in potential games [67]. this subsection with a proof of Theorem1. More specifically, given a nodal supply profile q, the N Proof of Theorem1. Let (q, θ) be a Nash equilibrium. Using modified cost functions {Cej(·)}j=1 are constructed in such the assumption that each producer’s cost function Cj(xj) is a manner as to reflect the best response conditions for strictly positive and strictly increasing over (0, ∞), we have all producers according to the stationarity conditions (Eq. that (16)–(17)) associated with problem (15). Since the modified R xj q xjCj(xj) − 0 Cj(z)dz cost functions are parametric in the ISO’s decision , we C (x ) ≤ C (x ) + (21) j j j j P X − q construct another optimization problem (18) to enable the k∈Ni,k6=j k i computation of the nodal supply profile q at a Nash equilib- = Cej (xj; qi) (22) rium. Towards this end, the modified nodal cost functions ! {G (·)}n xj ei i=1 are designed to ensure the equivalence between ≤ C (x ) 1 + . (23) j j P X − q the stationarity conditions for problem (18) and a certain k∈Ni,k6=j k i fixed point condition, which expresses the ISO’s decision q Let x∗ be an efficient production profile. It holds that: as a best response to the best response of producers given n the nodal supply profile q. We refer the reader to Appendix X X C for the complete proof of Proposition1. Cj(xj(q, θ)) i=1 j∈Ni n 3.3 Bounding the Efficiency Loss X X ≤ Cej(xj(q, θ); qi) (24) In Theorem1, we provide an upper bound on the worst-case i=1 j∈Ni allocative efficiency loss incurred at a Nash equilibrium. The n X X ∗ bound sheds light on the explicit role of market structure in ≤ Cej(xj ; qi) (25) determining the impact of producers’ strategic behavior on i=1 j∈Ni n ∗ ! market (in)efficiency at equilibrium. X X ∗ xj ≤ Cj(xj ) 1 + P (26) Xk − qi Theorem 1 (Price of Anarchy). Let Assumptions1-2 hold. i=1 j∈Ni k∈Ni,k6=j The price of anarchy (PoA) associated with the game n X X ∗ (N , X , π) satisfies ≤ Cj(xj )·  MS  i=1 j∈Ni j    (27) PoA ≤ 1 + max . (20) max j∈{1,...,N} RSIj − 1  min {Xj, qi }  1 + max    . Several important consequences can be deduced from j∈Ni,i∈V  P X − qmax  k∈Ni,k6=j k i Theorem1—the most of important of which relate to the explicit role played by market structure in determining the Here, inequality (24) follows from (21); inequality (25) fol- efficiency loss incurred at a Nash equilibrium. For instance, lows from the optimality of x(q, θ) for problem (15); inequal- the PoA bound in (20) reveals the inherent limitation of ity (26) follows from (23); and, inequality (27) follows from ∗ the market share index, by itself, as an accurate predictor the feasibility of x and q. It follows from inequality (27) of market power; and, provides a theoretical basis for the that the PoA at a Nash equilibrium is upper bounded by empirically observed effectiveness of the residual supply in-    max  dex in predicting the actual exercise of market power (as min {Xj, qi } PoA ≤ 1 + max   . measured by price markups relative to perfectly competitive j∈Ni,i∈V  P X − qmax  k∈Ni,k6=j k i levels [45], [59], [60]). In particular, the PoA bound ensures max max a low efficiency loss for electricity markets in which all The desired result follows, as MSj = min {Xj, qi } /qi RSI = (P X )/qmax participating power producers have large residual supply and j k∈Ni,k6=j k i by definition. indices—irrespective of their market shares. It is also worth mentioning that, when all producers are concentrated at a 3.4 Bounding the Price Markups single node, we recover as a special case the PoA bound es- In this section, we derive upper bounds on the nodal price tablished by Xu et al. [7] for single-node electricity markets. markups at a Nash equilibrium in terms of the Lerner index— Additionally, we note that the PoA bound in (20) hints a standard ex post measure of market power [68]. at the possibility of a Braess-like paradox, where an increase in a network’s transmission capacity can result in the (coun- Definition 5 (Lerner Index). Given a strategy profile (q, θ) ∈ terintuitive) increase in the aggregate cost of generation at X , the Lerner index of producer j ∈ Ni at node i ∈ V is Nash equilibrium. The argument behind such a claim is that defined as + an increase in a transmission line’s capacity can lead to an pi(q, θ) − ∂ Cj(xj(q, θ))/∂xj LIj(q, θ) := . increase in the maximum nodal supply at certain nodes in pi(q, θ) 9

It is straightforward to show that, under the additional Additionally, it follows from condition (16) in Proposi- assumption that (q, θ) ∈ XNE, the Lerner index of each tion1 that inequality (30) holds with equality if producer producer j is guaranteed to satisfy LIj(q, θ) ∈ [0, 1]. Ac- j’s cost function Cj is differentiable at xj(q, θ). This implies max cordingly, a Lerner index close to one (resp. zero) implies a the satisfaction of Eq. (29) if xj(q, θ) = qi < Xj. large (resp. small) price markup relative to the producer’s true marginal cost. The following corollary to Proposition 1 shows that the Lerner index of each producer is upper 3.5 Comparison to Historical Market Data bounded at equilibrium. It follows from (30) that—at a Nash equilibrium (q, θ)—the Corollary 1. Let Assumptions1-2 hold. At a Nash equilib- price at any node i ∈ V is upper bounded by rium (q, θ), the Lerner index of producer j ∈ Ni satisfies   + MSj ∂ Cj(xj(q, θ)) pi(q, θ) ≤ 1 + , (31) MSj RSIj − 1 ∂xj LIj(q, θ) ≤ (28) MSj + RSIj − 1 given any producer j ∈ Ni whose production capacity con- 3 if xj(q, θ) < Xj. Furthermore, if producer j ∈ Ni has a straint is nonbinding at Nash equilibrium. In this section, max differentiable cost function and xj(q, θ) = qi < Xj, then we evaluate the predictive accuracy of the price bound (31) 1 using market data from the Great Britain (GB) Electricity LIj(q, θ) = . (29) Pool for the winter of 1999-2000. During this time period, RSI j producers in the GB Electricity Pool were required to bid Corollary1 reveals the potential for large price markups supply functions in the form of nondecreasing step func- at nodes that have a dominant producer with a small resid- tions, and were remunerated for their cleared production ual supply index. We provide empirical evidence in support quantities according to a uniform market clearing price set of this claim in Section 3.5 using historical spot price data at the pool level. See [69], [70] for a detailed description of from the Great Britain Electricity Pool. the underlying market mechanism in place in Great Britain We also note that, in a related line of investigation, during this time period. Figure1 contains a scatter plot Newbery [46] employs a Cournot oligopoly model to char- obtained from [59] of spot electricity prices (£/MWh) versus acterize a producer’s Lerner index in terms of its RSI un- the residual supply index of the producer with the largest der a variety of assumptions on the market structure. In uncontracted production capacity during this time period. particular, under the assumption of symmetric producers, In order to calculate the price bound (31), we require Newbery shows the Lerner index of each producer to be estimates of the underlying producer’s true marginal cost inversely proportional to its RSI. This functional relationship and market share. If we adopt a simplifying assumption is similar in structure to our upper bound on the Lerner in- of linearity of the producer’s cost function, then the spot dex in Corollary1, despite the differences in the underlying prices in Figure1 imply an upper bound on the producer’s models of competition employed. We note, however, that true marginal cost of £8/MWh. As for the producer’s mar- implicit in the Cournot model, which Newbery treats, is the ket share, we do not have access to information on the requirement of nonzero demand elasticity. Consequently, his producer’s uncontracted production capacity or detailed results cannot be applied to the setting considered in this demand data for the time period under consideration. We, paper, which considers a perfectly inelastic demand model. therefore, estimate the producer’s market share according to a worst-case upper bound of MS ≤ 1. The combination of Proof of Corollary1. Let (q, θ) be a Nash equilibrium. It fol- these two approximations implies an upper bound on the lows from condition (16) in Proposition1 that spot prices of the form + ∂ Cej(xj(q, θ); qi)   p (q, θ) ≤ RSI i spot price ≤ 8 (£/MWh), (32) ∂xj RSI − 1 x (q, θ) < X if j j. Calculating the right derivative of the which is valid for all RSI ∈ (1, ∞). modified cost function yields We plot the estimated upper bound (32) against the ! + historical market data in Figure1. Notice that, with the xj(q, θ) ∂ Cj(xj(q, θ)) pi(q, θ) ≤ 1 + P . exception of a few outliers in the data, the a priori upper Xk − qi ∂xj k∈Ni\{j} bound (32) and the upper envelope of the observed spot (30) prices are in near agreement. The near agreement between It follows that the Lerner index of producer j satisfies our theoretical bound and the spot price data is particularly + striking in light of the apparent discrepancy between the ∂ Cj(xj(q, θ))/∂xj LIj(q, θ) = 1 − class of piecewise-constant supply functions employed in pi(q, θ) the GB Electricity Pool and the scalar-parameterized family 1 considered in this paper. This empirical observation lends ≤ 1 − .  1 + x (q, θ) P X − q some credence to the claim that the parametric family of j k∈Ni\{j} k i supply functions studied in this paper—although stylized in x (q, θ) = j . nature—preserves the key structural determinants of market x (q, θ) + P X − q j k∈Ni\{j} k i 3. We note that one such producer is guaranteed to exist at each The Lerner index bound in (28) follows, as it necessarily node given the satisfaction of Assumption2, i.e., that there is no pivotal max max holds that qi ≤ qi and xj(q, θ) ≤ min{Xj, qi }. producer in the market. 10

Spot price vs Residual Supply Index GB Winter 1999-2000

Spot price Upper bound (32) /MWh ￡

RSI%

Fig. 1: A scatter plot obtained from [59] of spot electricity prices versus the residual supply index (RSI) of the generator with the largest uncontracted production capacity in the Great Britain Electricity Pool during the 1999-2000 winter. The solid line corresponds to the price upper bound (32).

power and the mechanisms by which market power is ex- for each node i ∈ {1, 2}. This condition, in combination with ercised, e.g., through the economic withholding of capacity the assumption of linear production costs, guarantees the when (residual) supply is scarce. existence of a unique Nash equilibrium for the networked 5 supply function game. Finally, we assume that β2 > β1, i.e., that node 2 is more expensive than node 1. It follows 4 A TRANSMISSION EXPANSION PARADOX that the efficient production cost can be calculated as In this section, we restrict our attention to the setting of a N X ∗ + two-node power network, and provide a characterization of CostEff := Cj(xj ) = β1D + (β2 − β1)(d2 − c) . market structures under which a Braess-like paradox emerges j=1 due to the exercise of market power. Specifically, we char- acterize a range of scenarios in which the strengthening 4.2 Economic Inefficiency of Transmission Expansion of the network’s transmission line capacity results in the counterintuitive increase in the total cost of generation at the In what follows, we derive a necessary and sufficient con- Nash equilibrium.4 dition on the market structure under which transmission capacity expansion results in the increase in the total cost of generation at Nash equilibrium. We first characterize the 4.1 A Two-node Network locational marginal prices (LMPs) and nodal supply profile Consider a two-node power network with a nodal demand that result at the Nash equilibrium for the system under consideration. Let q = (q1, q2) be the nodal supply profile at profile given by d = (d1, d2) > 0, and a total demand of Nash equilibrium. It follows from Proposition1 that q is the D = d1 + d2. We denote the capacity of the transmission unique optimal solution to the following convex program: line connecting the two nodes by c ∈ R+. We assume that the producers that are common to a node are symmetric and 2 have linear cost functions. Specifically, we denote the pro- Z qi   X 1/Ni j ∈ N minimize βi 1 + dz duction capacity and cost function of each producer i 2 + q∈R+ 0 (Ni − 1)Ki/z − 1 at node i ∈ {1, 2} by Xj = Ki and Cj(xj) = (βixj) , i=1 (33) respectively. We enforce the satisfaction of Assumption2 subject to q1 + q2 = D

(i.e., that no producer be pivotal) by requiring that |q1 − d1| ≤ c.

Ki(Ni − 1) > 1 5. More specifically, the linearity of production costs implies the D differentiability of each modified cost function Cej . This, in combination with statement (iii) in Proposition1, guarantees the uniqueness of the 4. Braess’s original paradox revealed that the addition of new roads LMPs at a Nash equilibrium. The uniqueness of both the LMPs and the to a traffic network can manifest in the counterproductive effect of production profile at Nash equilibrium implies the uniqueness of Nash increasing drivers’ total commute time at equilibrium [71]. equilibrium. 11

Additionally, the LMPs at the unique Nash equilibrium can such counterintuitive market outcomes under transmission be computed according to Eq. (60) in AppendixC. More expansion. specifically, if there is positive production at node 2 at the It is also possible to extend this line of reasoning to estab- Nash equilibrium, i.e., q2 > 0, then the corresponding LMP lish similar conditions under which an increase in ‘competi- at each node i ∈ {1, 2} is given by tion’ at a node 2 (as measured by the number of producers at that node) results in the increased dispatch of the high   −1! Ki(Ni − 1) marginal cost generation at node 2, in place of the lower pi = βi 1 + Ni − 1 . (34) qi marginal cost generation at node 1—thereby increasing the total cost of generation at the Nash equilibrium. This seem- If instead q2 = 0, then the corresponding LMPs at each node ingly paradoxical behavior is in direct contrast to the more are identical and are given by commonly held belief that the market entry of additional   −1! producers serves to improve economic efficiency, in general. K1(N1 − 1) p = p = β 1 + N − 1 . (35) We note that Berry et al. first described such counterintuitive 1 2 1 1 D ‘network effects’ in their seminal paper [25], which employs a computational approach to the calculation of linear supply We let CostNE denote the aggregate production cost at Nash equilibrium. It is given by function equilibria in constrained transmission systems.

CostNE = β1q1 + β2q2. 4.3 Numerical Analysis In what follows, we investigate the behavior of the aggre- We consider a two-node power network with a nodal de- gate production cost at Nash equilibrium as a function of the mand profile given by d1 = d2 = 1, and set the number network’s transmission capacity c. In particular, we provide of producers at nodes 1 and 2 to be N1 = 3 and N2 = 10, an explicit characterization of the right derivative of CostNE respectively. We fix the marginal cost of producers at node with respect to c, and establish a necessary and sufficient 1 to be β1 = 1, and vary the marginal cost of producers condition under which this derivative is guaranteed to at node 2 between two values β2 ∈ {1.15, 1.45}. All pro- be strictly positive—that is, a condition under which the ducers are assumed to have identical production capacities aggregate production cost at the Nash equilibrium increases K1 = K2 = 0.51D. with the transmission capacity. β2 = 1.15 β2 = 1.45 Lemma 2. Let (p1, p2) denote the LMPs at the Nash equilib- 2.5 2.5 CostNE rium. Then 2.4 2.4 CostEff + 2.3 2.3 ∂ CostNE = (β − β ) · sgn(p − p ), (36) 2.2 ∂c 2 1 1 2 2.2 2.1 2.1 where sgn(0) = 0. Additionally, p1 > p2 if and only if 2 2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8  K (N − 1) −1 c c 1 + N 1 1 − 1 1 d − c β 1 > 2 . (a) (b)   −1 (37) K2(N2 − 1) β1 1.5 1.5 1 + N2 − 1 p1 d2 + c 1.4 1.4 p2 1.3 1.3 The proof of Lemma2 is deferred to AppendixE. The 1.2 1.2 necessary and sufficient condition in (37) sheds light on 1.1 1.1 the role of market and network structures in driving the 1 1 emergence of this Braess-like paradox. Loosely speaking, if 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 the network’s transmission capacity is sufficiently limited, c c and if the market power of producers (as measured by (c) (d) the residual supply index) at node 1 is sufficiently large relative to the market power of producers at node 2, then the Fig. 2: We fix two different values of β2, and vary the Nash equilibrium will result in LMPs that correspond to the transmission capacity c from 0 to 0.8. Figures2(a)-2(b) plot (inefficient) transmission of power from the high-marginal the efficient cost and the aggregate production cost at Nash cost node 2 to the low marginal cost node 1. For such market equilibrium. Figures2(c)-2(d) plot the nodal prices at Nash structures, a small increase in the network’s transmission equilibrium. capacity will induce an increase (resp. decrease) in the production at node 2 (resp. node 1), thereby increasing the The leftmost plots in Figure2 correspond to a marginal aggregate production cost at Nash equilibrium. cost of β2 = 1.15 at node 2. For this choice of marginal It is worth noting that Sauma and Oren [72] uncover cost, it is straightforward to show that condition (37) is an example of a similar transmission expansion paradox satisfied for all transmission capacity values c ∈ [0, 0.3]. in the context of a simultaneous-move networked Cournot Indeed, Figure2(a) shows the aggregate production cost at model. In contrast to their characterization, which is entirely Nash equilibrium to be strictly increasing over this range of numerical in nature, Lemma2 sheds light on how market capacity values, and constant for all other capacity values structure might induce market power that gives rise to greater than 0.3. It is also worth noting that Figure2(c) 12 corroborates the necessary and sufficient condition (37), preliminary results published as part of the 2016 IEEE Con- as the nodal prices satisfy p1 > p2 for all transmission ference on Decision and Control (CDC) [54]. This work was capacities less than or equal to 0.3 The rightmost plots in supported in part by the National Science Foundation under Figure2 correspond to a marginal cost of β2 = 1.45 at grants ECCS-1351621 and IIP-1632124, the Department of node 2. 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Weixuan Lin has been pursuing the Ph.D. de- gree in Electrical and Computer Engineering from Cornell University, Ithaca, NY, USA, since 2013. He received the B.S. degree in Electri- cal Engineering with high honors from Tsinghua University, Beijing, China, in 2013. His research interests include stochastic control, algorithmic game theory and online algorithms, with partic- ular applications to the management of uncer- tainty in distributed energy resources and the design and analysis of electricity markets. He is a recipient of the Jacobs Fellowship and the Hewlett Packard Fellowship.

Eilyan Bitar received the B.S. and Ph.D. de- grees in Mechanical Engineering from the Uni- versity of California at Berkeley in 2006 and 2011, respectively. He currently serves as an Associate Professor and the David D. Croll Sesquicentennial Faculty Fellow in the School of Electrical and Computer Engineering at Cornell University in Ithaca, NY, USA. Prior to joining Cornell in the Fall of 2012, he was engaged as a Postdoctoral Fellow in the department of Computing and Mathematical Sciences at the California Institute of Technology and at the University of California, Berkeley in Electrical Engineering and Computer Science, during the 2011-2012 academic year. His current research interests include stochastic control and game theory with applications to electric power systems and markets. Dr. Bitar is a recipient of the NSF Faculty Early Career Development Award (CAREER), the John and Janet McMurtry Fellowship, the John G. Maurer Fellowship, and the Robert F. Steidel Jr. Fellowship. 15

APPENDIX A node i ∈ {1, 2}. The production capacities are assumed to COMPETITIVE EQUILIBRIUM satisfy

Under the assumption of price-taking behavior, we establish N1K1 (N1 − 1)K1 (N2 − 1)K2 ≥ 1 > and > 1. both the existence and the efficiency of competitive equilibria, D D D which we define as follows. It follows that Assumption2 is violated, as RSIj < 1 for all Definition 6 (Competitive Equilibrium). The nodal price producers j ∈ N1. Define the production costs of producers profile p > 0 and producers’ bid profile θ ≥ 0 constitute a according to competitive equilibrium (CE) if producers are maximizing their ( + (xj) , if j ∈ N1 profits given the nodal prices p, i.e., Cj(xj) = + (β(t)x ) , if j ∈ N 


j 2 θj ∈ argmax pi · Sj θj, pi − Cj Sj θj, pi θj ≥ 0 where for each node i ∈ V and producer j ∈ Ni; and the market   1 + t/N1 clears at each node i ∈ V according to (N1−1)K1−t β(t) =   X 1 + (D−t)/N2 Sj(θj, pi) = qi, (N2−1)K2−(D−t) j∈Ni for some parameter t satisfying where the nodal supply profile q solves the ISO’s economic (N1 − 1)K1 dispatch (ED) problem   < t < (N1 − 1)K1. 1 + N2 (N2−1)K2 − 1 ( ) N1 D n X q ∈ argmin Gi(qi; θ) q ∈ X0 . This guarantees that β(t) > 1. Given these assumptions,

i=1 one can verify that the following strategy profile (q, θ) At a competitive equilibrium, each producer maximizes constitutes a Nash equilibrium for the game: its profit while taking its nodal price as given. In addition, q = (t, D − t) the market clears at each node in the network according     to a nodal price profile, which induces a nodal supply t/N1 qi θj = 1 + Ki − profile that solves the ISO’s ED problem. Next, we show (N1 − 1)K1 − t Ni that competitive equilibria exist, and that any competitive for all j ∈ N and i ∈ {1, 2}. The production profile at the equilibrium is efficient. i Nash equilibrium (q, θ) is therefore given by Proposition 2 (Existence and Efficiency of CE). Let Assump- ( t/N1, if j ∈ N1 tion1 hold, and assume that the ED problem (2) is strictly xj(q, θ) = feasible. There exists at least one competitive equilibrium. (D − t)/N2, if j ∈ N2. Furthermore, the production profile at any competitive equi- Also, the fact that N1K1 ≥ D implies that an efficient librium is efficient. production profile is given by We omit the proof of Proposition2, as it is straight- ( D/N , j ∈ N forward to establish equivalence between the conditions ∗ 1 if 1 xj = for competitive equilibrium in Definition6 and the (KKT) 0, if j ∈ N2. optimality conditions for the original ED problem (2). It The price of anarchy associated with this game, therefore, is important to mention that the guaranteed efficiency of satisfies competitive equilibrium is a consequence of the particular t + β(t)(D − t) nodal pricing mechanism that we employ—namely, loca- PoA ≥ , tional marginal pricing. We note that this is in contrast to the D formulation of [20], which analyzes linear supply function which implies that PoA → ∞ as t → (N1 − 1)K1. equilibrium in a transmission constrained power network under a uniform pricing mechanism. In the presence of trans- missions constraints, uniform pricing mechanisms do not APPENDIX C guarantee the existence or the efficiency of a competitive PROOFOF PROPOSITION 1 equilibrium, in general. The proof is divided into five parts. In part 1, we present necessary and sufficient optimality conditions for each pro- ducer’s profit maximization problem and the ISO’s eco- APPENDIX B nomic dispatch (ED) problem. We use these conditions in EXAMPLE:UNBOUNDED PRICEOF ANARCHY parts 2 and 3 to show that the production profile, x(q, θ), Consider a two-node power network with a nodal demand and nodal supply profile, q, at a Nash equilibrium (q, θ) profile given by d = (D/2,D/2), where D > 0. Assume are the unique optimal solutions to problem (15) and (18), that the transmission line has capacity at least c > D/2, and respectively. In part 4, we show that the nodal price pi(q, θ) that there are at least two producers at each node in the at each node i ∈ V at a Nash equilibrium (q, θ) satisfies network. Moreover, producers that are common to a node conditions (16)–(17). In part 5, we establish the existence of are assumed to have identical production capacities. That is, a Nash equilibrium by construction. Throughout the proof, we assume that Xj = Ki for each producer j ∈ Ni at each we will assume that there are at least two producers at each 16

node in the network, as this will serve to streamline the and the complementary slackness conditions exposition. It is straightforward to generalize the proof to µ ◦ (H(q − d) − c) = 0. accommodate scenarios in which Ni = 0 for certain nodes i ∈ V in the network. Also, it is straightforward to show that Part 1 (Optimality Conditions): Lemma3 provides a set P ! Xj θk of necessary and sufficient optimality conditions for each X k∈Ni Gi(qi; θ) = θj log P
θj k∈N Xk − qi producer’s profit maximization problem. Its proof is analo- j∈{Ni|θj >0} i gous to step 1 of the proof of [7, Thm. 4.1], and is, therefore, P θ > 0 G (q ; θ) = 0 P θ = 0 omitted for the sake of brevity. if j∈Ni j , and i i if j∈Ni j . Hence, each function Gi is differentiable in qi if inequality (42) is q ∈ X ∩ n Lemma 3. Let Assumptions1-2 hold, and let 0 R+. satisfied. A direct calculation shows that For each producer j ∈ Ni at each node i ∈ V, n !
X πj (q, θj, θ−j) ≥ πj q, θj, θ−j ∀ θj ∈ Xj ∇ Gi(qi; θ) = p(q, θ), (46) i=1 if and only if the following conditions are satisfied: where the vector p(q, θ) is defined according to Eq. (10). xj(q, θ) ∈ [0,Xj], (38) and Part 2 (Production Profile at Nash Equilibrium): Let (q, θ) " + # be a Nash equilibrium. We now prove that the production ∂ Cej pi(q, θ) ∈ 0, if xj(q, θ) = 0, (39) profile x(q, θ) (defined according to Eqs. (8)-(9)) is the ∂xj unique optimal solution to problem (15). First, it is straight- " − + # ∂ Cej ∂ Cej forward to show that the modified cost function Cej(xj; qi) pi(q, θ) ∈ , if xj(q, θ) ∈ (0,Xj), (40) is strictly convex in xj over [0,Xj], given the satisfaction ∂xj ∂xj of Assumptions1 and2. This—in combination with fact " − ! ∂ Cej that the feasible region of problem (15) is defined in terms pi(q, θ) ∈ , ∞ if xj(q, θ) = Xj, (41) ∂xj of linear constraints—implies that the KKT conditions (47)- (53) are necessary and sufficient for optimality. Specifically, a N where the production quantity xj(q, θ) is defined according to production profile x ∈ R is optimal for problem (15) if and Eqs. (8)-(9), and the nodal price p (q, θ) is defined according 2m i only if there exist Lagrange multipliers λ ∈ R and µ ∈ R+ to Eq. (10). The left and right partial derivatives of the such that the following conditions hold. modified cost function are calculated at: (i) Primal feasibility: − − ∂ Cej/∂xj := ∂ Cej(xj(q, θ); qi)/∂xj, Ax − d ∈ P, (47) + + ∂ Cej/∂xj := ∂ Cej(xj(q, θ); qi)/∂xj. xj ∈ [0,Xj] for j = 1,...,N. (48)

Lemma4 provides a set of necessary and sufficient (ii) Stationarity: optimality conditions for the ISO’s ED problem (6). + # ∂ Cej(xj; qi) Lemma 4. Let (q, θ) ∈ X , and assume that pi ∈ −∞, if xj = 0, (49) ∂xj X qi < Xj (42) " − + # j∈Ni ∂ Cej(xj; qi) ∂ Cej(xj; qi) pi ∈ , if xj ∈ (0,Xj), (50) for each node i ∈ V. Then, the nodal supply profile q is the ∂xj ∂xj " ! unique optimal solution to problem (6) if and only if there ∂−C (x ; q ) 2m ej j i exist multipliers λ ∈ R and µ ∈ R+ that satisfy pi ∈ , ∞ if xj = Xj, (51) ∂xj > p(q, θ) = λ1 − H µ, (43) n for each i ∈ V and j ∈ Ni, where the vector p ∈ R is µ ◦ (H(q − d) − c) = 0, (44) defined according to

where the vector p(q, θ) := (p (q, θ), . . . , p (q, θ)) is deter- > 1 n p := λ1 − H µ. (52) mined according to Eq. (10). Complementary slackness: Proof of Lemma4. Problem (6) is a convex program with (iii) linear constraints. It follows that the Karush-Kuhn-Tucker µ ◦ (H(Ax − d) − c) = 0. (53) (KKT) conditions are both necessary and sufficient for opti- mality. That is to say, q ∈ X0 is optimal if and only if there We now employ Lemmas3 and4 to show that the KKT 2m exist Lagrange multipliers λ ∈ R and µ ∈ R+ associated conditions are satisfied at x = x(q, θ). 1>(q − d) = 0 H(q − d) ≤ c with the constraints and , Primal feasibility: Since (q, θ) is a Nash equilibrium, it respectively, which satisfy the stationarity conditions follows from Lemma3 that conditions (38)-(41) are satisfied n ! for each j ∈ {1,...,N}. The combination of condition (38) X > ∇ Gi(qi; θ) − λ1 + H µ = 0, (45) and the fact that Ax(q, θ) = q ∈ X0 implies the satisfaction i=1 of the primal feasibility conditions (47)-(48). 17

n Stationarity: Inequality (38) and Assumption2 together (i) Primal feasibility: q ∈ X0 ∩ R+. guarantee that (ii) Stationarity: max X 0 ≤ qi ≤ qi < Xj − max Xk (55) + # k∈Ni ∂ G (q ) j∈N ei i i pi ∈ −∞, if qi = 0, (58) P ∂qi for each node i ∈ V. It follows that qi < j∈N Xj for i " − + # each node i ∈ V. And, since (q, θ) is a Nash equilibrium, ∂ Gei(qi) ∂ Gei(qi) pi ∈ , if qi > 0, (59) it follows from Lemma4 that there exist multipliers λ ∈ R ∂qi ∂qi 2m and µ ∈ R+ that satisfy Eqs. (43) and (44). The satisfaction of Eq. (43), in combination with the conditions (39)–(41), im- for each node i ∈ V, where plies that (x(q, θ), λ, µ) satisfy desired the set of stationarity p = λ1 − H>µ. (60) conditions (49)–(52). (iii) Complementary slackness: Complementary slackness: Since (q, µ) satisfy Eq. (44) and q = Ax(q, θ), it follows that µ ◦ (H(q − d) − c) = 0. (61) µ ◦ (H(Ax(q, θ) − d) − c) = 0. (56) The assumption that (q, θ) is a Nash equilibrium implies n that q ∈ X0 ∩ R+ (cf. the inequality in (55)). We now Part 3 (Nodal Supply Profile at Nash Equilibrium): Let (q, θ) establish the existence of Lagrange multipliers λ ∈ R and 2m be a Nash equilibrium. We now prove that the nodal supply µ ∈ R+ such that the stationarity and complementary profile q is the unique optimal solution to problem (18). slackness conditions are satisfied at At the heart of our proof is the following technical lemma, + # ∂ Gei(qi) which establishes the strict convexity of the function Gei over pi(q, θ) ∈ −∞, if qi = 0, (62) max ∂qi a superset of [0, qi ], and characterizes its left and right " − + # derivatives. Its proof can be found in AppendixD. ∂ Gei(qi) ∂ Gei(qi) pi(q, θ) ∈ , if qi > 0, (63) Lemma 5. Let Assumption1 hold. For each node i ∈ V, ∂qi ∂qi define the constant for each node i ∈ V, and X Qi := Xj − max Xk, p(q, θ) = λ1 − H>µ, k∈Ni (64) j∈Ni µ ◦ (H(q − d) − c) = 0. (65) and define the set Λi(z) according to Eq. (54) for each z ∈ Given the assumption that (q, θ) is a Nash equilibrium, it [0,Qi). All of the following statements are true: follows from Lemma3 that conditions (39)–(41) are satisfied (i) The set Λi(z) is compact for each z ∈ [0,Qi). for each j ∈ Ni and i ∈ V. It is straightforward to verify p (q, θ) ∈ Λ (q ) i ∈ V (ii) The function Gei(qi) is strictly convex on [0,Qi). that this implies that i i i for all . It follows from Lemma5 that conditions (62) and (63) are (iii) The left and right derivatives of the function Gei(qi) both satisfied. Finally, Lemma4 guarantees the existence 2m satisfy Lagrange multipliers λ ∈ R and µ ∈ R+ such that Eqs. − (64)–(65) are satisfied. ∂ Gei(z) = inf Λi(z) ∀z ∈ (0,Qi), ∂qi Part 4: (Nodal Price at Nash Equilibrium): Let (q, θ) be a + Nash equilibrium. The fact that the nodal price pi(q, θ) at ∂ Gei(z) = sup Λ (z) ∀z ∈ [0,Q ). each node i ∈ V satisfies conditions (16)–(17) immediately ∂q i i i follows from conditions (39)–(41) in Lemma3, as the left derivative of the modified cost function Cej(xj; qi) evaluated Assumption2 implies that any feasible solution q ∈ X0 ∩ − n at xj = 0 satisfies ∂ Cej(0; qi)/∂xj = 0. R+ to problem (18) is guaranteed to satisfy Part 5 (Construction of a Nash Equilibrium): Before pro- 0 ≤ q ≤ qmax < Q (57) i i i ceeding with the proof, we first provide a sketch of the main for each node i ∈ V. This implies that the objective function arguments employed. We establish the existence of a Nash of problem (18) is strictly convex over its feasible region, equilibrium by construction. We do so, in part 5-A, by first n which is defined by linear constraints. Therefore, q is the constructing a nodal supply profile q ∈ R+ as the unique op- unique optimal solution to problem (18) if and only if there timal solution to problem (18). Next, we solve an equivalent 2m exist Lagrange multipliers λ ∈ R and µ ∈ R+ such that reformulation of problem (15) to obtain a production profile N n (q, λ, µ) satisfy the following KKT conditions. x ∈ R+ , and define a nodal price vector p ∈ R+ according to a linear combination of its optimal Lagrange multipliers.

 

 N X n o  Λi(z) := λ ∈ R ∃x ∈ R such that xj = z and xj ∈ argmin Cej(xj; z) − λxj ∀ j ∈ Ni (54) xj ≤Xj  j∈Ni  18

Using the resulting production quantities and nodal prices, Using this fact, it is straightforward to verify that the nodal we construct a producer strategy profile according to prices p(q, θ) and production quantities x(q, θ) induced by (q, θ) satisfy θj := pi(Xj − xj) pi(q, θ) = pi ∀ i ∈ V, (71) for each producer j ∈ Ni and node i ∈ V. In part 5- B, we complete the proof by showing that the necessary xj(q, θ) = xj ∀ j ∈ {1,...,N}, (72) and sufficient conditions for Nash equilibrium established where recall that pi(q, θ) is defined according to Eq. (10), in Lemmas3 and4 are indeed satisfied by the pair (q, θ) as and xj(q, θ) is defined according to Eqs. (8)–(9). constructed. To complete the proof, it suffices to show that the Part 5-A (Constructing a Candidate NE): Let q be the necessary and sufficient conditions for Nash equilibrium unique optimal solution to problem (18). Consider the fol- established in Lemmas3 and4 are satisfied by the pair (q, θ) lowing relaxation to problem (15), where we have dropped as constructed. This is immediate to see upon examination the nonnegativity constraints on the production quantities. of Eqs. (67)–(72). This completes the proof that (q, θ) is a n Nash equilibrium. X X minimize Cej(xj; qi) N x∈R i=1 j∈Ni (66) APPENDIX D subject to Ax − d ∈ P, PROOFOF LEMMA 5

xj ≤ Xj, j = 1,...,N. Proof of statement (i): Fix i ∈ V and z ∈ [0,Qi). Consider the Assumptions1 and2 guarantee that each modified cost following convex optimization problem: C (x ; q ) X function ej j i is strictly convex and strictly increasing minimize Cej(xj; z) N x [0,X ] x ≤ 0 x∈R in j over j , and is equal to zero for all values j . j∈Ni This guarantees that problem (66) has a unique optimal X subject to x = z solution that is nonnegative elementwise, which, in turn, j (73) j∈N implies the optimality of this solution for problem (15). i N xj ≤ Xj, if j ∈ Ni Let x ∈ R+ be the unique optimal solution to problem (66), which is a convex program with linear constraints. It xj = 0, if j ∈ {1,...,N}\Ni, follows that there exist Lagrange multipliers λ ∈ R and µ ∈ 2m It follows from Assumption1 that the modified cost function R+ such that the following conditions are satisfied. Cej is strictly convex and strictly increasing over [0,Xj], and (i) Stationarity: For each i ∈ V and j ∈ Ni, satisfies Cej(xj; z) = 0 for xj ≤ 0. This guarantees that " − + # problem (73) admits a unique optimal solution x(z) that ∂ Cej(xj; qi) ∂ Cej(xj; qi) e pi ∈ , if xj < Xj (67) is non-negative element-wise. Notice that the set Λi(z) is ∂xj ∂xj the set of optimal Lagrange multipliers associated with the " − ! P x = z ∂ Cej(xj; qi) nodal power balance constraint j∈Ni j for problem pi ∈ , +∞ if xj = Xj (68) (73). It can, therefore, be expressed as ∂xj ( ∂−C (x (z); z) ∂+C (x (z); z) where ej ej ej ej Λi(z) = λ ≤ λ ≤ , ∂xj ∂xj p := 1λ − H>µ. (69) ) (ii) Complementary slackness: ∀j ∈ Ni . µ ◦ (H(q − d) − c) = 0. (70) The above expression can be simplified to The vector p that we specify in Eq. (69) will play the role of " ( − ) ( + )# a ‘nodal price vector’ in constructing producers’ strategy ∂ Cej(xej(z); z) ∂ Cej(xej(z); z) Λi(z) = max , min . profile. Specifically, we construct the producers’ strategy j∈Ni ∂xj j∈Ni ∂xj profile θ according to (74) θ := p (X − x ) j i j j It follows that set Λi(z) is compact for every z ∈ [0,Qi).

for each producer j ∈ Ni and node i ∈ V. As constructed, Proof of statement (ii): It suffices to show that gi(z) is the producers’ strategy profile θ is guaranteed to be non- strictly increasing in z over (0,Qi). Fix 0 < z < y < Qi. We negative elementwise, as the inequalities (67)–(68) guarantee now show that gi(z) < gi(y). First note that gi(z) satisfies that pi ≥ 0 for each i ∈ V, since each modified function Cej ( − ) is non-decreasing over (−∞,Xj]. ∂ Cej(xej(z); z) gi(z) = max = inf Λi(z), (75) j∈Ni ∂x Part 5-B (Checking Necessary and Sufficient NE Conditions): j N n Recall that x ∈ R+ and q ∈ R+ are the unique optimal where solutions to problems (66) and (18), respectively. It is not − − ! ∂ Cej(xj(z); z) ∂ Cj(xj(z)) xj(z) difficult to show that x and q are related according to e = e 1 + e . ∂x ∂x P X − z j j k∈Ni,k6=j k Ax = q. (76) 19

for each j ∈ Ni. Additionally, 0 < z < y < Qi implies that function of problem (78) is strictly convex in x and jointly there exists j0 ∈ Ni, such that continuous in (x, z) for all z ∈ (0,Qi). It follows from Berge’s maximum theorem in [73, p. 116] that the unique 0 ≤ xj (z) < xj (y) ≤ Xj. e 0 e 0 parametric optimizer xe(z) of problem (78) is continuous in This implies the following chain of inequalities: z on (0,Qi). In particular, this implies the left-continuity of xej(z), which completes the proof. ∂+C (x (z); z) ej0 ej0 gi(z) = inf Λi(z) ≤ sup Λi(z) ≤ ∂xj 0 APPENDIX E − ∂ Cej (xj (y); z) < 0 e 0 PROOFOF LEMMA 2 ∂xj0 ! The crux of the proof centers on the derivation of a closed- ∂−C (x (y)) x (y) = j0 ej0 1 + ej0 form expression for the optimal solution of problem (33). We ∂x P X − z j0 k∈Ni,k6=j0 k first eliminate the decision variable q2 through substitution ! q = D − q ∂−C (x (y)) x (y) of the power balance constraint 2 1, which yields < j0 ej0 1 + ej0 the equivalent reformulation of problem (33) as: ∂x P X − y j0 k∈Ni,k6=j0 k Z q1   − 1/N1 ∂ Cej (xj (y); y) minimize β1 1 + dz = 0 e 0 q ∈ (N − 1)K /z − 1 ∂x 1 R 0 1 1 j0 D−q   Z 1 1/N2 ≤ gi(y). + β2 1 + dz 0 (N2 − 1)K2/z − 1 + + Here, the first line follows from a combination of Eqs. (74) subject to (d1 − c) ≤ q1 ≤ D − (d2 − c) . and (75); the second line follows from the strict convexity of (79) the function Cej(xj; z) in xj over [0,Xj]; the third and the It will be notationally convenient to denote the projection b fifth lines follow from Eq. (76); and the fourth line follows operator onto a closed interval [a, b] ⊆ R according to [·]a. It from the fact that 0 < z < y < Qi. This finishes the proof of is straightforward to show that the unique optimal solution statement (ii). to problem (79) is given by

D−(d −c)+ Proof of statement (iii): It will be convenient to define the q = [q ] 2 , (80) 1 e1 d1−c function where q1 is the unique solution to the following first-order h (z) := sup Λ (z). e i i condition on the open interval (0, (N1 − 1)K1): Using the fact that convex functions defined over a bounded   −1! K1(N1 − 1) interval are differentiable at all but countably many points β1 1 + N1 − 1 q1 in that interval, it is straightforward to show that hi(z) = (81)   −1! gi(z) for all but countably many z ∈ [0,Qi). This implies K2(N2 − 1) that = β2 1 + N2 − 1 . D − q1 Z qi Z qi Gei(qi) = gi(z)dz = hi(z)dz. (77) It follows from Eq. (80) that 0 0 + Cost = β D + (β − β )[q ]D−(d2−c) . Consequently, statement (iii) is true if hi(z) is right- NE 2 1 2 e1 d1−c continuous on [0,Q ) and g (z) is left-continuous on (0,Q ). i i i Since q does not depend on the transmission capacity c, it We only prove the left-continuity of the function g (z), as the e1 i holds that proof of right-continuity of h (z) is analogous. i  First, it follows from Eq. (76) and the convexity of Cej that + β2 − β1 if qe1 < d1 − c, ∂ CostNE  + gi(z) is left-continuous if xej(z) is left-continuous in z for = β1 − β2 if qe1 > D − (d2 − c) and c < d2, ∂c  each j ∈ Ni. Recall that for any z ∈ (0,Qi), the parametric 0 otherwise. optimizer xe(z) is the unique optimal solution to problem (73), and is guaranteed to be nonnegative elementwise. This To complete the proof of the first part of Lemma2, it suffices implies that xe(z) is also the unique optimal solution of the to show that the following two conditions hold: following convex program: p1 > p2 ⇐⇒ qe1 < d1 − c, (82) X + minimize Cej(xj; z) p1 < p2 ⇐⇒ q1 > D − (d2 − c) and c < d2. (83) N e x∈R j∈Ni X To show that condition (82) holds, first recall the explicit subject to x = z j (78) formulae for the LMPs at Nash equilibrium in Eqs. (34)-(35). j∈Ni It follows that p1 > p2 if and only if 0 ≤ xj ≤ Xj, if j ∈ Ni   −1! x = 0, j ∈ {1,...,N}\N . K1(N1 − 1) j if i β1 1 + N1 − 1 q1 The feasible region of problem (78) is compact and con- (84)   −1! tinuous (i.e., both upper and lower hemicontinuous) in K2(N2 − 1) >β2 1 + N2 − 1 , the parameter z over (0,Qi). Additionally, the objective D − q1 20

where q1 is specified according to Eq. (80). It follows that

p1 > p2 ⇐⇒ qe1 < q1, since qe1 is the unique solution to Eq. (81). Furthermore, it holds that

qe1 < q1 ⇐⇒ qe1 < d1 − c, D−(d −c)+ since q = [q ] 2 . This proves that (82) holds. The 1 e1 d1−c proof that (83) holds is analogous, and is, therefore, omitted. To prove the second part of Lemma2, note that the previous arguments also imply that p1 > p2 if and only if the inequality in (84) is satisfied for q1 = d1 − c. Plugging q1 = d1 − c into (84) shows that p1 > p2 if and only if the inequality in (37) holds.