A Capacity-Price Game for Uncertain Renewables Resources
11 A Capacity-Price Game for UncertainRenewables Resources
Pan Li, Shreyas Sekar,
Member, IEEE, and Baosen Zhang,
Member, IEEE <-this
Abstract —Renewable resources are starting to constitute a growing portion of the total generation mix of the power system. A keydifference between renewables and traditional generators is that many renewable resources are managed by individuals, especially inthe distribution system. In this paper, we study the capacity investment and pricing problem, where multiple renewable producerscompete in a decentralized market. It is known that most deterministic capacity games tend to result in very inefficient equilibria, evenwhen there are a large number of similar players. In contrast, we show that due to the inherent randomness of renewable resources,the equilibria in our capacity game becomes efficient as the number of players grows and coincides with the centralized decision fromthe social planner’s problem. This result provides a new perspective on how to look at the positive influence of randomness in a gameframework as well as its contribution to resource planning, scheduling, and bidding. We validate our results by simulation studies usingreal world data.
Index Terms —Capacity game, Nash equilibrium, renewables generation (cid:70)
NTRODUCTION
Distributed renewable resources are starting to play an in-creasingly important role in energy generation. For example,the installation of photovoltaic (PV) panels across the worldhas grown exponentially during the past decade [1]. Theserenewable resources tend to be different from traditionallarge-scale generators as they are often spatially distributed,leading to many small generation sites across the system.The proliferation of these individual renewable generators(especially PV) has allowed for a much more flexible system,but also led to operational complexities because they areoften not coordinated [2]. Recently, there has been strongregulatory and academic push to allow these individualgenerators to participate in a market, hoping to achieve amore efficient and streamlined management structure [3],[4]. Therefore, in this paper we study an investment gamewhere individual firms decide their installation capacities ofPV panels and compete to serve the load.Currently, there are several lines of fruitful researchon the investment of solar energy resources. The commonchallenge in these works is to address the pricing of solarenergy, since once installed, power can be produced at nearzero operational cost. In [5], feed-in tariffs (fixed prices) areused to guide the investment decisions. In [6], [7], risksabout future uncertainties in prices are taken into account,although these prices are assumed to be independent of theinvestment decisions. Instead of exogenously determinedfeed-in tariffs, [8], [9], [10] study incentive based pricing, • Pan Li is with Facebook Inc., Menlo Park, CA. This work was done whenshe was with the University of Washington. E-mail: [email protected]. • Shreyas Sekar is with Harvard University, Boston, MA. This workwas done when he was with the University of Washington. .E-mail:[email protected]. • Baosen Zhang is with the Department of Electrical and Computer En-gineering, University of Washington, Seattle, 98195. .E-mail: [email protected]. • This work is partially supported by NSF grant CNS-1544160. arguing that the price of solar energies should match theirmarket value, which is the revenue that those resourcescan earn in markets, without the income from subsidies.However, the investment question of how to decide thecapacity of each solar installation is not considered.A common assumption made in existing studies is thatan operator (or utility) makes centralized decisions aboutthe capacity of the solar installation at different sites [11],[12] as well as the market clearing price upon aggregatingthe bids from individual producers. On the other hand,since small PV generation is mostly privately owned, it isarguably more realistic to consider an environment whereboth the investment and the pricing decisions are made in adecentralized manner. In this market setting, the PV ownerscompete by making their own investment and price biddingdecisions, based on the information they have, as opposedto a centralized decision made by a single operator. In thispaper, we are interested in understanding these strategicdecisions, particularly in the electrical distribution system.The competition between individual producers in a de-centralized market for electricity is normally studied eithervia the Cournot model or the Bertrand model [13]. In theformer, the producers compete via quantity, while in thelatter they compete via price. In this work, we adopt theBertrand competition to model price bidding since it is amore natural process in the distribution system, where thereis no natural inverse demand function (required by theCournot competition model) [14], [15]. Then the investmentgame becomes a two-level game as shown in Fig. 1b. Forany given capacities, the producers compete through theBertrand model to determine their prices to satisfy thedemand in the system. Then the outcome of this game feedsinto an upper level capacity game, where each producerdetermines its investment capacities to maximize its expectedprofit . Fig. 1 depicts a detailed comparison between this two-level Bertrand game and a more centralized marketplacewhere the utility selects the clearing price. a r X i v : . [ m a t h . O C ] O c t Energy producers submit aper-unit bid and a supply capThe designer selects k buyers having the lowestbids so that the total supplymatches the demandThe market clearing priceequals the k + 1 th lowestbid. Each of the k buyersreceives a per-unit paymentequal to the clearing price. (a) A description of a centralized mechanism where a num-ber of producers submit bids and a central authority sets themarket clearing price so that the overall supply coincideswith demand. Energy producers determineinvestment capacity tomaximize expected profit.Given capacity, each producerbids according to a Bertrandmodel to maximize revenue.Buyers purchase sequentiallystarting with the lowest biduntil total demand is met optimal revenue (b) An illustration of the two level game between a groupof solar energy producers. In the higher level, the producersdetermine their investment capacity to maximize expectedprofit, and the lower game determines the pricing of solarenergy through Betrand competition.
Figure 1: Centralized vs. Decentralized market mechanisms.This type of two-level game was studied in [16] in thecontext of communication network expansion. They showedthat Nash equilibria exist, but the efficiency of any ofthese equilibria are bad compared to the social planner’s(or operator’s) solution. More precisely, as the number ofplayers grows, the social cost of all of the equilibria growwith respect to the cost of the social planner’s problem.Therefore, instead of increasing efficiencies, competitioncan be arbitrarily bad. A similar intuition has existed intraditional power system investment problems, where themarket power of the generators is highly regulated andclosely monitored [17].In this paper, we show that contrary to the result in[16], the investment game between renewable producers leads toefficient outcomes under mild assumptions.
More precisely, 1)the investment capacity decisions made by the individualproducers match the capacity decisions that would be madeby a social planner; 2) as the number of producers increases,the equilibria of the price game approaches a price level thatallows the producers to just recover their investment costs.The key difference comes from the fact that renewables are inherently random . Therefore instead of trying to exploit the“corner cases” in a deterministic setting as in [16], [17],the uncertainties in renewable production naturally inducesconservatism into the behavior of the producers, leading toa drastic improvement of the Nash equilibria in terms ofefficiency. Therefore, uncertainty helps rather than hindersthe efficiency of the system.To analyze the equilibria of the game, our work buildson the results in [18]. In [18], the authors discuss the pricebidding strategies in markets with exactly two renewableenergy producers. They show that a unique mixed pric-ing strategy always exists given that the capacity of thoseproducers are fixed beforehand. They extend it to a storagecompetition problem in later work [19]. However, this workdid not address the strategic nature of the capacity invest-ment decisions, nor did it consider markets with more than two producers.In our setting, we explicitly consider the joint competi-tion for capacity considering each player’s investment cost,as well as the bidding strategy to sell generated energy. Thisproblem is neither studied in traditional capacity investmentgames (randomness is not considered) [20], [21] nor incompetition of renewable resources (investment strategy isconsidered) [8], [24]. To characterize the Nash equilibria inthe two level capacity-pricing game, we consider two per-formance metrics. The first is social cost, which is the totalcost of a Nash equilibrium solution with respect to the socialplanner’s objective. The second is market efficiency, whichmeasures the market power of the energy producers. As acomparison, the results in [16] show that in a deterministiccapacity-pricing game, as the number of producers grows, neither the social cost nor the efficiency improves at equilibrium .In contrast, we show that a little bit of randomness leadsto improvements on both metrics. Specifically, we make thefollowing two contributions:1) We consider a two level capacity-pricing game be-tween multiple renewable energy producers withrandom production. We show that contrary to com-monly held belief, randomness improves the qualityof the Nash equilibria.
2) We explicitly characterize the Nash equilibria andshow that the social cost and efficiency improve asthe number of producers grows.The rest of the paper is organized as follows. Section2 motivates the problem set up and details the modelingof both the decentralized and centralized market. Section 3formally introduces the evaluation metrics for our setting.Section 4 presents the main results of this paper, i.e., the
1. The work in [22], [23] studies an investment game where thedemand curve is uncertain, but under a very different context thanours2. This is conceptually similar to the results obtained in [25], whererandomness increases the efficiency of Cournot competition. relationship between the proposed decentralized marketand the social planner’s problem, and the analysis on theefficiency of the game in the decentralized market. Proofsfor the main theorems are left in the appendices for inter-ested readers. The simulation results are shown in Section 5followed by the conclusion in Section 6.
ECHNICAL PRELIMINARIES
Traditionally, power systems are often built and operatedin a centralized fashion. The system operator acts as thesocial planner by aggregating the producers and makes cen-tralized decisions on investment and scheduling (as shownin Fig. 2a). The goal of the social planner is to maximizethe overall welfare of the whole system— this includesoptimizing the costs incurred due to the investment andinstallation, and the cost paid by the consumers. (a) Centralized. (b) Decentralized.
Figure 2: Centralized vs. decentralized market setup.However, as distributed energy resources (DERs) start todisperse across the power distribution network, the central-ized setup becomes difficult to maintain and manage. DERssuch as rooftop PV cells are small, numerous, and owned byindividuals, allowing them to act as producers and choosetheir own capacities and prices. Consequently, managingthese resources through a decentralized market (as shownin Fig. 2b) is starting to gain significant traction in the powerdistribution system.Several issues arise in a decentralized market. Chiefamong them is that it is not clear whether the decentral-ized market achieves the same decision as if there were acentral planner maximizing social welfare. The competitionbetween energy producers is suboptimal if the followingoccurs: • If the investment decisions by the competing produc-ers deviate from the social planner’s decision: thismeans that the competition is sub-optimal when itcomes to finding a socially desirable investment plan. • If the bidding strategy leads to a higher paymentfrom electricity consumers than that from the socialplanner’s decision, it means that the energy resourceproducers are taking advantage of the buyers and themarket is not efficient.Both of these adverse phenomena can happen in decen-tralized markets in the absence of uncertainty [16], [26], [27], even when there are a large number of individualplayers. However, the rest of this paper shows that neitherof them occur in a decentralized market with renewablesresources having random generation. We show that theinherent uncertainty in the production naturally improvesthe quality of competition. We start by formally introducingthe game in the next section.
Throughout this paper, we denote some important terms bythe following: • Z i : Random variable representing the output of pro-ducer i , scaled between 0 and 1. The moments of Z i are denoted as E Z i = µ i , E ( Z i − E Z i ) = σ i and E | Z i − E Z i | = ρ i . • C i : Capacity of producer i . • D : Total electricity demand in the market • N : Number of producers in the market. • γ i : Investment cost for unit capacity for producer i . • ξ : Efficiency of the game equilibrium. • x − i : The quantities chosen by all other producersexcept i , that is x − i = [ x , . . . , x i − , x i +1 , . . . , x N ] . • ( x ) + := max( x, . • ( x ) − := min( x, . Renewable Production Model:
When producer i invests ina capacity of C i , its actual generation of energy is a randomvariable given by C i Z i ≤ C i . That is, due to the randomnessassociated with renewables, its realized production may notequal its maximum capacity.We make the following assumption on the Z i ’s: [A1] We assume that the random variable Z i has support [0 , and its density function is bounded and continuous onits domain. This assumption is mainly made for analyticalconvenience and captures a wide range of probabilisticdistributions used in practice, e.g., truncated normal distri-bution and uniform distribution. Furthermore, we assumethat Z i is not a constant, so E ( Z i − E Z i ) = σ i > .We also make the following assumptions on demand D : [B1] We assume that the random variable D is non-negativeand that D is bounded, i.e., D min ≤ D ≤ D max where D min D max is a constant bounded away from zero. Consider N renewable producers who compete in a de-centralized market. Each producer needs to decide twoquantities: capacity (sizing) and the corresponding everydayprice bidding strategy. To make this decision, each pro-ducer needs to take into consideration the fact that largercapacities lead to higher investment costs but may alsoresult in enhanced revenue due to increased sales. If theinvested capacity is low, then the investment cost is lowbut the producer risks staying out of the market becauseof less capability to provide energy. Therefore competitionrequires non-trivial decision making by the decentralizedstakeholders. In this paper, we consider both cases whereeach producer has either same or different investment cost.To start with, let us assume γ i = γ for all i . This assumptionis true to the first order since the solar installation cost inan area is roughly the same for all the consumers. A more general case with different γ i ’s is left to Section 4.6. Since theproducers need to compete for capacity based on revenue(which is determined by optimal bidding), we refer to thecapacity competition—how much to invest—as the capacitygame and the pricing competition, i.e., how much to bid, asthe price sub-game . To generalize the scenario we assume thatthe price bidding is a one-time action that approximates anamortized version of a series of continuous bidding actions. The ultimate decision for the producers is to determine theoptimal capacity to invest in. Suppose that the capacity isdenoted by C i for each producer i , then each producer’sobjective is to maximize its profit, which is specified as:(Profit) π i ( C i , C − i ) − γC i , ∀ i, (1)where π i ( C i , C − i ) is the payment (revenue) from con-sumers to producer i when its capacity is fixed at C i andthe others’ capacities are fixed at C − i . This payment isdetermined by the price sub-game given that a capacitydecision is already made, i.e., C = [ C , C , ..., C N ] : we leavea detailed discussion of the revenue and the price sub-gameto Section 2.5. The term γC i represents the investment cost.Since we are in a game-theoretic scenario, the appropri-ate solution concept is that of a Nash equilibrium. Specifi-cally, a capacity vector C (cid:5) = [ C (cid:5) , C (cid:5) , ..., C (cid:5) N ] is said to be aNash equilibrium if: C (cid:5) i = arg max C i ≥ π i ( C i , C (cid:5)− i ) − γC i , ∀ i, (2)The Nash equilibrium shown in (2) is interpreted as thefollowing: each producer i chooses a capacity C i such thatgiven the optimal capacity strategy of the others, there isno incentive for this producer to deviate from this capacity C i . Note that while choosing its capacity, each producer im-plicitly assumes that its resulting revenue is decided by thesolution obtained via the price sub-game. Note that for eachPV producer to make a decision according to (2), it needs toknow the distribution of demand D , where random variable D can be seen as discounted back from infinite horizon sowe do not explicitly model D t in this manuscript. It alsoneeds to know the distribution of solar radiation Z i , andinvestment price γ as shown in next Section. Distributionof D can be obtained through historical data, Z i can be es-timated based on geographical differences, and investmentprice should be publicly available given that PV producersuse similar materials to fabricate panels. Therefore each PVplayer can independently play its strategy without accessingprivate information of others. In this section, we explicitly characterize the payment func-tion π i ( C i , C − i ) at the equilibrium solution of the pricesub-game for a fixed capacity vector ( C i ) ≤ i ≤ N . The pro-ducers now compete to sell energy at some price p i . Thisis known as the Bertrand price competition model, wherethe consumer prefers to buy energy at low prices. In thismodel, consumers resort to buying at a higher price onlywhen the capacity of all the lower-priced producers areexhausted. Suppose that the profit for producer i when the producers bid at p = [ p , p , ..., p N ] = [ p i , p − i ] is denotedby π i ( C i , C − i , p i , p − i ) . We make the following assumptionabout the prices: [A2] The customers have the options to buy energy at unitprice from the main grid.This assumption follows the current structure of a distri-bution system, where customers have access to the maingrid at a fixed price, and here we normalize the price to . Equivalently, this can be thought as the value of the lostload in a microgrid without a connection to the bulk electricsystem [28].As shown in [16], [18], there is no pure Nash equilibriumon p for the price sub-game. Intuitively, this means that noplayer can bid at a single deterministic price and achieve themost revenue, since the other players can undercut by a tinyamount and sell all their generation. Therefore no playersettles on a pure strategy. Such a situation particularly ariseswhere each producer is small ( C i ≤ D, ∀ i ), but the aggregateis large ( (cid:80) i C i > D , where D denotes the total demand inthe market).However, there exists a mixed Nash equilibrium on price p ,where the optimal bids follow a distribution such that thebids of each DER are independent of the rest. Informally,this implies that each producer i draws its price p i froma distribution P ∗ i , which maximizes its expected revenuegiven the distributions of the other producers. For exam-ple, the price distribution of a two player Bertrand modelis given in [18]. For our purpose, the exact form of theoptimal price distribution is not of particular interest. Thequantity of interest is the form of the revenue function ,i.e., the expected payment, resulting from this random pricebidding. Let us denote the expected payment to producer i based on the optimal random price by π i ( C i , C − i ) = E p ∼P ∗ × ... ×P ∗ N π i ( C i , C − i , p i , p − i ) . Proposition 1 character-izes the optimal payment to each producer: Proposition 1.
Given any solution ( C , C , . . . , C N ) having C ≤ C ≤ . . . ≤ C N , the expected payment received byproducer i in the equilibrium of the pricing sub-game is givenby: π i ( C , . . . , C N ) = π N ( C , . . . , C N ) C i E [min( Z i , DC i )] C N E [min( Z N , DC N )] . (3) Moreover, the expected payment received by producer N with thelargest capacity investment is given by: π N ( C N , C − N ) = E D E Z N ,Z − N min { ( D − (cid:88) j (cid:54) = N Z j C j ) + , Z N C N } . (4)As a consequence of Proposition 1, we have that if thecapacities are symmetric at equilibrium, i.e., C = C = · = C N , and Z i ’s are identically distributed for all ≤ i ≤ N ,the expected payment of producer i is given by: π i ( C i , C − i ) = E D E Z i ,Z − i min { ( D − (cid:88) j (cid:54) = i Z j C ) + , Z i C i } . A complete proof of Proposition 1 is deferred to theAppendix. Let us now understand Proposition 1 for thesymmetric investment solution. Equation (4) denotes the payment received by producer i when it bids determinis-tically at price p i = 1 and all of the other producers bidaccording to their mixed pricing strategy. By assumptionA2, this player bids at the highest possible price. Then theamount of energy sold equals the minimum of the leftover-demand from the market ( E ( D − (cid:80) j (cid:54) = i Z j C j ) + ) and theplayer’s actual production ( C i Z i ). Since p i = 1 belongs tothe support of the mixed pricing strategy adopted by thisplayer, one can use well known properties of mixed Nashequilibrium [16], [18] to argue that producer i ’s paymentat this price equals the expected payment received by thisproducer at the equilibrium for the pricing sub-game. VALUATION METRICS
One essential characteristic of a game is its cost as comparedto a centralized decision. In this section, we present thebenchmark cost that we consider; in particular, we focus onthe social cost minimization achieved by a social plannercontrolling the producers. In Section 3.2 we give moredetails on the definition of game efficiency as compared tothis benchmark.Suppose that these producers are managed by a socialplanner in a centralized manner. The purpose of the socialplanner is to fulfill demand while minimizing the total costby deciding the investment capacities of the producers.Since we approximate the continuous actions of energybuying and selling from PV producer to a one-time actionas in Section 2.3, the social planner thus wants to minimize social cost in the following form: C ∗ = arg min C i ≥ , ∀ i ∈{ , ,...,N } N (cid:88) i =1 γC i + E { ( D − N (cid:88) i =1 Z i C i ) + } , (5)where C ∗ = [ C ∗ , C ∗ , ..., C ∗ N ] is the optimal capacity de-cision from the social planner for each producer i , andexpectation is over all randomness, i.e., Z i ’s and D . In whatfollows, we adopt this routine if not specifcied otherwise.The social cost presented in (5) is composed of two terms.The first term is the total investment cost which is linear inthe capacities, and the second term is the imbalance costin buying energy from electricity grid if the renewablescannot satisfy the demand. These two terms represent thetradeoff between investing energy resources and buyingenergy from conventional generators in order to meet theelectricity demand. Given the definition of the equilibrium solutions due toboth price and capacity competition, a natural question is toevaluate the performance of the decentralized market: i.e., does competition result in efficiency? . As mentioned previously,we measure this efficiency via two metrics: the social costof the decentralized capacity investment compared to thatachieved by the social planner, and the total investment costcompared to the payments made by the demand.
Example.
Let us consider a one-player case with determin-istic demand D , where there is only one producer partici-pating in the electricity market. We further assume that therandom output of this plant follows a uniform distribution, i.e., Z ∼ unif (0 , . Suppose that γ < , otherwise thereis no incentive to enter the market. The social planner’soptimization is reduced to : C ∗ = arg min C γC + E ( D − Z C ) + , (6)where C ∗ = (cid:113) D γ . In this case, the total investment cost is D (cid:113) γ : in a centralized scenario, one can imagine that this isthe price charged by the social planner to the demand, andthus there is no ‘markup’.Let us now take a look at the decentralized market. Sincethere is only one producer, the decentralized investmentstrategy clearly coincides with that of the social planner.The payment from the demand to the producer as per (4)is E min { D, Z C (cid:5) } = D (1 − (cid:113) γ ) > D (cid:113) γ (when γ < ).This suggests that the producer is exploiting its marketpower to considerably improve its profit and the benefitsof renewables are not being transferred to the consumers. Market Efficiency
As noticed in the above example, inef-ficiency arises due to the high prices felt by the demandin the decentralized market. Formally, we define marketefficiency as the ratio between the investment cost paidby the producers to the total payment received by theproducers at any equilibrium of the capacity price game.Therefore, efficiency takes the following form: ξ = γ (cid:80) Ni =1 C (cid:5) i (cid:80) Ni =1 π ( C (cid:5) i , C (cid:5)− i ) . (7)A “healthier” game should achieve a higher ξ that is asclose as to 1. This means that the producers should bid atthe prices that cover their investment cost, so that biddingis efficient and does not take advantage of the electricityconsumers. A particularly interesting question is whethercompetition leads to increased efficiency as the number ofproducers in the market increases. We formalize this notionbelow. Definition 1.
We define the efficiency of a Nash equilibriumin a capacity game illustrated in (1) by ξ . The capacity gameis asymptotically efficient when ξ → as N → ∞ for everyNash equilibrium. Now the question of interest is 1) whether uncertainty ingeneration deteriorates or improves the market efficiency ofthe game, and 2) whether efficiency increases as the numberof players in the game increases. In the following sections,we will see that without randomness in the generation,the producers are able to charge a relatively high price forenergy, which makes the game less efficient. Interestingly,when producer’s generation becomes uncertain, the gamebecomes more efficient as more producers are involved inthe decentralized market.
Inefficiency due to Social Cost:
When there are multipleproducers, it is possible that even the investment decisionsmay not coincide with that of the social planner. Therefore,a second source of inefficiency is the social cost due to thecapacity investment, as defined in (5). More concretely, wecompare the social cost of the equilibrium solution ( C (cid:5) i ) with that of the social cost of the planner’s optimal capacity ( C ∗ i ) — clearly, the latter cost is smaller than or equal to theformer. Before moving on to the main results, we highlight the(in)efficiency of the equilibrium in the deterministic versionof the capacity game, i.e., one without production uncer-tainty where Z i = 1 with probability one. Understandingthe inefficiency of this deterministic game is the startingpoint for us to better gauge the effects of uncertainty ininvestment games.We begin with the social planner’s problem with fixed D , which in the absence of uncertainty can be formulated asfollows: min C i ≥ , ∀ i ∈{ , ,...,N } γ N (cid:88) i =1 C i + ( D − N (cid:88) i =1 C i ) + . (8)Every solution with non-negative capacities that satisfies (cid:80) Ni =1 C ∗ i = D optimizes the above objective — this includesthe symmetric solution C ∗ = C ∗ = · · · = C ∗ N = DN . Movingon to the decentralized game with deterministic energygeneration, we can directly characterize the equilibrium so-lutions using the results from [16]. Specifically, by applyingProposition 13 in that paper, we get although there are mul-tiple equilibrium solutions, every such solution ( C i ) ≤ i ≤ N satisfies ( i ) (cid:80) Ni =1 C i = D , and ( ii ) π i ( C i , C − i ) = C i .The second result implies that at every equilibrium, eachproducer charges a price that is equal to the electricity priceof one from the main grid. Finally, by applying (7), we cancharacterize the efficiency in terms of the investment cost γ : ξ = γDD = γ. (9) Why is this result undesirable?
First note that when γ < , (9) implies that the deterministic game is inefficient at every Nash equilibrium . In fact, using the results from [16],one can deduce that the system is inefficient even whendifferent producers have different investment costs. Perhapsmore importantly, the costs of investment as well as themarket price of renewable energy have dropped consistentlyover the past decade and are expected to continue doing soin the future [29], [30], [31]. In this context, Equation (9)has some stark implications, namely that as γ (the investmentprice) drops in the long-run, the efficiency actually becomes worse( ξ → as γ → ) , i.e., the improvements in renewabletechnologies do not benefit the electricity consumers. AIN RESULTS
To better illustrate the results, In this section, we first assumethat γ i ’s are the same across all producers and characterizethe capacity decision from the social planner’s problem. Wethen illustrate the relationship between the decentralizedmarket, and the social planner’s problem in the centralizedmarket. We also give a thorough analysis on the efficiency ofthe decentralized market. We begin by considering the casewhere the capacity generated by the producers are indepen-dent of each other and then move on to the correlated case.In the end of this section, we extend the result to asymmetric γ i ’s. All of the proofs from this section can be found in theappendix. An immediate observation of the socially optimal capacityas described in (5) is that if the randomness is independentand identical across different producers, the socially optimalcapacity is symmetric:
Theorem 1.
If the random variables Z i are i.i.d. and satisfyassumption A1 and γ i = γ, ∀ i , then the optimal capacity obtainedby (5) is symmetric, i.e., C ∗ = C ∗ = · · · = C ∗ N = C ∗ . Theorem 1 states that when the investment cost per unitcapacity is the same across all producers, and the randomvariable is i.i.d., then the optimal decision for the socialplanner is to treat all producers equally and invest thesame amount of capacity for each producer. In reality, therandomness due to renewable sources can be correlated andSection 4.4 shows that Theorem 1 stills holds under someconditions on the nature of the correlation.
Now that we have captured the structure of the sociallyoptimal capacity decision, we want to address the issueof whether or not the capacity price game admits Nashequilibrium solutions in the decentralized market. A secondquestion concerns the social cost of Nash equilibria whencompared to the optimum investment decision adopted bya social planner. As discussed in Section 3.2, one of thetwo sources of inefficiency in decentralized stems from thefact that the social cost of equilibrium solutions may belarger than that of the central planner’s solution. We presentTheorem 2 which addresses both of these questions byproving the existence of a Nash equilibrium that coincideswith the socially optimal capacity decision. Characterizingother Nash equilibriums is left to Theorem 3 in Section 4.3.
Theorem 2.
There is a Nash equilibrium that satisfies (2) , whichalso minimizes the social cost. That is, ( C ∗ , C ∗ , . . . , C ∗ ) is aNash equilibrium. Therefore, existence is always guaranteed in our setting.More importantly, Theorem 2 provides an interesting rela-tionship between the centralized decision that minimizessocial cost, and the decentralized decision where producersseek to maximize profit. It states that the game yields asocially optimal capacity investment solution as if therewere a social planner controlling the producers. In addition,as we will show later in Section 4.5, this Nash equilibrium isthe unique symmetric equilibrium in the capacity game. Forthe following sections, we use C ∗ to denote both the sociallyoptimal capacity decision and this Nash equilibrium. Although the capacity price game studied this work admitsa Nash equilibrium that minimizes the social planner’sobjective, there may also exist other equilibria that resultin sub-optimal capacity investments. How do these (po-tential) multiple equilibria look like from the consumers’perspective, i.e., is the price charged to consumers largerthan the investment? In this section, we show a surprisingresult: the two-level capacity-pricing game is asymptoticallyefficient. That is, as N → ∞ , the total payment made to the producers approaches the investment costs for everyNash equilibrium . The reason for this startling effect is thatas the number of producers competing against each otherin the market increases, with the presence of uncertainty,the market power of these producers decreases and theefficiency of the game equilibrium increases. We first presentour main theorem with i.i.d. generation. Theorem 3.
Let ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) denote any Nash equilibriumsolution in an instance with N producers and N > DminDmax γ ,where γ i = γ, ∀ i . Then, as long as the Z i ’s are i.i.d and satisfyassumption A1, we have that: N (cid:88) i =1 π i ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) ≤ γ N (cid:88) i =1 C (cid:5) i + αN − c , where α, c > are constants that are independent of N .Therefore, as N → ∞ , ξ → , where ξ denotes the marketefficiency due to any Nash equilibrium solution.
Combining Theorems 2 and 3 yields that if we restrictthe game to only have the symmetric equilibrium, thenthe equilibrium minimizes the social cost and the gameis asymptotically efficient. Moving beyond the symmetricequilibrium, Theorem 3 states that any Nash equilibrium ob-tained from the capacity game is efficient, that the collectedpayment (revenue) tends to exactly cover the investmentcost. This further suggests that the capacity game describedin (1) elicits the true incentive for the producers to generateenergy.
In reality, renewable generation due to multiple entities ina power distribution network is usually correlated witheach other because of geographical adjacencies. We assumethat the randomness of each producer’s generation can becaptured as an additive model written as the following: Z i = ˆ Z i + ¯ Z. (10)The model in (10) captures the nature of renewable genera-tion. We can interpret ¯ Z as the shared random variable fora specific region. For example, the average solar radiationfor a region should be common to every PV output in thatregion. On the other hand, ˆ Z i can be seen as the individual-level random variable for the particular location of each PVplant i , and this random variable can be seen as i.i.d. acrossdifferent locations.For analytical convenience, we make the following as-sumptions on Z i : [A3] Both ¯ Z and ˆ Z i in (10) satisfy assumption A1, the ˆ Z i ’sare i.i.d, and are independent of ¯ Z for all i .If the correlation is captured as in (10), the optimalcapacity decision is still symmetric, i.e., C ∗ i = C ∗ j , ∀ i (cid:54) = j is a valid solution to (5). This is stated in Theorem 4. Theorem 4.
If the random variable Z i is captured as in (10) andassumption A3 is satisfied, then the optimal capacity vector thatminimizes the planner’s social cost is symmetric, i.e., C ∗ = C ∗ = · · · = C ∗ N = C ∗ when γ i = γ, ∀ i . In addition, note that Theorem 2 does not require the i.i.dassumption on Z i . Therefore, we infer that the symmetric solution that minimizes social cost is a Nash equilibriumeven when the generation is correlated. In what follows, wefurther show that correlation does not tamper the efficiencyof any Nash equilibria in the capacity game. Theorem 5.
Suppose that ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) denotes any Nashequilibrium solution in an instance with N producers and N > γ , where γ i = γ, ∀ i . Then, as long as the random variable Z i , iscaptured in (10) , and assumption A3 is satisfied, we have that: N (cid:88) i =1 π i ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) ≤ γ N (cid:88) i =1 C (cid:5) i + αN − c , where α, c > are constants that are independent of N . Theorem 5 extends the statement in Theorem 3 fromi.i.d. random variables to correlated random variables. Thisindicates that if the randomness of each producer is cap-tured by an additive model interpreted as the sum ofshared randomness and individual-level randomness, thenthe decentralized market is efficient and that both producersand electricity users benefit from this market.
Although our setting could admit many equilibrium solu-tions, we know that one of these solutions must alwaysbe symmetric, i.e., every producer has the same investmentlevel. This solution is of particular interest as it minimizesthe social cost. We now show that the symmetric Nashequilibrium C ∗ , C ∗ , . . . , C ∗ N is unique in Theorem 6. Theorem 6.
Under assumption A1, the symmetric Nash equilib-rium in the capacity game (1) is unique.
Theorem 6 states that there is only one symmetric Nashequilibrium in the capacity game. This indicates that if thedecentralized market is regulated such that each producerbehaves similarly in the presence of uncertainty, then it isguaranteed that the competition is both efficient and sociallyoptimal in the investment decision.
We now extend our results to a more general case whenPV providers’ investment price is different. This would bethe case when installation is dispersed cross areas whereunit installation cost is not the same. We consider an ar-bitrary equilibrium solution of the two-level game withasymmetric investment costs, ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) , such that C (cid:5) ≤ C (cid:5) ≤ . . . ≤ C (cid:5) N . This assumption is clearly withoutloss of generality. Moreover, we assume that γ i representsthe investment cost of the PV whose capacity at equilibriumis given by C (cid:5) i . Let γ min = min i γ i . We start with Theorem7, which guarantees that the game always admits a Nashequilibrium. Theorem 7.
Given an instance of the PV game in (1) , but withasymmetric investment costs γ ≥ γ ≥ · · · ≥ γ N , there alwaysexists a Nash equilibrium for the capacity game as long as thedistributions ( Z i ) Ni =1 are identical. Detailed proof is left in Appendix G. With this guarantee,we now introduce the result on the efficiency of the gamewith asymmetric γ i ’s. Theorem 8.
Suppose that ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) denotes any Nashequilibrium solution in an instance with N producers, asymmetricinvestment costs ( γ , . . . , γ N ) and N > DminDmax γ min . Then,as long as the random variable Z i , is captured in (10) , andassumption A3 is satisfied, we have that: N (cid:88) i =1 π i ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) ≤ N (cid:88) i =1 γ max C (cid:5) i + αN − c , where α, c > are constants that are independent of N and γ max = max i γ i . Since the actual investment cost incurred by theproviders is (cid:80) Ni =1 γ i C (cid:5) i , the above theorem implies that theratio of the total payment to the investment costs (e.g., theprice of anarchy) is at most γ max γ min . This allows us to quantifythe efficiency at equilibrium. The efficiency is thus quan-tified by how much the largest invest price γ max deviatesfrom each of the γ i ’s. When the investment prices are notvery different across producers which should be the realisticcase, the efficiency is not very far away from the case when γ i ’s are the same. IMULATION
In this section, we validate the statements by providingsimulation results based on both synthetic data and real PVgeneration data. We use the symmetric Nash equilibrium asthe solution of interest in our simulations.
Let us assume that the generation distribution is uniform,i.e., Z i ∼ unif (0 , . Suppose that the investment price isthe same for all players, i.e., γ = 0 . , then following theanalysis in Section 4, we know that the optimal capacitysatisfies C ∗ = C ∗ . Assuming that the demand is uniformlydistributed between 0.75 and 1.25, we solve the social opti-mization in (5) with equal investment price γ . The optimalsolution leads to a total capacity of C ∗ tot = C ∗ + C ∗ = 1 . ,where C ∗ = C ∗ = 0 . . The result is shown in Fig. 3. C C S o c i a l c o s t Figure 3: Social cost with respect to total capacity wheninvestment price is the same.To verify that C ∗ = C ∗ = 0 . is indeed a symmetricNash equilibrium, we vary the capacity from C ∗ and study how player ’s profit changes. The analysis for player 2 pro-ceeds in the same way because of symmetry. We show theresult of optimality for player 1 in Fig. 4 in terms of profit,with a fixed capacity for player 2 where C = C ∗ = 0 . . C P r o f it f o r p l a y e r C = 0.86 Figure 4: Profit for player 1 when its capacity deviates from C ∗ .As can be seen from Fig. 4, the profit for player 1—when the other player’s capacity is fixed at C ∗ —peaks at C = C ∗ . By symmetry, we can argue that player ’s profitis maximized at C ∗ when player ’s capacity remains fixed.Therefore, ( C ∗ , C ∗ ) is indeed a Nash equilibrium as neitherplayer has any incentive to deviate from its investmentstrategy. In other words, the socially optimal capacity is alsoa Nash equilibrium for the game shown in (1). To illustrate that the Nash equilibrium is efficient withrespect to the metric defined in (7), we need to show thatthe payment collected from users in the game exactly coversthe investment costs of the producers when the numberof producers increases. Since the computational complex-ity grows exponentially with the number of producers (ifthey are not identical), the simulation is unachievable forlarge number of asymmetric producers. For illustrationpurposes we simulate the capacity game with identicalplayers ( γ i = 0 . , ∀ i ) with i.i.d. generation (uniform dis-tribution). We then compute the efficiency ξ when thereare , , , , , , players in the game. Theresults are shown in Fig. 5. Number of players in the game E ff i c i e n c y ξ Figure 5: Efficiency of the symmetric Nash equilibrium inthe game as a function of number of players.In Fig. 5, we see that the efficiency is growing with thenumber of players in the game. We therefore infer that thecompetition is healthy as the producers only bid their truecosts and do not exploit the consumers of electricity.
In this section, we simulate the efficiency of the gameequilibrium using a real PV generation profile obtained from the National renewable energy laboratory [32]. Our datacomes from distributed PVs located in California with a 5minute resolution. Typical PV profiles after normalizationare shown in Fig. 6. From Fig. 6, we see that the random-ness of PV generation from different locations is stronglycorrelated. The correlation between those PV profiles is alsosymmetric across different PV plants, as shown in Fig. 7.
Time (hour) N o r m a li ze d P V g e n e r a ti on Time (hour) N o r m a li ze d P V g e n e r a ti on Figure 6: PV generation profile in different locations.
Correlation between PV generation
Index of PV plant I nd e x o f P V p l a n t Figure 7: Correlation of PV generation in different locations.A lighter color (yellow) represents stronger correlation anddark colors (blue) represent weak correlation.We then use these PV profiles to obtain the game equilib-rium as we vary the number of PV participants. The resultis shown in Table 1, with the assumption that γ = 0 . .As we can see from Table 1, in the absence of random-ness when the producers are assumed to generate energydeterministically, the efficiency is the investment price asdescribed in Equation (9). The efficiency of the game withuncertainty improves as the numbers of producers in themarket increases.Table 1: Game efficiency with different number of producers,when investment price is . and demand D = 5 . Number of producers 5 30 120
Efficiency of deterministic producers 0.15 0.15 0.15Efficiency of random producers 0.83 0.96 0.98
Table 2: The ratio between total capacity and market de-mand , i.e., (cid:80) i C ∗ i /D , when investment price is . anddemand D = 5 . Number of producers 5 30 120 (cid:80) i C ∗ i /D with deterministic producers 1 1 1 (cid:80) i C ∗ i /D with random producers 1.26 1.32 1.30 In addition, in a deterministic game, the total capacityis always the same as the market demand because there is no randomness in generation. In the capacity game withuncertainty, since each producer faces randomness in itsown production as well as the random generation from theother producers, the total invested capacity is greater thandemand as illustrated in Table. 2. This means that in thecapacity game with uncertainty, the total capacity exceedsdemand elicits competition among producers.
ONCLUSIONS AND F UTURE W ORK
In this paper, we consider a scenario where many dis-tributed energy resources compete to invest and sell energyin a decentralized electricity market especially when uncer-tainty is present. Each energy producer optimizes its profitby selling energy. We show that such a competitive gamehas a Nash equilibrium that coincides with the solutionfrom a social welfare optimization problem. In addition, weshow that all Nash equilibria are efficient, in the sense thatthe collected payment to the energy producers approachestheir investment costs. Our statement is validated both bytheoretical proofs and simulation studies. Finally, we showthat in systems where the investment costs are asymmetric,the inefficiency is at most the ratio of the maximum andminimum investment costs. Of course, in many situations,one can expect different producers have access to similartechnologies and therefore, approximately similar invest-ment costs. This can lead to small and bounded levels ofinefficiency in practice.Our work raises a number of intriguing possibilities forfuture research in this area. First, it is worth noting thatthe setting studied in this paper admits a multiplicity ofequilibrium solutions; we side-step this issue by provingthat all of the equilibria are efficient from the perspectiveof the consumers. In future work, it may be beneficialto study these equilibria in greater detail, particularly thesolutions that are more likely to be formed in practiceand understand how they compare to the optimal capacityinvestment strategy. Secondly, this work focuses on theproperties of a very specific auction mechanism where eachproducer sells energy at the exact bid price (i.e., pay-as-bid auction). Understanding how other commonly proposedauction formats—e.g., Cournot mechanism [25], [33], clockauctions [34] —behave under uncertainty is interesting froma design perspective.Finally, the central theme in this work—that of lever-aging uncertainty to improve outcomes—is applicable to awide range of systems with some uncertainty in demandor supply. This could include markets in domains suchas transportation, communication, cloud computing, etc.Characterizing broad conditions under which uncertaintyhelps or harms systems with self-interested agents is indeedan important avenue for future research. R EFERENCES [1] G. Timilsina, L. Kurdgelashvili, and P. Narbel, “Solar energy:Markets, economics and policies,” vol. 16, no. 1, pp. 449–465, 2012.[2] C.-J. Winter, R. L. Sizmann, and L. L. Vant-Hull,
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Pan Li received the B.S. degree in Electrical En-gineering, M.S. degree in Systems Engineeringfrom Xi’an Jiaotong University, Xi’an, China, in2011 and 2014, respectively; and she receivedher Ph.D. degree in Electrical Engineering fromUniversity of Washington, Seattle in 2018. Shealso holds the engineering diploma from EcoleCentrale de Lille, France. Her research inter-ests included machine learning applications indemand response programs and in distributionnetworks. She is now research scientist at Face-book, Inc..
Shreyas Sekar is a Postdoctoral researcher inElectrical Engineering at the University of Wash-ington, Seattle. He received his Ph.D. in Com-puter Science from Rensselaer Polytechnic In-stitute in and a B.Tech in Electronics andCommunications Engineering from the Indian In-stitute of Technology, Roorkee in . His re-search interests are in computational economicsand algorithmic game theory. He is a recipient ofthe
Robert McNaughton prize for the bestgraduate student in CS at Rensselaer Polytech-nic Institute.
Baosen Zhang received his Bachelor of AppliedScience in Engineering Science degree fromthe University of Toronto in 2008; and his PhDdegree in Electrical Engineering and ComputerSciences from University of California, Berkeleyin 2013.He was a Postdoctoral Scholar at StanfordUniversity, affiliated with the Civil and Environ-mental Engineering and Management & ScienceEngineering. He is currently an Assistant Profes-sor in Electrical and Computer Engineering atthe University of Washington, Seattle, WA. His research interests arein power systems and cyberphysical systems. A PPENDIX AP ROOF FOR P ROPOSITION We know that the producers adopt a mixed pricing strat-egy in the equilibrium for the pricing sub-game. Let l i , u i denote the lower and upper support of the distributioncorresponding to the mixed strategy of producer i . Fromprevious results [16], [18] and assumption A2, we know that l i = l j and u i = u j = 1 for all i, j . Therefore, let p denotethe common lower support (price) for every producer, it isclaimed in [18] that only one producer can have an atom andit must be at upper support. Using the basic properties ofmixed strategy equilibria (e.g., see [16]), we can infer that thetotal payment received by any producer i equals its paymentwhen this producer bids a deterministic price of p and all ofthe other producers bid according to their mixed strategiesin the pricing sub-game equilibrium. Explicitly writing thisout, we get π i ( C , C , . . . , C N ) = p E [min( C i Z i , D )] = pC i E [min( Z i , DC i )] (11) π N ( C , C , . . . , C N ) = p E [min( C N Z N , D )] = pC N E [min( Z i , DC N )] . Indeed, observe that when any one player selects a priceof p , all of the capacity generated by this player must besold because no other player can bid below this price andthe probability that other players bid exactly at this pricecan be ignored due to the lack of atoms. Using the secondequation above, we can explicitly characterize p in termsof the payment received by the producer with the highestcapacity investment, i.e., p = π N ( C , C , . . . , C N ) C N E [min( Z i , DC N )] . Substituting the above into Equation 11 gives us: π i ( C , C , . . . , C N ) = π N ( C , C , . . . , C N ) C i E [min( Z i , DC i )] C N E [min( Z N , DC N )] . In order to complete the proof, we need to show that π N ( C N , C − N ) = E [min { ( D − (cid:80) j (cid:54) = N Z j C j ) + , Z N C N } ] . Theproof utilizes techniques very similar to those adoptedin [16], so we only sketch the details and highlight the keydifferences. As argued before, in the equilibrium for thepricing sub-game, each producer i plays a mixed strategywith prices in the range [ p i , . We first claim that in theequilibrium at most one player has an atom at the uppersupport price and this can only be producer N . The first partof the claim can be proved similar to the proof of Lemma in [16], namely that if multiple producers have an atom atthe upper support, then at least one producer can deviate tohaving an atom at price − (cid:15) instead and strictly improveits profit. For the second part, assume by contradiction thatanother producer (say i ) has atom at the upper support.By assumption, only producer i has an atom at the uppersupport. So, when it bids a deterministic price of p i = 1 , theconsumers would first consume from the lower priced pro-ducers ( j (cid:54) = i ) and only the leftover demand would be satis-fied by i . Therefore, we can infer that the payment receivedby i equals: π i ( C ) = E [min { ( D − (cid:80) j (cid:54) = i Z j C j ) + , Z i C i } . ] Then, as per (3), we have that π N ( C ) = π i ( C ) C N G N C i G i , where G j = E [min( Z j , DC j )] for all j . We have that: π N ( C ) = E [min { ( D − (cid:88) j (cid:54) = i Z j C j ) + , Z i C i } ] C N G N C i G i < E [min { ( D − (cid:88) j (cid:54) = N Z j C j ) + , Z N C N } ] G N G i ≤ E [min { ( D − (cid:88) j (cid:54) = N Z j C j ) + , Z N C N } ] . (12)The strict inequality comes from the fact that C i < C N and that Z i , Z N are identically distributed (conditionalon Z − i,N ) . The final inequality comes from the fact that G N ≤ G i since C i < C N . However, (12) is a lower boundon the producer N ’s payment if it deviates to bidding a de-terministic price of p N = 1 , and so, we have a contradictionsince producers cannot strictly improve their payments bydeviating at equilibrium. In summary, we have that onlyproducer N can have an atom at the upper support. As weargued before, since no other producer as an atom at theupper support, when producer N bids p N = 1 , it only getsthe leftover demand and therefore, its payment is given by π N ( C ) = E [min { ( D − (cid:80) j (cid:54) = N Z j C j ) + , Z N C N } . ] A PPENDIX BP ROOF FOR T HEOREM Suppose that at the optimum solution that minimizes Equa-tion (5), the aggregate capacity investment by the producersis C ∗ tot , and let C ∗ = C ∗ = . . . = C ∗ N = C ∗ tot N ≥ . Then, inorder to prove Theorem (1), it is sufficient to show that forany capacity C , C , . . . , C N ≥ with (cid:80) Ni =1 C i = C ∗ tot , thefollowing equation is satisfied: E [( D − N (cid:88) i =1 Z i C ∗ i ) + ] ≤ E [( D − N (cid:88) i =1 Z i C i ) + ] . In fact, using the transformation that for any capac-ity vector ( C (cid:48) , . . . , C (cid:48) N ) , E [( D − (cid:80) Ni =1 Z i C (cid:48) i ) + ] = E D − E [min( D, (cid:80) Ni =1 Z i C (cid:48) i )] , the above equation can be rewrittenas: E [min( D, N (cid:88) i =1 Z i C ∗ i )] ≥ E [min( D, N (cid:88) i =1 Z i C i )] . (13)So, to prove Theorem 1, it is indeed sufficient to proveEquation (13). To prove this, we introduce Proposition 2 andProposition 3. Proposition 2.
Let us consider the following definitions: • X := Z C N + Z C N + . . . + Z N C N N , • X := Z C N + Z C N + . . . + Z N C N • . . . • X N := Z C N N + Z C N + . . . + Z N C N − N That is: X i = N (cid:88) j =1 Z j C i + j − N , (14) where i + j − is computed modulo N . Then: E [min( D, N (cid:88) i =1 X i )] ≥ N (cid:88) i =1 E [min( DN , X i )] . (15) Proof.
We prove the inequality by proving that inequalityholds for each realization of X = x , X = x , . . . , X N = x N and each realization of D = d . Denote the whole indexset by N . In each realization, there are the following fourscenarios: • Each of x i is smaller than dN . In this case, min( d, (cid:80) Ni =1 x i ) = (cid:80) Ni =1 x i , and min( dN , x i ) = x i .So equality holds. • Each of x i is bigger than dN . In this case, min( d, (cid:80) Ni =1 x i ) = d , and min( dN , x i ) = dN . Soequality again holds. • x j , j ∈ J ⊆ N is bigger than dN , the rest aresmaller than dN , but (cid:80) Ni =1 x i ≤ d . In this case, min( d, (cid:80) Ni =1 x i ) = (cid:80) Ni =1 x i . The RHS of (15) reducesto dN |J | + (cid:80) i ∈N \J x i ≤ (cid:80) Ni =1 x i . Therefore, theinequality holds. • x j , j ∈ J ⊆ N is smaller than dN , the rest are biggerthan dN , but (cid:80) Ni =1 x i ≥ d . The RHS of (15) reduces to dN ( N −|J | )+ (cid:80) j ∈J x j ≤ d . Therefore, the inequalityholds.In all cases, we have that: E [min( D, N (cid:88) i =1 X i )] ≥ N (cid:88) i =1 E [min( DN , X i )] . (16) Proposition 3.
With the assumptions in Proposition 2, we have: E [min( DN , X i )] = E [min( DN , X )] , ∀ i ∈ N . (17)To prove Proposition 3, we introduce Fact 1. It is basedon exchangeability of independent random variables, andthat Z i ’s’ are i.i.d. copies. We refer to [35] for interested usersand omit the proof here. Fact 1. If Z i ’s are i.i.d. random variables, then: f ( z , z , . . . , z N ) = f ( z S , z S , . . . , z S N ) , (18) where S , S , . . . , S N is a permutation of , , . . . , S N , and f ( · ) is the density function. Now we proceed to prove Proposition 3.
Proof of Proposition 3.
Using Fact 1, we show that E [min( DN , X i )] is the same for all i : E [min( D/N, X i )] = E [min( D/N, N (cid:88) j =1 Z j C i + j − N )] ( a ) = E [min( D/N, N (cid:88) j =1 Z i + j − C i + j − N )] ( b ) = E [min( D/N, N (cid:88) k =1 Z k C k N )]= E [min( D/N, X )] , (19) where ( a ) is based on the observation that [ i, i + 1 , . . . , i + N − , mod N, ∀ i is a permutation of [1 , , . . . , N ] , and ( b ) is the result of rearranging C i + j − .Now, we are ready to prove (13). Let ( X i ) Ni =1 be asdefined in the statement of Proposition 2. The LHS of (13)can be written as: E [min( D, N (cid:88) i =1 Z i C ∗ i )] = E [min( D, (cid:88) i Z i C ∗ tot /N )] ( a ) = E [min( D, (cid:88) i Z i (cid:80) j C ∗ j N )] ( b ) = E [min( D, (cid:88) i Z i X i )] ( c ) ≥ N (cid:88) i =1 E [min( DN , X i )] ( d ) = N (cid:88) i =1 E [min( DN , X )]= N E [min( DN , X )]= E [min( D, N (cid:88) i =1 Z i C i )] , (20)where we decompose C ∗ tot into individual C ∗ j ’s in ( a ) anduse definition of X i in Proposition 2. Inequality ( c ) is againbased on Proposition 2 and ( d ) is based on Proposition 3.This concludes the proof. A PPENDIX CP ROOF FOR T HEOREM Suppose that there are N producers in the market, and sup-pose that the optimal capacity from solving (5) is denotedby C ∗ , C ∗ , ..., C ∗ N , we argue that C ∗ , C ∗ , ..., C ∗ N is a Nashequilibrium for the capacity game in (1).We prove the equilibrium for player 1, and the sameargument holds for any of the rest players. To show this, we rewrite C ∗ as the following: C ∗ = arg min C γC + γ N (cid:88) i =2 C ∗ i + E { ( D − N (cid:88) i =2 Z i C i − Z C ) + } = arg min C γC + D − E min { D, N (cid:88) i =2 Z i C i + Z C } = arg min C γC − E min { D − N (cid:88) i =2 Z i C i , Z C } = arg min C γC − E min { ( D − N (cid:88) i =2 Z i C i ) + , Z C }− min E { ( D − N (cid:88) i =2 Z i C i ) − , Z C } = arg min C γC − E min { ( D − N (cid:88) i =2 Z i C i ) + , Z C }− E ( D − N (cid:88) i =2 Z i C i ) − = arg max C E min { ( D − N (cid:88) i =2 Z i C i ) + , Z C } − γC = C (cid:5) , (21)which characterizes the optimal solution to the game de-picted in (1). A PPENDIX DP ROOF FOR T HEOREM Similar to the proof for Theorem 1, we need to show that(13) is true when Z i = ¯ Z + ˆ Z i as given in (10). The proofboils down to show that Proposition 2 and Proposition 3 aretrue under such asssumption on correlation. Note that theproof for Proposition 2 does not require that Z i ’s to be i.i.d.,therefore naturally carries over. To show Proposition 3, weneed Lemma 1. Then these two propositions validate (13),which concludes the proof. Lemma 1. If Z i = ¯ Z + ˆ Z i as (10) , then: f ( z , z , . . . , z N ) = f ( z S , z S , . . . , z S N ) , where S , S , . . . , S N is a permutation of , , . . . , S N , and f ( · ) is the density function.Proof. f ( z , z , . . . , z N )= f (¯ z + ˆ z , ¯ z + ˆ z , . . . , ¯ z + ˆ z N )= (cid:90) ¯ z f ( { ¯ z + ˆ z , ¯ z + ˆ z , . . . , ¯ z + ˆ z N }| ¯ Z = ¯ z ) f ( ¯ Z = ¯ z ) d ¯ z ( a ) = (cid:90) ¯ z f (¯ z + ˆ z , ¯ z + ˆ z , . . . , ¯ z + ˆ z N ) f ( ¯ Z = ¯ z ) d ¯ z ( b ) = (cid:90) ¯ z f (¯ z + ˆ z S , ¯ z + ˆ z S , . . . , ¯ z + ˆ z S N ) f ( ¯ Z = ¯ z ) d ¯ z = f ( z S , z S , . . . , z S N ) , where ( a ) is based on the assumption that ¯ Z is independentof ˆ Z i (assumption A3), and ( b ) is based on Fact 1. A PPENDIX EP ROOF FOR T HEOREMS AND T HEOREM E.1 Berry-Esseen Theorem
The following Lemma is useful to facilitate the proofs ofTheorem 3 and Theorem 5. It relates the behavior of themean of independent random variables to a standard Gaus-sian distribution in terms of CDF.
Lemma 2 (Berry-Esseen Theorem) . There exists a positiveconstant α , such that if X , X , . . . , X N , are independentrandom variables with E ( X i ) = 0 , E ( X i ) = σ > , and E ( | X i | ) = ρ i < ∞ , and if we define S N = (cid:80) i X i √ (cid:80) i σ i , then F N ,the cumulative distribution function of S N is close to Φ , the CDFof the standard Gaussian distribution. This is mathematicallyinterpreted as: | F N ( x ) − Φ( x ) | ≤ αψ, (22) where ψ = ( (cid:80) i σ i ) − max ≤ i ≤ N ρ i σ i . E.2 Some useful lemmas
Before the detailed proof, let us visit some useful propo-sitions and lemmas that assist the proofs for Theorem 3and Theorem 5. In what follows, we assume without loss ofgenerality that given any solution ( C , C , . . . , C N ) , it mustbe the case that C ≤ C ≤ . . . ≤ C N . Proposition 4.
The partial derivative of π N ( C − N , C N ) = E (min( D − (cid:80) N − j =1 C i Z i ) + , C N ) is: ∂π N ( C − N , C N ) ∂C N = E { ( D − N − (cid:88) j =1 C j Z j ) + ≥ C N Z N Z N } , (23) and for all i (cid:54) = N . ∂π N ( C − N , C N ) ∂C i = E { ≤ ( D − N − (cid:88) j =1 C j Z j ) ≤ C N Z N } ( − Z i ) . (24)Based on Proposition 4, we have the following lemmaon the optimality of the symmetric Nash equilibrium of thegame. Lemma 3.
If the invested capacity at Nash equilibrium is sym-metric, i.e., C = C = · · · = C N = C , then: • γ = E (cid:104) (cid:110) ( D − C (cid:80) Nj (cid:54) = i Z j ) + ≥ CZ i (cid:111) Z i (cid:105) , where {·} is the indicator function and takes value 1 if theargument is true, otherwise takes value 0. • N C ≤ E D/γ .Proof.
We begin by proving the first part. Suppose thatcapacities are the same, i.e., C = C = · · · = C N = C ,the profit for each producer in the capacity game is: E min ( D − C N (cid:88) j (cid:54) = i Z j ) + , CZ i − γC. (25)The optimality of a player i in the game is captured asthe following: ∂∂C i E min ( D − C N (cid:88) j (cid:54) = i Z j ) + , C i Z i − γ = 0 . (26) Differentiating in the expectation with respect to each indi-vidual C i and based on Proposition 4, we have: γ = E ( D − C N (cid:88) j (cid:54) = i Z j ) + ≥ CZ i Z i , (27)where {·} is the indicator function and takes value 1 if theargument is true, otherwise takes value 0.Then we proceed to prove the second part, i.e., N C ≤ E D/γ .When there is a social planner making centralized deci-sion as described in (5), the total payment from the electric-ity consumers in the system is min C i , ∀ i =1 , ,...,N E (cid:34) ( D − N (cid:88) i =1 C i Z i ) + (cid:35) + γ N (cid:88) i =1 C i , (28)where each site should have the same optimal investedcapacity C = C = · · · = C N = C as discussed in Section4. This also coincides with the symmetric Nash equilibriumin the capacity game.To show that N C is a bounded by a constant, assumingdifferentiability and based on (28), we know γN = E (cid:34) D ≥ C N (cid:88) i =1 Z i ) N (cid:88) i =1 Z i (cid:35) (29) = E (cid:34) N (cid:88) i =1 Z i ≤ D/C ) N (cid:88) i =1 Z i (cid:35) (30) ≤ E (cid:34) N (cid:88) i =1 Z i ≤ D/C ) D/C (cid:35) (31) ≤ E D/C, (32)rearranging, we get
N C ≤ E D/γ .Based on Lemma 3, we now present two lemmas onarbitrary Nash equilibria of the game in Lemma 4 andLemma 5.
Lemma 4.
Given any equilibrium solution of the two-level game ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) , it must be the case that C (cid:5) ≤ E DγN and C (cid:5) N ≤ E Dγ .Proof. Assume by contradiction that the first part of theabove statement is not true and there exists an equilibriumsolution ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) such that E DγN < C (cid:5) ≤ . . . ≤ C (cid:5) N .Recall the formula for the payment received by producer N ,i.e., π N ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) = E [( D − (cid:80) i (cid:54) = N C (cid:5) i Z i ) + , C (cid:5) N Z N )] .Since this is an equilibrium solution, the derivative of thispayment must equal the investment cost γ . More specif-ically, using the expression for the derivative that waspreviously derived in Equation (27), we have that (cid:18) ddC π N ( C (cid:5) , C (cid:5) , . . . , C ) (cid:19) C = C (cid:5) N = γ = ⇒ E ( D − (cid:88) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N Z N = γ. Now, since the symmetric equilibrium solution ( C ∗ , . . . , C ∗ ) also satisfies this condition, we have that: E ( D − (cid:88) i (cid:54) = N C ∗ Z i ) + ≥ C ∗ Z N Z N = γ. Further, recall that in the symmetric equilibrium C ∗ ≤ E DγN as derived in Equation (32). Since C (cid:5) > E DγN , this impliesthat C (cid:5) > C ∗ . Finally, let E denote the set of events (cid:110) ( D − (cid:80) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N (cid:111) and let E ∗ denote theevents satisfying (cid:110) ( D − (cid:80) i (cid:54) = N C ∗ Z i ) + ≥ C ∗ Z N (cid:111) . Since C ∗ < C (cid:5) ≤ C (cid:5) ≤ . . . ≤ C (cid:5) N , it is not hard to deduce that E ⊂ E ∗ . In-deed for any (non-zero) instantiation ( Z , . . . , Z N ) when ( D − (cid:80) i (cid:54) = N C ∗ Z i ) + > , we have that ( D − (cid:80) i (cid:54) = N C (cid:5) i Z i ) + < ( D − (cid:80) i (cid:54) = N C ∗ Z i ) + and C (cid:5) N Z N > C ∗ Z N . Therefore, we getthat: E ( D − (cid:88) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N Z N < E ( D − (cid:88) i (cid:54) = N C ∗ Z i ) + ≥ C ∗ Z N Z N = γ, which is a contradiction.Next, we prove that at equilibrium C (cid:5) N ≤ E Dγ . Theproof is somewhat similar and once again, proceeds bycontradiction. Suppose that C (cid:5) N > E Dγ . Now, we have: γ = E ( D − (cid:88) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N Z N ≤ E [ { D ≥ C (cid:5) N Z N } Z N ] < E (cid:20) (cid:26) D ≥ E Dγ Z N (cid:27) Z N (cid:21) = E (cid:20) (cid:26) γ ≥ E DD Z N (cid:27) Z N (cid:21) ≤ E (cid:20) (cid:26) γ ≥ E DD Z N (cid:27) D E D γ (cid:21) = 1 E D γ E { (cid:26) γ ≥ E DD Z N (cid:27) D }≤ γ, which is an obvious contradiction. Lemma 5.
Suppose that C i ≤ C j ≤ E Dk for some k ≤ . Then,we have that: E [min( Z j , DC j )] ≤ E [min( Z i , DC i )] . E [min( Z j , DC j )] ≥ qk E [ Z i ] . Proof.
The first part is easy to see. For any instantiation of Z j and D , we have that min( Z j , D/C j ) ≤ min( Z j , D/C i ) since C i ≤ C j . Taking the expectation, and changing the variable
3. For our purposes, an event is a tuple of instantiations of thei.i.d random variables ( Z , Z , . . . , Z N ) and D satisfying the requiredcondition from Z j to Z i (these variables have the same marginaldistribution due to either i.i.d assumption, or follows (10)),we get that: E [min( Z j , DC j )] ≤ E [min( Z j , DC i )] = E [min( Z i , DC i )] . The second part of the lemma can be proved as follows: onceagain fix any instantiation of Z j , we have that: E D | Z j min( Z j , DC j ) ≥ E DD | Z j { D ≥ E D } min( Z j , DC j ) ≥ q min( Z j , k ) ≥ qk min( Z j , qkZ j (33)Taking the expectation, we get the required result.Last, we present a lemma on the bound for integratingon a standard Gaussian distribution. Lemma 6.
Let Φ( · ) denote the CDF for standard Gaussiandistribution, i.e., zero mean and unit variance. Then: Φ( x ) − Φ( y ) ≤ √ π ( x − y ) , x > y. (34)Lemma 6 is a direct observation based on the densityfunction f ( x ) of standard Gaussian random variable, i.e., f ( x ) = √ π e − x which has a maximum value of √ π . E.3 Proof for Theorem 5 and Theorem 3
Now we proceed to prove Theorem 5 and Theorem 3. Notethat Theorem 3 is a special case of 5 when ¯ Z = 0 , in thefollowing proof, we assume that Z i = ¯ Z + ˆ Z i as in (10).To avoid lengthy notation, let us define G i = E [min( Z i , D/C (cid:5) i )] for all ≤ i ≤ N . Consider the paymentreceived by the producer with the smallest investment,which happens to be C (cid:5) . As per Equation (3), this equals: π ( C (cid:5) , . . . , C (cid:5) N ) = π N ( C (cid:5) , . . . , C (cid:5) N ) C (cid:5) G C (cid:5) N G N = π N ( C (cid:5) , . . . , C (cid:5) N ) C (cid:5) E [ Z ] C (cid:5) N G N . (35)The second equation comes from our assumption that N > DminDmax γ . Therefore by Lemma 4, we have that C ≤ D and G = E [min( Z , E [ Z ] . In what follows, wewill continue to use G = E [ Z ] for consistency but re-mark that G is a constant that is independent of C (cid:5) . Thetotal profit made by this producer is π ( C (cid:5) , . . . , C (cid:5) N ) − γC (cid:5) . Since this is an equilibrium solution, we have that (cid:0) ddC π ( C, C (cid:5) , . . . , C (cid:5) N ) (cid:1) C = C (cid:5) = γ . Expanding the differen-tiation term, we get that: G C (cid:5) N G N (cid:32) π N ( C (cid:5) , . . . , C (cid:5) N ) − C (cid:5) E [ ≤ D − (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N Z (cid:33) = γ. (36) In the above equation, we usedthe fact that: ddC π N ( C, . . . , C (cid:5) N ) = E [ (cid:110) ≤ D − CZ + (cid:80) N − i =2 C (cid:5) i Z i ≤ C (cid:5) N Z N (cid:111) − Z ] .Rearranging Equation (36), we get an upper bound for thepayment made to producer N , namely π N ( C (cid:5) , . . . , C (cid:5) N ) = γ C (cid:5) N G N G + C (cid:5) E [ { ≤ D − (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N } Z ] . (37)Fix some constant κ . The rest of the proof proceeds intwo cases:(Case I: (cid:80) N − i =1 C (cid:5) i C (cid:5) N ≤ κN )Intuitively, this refers to the case where the investmentsare rather asymmetric—i.e., the investment by the ‘largerproducers’ is significantly bigger than that by the ‘smallerproducers’. Note that in Equation (37), Z ≤ . Therefore,we get: π N ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ C (cid:5) N G N G + C (cid:5) E [ { ≤ D − (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N } ]= γ C (cid:5) N G N G + C (cid:5) P r ≤ D − (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N ≤ γ C (cid:5) N G N G + C (cid:5) . Recall from Lemma 4 that in any equilibrium solution,we must have that C (cid:5) ≤ E DγN . Substituting this above, weget that π N ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ C (cid:5) N G N G + E DγN .
Next, observe that for any i (cid:54) = N , we can apply Proposi-tion 1 to obtain an upper bound on its profit, namely that: π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ π N ( C (cid:5) , . . . , C (cid:5) N ) C (cid:5) i G i C (cid:5) N G N ≤ (cid:18) γ C (cid:5) N G N G + E DγN (cid:19) C (cid:5) i G i C (cid:5) N G N = γ C (cid:5) i G i G + E DC (cid:5) i G i γN C (cid:5) N G N ≤ γC (cid:5) i + E DC (cid:5) i E [ Z i ] qγN C (cid:5) N γ E [ Z N ] (38) = γC (cid:5) i + E DC (cid:5) i qγ N C (cid:5) N . (39)Equations (38) and (39) were derived using Lemma 5,namely we used the simple properties that ( i ) G i ≤ G , ( ii ) G i ≤ E [ Z i ] , and ( iii ) G N ≥ qγ E [ Z N ] , and finally the factthat E [ Z i ] = E [ Z N ] .Summing up (39) over all i including i = N , we get that N (cid:88) i =1 π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ N (cid:88) i =1 C (cid:5) i + E DN qγ (cid:32) N (cid:88) i =1 C (cid:5) i C (cid:5) N (cid:33) . Of course, as per our assumption, we have that (cid:80) N − i =1 C (cid:5) i C (cid:5) N ≤ κN . Substituting this above, we get that N (cid:88) i =1 π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ N (cid:88) i =1 C (cid:5) i + E DN qγ (cid:16) κN + 1) (cid:17) . ≤ γ N (cid:88) i =1 C (cid:5) i + E Dqγ (cid:18) κN − + 1 N (cid:19) . This proves the theorem statement for the case where (cid:80) N − i =1 C (cid:5) i C (cid:5) N ≤ κN . Note that the N term can be incor-porated into the constant α , without affecting any of theasymptotic bounds.(Case II: (cid:80) N − i =1 C (cid:5) i C (cid:5) N > κN )Let us go back to Equation (36) and consider the term B = E [ (cid:110) ≤ D − (cid:80) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N (cid:111) Z ] . We willnow obtain a tighter upper bound on this quantity condi-tional upon (cid:80) N − i =1 C (cid:5) i C (cid:5) N > κN . First note that applying theCauchy-Schwarz inequality, we can get a lower bound onthe sum-of-squares, i.e., N − (cid:88) i =1 ( C (cid:5) i C (cid:5) N ) ≥ N − N − (cid:88) i =1 C (cid:5) i /C (cid:5) N ) ≥ κ N / N − ≥ κ N . (40)The final simplification comes from the fact that N ≥ (since we assumed N > γ ≥ ). Next, we have that: B ≤ E [ ≤ D − (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N ]= P r ·| D D − C (cid:5) N Z N ≤ (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ D = E D Pr ·| D DC (cid:5) N − Z N ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N Z i ≤ DC (cid:5) N ≤ E D Pr ·| D DC (cid:5) N − ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N Z i ≤ DC (cid:5) N . Using the fact that Z i = ˆ Z i + ¯ Z , we can write out theprobability more explicitly: E D Pr ·| D DC (cid:5) N − ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N Z i ≤ DC (cid:5) N = E D Pr ·| D DC (cid:5) N − ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N ( ˆ Z i + ¯ Z ) ≤ DC (cid:5) N . (41)Let us denote Z = (cid:80) i (cid:54) = N C (cid:5) i C (cid:5) N ¯ Z and Z (cid:48) = (cid:80) i (cid:54) = N C (cid:5) i C (cid:5) N ˆ Z i .From the assumption A3 along with (10) we know that ¯ Z is independent of ˆ Z i , therefore Z is independent of Z (cid:48) .The probability in (41) can be written as an integral over the possible values for Z with f Z ( · ) being the probabilitydensity function for the random variable Z . So, we have: E D Pr ·| D DC (cid:5) N − ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N Z i ≤ DC (cid:5) N = E D Pr ·| D (cid:18) DC (cid:5) N − ≤ Z + Z (cid:48) ≤ DC (cid:5) N (cid:19) ( a ) = E D (cid:90) DC (cid:5) N f Z ( z ) P r (cid:18) DC (cid:5) N − ≤ Z + Z (cid:48) ≤ DC (cid:5) N | Z = z (cid:19) dz ( b ) = E D (cid:90) DC (cid:5) N f Z ( z ) P r { DC (cid:5) N − − z ≤ Z (cid:48) ≤ DC (cid:5) N − z } dz, (42)where ( a ) uses the property of conditional probability, and ( b ) is based on the fact that Z and Z (cid:48) are independent.Since Z (cid:48) = (cid:80) i (cid:54) = N C (cid:5) i C (cid:5) N ˆ Z i , where ˆ Z i ’s are i.i.d. ran-dom variables and E ˆ Z i = ˆ µ , E ( ˆ Z i − E ˆ Z i ) = ˆ σ > , E | ˆ Z i − E ˆ Z i | = ˆ ρ < ∞ , this variable has a mean µ (cid:48) =ˆ µ (cid:80) N − i =1 C (cid:5) i /C (cid:5) N ≥ ˆ µκN . Similarly, the variance of Z (cid:48) canbe written as ( σ (cid:48) ) = ˆ σ (cid:80) N − i =1 ( C (cid:5) i C (cid:5) N ) . Now, applying Equa-tion (40), we get the following lower bound for variance: ( σ (cid:48) ) ≥ σ κ N . (43)Note that C (cid:5) i C (cid:5) N ˆ Z i are independent random variables be-cause ˆ Z i ’s are i.i.d.. What is more, we know that C (cid:5) i C (cid:5) N ˆ Z i hasmean ¯ µ i = C (cid:5) i C (cid:5) N ˆ µ , non negative variance ¯ σ i = ( C (cid:5) i C (cid:5) N ) ˆ σ and finite centered third moment ¯ ρ i = ( C (cid:5) i C (cid:5) N ) ˆ ρ . Denote S N = Z (cid:48) − (cid:80) i ¯ µ i √ (cid:80) i ¯ σ i and let F N denote the CDF of S N . Werewrite the probability of interest as the following: Pr ·| D DC (cid:5) N − − z ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N ˆ Z i ≤ DC (cid:5) N − z = Pr ·| D (cid:18) DC (cid:5) N − − z ≤ Z (cid:48) ≤ DC (cid:5) N − z (cid:19) = Pr ·| D DC (cid:5) N − − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i ≤ Z (cid:48) − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i ≤ DC (cid:5) N − z (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i = F N DC (cid:5) N − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i − F N DC (cid:5) N − − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i . (44)We can now apply the Berry-Esseen Theorem from Lemma 2 to get an upper bound:
P r ·| D DC (cid:5) N − − z ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N ˆ Z i ≤ DC (cid:5) N − z = F N DC (cid:5) N − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i − F N DC (cid:5) N − − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i ( a ) ≤ Φ DC (cid:5) N − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i − Φ DC (cid:5) N − − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i +2 α ( (cid:88) i ¯ σ i ) − max i ¯ ρ i ¯ σ i ( b ) ≤ Φ DC (cid:5) N − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i − Φ DC (cid:5) N − − z − (cid:80) i ¯ µ i (cid:113)(cid:80) i ¯ σ i +2 α (2¯ σ κ N − ) max i ( C (cid:5) i C (cid:5) N ¯ ρ ¯ σ ) ( c ) ≤ √ π (cid:113)(cid:80) i ¯ σ i + 2 α (2¯ σ κ N − )( ¯ ρ ¯ σ ) ( d ) ≤ κ (cid:48) N − , (45)where Φ( · ) is the CDF for standard Gaussian distribution.Inequality ( a ) applies Berry Esseen Theorem to F N ( x ) at x = DC (cid:5) N − z − (cid:80) i ¯ µ i √ (cid:80) i ¯ σ i and x = DC (cid:5) N − − z − (cid:80) i ¯ µ i √ (cid:80) i ¯ σ i . Inequality ( b ) is based on (43). Inequality ( c ) is based on Lemma 6.Inequality ( c ) also depends on the fact that C (cid:5) i C (cid:5) N < , ∀ i .Lastly, ( d ) uses the fact that (cid:80) i ¯ σ i ≤ √ σ κ N from (43),and rewrites the constants into κ (cid:48) for brevity.Plugging the above upper bound back to (42), we getthat: P r DC (cid:5) N − ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N Z i ≤ DC (cid:5) N ≤ κ (cid:48) N − (cid:90) D/C (cid:5) N f Z ( z ) dz ≤ κ (cid:48) N − . (46)If ¯ Z = 0 , then Z i ’s are i.i.d. random variables, then Z = 0 and the bound naturally carries over, as stated inthe following corollary. Corollary 1. If Z i ’s are i.i.d. random variables, then it is a specialcase of correlated Z i ’s in Assumption A3 when ¯ Z , and the upperbound in (45) is valid for i.i.d. Z i ’s, i.e.: P r ·| D DC (cid:5) N − ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N Z i ≤ DC (cid:5) N ≤ κ (cid:48) N − . (47)Now we can complete the proof. Going back to (37), andsubstituting the above upper bound to get that: π N ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ C (cid:5) N G N G + C (cid:5) κ (cid:48) N − ≤ γ C (cid:5) N G N G + E DγN κ (cid:48) N − . (48) Next, we upper bound the aggregate payment made tothe producers using Proposition 1. Recall that G i ≤ G forall i and that G N ≥ γ E [ Z N ] as per Lemma 5. We now getthat: N (cid:88) i =1 π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ N (cid:88) i =1 π N ( C (cid:5) , . . . , C (cid:5) N ) ¯ C i G i C (cid:5) N G N ≤ N (cid:88) i =1 (cid:18) γ C (cid:5) N G N G + DG i γN G N κ (cid:48) N − (cid:19) ¯ C i G i C (cid:5) N G N ≤ γ N (cid:88) i =1 C (cid:5) i G i G + E Dqγ N κ (cid:48) N − N (cid:88) i =1 ¯ C i C (cid:5) N ≤ γ N (cid:88) i =1 C (cid:5) i + E Dqγ N κ (cid:48) N − × N = γ N (cid:88) i =1 C (cid:5) i + E Dqγ κ (cid:48) N − . In the penultimate equation, we used the fact that C (cid:5) i ≤ C (cid:5) N and so, trivially, (cid:80) Ni =1 C (cid:5) i C (cid:5) N ≤ N . This completesthe proof of the second case, and hence, the theorem. A PPENDIX FP ROOF FOR A SYMMETRIC G AMMA
Before stating the main result in this section, we extendLemma 4 to arbitrary investment costs to obtain an upperbound on C (cid:5) . Lemma 7.
Given any equilibrium solution ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) of the two-level game with asymmetric costs ( γ , γ . . . , γ N ) , itmust be the case that C (cid:5) ≤ E Dγ min N .Proof. Assume by contradiction that the above statementis not true and there exists an equilibrium solution ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) such that E Dγ min N < C (cid:5) ≤ . . . ≤ C (cid:5) N .Proceeding similarly to the proof of Lemma 4, we can dif-ferentiate the profit made by the producer with the highestcapacity ( C (cid:5) N ) to get the following: (cid:18) ddC π N ( C (cid:5) , C (cid:5) , . . . , C ) (cid:19) C = C (cid:5) N = γ N ≥ γ min = ⇒ E ( D − (cid:88) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N Z N ≥ γ min . (49)Next, consider a ‘new instance’ of the capacity-pricegame with N providers, all of whom have a symmetricinvestment cost equal to γ min . From Theorem 2, we knowthat instances with symmetric investment costs admit sym-metric equilibrium solutions. Let ( C (cid:48) , . . . , C (cid:48) ) denote thesocially optimal investment strategy for this new instancewith symmetric investment cost γ min , which is also an equi-librium solution. Clearly, for this symmetric equilibrium,the derivative of profit must also equal the investment cost.Therefore: (cid:18) ddC π N ( C (cid:48) , C (cid:48) , . . . , C ) (cid:19) C = C (cid:48) = γ min = ⇒ E ( D − (cid:88) i (cid:54) = N C (cid:48) Z i ) + ≥ C (cid:48) Z N Z N = γ min (50)Further, recall that in the symmetric equilibrium forthe instance where all investment costs are γ min , it mustbe that C (cid:48) ≤ E Dγ min N as derived in Equation (32). Since C (cid:5) > E Dγ min N in our original instance, this implies that C (cid:5) > C (cid:48) . Finally, let E denote the set of events (cid:110) ( D − (cid:80) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N (cid:111) and let E (cid:48) denote theevents satisfying (cid:110) ( D − (cid:80) i (cid:54) = N C (cid:48) Z i ) + ≥ C (cid:48) Z N (cid:111) . Since C (cid:48) < C (cid:5) ≤ C (cid:5) ≤ . . . ≤ C (cid:5) N , it is not hard to deduce that E ⊂E (cid:48) . Indeed for any (non-zero) instantiation ( Z , . . . , Z N ) ,we have that ( D − (cid:80) i (cid:54) = N C (cid:5) i Z i ) + < ( D − (cid:80) i (cid:54) = N C (cid:48) Z i ) + and C (cid:5) N Z N > C (cid:48) Z N . Combining (49) and (50) alongwith the fact that (cid:110) ( D − (cid:80) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N (cid:111) < (cid:110) ( D − (cid:80) i (cid:54) = N C (cid:48) Z i ) + ≥ C (cid:48) Z N (cid:111) , we get that: γ min ≤ E ( D − (cid:88) i (cid:54) = N C (cid:5) i Z i ) + ≥ C (cid:5) N Z N Z N < E ( D − (cid:88) i (cid:54) = N C (cid:48) Z i ) + ≥ C (cid:48) Z N Z N = γ min , which is a contradiction.Now we can proceed to the proof of Theorem 8. Proof.
The proof is very similar to that of Theorem 5, so weonly sketch the new arguments. As with the previous proof,suppose that G i = E [min( Z i , D/C (cid:5) i )] for all ≤ i ≤ N .Since N > DminDmax γ min , we can invoke Lemma 7 to get that C (cid:5) ≤ E D .Consider the payment received by producer whosecapacity happens to be C (cid:5) . As per Equation (3), this equals: π ( C (cid:5) , . . . , C (cid:5) N ) = π N ( C (cid:5) , . . . , C (cid:5) N ) C (cid:5) G C (cid:5) N G N The total profit made by this producer is π ( C (cid:5) , . . . , C (cid:5) N ) − γ C (cid:5) . Since this is an equilibrium solution, we have that (cid:0) ddC π i ( C, . . . , C (cid:5) N ) (cid:1) C = C (cid:5) = γ . Proceeding similarly to theproof of Theorem 4 bearing in mind that G is independentof C , we get that: π N ( C (cid:5) , . . . , C (cid:5) N )= γ C (cid:5) N G N G + C (cid:5) E [ { ≤ D − (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N } Z ] . (51)Fix some constant κ . The rest of the proof proceeds intwo cases:
4. For our purposes, an event is a tuple of instantiations of thei.i.d random variables ( Z , Z , . . . , Z N ) and D satisfying the requiredcondition. (Case I: (cid:80) N − i =1 C (cid:5) i C (cid:5) N ≤ κN )Since, Z ≤ , we can manipulate (51) to get: π N ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ C (cid:5) N G N G + C (cid:5) P r ≤ D − (cid:88) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N ≤ γ C (cid:5) N G N G + C (cid:5) . Recall from Lemma 7 that in any equilibrium solution, wemust have that C (cid:5) ≤ Dγ min N . Substituting this above, we getthat π N ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ C (cid:5) N G N G + E Dγ min N .
Next, observe that for any i (cid:54) = N , we can apply Proposi-tion 1 to obtain an upper bound on its profit, namely that: π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ π N ( C (cid:5) , . . . , C (cid:5) N ) C (cid:5) i G i C (cid:5) N G N ≤ (cid:18) γ C (cid:5) N G N G + E Dγ min N (cid:19) C (cid:5) i G i C (cid:5) N G N ≤ γ C (cid:5) i + E DC (cid:5) i E [ Z i ] γ min N C (cid:5) N γ N E [ Z N ] ≤ γ C (cid:5) i + E DC (cid:5) i γ N C (cid:5) N . (52)Recall that γ N ≥ γ min .Summing up Equation 52 over all i including i = N , weget that N (cid:88) i =1 π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ N (cid:88) i =1 C (cid:5) i + E DN γ (cid:32) N (cid:88) i =1 C (cid:5) i C (cid:5) N (cid:33) . Of course, as per our assumption, we have that (cid:80) N − i =1 C (cid:5) i C (cid:5) N ≤ κN . Substituting this above, we get that N (cid:88) i =1 π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ N (cid:88) i =1 C (cid:5) i + E Dγ (cid:18) κN − + 1 N (cid:19) ≤ γ max N (cid:88) i =1 C (cid:5) i + E Dγ (cid:18) κN − + 1 N (cid:19) . This proves the theorem statement for the case where (cid:80) N − i =1 C (cid:5) i C (cid:5) N ≤ κN .(Case II: (cid:80) N − i =1 C (cid:5) i C (cid:5) N > κN )Let us go back to Equation (51) and consider the term B = E [ (cid:110) ≤ D − (cid:80) i (cid:54) = N C (cid:5) i Z i ≤ C (cid:5) N Z N (cid:111) Z ] . By pro-ceeding identically as we did in the proof of Theorem 5and applying the Cauchy-Scwarz inequality, we obtain thefollowing bound: B ≤ Pr DC (cid:5) N − ≤ (cid:88) i (cid:54) = N C (cid:5) i C (cid:5) N Z i ≤ DC (cid:5) N ≤ κ (cid:48) N − , . (53) where κ (cid:48) is some constant. Now we can complete theproof. Going back to (51), and substituting the above upperbound to get that: π N ( C (cid:5) , . . . , C (cid:5) N ) ≤ γ C (cid:5) N G N G + C (cid:5) κ (cid:48) N − ≤ γ C (cid:5) N G N G + E DγN κ (cid:48) N − . (54)Next, we upper bound the aggregate payment made tothe producers using Proposition 1. Recall that G i ≤ G forall i and that G N ≥ γ min E [ Z N ] as per Lemma 5. We nowget that: N (cid:88) i =1 π i ( C (cid:5) , . . . , C (cid:5) N ) ≤ N (cid:88) i =1 π N ( C (cid:5) , . . . , C (cid:5) N ) ¯ C i G i C (cid:5) N G N ≤ N (cid:88) i =1 (cid:18) γ C (cid:5) N G N G + E DG i γ min N G N κ (cid:48) N − (cid:19) ¯ C i G i C (cid:5) N G N ≤ γ N (cid:88) i =1 C (cid:5) i G i G + E Dqγ
N κ (cid:48) N − N (cid:88) i =1 ¯ C i C (cid:5) N ≤ γ N (cid:88) i =1 C (cid:5) i + E Dγ κ (cid:48) N − . ≤ γ max N (cid:88) i =1 C (cid:5) i + E Dqγ κ (cid:48) N − . In the penultimate equation, we used the fact that C (cid:5) i ≤ C (cid:5) N and so, trivially, (cid:80) Ni =1 C (cid:5) i C (cid:5) N ≤ N . This completesthe proof of the second case, and hence, the theorem. A PPENDIX GP ROOF FOR T HEOREM To prove that there is only one symmetric Nash equilibrium,we need to show that there is a unique C such that C = C = · · · = C N = C which minimizes the total profit in thegame: N E min ( D − C N (cid:88) j (cid:54) = i Z j ) + , CZ j − γCN. (55)The optimality condition on C to minimize this paymentis shown in (27).Showing a unique C maximizes (55) is equiva-lent to showing that the right hand side that in-volves C in (27) has only one intersection with γ , i.e., E (cid:104) (cid:110) ( D − C (cid:80) Nj (cid:54) = i Z j ) + ≥ CZ i (cid:111) Z i (cid:105) is monotonic with re-spect to C . This can be seen from the fact that as C increases, ( D − C (cid:80) Nj (cid:54) = i Z j ) + decreases for each realization of Z j and CZ i increases by each realization of Z i . Therefore the terminside the expectation is monotonically decreasing as C increases. A PPENDIX HP ROOF FOR T HEOREM Proof.
We use a standard equilibrium existence result forgames with continuous strategy spaces originally proposedby Debreu [36]. We refer the reader to [37] for a moreaccessible restatement of the previous result which we willutilize in this proof. In particular, we will prove a slightlystronger claim than the theorem statement, namely thatthere always exists an equilibrium ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) suchthat C (cid:5) N = max { C (cid:5) , C (cid:5) , . . . , C (cid:5) N } —i.e., the maximum ca-pacity investment at equilibrium is by the player with thesmallest investment cost γ N .Let us begin by considering a slightly restricted ver-sion of our original PV game, where we only allow forinvestment strategies C = ( C , C , . . . , C N ) such that C N ≥ C , . . . , C N − —i.e., the feasible strategies of thisgame only include those where PVs { , , . . . , N − } investcapacities smaller than or equal to those by PV N . Recallthat for all i ≤ N , PV i ’s profit is given by: π i ( C ) − γ i C i ,where: π i ( C ) = E (cid:2) min { ( D − (cid:88) j (cid:54) = i Z j C j ) + , Z N C N } (cid:3) E [min( C i Z i , D )] E [min( C N Z N , D )] . (56)Note that the expectations are over the randomness inboth the demand D and the production ( Z i ) Ni =1 . As per thestatement in [37], this restricted version of our game admitsa Nash equilibrium as long as the following conditionsare met: ( i ) The strategy space available to each player isa compact and convex subset of the Euclidean space andis also upper and lower hemicontinuous and non-emptyvalued; ( ii ) π i ( C ) − γ i C i is continuous in C and quasi-concave in C i for all i . It is important to note that the resultby [37] allows for the strategy of PV i to be a function of C − i as is the case in our restricted game.For each player i < N , the strategy space available tothis player i is C ∈ [0 , C N ] . For player N its strategy spaceis [max N − i =1 C i , C max ] , where C max is some finite upperbound on the maximum possible investment by a player.Clearly, each player’s strategy space is convex, compact,continuous, and non-empty valued. It only remains for usto prove that the profit function is continuous and quasi-concave. From (56), it is not hard to observe that all thethree components that make up π i ( C ) are continuous in allits inputs, i.e.,1) E (cid:2) min { ( D − (cid:80) j (cid:54) = i Z j C j ) + , Z N C N } (cid:3) E [min( C i Z i , D )] E [min( C N Z N , D )] are all continuous in the vector of capacities C . Therefore, itfollows that π i ( C ) − γ i C i is continuous in C for all i .To prove quasi-concavity, we need to show that π i ( λC (1) i + (1 − λ ) C (2) i , C − i ) − γ i ( λC (1) i + (1 − λ ) C (2) i ) ≥ min { π i ( C (1) i , C − i ) − γ i C (1) i , π i ( C (2) i , C − i ) − γ i C (2) i } , for arbitrary C (1) i , C (2) i ≤ C N and all λ ∈ [0 , . Withoutloss of generality, fix some C (1) i ≤ C (2) i ≤ C N and λ ∈ [0 , .In order to prove quasi-concavity via the above inequality,it is sufficient to prove that the derivative of the PV profit, i.e., ddC i ( π i ( C i , C − i ) − γ i C i ) is monotonically decreasing in C i . Indeed, suppose that ˜ C i denotes the point where ddC i ( π i ( C i , C − i ) − γ i C i ) C i = ˜ C i = 0 . Then, the derivative being monotonically non-increasingimplies that the function π i ( C i , C − i ) − γ i C i is increasingfrom C i = 0 to C i = ˜ C i and decreasing from C i = ˜ C i to C i = C N . This naturally implies quasi-concavity. In orderto prove the required condition on the derivative, considerdifferentiating π i ( C ) with respect to C i . This gives us: ddC i π i ( C i , C − i ) = 1 C N G N (cid:16) π N ( C ) ddC i E [min( C i Z i , D )] − E [min( C i Z i , D )]) E [ { ≤ D − (cid:88) i (cid:54) = N C i Z i ≤ C N Z N } Z i ] (cid:17) , where G N = E [min { Z N , DC N } ] . In the above differentia-tion, we used (56) to obtain the expression for π i ( C ) andProposition 4 for the expression ddC i π N ( C ) . The fact that ddC i ( π i ( C i , C − i ) − γ i C i ) is non-increasing in C i comes fromthe observations below:1) π N ( C ) = E [min(( D − (cid:80) j (cid:54) = N C i Z i ) + , C N Z N )] isnon-increasing in C i .2) ddC i E [min( C i Z i , D )] = E [ { D ≥ C i Z i } Z i ] is non-increasing in C i .3) E [min( C i Z i , D )]) is non-decreasing in C i and there-fore, − E [min( C i Z i , D )]) is non-increasing in thesame argument.4) E [ (cid:110) ≤ D − (cid:80) i (cid:54) = N C i Z i ≤ C N Z N (cid:111) Z i is non-decreasing in C i and thus, its negative is non-increasing in the same argument.To sum up, we have shown that ddC i π i ( C ) and therefore, ddC i ( π i ( C ) − γ i C i ) is non-increasing in C i , which in turnimplies quasi-concavity. We can then apply the equilibriumexistence result from [36], [37] to show that the restrictedversion of our original PV game where player N has thehighest capacity investment always admits a Nash equilib-rium. To complete the proof, we need to show that the Nashequilibrium for this restricted game is also an equilibriumfor our original PV game with no restrictions on the playerstrategies. Suppose that C (cid:5) = ( C (cid:5) , C (cid:5) , . . . , C (cid:5) N ) denotes anequilibrium for the restricted game and assume by contra-diction that this is not an equilibrium for the original game.Then, only one of two possibilities can occur: ( i ) either thereexists player i < N , who can strictly improve its profit byincreasing its capacity investment to C (cid:48) i > C (cid:5) N or ( ii ) player N can strictly increase its profit by lowering its capacityinvestment to C (cid:48) N < max i (cid:54) = N C (cid:5) i .Consider the first case. We will prove that if there existssuch a player i , then it must also be true that player N can increase its capacity investment and consequentially, itsprofit which contradicts the fact that C (cid:5) is an equilibriumfor the restricted game. If such an improving strategy existsfor player i , it must definitely be the case that the righthand derivative of its profit at C i = C (cid:5) N must be strictlypositive—this is because the derivative of player i ’s profit is monotonically non-increasing in C i when C i ≥ C (cid:5) N . That is,we have that: ddC i ( π i ( C i , C (cid:5)− i )) C i =( C (cid:5) N ) + = ddC i ( E [min(( D − (cid:88) j (cid:54) = i C (cid:5) j Z j ) + , C i Z i )]) C i = C (cid:5) N > γ i . Note that the derivative in the right hand side aboveequals − E { ≤ ( D − (cid:88) j (cid:54) = i C (cid:5) j Z j ) ≤ C (cid:5) N Z i } Z i by Proposition 4. However, since γ i ≥ γ N , this in turnimplies that ddC N ( π N ( C N , C (cid:5)− N )) C N =( C (cid:5) N ) = − E { ≤ ( D − (cid:88) j (cid:54) = N C (cid:5) j Z j ) ≤ C (cid:5) N Z N } Z N > γ N . Recall by our assumption that Z N and Z i are identicallydistributed conditional on ( Z j ) j (cid:54) = i,N . In words, this meansthat player N can also increase its capacity and its profit.This is of course a contradiction of the fact that C (cid:5) is anequilibrium for the restricted game.In the second case, suppose that player N can lowerits capacity and improve profits. The proof in this case isquite analogous to the previous case. Indeed, if player N canlower its capacity to C (cid:48) N < max j (cid:54) = N C (cid:5) j and increase profits,then one can use monotonicity arguments similar to beforeto show that the player i = arg max j (cid:54) = N C (cid:5) j can also lowerits capacity investment and improve profits. Once again, acontradiction of the fact that C (cid:5) is an equilibrium for therestricted game. This completes our proof that there existsa Nash equilibrium C (cid:5)(cid:5)