Equilibria on the Day-Ahead Electricity Market

Equilibria on the Day-Ahead Electricity Market Margarida Carvalho INESC Porto, Portugal Faculdade de Ciências, Universidade do Porto, Portugal argarida.carvalho@dcc.fc.up.pt João Pedro Pedroso INESC Porto, Portugal Faculdade de Ciências, Universidade do Porto, Portugal jpp@fc.up.pt Technical Report Series: DCC-01-06 Departaento de Ciência de Coputadores Faculdade de Ciências da Universidade do Porto Rua do Capo Alegre, 101/1055, 4169-007 PORTO, PORTUGAL Tel: 0 40 900 Fax: 0 40 950 http://www.dcc.fc.up.pt/pubs/

Equilibria on the Day-Ahead Electricity Market Margarida Carvalho a,b, João Pedro Pedroso a,b a Faculdade de Ciências, Universidade do Porto, Rua do Capo Alegre, 4169-007 Porto, Portugal b INESC TEC, Rua Dr. Roberto Frias 378, 400-465 Porto, Portugal Abstract In the energy sector, there has been a transition fro onopolistic to oligopolistic situations pool arkets; each tie ore copanies optiization revenues depend on the strategies of their copetitors. The arket rules vary fro country to country. In this work, we odel the Iberian Day-Ahead Duopoly Market and find exactly which are the outcoes Nash equilibria of this auction using gae theory. Keywords: Duopoly, Nash Equilibria, Day-Ahead Market 1. Introduction Over the last years the way electricity is produced and delivered has changed considerably. Market echaniss were ipleented in several countries and electricity arkets are no longer vertically integrated. Nowadays, in any countries the electricity arkets are based on a pool auction to purchase and sell power. Producing copanies offer electricity in a arket and the buyers subit acquisition proposals. An electricity pool arket is characterized by a single price arket clearing price for electricity paid to all the proposals accepted in the arket. The way in which the arket clearing price is deterined varies fro one country to another: the last accepted offer, the first rejected offer, ultiple unit Vickey see Anderson and Xu 004, Son et al. 004 and Zieran et al. 1999. The Iberian arket uses the last accepted offer echanis, which sees to provide a copetitive price and an appropriate incentive for investent and new entry. Indeed, ost pool arkets use the last accepted block rule see Son et al. 004. To analyze the copanies behavior in the electricity arket, gae theory has been used as a generalization of decision theory see Singh 1999. The concept of Nash equilibriu NE for a gae is used as a solution. The NE leads to the strategies that axiize the copanies profit see for exaple Hasan and Galiana 010, Hobbs et al. 000 and Son Eail addresses: argarida.carvalho@dcc.fc.up.pt Margarida Carvalho, jpp@fc.up.pt João Pedro Pedroso

and Baldick 004. This eans that in a NE nobody has advantage in oving unilaterally fro it. As stressed in Baldick 006, for a odel to be tractable it ust abstract away fro at least soe of the detail. Solvable electricity arket odels do not use, for exaple, transition constraints. First it is necessary to understand the effect changes have on arket rules or structures. In Anderson and Xu 004, the Australian power arket is considered in detail, as we will do here for the Iberian case. There are soe crucial differences between the forulation in Anderson and Xu 004 and ours, such as the pool auction structure and the deand shape. In this context, the authors of Son et al. 004 analyze the arket equilibria with the first rejected echanis and the pay-as-bid pricing. Here, the aount supplied by each generator is a discrete value; this also differs fro our odel. In Lee and Baldick 003 an electricity arket with three copanies is forulated, and the space of strategies is discretized in order to find a NE. Recently, in an attept to predict arket prices and arket outcoes, ore coplex odels have been used. However, any ties that does not allow the use of analytical studies. Thus, techniques fro evolutionary prograing see Barforoushi et al. 010 and Son and Baldick 004 and atheatical prograing see Hobbs et al. 000 and Pereira et al. 005 have been used in these new odels. In our Iberian arket duopoly odel, we do not consider network constraints. We will provide a detailed application of non cooperative gae theory in our forulation of the electricity arket. To the best of our knowledge, such theoretical treatent has not been considered before. In our forulation, deand elasticity will be a paraeter as this is a realistic approach that only has been considered in the siulation of arkets, but not in a theoretical way. Apart of being the first detailed approach of the Iberian arket, it points out the existence of NE in pure strategies in all the instances of this gae, showing how the NE are conditioned by capacity, production costs and elasticity paraeters. Soe potential properties about the oligopoly case are also highlighted. In Section, the Iberian duopoly arket odel will be presented and the concept of NE will be foralized. In Section 3, soe rearks will be ade which allow us to find the NE in a constructive way. Section 4 concludes this detailed NE classification.. Iberian Duopoly Market Model In this section we will begin by explaining our odel and fixing notation. Then, gae theory will be introduced with the ai of giving us tools to analyze this arket. We set up a odel for a gae with two producing firs, which will represent the players, labeled as Fir 1 and Fir. It is assued that each Fir i owns a generating unit with arginal cost c i and capacity E i > 0. Both firs subit siultaneously a bid to the arket, using a pair q i, p i S i = 0, E i ] 0, b] for Fir i, where q i is the proposal quantity, p i is the bid price and S i is the space of strategies. Fir i s payoff Π i q 1, p 1, q, p is a function that depends on the strategic choices of the rival fir and its own. The deand is a segent P = Q + b characterized by the real constants < 0 and b > 0. It is assued that deand, the firs arginal costs and capacities are known 3

EQUILIBRIA ON THE DAY-AHEAD DUOPOLY MARKET 3 b = 1. $/MWh 0.6 Fir 3 supply curve Pd = 0.4 0. Fir Fir 1 MWh 100 Qd = 160 190 40 Figure.1. Dispatch Figure.1: Econoic Dispatch. Iberian Duopoly Market Model In this section we will begin by explaining our odel and fixing notation. Then, gae theory will be introduced with the ai of giving us tools to analyse this arket. We set up a odel for a gae with two producing firs, which will represent the players, labelled as Fir 1 and Fir. It is assued that each Fir i owns a generating unit with arginal cost ci and capacity Ei > 0. Both firs subit siultaneously a bid in the arket, through a pair qi, pi Si = 0, Ei] 0, b] for fir i, where qi is the proposal quantity, pi is the bid price and Si is the space of strategies. Fir i s payoff Πi q1, p1, q, p is a function that depends on the strategic choices of the rival fir. The deand is a segent P = Q + b characterized by the real constants < 0 and b > 0. It is assued that the deand and firs arginal costs and capacities are known by all agents. These assuptions can be justified by the knowledge of inforation about the technology available for each fir, fuel costs, and precise forecasts of deand. Without loss of generality let c1 < c < b. Once, the arket operator has the firs proposals and the deand, it finds the intersection between the deand representation and the supply curve, which gives the arket clearing price Pd and quantity Qd. For exaple, suppose that = 1, b = 1. and Fir 1 00 bid 90, 0.4, Fir bid 100, 0. and Fir 3 bid 60, 0.6. The Market Operator organizes the proposals by ascending order of prices which gives the supply curve, see Figure.1. Then, the Pd and Qd are found. The accepted bids are the ones in the left side of Qd. by all agents. These assuptions can be justified by the knowledge of inforation on the technology available for each fir, fuel costs, and precise deand forecasts. Without loss of generality c 1 < c < b. Once, the arket operator has the firs proposals and the deand, it finds the intersection between the deand representation and the supply curve, which gives the arket clearing price P d and quantity Q d. For exaple, suppose that = 1, b = 6 and 00 5 Fir 1 bid 90, 0.4, Fir bid 100, 0. and Fir 3 bid 60, 0.6. The Market Operator organizes the proposals by ascending order of prices which gives the supply curve, see Figure.1. Then, P d and Therefore Q d the are revenue found. of fir i is given The by: accepted bids are the ones in the left side of Q d. Therefore the revenue of Fir i is given by: Π i = P d c i g i where g i q i is the accepted quantity in the econoic dispatch. Note that with this structure linear deand and linear production cost the firs profit is concave and thefore the optiu strategies are an extree point or a stationary point. As a tie breaking rule, we proportionally divide the quantities proposed by the firs declaring the sae price, in case the total quantity is not fully required. If the deand segent intersects the supply curve in a discontinuity, all the proposals with prices below the intersection are fully accepted and the arket clearing price is given by the last accepted proposal. Now we are able to define a NE in pure strategies. Definition 1. In a gae with n players a point s NE = s NE 1, s NE,..., s NE n, where s NE i specifies a strategy over the set of strategies of player i, is a Nash Equilibriu if i {1,,..., n}: Π i s NE Π i si, s NE i s i S i where Π i is the utility of player i, S i is his space of strategies and s NE i 4 is s NE except s NE i.

Our task is to classify all the NE in this odel when players choose their actions deterinistically. This is a hard task because the classical approach of reaction functions is inadequate, since the payoff functions are non-continuous. During this work, besides the NE, it was observed that ɛ-equilibria are also likely to be arket outcoes. Definition. In a gae with n players a point s NEɛ = s NEɛ 1, s NEɛ,..., s NEɛ n, where s NEɛ i specifies a strategy over the set of strategies of player i, is an ɛ-equilibriu if i {1,,..., n} and for a real non-negative paraeter ɛ: Π i s NE ɛ Πi si, s NEɛ i ɛ si S i. In our duopoly case, ɛ is infinitesial and just one of the firs has an ɛ advantage in changing its strategy. Thus, we also found ɛ-equilibria. 3. Nash Equilibria Classification The goal of this section is to describe Nash equilibria that ay arise in the duopoly case, characterizing Nash equilibria in ters of the paraeters: c 1, c, E 1, E, and b. For that purpose, we first eliinate soe cases where there cannot be NE. Recall the assuption c 1 < c < b. Lea 3. In the duopoly arket odel no Nash equilibriu in pure strategies has a tie in the proposals prices, which eans that p 1 = p. Proof. In the tied case both firs bid p 1 = p = P d which iplies Π tied Q d 1 = P d c 1 q 1 and Π tied = P d c q 1 + q q new 1 = q Q d q 1 + q with q 1 + q > Q d. Note that if q 1 + q = Q d, the firs proposals are totally accept. If P d < c 1, then Π tied 1 < 0 an thus, Fir 1 has incentive to change its strategy to p 1 = c 1, since its payoff will increase to zero. If P d = c 1 < c, then Π tied < 0, thus Fir has incentive to change its behavior as Fir 1 did in the previous case. If P d > c 1, Fir 1 has stiulus to choose p new 1 < P d = p and { E 1 if E 1 < P d b P d b ε otherwise with ε > 0 infinitesial. The reason is that with this new choice Fir 1 has an accepted quantity g 1 = q1 new Pd b Q > q d 1 q 1 +q = q 1 q 1 +q and the arket clearing price is aintained, up to an infinitesial quantity. Therefore, revenue of Fir 1 increased. Since there is always at least one fir that benefits fro changing its behavior unilaterally, this cannot be an equilibriu. 5

Note that this lea can be easily generalized for the oligopoly case, as long as the arginal costs are different for all firs. We have just eliinated fro the possible set of NE the cases with a tie in prices. Another particular situation occurs when the deand intersects the supply curve in a discontinuity. This possibility can also be discarded fro the potential NE. Lea 4. In the duopoly arket odel, a Nash equilibriu in pure strategies always intersects the supply curve. Proof. Let us prove the lea by contradiction. In the duopoly arket odel, suppose that there is an equilibriu such that the deand curve intersects the supply curve in a discontinuity. It suffices to note that the fir with the last proposal being accepted takes advantage increasing the price of this proposal unilaterally up to the intersection point, because the arket clearing price increases and the arket clearing quantity is aintained. Therefore, this leads to a contradiction, since it was assued that there was an equilibriu. Potential equilibria will have: Fir 1 onopolizing the arket with P d = p 1 < p = c, Fir 1 deciding P d = p 1 > p or Fir deciding P d = p > p 1. Fir never onopolizes the arket since it is the less copetitive copany c > c 1. The proposition below suarizes the interesting strategies in an equilibriu, that is, the potential equilibria. In the following sections we will evaluate under which conditions they are an equilibriu. Proposition 5. In the duopoly arket odel the equilibria have 1. Fir 1 deciding the arket clearing price and both firs producing; this eans P d = p 1 > p, in which case Fir plays the largest possible quantity, as long as p 1 reains greater than c, and Fir 1 ay play: a the duopoly optiu q 1 1 b q, p 1 = c 1+b+q stationary point; b the duopoly optiu q 1 = E 1, p 1 = E 1 + q + b extree point;. Fir 1 onopolizing; which eans P d = p 1 < p = c and in this case Fir 1 ay play: stationary point; a the onopoly optiu q 1 1 b, p 1 = c 1+b b the onopoly optiu q 1 = E 1, p 1 = E 1 + b extree point; c a price close to Fir s arginal cost q 1 ε b,, p 1 = c ε or equivalently q 1 = c b ε, p 1 < c for ε > 0 arbitrary sall; ; 3. Fir deciding the arket clearing price and both firs producing; this eans P d = p > p 1, in which case Fir 1 plays the largest possible quantity, as long as P d = p, and Fir ay play: a the duopoly optiu q b q 1, p = c +b+q 1 stationary point; b the duopoly optiu q = E, p = E + q 1 + b extree point. 6

Proof. Let us start with the siplest case: Consider an equilibriu with P d = p 1 < p = c. The optial strategy in the onopoly case is: p1 b axiize Π 1 q 1, p 1 = p 1 c 1 A stationary point is at subject to p 1 0, b] Π 1 p 1 = 0 p 1 = c 1 + b with Π 1 < 0, q p 1 p 1 b = c 1 b E 1 1 and c 1+c < c. When E 1 < c 1 b the onopoly optiu is an extree point: q 1 = E 1 and p 1 = E 1 + b. Therefore, Fir 1 s best strategy is one of the just derived if c > E 1 + b > c 1+b. Otherwise, for Fir 1 to onopolize, it has to bid a price below c. In this case, the strategy with the highest payoff is q 1 ε b, p 1 = c ε, for ε arbitrarily sall. To ake this bid Fir 1 s capacity ust satisfy E 1 ε b. Equivalently, Fir 1 could bid q 1 b ε, p 1 < c, allowing Fir to decide P d = c, but Fir s participation in the arket would be as sall as ε. 1 Considerer an equilibriu with P d = p 1 > p. In this case, Fir is playing a quantity bid as high as possible, as long as p 1 > c. Now, we just have to see which is the best strategy for Fir 1 when P d = p 1 > p which iplies q < p 1 b : p1 b axiize Π 1 q 1, p 1, q, p = q p 1 c 1 A stationary point is at Π 1 p 1 subject to p 1 0, b] = 0 p 1 = c 1 + b + q which iplies q 1 q = c 1 b q or if E 1 < c 1 b q p 1 = E 1 + q + b extree point. The bid quantity for Fir akes sense since once P d = p 1 is fixed Fir s profit increases with q. 3 Copletely analogous to the above case. p 1 b In Figure 3.1, the above proposition is suarized. Using this proposition, we will copute the conditions in which the above equilibria exist; this eans that neither of the firs will have advantage in unilaterally oving fro the chosen strategies. Note that g 1 = c 1 b and p 1 = c 1+b is the onopoly optiu. Therefore, we will use these two values to start our equilibria classification. Figure 3. represents the initial division of the space of paraeters which will allow us to start a classifying the equilibria that ay occur in this arket. Before the coputation of equilibria, we prove the following theore. 7

54 CHAPTER 4. IBERIAN MARKET MODEL Potential equilibria Fir 1 onopolizes Both firs participate in the arket EQUILIBRIA ON THE DAY-AHEAD DUOPOLY MARKET 7 Paraeters c1, c, E1, E,, b Case a: The onopoly optiu is a stationary point. Case b: The onopoly optiu is an extree point. Fir 1 decides Pd = p1 > p Fir decides Pd = p > p1 E1 < c1 b E1 1 b Case c: The duopoly optiu is a stationary point. Case d: The duopoly optiu is an extree point. Case e: The duopoly optiu is a stationary point. c1+b Both firs participate in the arket c1+b < c Fir 1 produces at full capacity c1+b Fir becoes copetitive c1+b < c Fir 1 onopolizes Figure 4.3: Potential equilibria. Figure 3.1: Potential equilibria. which iplies q1 p 1 b c1 b q q = or if E1 < c1 b q p 1 = E1 + q + b extree point. The bid quantity for Fir akes sense since once Pd = p 1 is fixed Fir s profit increases with q. Strategies 3.4. Fir 1: s NE 1 = Figure 3.1. Decision tree Figure 3.: Decision tree. ] c1 b q1, E1, c1 + b Theore 3 Copletely analogous 6. to There the above case. is always an equilibriu in the or Iberian duopoly arket odel. The equilibriu is a Nash equilibriu or an ɛ-equilibriu with ɛ infinitesial. 3.1 Fir : s NE = q 0, E], p c, b] 3. Fir : s NE = 0, p 0, b] 3.3 In Figure 4.3, the above proposition is suarized. With these bids, the arket clears with price Pd = c1+b < c and quantity Qd = c1 b. Fir 1 is at onopoly s optiu, and Fir has no influence in the arket. Using this proposition, we will copute the conditions in which the above equilibria 3.. Fir 1 produces at full capacity: E1 < Proof. exist; this eanssuppose that neither of the that firs will there have advantage is aninalgorith unilaterally ovinga which given the proposals firs c1 b q and 1, c1+b p 1 <, c. q, p, fro the chosen strategies. Recall fro the last section that g1 = c1 b c1+b and p1 = is was able the onopoly to output optiu. Thus the we will strictly use these twobest values toreaction start our equilibria for each of Letthe. us start by considering Therefore, E1 + b < Ac, 1 ie, qfir 1, p cannot 1, q enter, p in the = arket with classification. Figure 4.4 represents the initial division of the space of paraeters positive profit. Note that c b c1 b < E1 <. In this case, 1 will onopolize the electricity arket, although its capacity is less than the optiu c1 b. The Nash equilibriu q which 1, p will 1 and A q 1, p 1, q, p = q allow us to start a classification of the equilibria that, p ay are the bids that axiize the profit for Fir 1 and occur in this is given by: Fir arket., respectively, given q 1, p 1, q, p. Strategies 3.5. Our goal is to prove that if we iteratively apply algorith A for Firsoe 1: s NE 1 = initial E1, E1 + b proposals, 3.4 it will find a fixed point of A. This is A 1 q 1, p 1, q, p = q 1, p 1 Fir and : seeaequation q 1 3., por 1 3.3, q, p = q, p, which is an equilibriu of the gae. Let the initial bids be q 1, p 1, q, p = E 1, p 1, E, c, where p 1 is the best bid price for Fir 1 such that it is lower than c. Reeber, as entioned in Proposition 5, that in an equilibriu the firs bid their entire production capacity. Applying A 1 E 1, p 1, E, c, we can obtain: 1. A 1 E 1, p 1, E, c = E 1, p 1, eaning that Fir 1 is already aking its best proposal according to Fir s strategy. Now, we apply A E 1, p 1, E, c to see if Fir has advantage in increasing its price bid. a If A E 1, p 1, E, c = E, c, then E 1, p 1, E, c is a fixed point of A and thus it is an equilibriu. Case of Proposition 5 describes this situation. b If A E 1, p 1, E, c = E, p, where p is equal to the one described in case 3 of Proposition 5, then E 1, p 1, E, p is an equilibriu. Fir 1 does not have advantage in changing fro p 1 to the price of case 1 in Proposition 5, since if it has, Fir 1 had done that in the previous step.. A 1 E 1, p 1, E, c = E 1, p 1, where p 1 is equal to the one of case 1 in Proposition 5. Fir 1 had advantage in increasing its price bid to p 1. Coputing A E 1, p 1, E, c we obtain: 8

a A E 1, p 1, E, c = E, c. Fir keeps its bid which iplies that E 1, p 1, E, c is an equilibriu. b A E 1, p 1, E, c = E, p, where p is equal to the one of case 3 in Proposition 5. Since A, and in particular A, only changes to bids that strictly increase profit, we can conclude for this case that p 1 < p. By Proposition 5, p 1, is the best strategy to Fir 1 when it decides on P d and Fir is biding a price lower than p 1. Therefore, A 1 E 1, p 1, E, p = E 1, p 1 which eans that E 1, p 1, E, p is an equilibriu. In AppendixI the existence of equilibria is proven, but using a erge of the cases presented in the following sections and algebraic arguents. 3.1. Fir 1 onopolizes: E 1 1 b and c 1+b < c Fir 1 onopolizes the arket, which eans that its capacity is high enough, and its arginal cost low enough, to keep Fir out of the arket. This is the case a of Figure 3.1. In this case, the Nash equilibriu is given by: Strategies 7. ] Fir 1: s NE c1 b 1 = q 1, E 1, c 1 + b 3.1 or Fir : s NE = q 0, E ], p c, b] 3. Fir : s NE = 0, p 0, b] 3.3 With these bids, the arket clears with price P d = c 1+b < c and quantity Q d = c 1 b Fir 1 is at the onopoly s optiu, and Fir has no influence on the arket. 3.. Fir 1 produces at full capacity: E 1 < c 1 b and c 1+b < c We are going to have cases b, e and d of Figure 3.1 as equilibria. Let us start by considering that E 1 + b < c, eaning that, Fir cannot enter the arket with positive profit. Note that c b < E 1 < c 1 b. In this case, Fir 1 will onopolize the electricity arket, although its capacity is lower than the optiu c 1 b. The Nash equilibriu is given by:. Strategies 8. Fir 1: s NE 1 = E 1, E 1 + b 3.4 Fir : see Equations 3. or 3.3 9

As before, Fir 1 does not have an incentive to change its strategy, as this will ean that q 1, p 1 = E 1, E 1 + b is not an optiu. Let us now consider E 1 + b. Fir 1 cannot onopolize the arket and, in this case, the arginal cost of Fir is lower than the onopoly price E 1 + b, i.e., E 1 + b. Moreover, in an equilibriu for this case, Fir 1 never decides the price, that is., P d = p > p 1. Otherwise, by Proposition 5, if P d = p 1 > p, Fir 1 would be bidding the duopoly optiu price P d = p 1 = c 1+b+q requiring: p 1 = c 1 + b + q > c but this would iply a negative quantity for Fir, which is absurd: q < c c 1 b < 0. The inequality c c 1 b < 0 is equivalent to c > c 1+b, which holds by assuption. Therefore, we discarded the possibility of Fir 1 deciding P d = p 1 > p case c of Figure 3.1. The cases d and e in Figure 3.1 reain as potential NE, which we will discuss below. Suppose that Fir 1 s bidding price is p 1 < c p = P d. The best reaction for Fir is the duopoly optiu in Proposition 5: p = c +b+q 1 and the corresponding quantity is q b q 1. Indeed p : p = c + b + q 1 + b + E 1 E 1 + b therefore, the bid price p akes sense, since p > c. Furtherore, Fir 1 bids q1 = E 1 and p 1 < c, otherwise it would have advantage in increasing its quantity and this would not be an equilibriu. Fir does not have incentive to change its ove q b E 1, p = c +b+e 1, as this is the optiu stationary point when p 1 < c. In order to have an equilibriu, neither of the firs can benefit fro unilaterally changing their behavior. When Fir decides P d = p > p 1, this fir will not benefit fro changing its behavior, because that would ean decreasing the price p, p 1 > p, and this does not increase Fir s profit, since p 1 is lower than c. May Fir 1 be encouraged to reconsider its strategy? In other words, would Fir 1 be interested in increasing price p 1? The answer is no, because as we have seen at the beginning: if Fir 1 picks the price P d = p 1 > p it will be p 1 = c 1+b+q but p 1 < c p. Therefore, now we just have to distinguish the duopoly optiu as a stationary point and the duopoly optiu as an extree point: In the case E b E 1, Fir s optiu is a stationary point. As we assue E 1 + b > c, Fir 1 is not onopolizing, and the Nash Equilibriu is given by 10

Strategies 9. Fir 1: s NE 1 = E 1, p 1 0, c 3.5 ] Fir : s NE c b E 1 = q, E, c + b + E 1 3.6 For E 1 + b = c and ε, ε > 0 arbitrary sall, the equilibriu is given by Strategies 10. Fir 1: s NE 1 = E 1 ε, p 1 0, c 3.7 Fir : s NE = q ε, E ], c 3.8 In the case E < c b E 1 both firs bid at full capacity. Here, Fir does not have enough capacity to produce the quantity c b E 1 required, so q = E and p = q 1 + E + b. The best reaction quantity for Fir 1 is q1 = E 1 as before. Fir is playing q = E, p = E 1 + E + b which, by construction, is the best reaction when Fir 1 plays q 1, p 1 = E 1, p 1 < c. These strategies are a Nash equilibriu. Strategies 11. Fir 1: s NE 1 = E 1, p 1 0, E 1 + E + b] 3.9 Fir : s NE = E, E 1 + E + b 3.10 Clearly, we can invert the prices of each fir, reaching the Nash equilibriu: Strategies 1. Fir 1: s NE 1 = E 1, E 1 + E + b 3.11 Fir : s NE = E, p 0, E 1 + E + b] 3.1 The conclusions of this section are suarized in the decision tree of Figure 3.3. 3.3. Fir becoes copetitive: E 1 1 b and c 1+b In this case, we will have cases c, b and e of Figure 3.1 as equilibria. 11

E 1 < c1 b c1+b and < c E 1 + b < c E 1 + b Fir 1 onopolizes the arket see the NE 3. to 3.4 Both Firs participate in the Market E < c b E1 E b E1 see the NE 3.9, 3.10, 3.11 and 3.1 E 1 + b > c E 1 + b = c see the NE 3.5 to 3.6 see the NE 3.7 to 3.8 Figure 3.3: Equilibria for Section 3.. Figure 3.3: Equilibria for Section 3.. 3.3.1. Fir 1 decides P d = p 1 > p We are interested in finding under which conditions case c in Figure 3.1 is a NE. Clearly p 1 > c and therefore, we assue p = c. Under the results of Proposition 5, Fir 1 ust be bidding the stationary point p 1 = c 1+b+q and q 1 1 b q. This requires p 1 = c 1+b+q and thus, we ust have q c c 1 b. Siilarly, the quantity produced 8 by Fir 1 c 1 b q ust be positive. This leads to the inequality q c 1 b, which is weaker than the forer one, since b > c > c 1. The best strategy for Fir is: q = { c c 1 b ε if E c 1 b E otherwise. 3.13 Note that the quantity q entioned above is positive as long as c c 1+b, which holds. Therefore, Fir is playing the largest possible quantity, as stated in Proposition 5. For the sake of siplicity, let us first consider the case of Fir 1 playing q1 = c 1 b q, rather than q1 > c 1 b q. Hence, q 1 c 1 b which by assuption is lower than E 1 and therefore, it akes sense to bid this quantity at price p 1. For the profile of strategies s 1 = q1, p 1 and s = q, c to be an equilibriu, it is required that neither fir changes its strategy. If Fir has an incentive to choose another strategy it would be p > p 1. The question now is whether there is a strategy s = q, p such that Π q, p, q1, p 1 > Π q, p, q1, p 1. When Fir increases the price, the best strategy is to choose p = 1

c +b+q 1 = b+c 1+c q 4. In order to have p > p 1, the following inequality ust hold q > c c 1 b. 3.14 3 Inequality 3.14 depends on our instance of the proble. Notice that c c 1 b > c c 1 b, 3 so Inequality 3.14 holds whenever E > c c 1 b. Our goal is to see under which conditions 3 Fir does not change the price to p. In this context, if E c c 1 b then q = E and Fir does not have advantage in 3 picking a strategy different fro E, c. Suppose E > c c 1 b 3 that is larger than b + c1 + c q = 4 ; then p = b+c 1+c q > p 4 1 = c 1+b+q c1 + b + q Π q, c, q 1, p 1 = p b Π q 1, p, q1, p 1 = c q c b + c1 + c q 4b 4. Can this be the case c 1 b q? The answer is no. This proof falls naturally, see in AppendixA. Consequently, we are only interested in the case where q = E c c 1 b. 3 Now, we have to study under which conditions Fir 1 does not have advantage in picking p 1 < p = c. If E 1 < c b then Fir 1 bids p 1 < c and q 1 = E 1. Otherwise, if E 1 b, Fir 1 bids p 1 < c and q 1 = c b ε or p 1 = c ε and q 1 ε b. Obviously in the second case E 1 b the Fir 1 s profit is higher, so we use it to copare with its profit of our possible NE: Π 1 q1, p c1 + b + E c1 b E 1, E, c = c 1 = b + E c 1. 4 So when E 0, in Π 1 q 1, p 1, E, c li ε 0 Π 1 q 1, p 1, E, c see AppendixB. Thus, when Equation 3.15 holds and E 1 b Strategies 13. c 1 b + ] c 1 c c b, c c 1 b 3 we have the equilibriu: ] Fir 1: s NE c1 b E 1 = q 1, E 1, c 1 + b + E 13 3.15 3.16

E 1 1 b c1+b and Fir 1 decides P d = p 1 > p E 1 < c b E 1 b Fir 1 never onopolizes the arket Fir 1 can onopolize the arket E 1 b+c c 1 9c 1 c E 1 < b+c c 1 9c 1 c c b+4c 1 5 c < b+4c 1 5 A B B < A A B A > B E A E > A E > B E B E A E > A E > B E B see the NE 3.16 to 3.17 No NE under these conditions see the NE 3.16 to 3.17 No NE under these conditions see the NE 3.16 to 3.17 Figure 3.4: Fir 1 decides P d = p 1 > p. Figure 3.4: Fir 1 decides P d = p 1 > p. Fir : s NE = E, p 0, c ] 3.17 Note that we have relaxed Fir s biding price. This is possible because the profits do not depend on it. When E 1 < c b and Equation 3.18 holds and we also have the above equilibriu see 10 Equations 3.16 and 3.17. c 1 b + ] E 1 c 1 c E 0, in, c c 1 b 3.18 3 Let A = c 1 b+ E 1 c 1 c, A = c 1 b+ c 1 c c b and B = c c 1 b. We built the 3 decision tree in Figure 3.4. In order to verify that all the decisions in the tree ake sense, that is, that all the regions in its leafs are non-epty, let us observe the following: c 1 b b+c c 1 9c 1 c when c c 1+b ; b+c c 1 9c 1 c < c b when c ] 4c 1 +b, c 1+b 5. This supports the fact that the decision tree akes sense or, in other words, that decisions do not lead us to epty spaces. Reeber, that we used q1 = c 1 b q. In this case, the capacity of Fir had to be lower than c c 1 b, otherwise, Fir would change its strategy with benefit. 3 14

It could be possible to find ore general conditions for the Nash equilibriu with Fir 1 deciding P d = p 1 > p. We did not try the strategy q1 > c 1 b q. Note that with bids s 1, s = q1, p 1, q, p = q1 > c 1 b q, c 1+b+q, q, c, Fir 1 only produces the quantity g 1 = c 1 b q. How- ever Fir 1 ay play a quantity q 1 larger than g 1, in order to reduce Fir s incentive in changing the price p. So, our goal is to find a lower bound for q1 when E c 1 b. 3 Let us see how larger q 1 has to be. If Fir increases the price c to p = c +b+q 1 this iplies: c + b + q1 > p 1 = c 1 + b + q q1 < c 1 c and the new quantity dispatched should be positive: + q < c b c +b+q1 b q1 > 0 q1 < c b therefore q1 < c 1 c + q is the strongest condition until now. We still have to add the condition that leads Fir to a higher profit is larger than Π q1, p 1, p b q 1, p c + b + q = 1 Π q 1, p 1, q, c = q c 1 + b + q c c c b q 1 when q1 < c b+ K, see AppendixC. If q 1 b+ K Fir does not have advantage in changing its strategy. Now, we erely need the conditions for Fir 1 to keep the strategy q1, p 1. Proceeding as before: 1. Let E 1 b. Fir 1 does not change its strategy q 1 > c b+ K, p 1 if q 0, c 1 b + c b c 1 c ] c 1 b c b c 1 c,. Note that and c 1 b c b c 1 c c b c 1 c }{{} <0 c 1 b + c b c 1 c > c c 1 b < c c 1 }{{} >0 < c c 1 b 15

thus The Nash equilibriu is given by: c + c 3c 1 + b c 1 b c 1 > 0 c ] c 1, c 1 + b q = E c 1 b + c b c 1 c. Strategies 14. Fir 1: s NE 1 = q 1, c 1 + b + E c b + ] K with q 1, E 1 3.19 Fir : see Equation 3.17. Let E 1 < c b. Fir 1 does not change its strategy if q 0, c 1 b + E 1 c 1 c ] c 1 b E 1 c 1 c,. Note that: and c 1 b E 1 c 1 c c 1 b + E 1 c 1 c > c c 1 b < c c 1 b c c 1, c 1 E 1 ] and c 1 E 1 > c 1 c 1 b = c 1 + b c 1, c ] 1 + b c 1, c 1 E 1 ] thus for q = E < c 1 b + E 1 c 1 c we have the Nash equilibriu of 3.19 and 3.17. The decision tree is in Figure 3.5. 16

E 1 1 b c1+b and and E c1 b 3 Fir 1 decides P d = p 1 > p E 1 < c b E 1 b E A E 1 b+ K E > A E 1 < c b+ K E > A E A See the NE 3.19 and 3.17 No NE under these conditions No NE under these conditions see the NE 3.19 and 3.17 4. Conclusions 5. Acknowledgents Figure 3.5: Fir 1 decides P d = p 1 > p. Figure 3.5: Fir 1 decides P d = p 1 > p. 3.3.. Fir 1 onopolizes This work was supported by a INESC Porto fellowship in the setting of the Optiization Now we areinterunit goingline. to establish the conditions that ake Fir 1 onopolize the arket as an equilibriu case b in Figure 3.1. Here, the arket outcoe is an ɛ-equilibriu. AppendixA. Fir changes its strategy Notice that in this case, p 1 < c otherwise, Fir would have advantage in participating in the arket and it is required E Assue that E 1 1 b, c1+b and E > c c1 b 3, and that Fir 1 is playing q1 = c1 b q, p 1 = c1+b+q. By Equation 3.13, since E > c c1 b 3 then q 1 b c 1 b < c 1 b, otherwise the deand is > c c1 b 3. Let us prove that Fir will change not intersecteditsby strategy Firq, p1 s = cbid. to q Hence = p b the q 1, p = best bid b+c1+c q for 4 : Fir 1 is q 1, p 1 = c ε b, c ε with ε > 0. Fir bids q, p Π= qe, p, q, c 1, p. 1 < ΠAny q, p other, q1, p strategy fro Fir leads to the 1 sae or less profit. q c 1 + b + q c < 1 16 b + c 1 c q q c b c 1 3 for this reason, Fir has benefit in odifying its strategy. 1. Consider E b. If 9 Fir 1 chooses p q + 3q c 1 + b c + 1 1 > c c 1 +, the b c produced quantity is zero, and < 0 therefore, Fir 1 will not choose to ake a bid different fro p 1 = c ε. Hence the equilibriu is: Strategies 15. Fir 1: s NE 1 = q1 1 c ε b ], E 1, c ε 3.0 or Strategies 16. Fir : s NE = E, c 3.1 Fir 1: s NE c b 1 = ε, p 1 c 1, c ε] 3. Fir : see Equation 3.1 17

. Consider E < c b. Then Fir 1 can bid p 1 > c, producing a non zero quantity. If Fir 1 has incentive to change its strategy it will be to: c1 b E q 1, p 1 =, c 1 + b + E. However, for this new bid to ake sense, we need p 1 = c 1 + b + E > c, which depends on the instance of our proble. Reark: c c 1 b < c b c > c 1. Using this the following cases are possible. a Consider E < c c 1 b p 1 = c 1+b+E > c. Is p1 b Π 1 E c1 + b + E c1 b E, p 1, E, c = c 1 higher than li ε 0 Π 1 c ε b c b, c ε, E, c = c c 1 Not when Equation 3.3 holds see AppendixD for a proof. c 1 b + ] c 1 c c b E, c c 1 b. 3.3 Hence, the equilibriu is given by: Strategies 17. Fir 1: see Equation 3.0? or Strategies 18. Fir : see Equation 3.1 Fir 1: see Equation 3. Fir : see Equation 3.1 b Consider E c 1 b p 1 = c 1+b+E c. In this case, Fir 1 does not have stiulus to change its behavior and hence, the equilibriu is given by Equations 3.0 to 3.. Let B = c c 1 b Since, with the assuptions of this section, c b tree akes sense.. We have the decision tree of Figure 3.6 corresponding to this case. A yields, the decision 18 c 1 b

E 1 1 b c1+b and Fir 1 onopolizes E b See the NE 3.0 to 3. E 1 b E c1 b E < c b E 1 < c b No NE under these conditions E < c c1 b See the NE 3.0 to 3. E A See the NE 3.0 to 3. E < A No NE under these conditions Figure 3.6: Fir 1 onopolizes. AppendixB. Fir 1 does not change its strategy Assue that E 1 1 b, c1+b, E c c1 b 3 and E 1 b, and that Fir is playing q = E, p = c. Let us prove in which conditions Fir 1 does not change its strategy q 1 = c1 b E, p 1 = c1+b+e to q1 = c ε b, p 1 = c ε : c1 b+ c bc1 c Figure 3.6: Fir 1 onopolizes. 3.3.3. Fir decides P d = p > p 1 Finally, we will search for equilibria where Fir decides the arket clearing price, in Π 1 q1, p 1, q, p li Π 1 q 1, p 1, q ε 0, p other words, p 1 < p = P d case d and e of Figure 3.1. b + E c 1 Note that we have to ipose E c b 1 < c b, due to the fact that if E li c c 1 4 ε 0 ε 1 b, Fir 1 has incentive to onopolize the arket 1 since it has sufficient b c 1 + E b c 1 + E 4 capacity c for that purpose. So, b c c 1 > 0 we consider c 1 b E 1 < c b. E 1 4 E 1 b c 1 1 4 b c 1 c b Let us assue p 1 < c. Fro Proposition 5, the best c strategy c 1 = for 0 Fir is p = c +b+q1 and q b q1, where q 1 = E 1. E = c 1 b ± c b c 1 c For this strategy there are the following requireents the price of the duopoly optiu In what follows it is study this solution: akes sense and Fir has the production capacity of playing the stationary point of the c1 b c bc1 c > c1 b+ c bc1 c < ; duopoly optiu: 1. p = c +b+e 1 E 1 c b, which holds by our assuption;. E b E 1 < 0 c 1 c c b > b c1 c 1 c c b > b c1 4 c +c c 1 + b c 1 b b c1 4 > 0, note that c +c c 1 + b c 1 b b c1 4 = 0 c = c1+b thus, 0 c1 b+ c bc1 c c1 b c bc1 c which depends on the instance of our proble. We have to study each of these cases. ; 1. Suppose E b E 1, then Fir plays the stationary optiu q b E 1, p = c + b + E 1 14 and Fir 1 E 1, p 1 < c. Obviously Fir will not have incentive in reconsidering another proposal, but Fir 1 ay have advantage in choosing a higher price p 1 = c 1+b+q > p, such that Π 1 q1, p 1 < c, q, p < Π p1 b 1. Is p1 = c 1+b+q c +b+e 1? q, p 1 > p, q, p 19 > p =

c 1 + b + q > c + b + E 1 q < c c 1 + E 1 which depends on the instance of our proble. Thus, these cases have to be considered separately: a Suppose c b E 1 c 1 +E 1 E 1 c 1 c b. However, E 3 1 1 b c 1 c b c 3 1+b, which by assuption does not hold. So this case never happens. b Suppose E 1 > c 1 c b and q 3 = c b E 1, then Fir 1 has benefit in increasing its proposal price since: p1 b Π 1 q, c 1 + b + q, q, p > Π 1 E 1, p 1 < c, q, p 1 c Suppose E 1 > c 1 c b 3 1 16 c 1 c b + E 1 > c + b + E 1 c 1 E E 1 c 1 b c. 3, then as we already did before, there is and q > c b E 1 an equilibriu if E 1 b+ K 1, where K 1 = E1+ c b + 4c 1 E 1 : Strategies 19. Fir 1: see Equation 3.5 Fir : s NE c1 b + ] K 1 = q, E, c + b + E 1 3.4. Suppose E < c b E 1. In this case q 1, p 1 = E 1, p 1 < c and q, p = E, E 1 + E + b. Fir will not change this behavior, so let us see when Fir 1 has advantage in increasing p 1 to p 1. For the purpose we need and E > c 1 b p 1 = c 1 + b + E > p = E 1 + E + b E > c 1 b E 1 E 1 is true, since E > 0 and c 1 b E 1 0 E 1 1 b. So, p 1 = c 1+b+E > p = E 1 + E + b. Is p1 b Π 1 E, c 1 + b + E, E, p higher than = 1 4 c 1 + b + E Π 1 E 1, p 1 < c, E, p = E 1 E 1 + E + b c 1? Yes, see AppendixE. Therefore Fir 1 has stiulus in changing its strategy. In short, we can suarize this Nash equilibria with the decision tree of Figure 3.7. 0

E 1 1 b c1+b and Fir decides P d = p > p 1 E 1 b+ K1 E b E1 E 1 < c b No NE under these conditions E < c1 b+ K1 E < c b E1 E 1 b No NE under these conditions See the NE 3.5 and 3.4 No NE under these conditions Figure 3.7: Fir decides P d = p > p 1.. Suppose E < c b E1. In this case q1, p 1 = E 1, p 1 < c and q, p = E, E 1 + E + b. Fir will not change this behavior, so let us see when Fir and 1 has c 1+b advantage in increasing p 1 to p 1. For the purpose we need p 1 = c 1 + b + E > p = E 1 + E + b E > c 1 b < c E 1 b and E > c1 b E 1 is true, since E > 0 and higher than Π 1 E 1, p 1 < c, E, p = E 1 E 1 + E + b c 1? Yes, see Appendix AppendixE. Therefore Fir 1 has stiulus in changing its strategy. In short, we can suarize this Nash equilibria with the decision tree of Figure 3.7. 4. Conclusions 5. Acknowledgents Figure 3.7: Fir decides P d = p > p 1. 3.4. Both firs participate in the arket: E 1 < c 1 b The NE of this section are going to be the ones fro cases c, d and e of Figure 3.1. Since E 1 < c 1 b, Fir 1 does not have capacity to onopolize the arket. c 1 b E 1 0 E 1 1 b 3.4.1. Fir decides P d = p > p 1. We start with the case So, p 1 in which P = c1+b+e > p = E 1 d = + E p > p + b. 1 case e of Figure 3.1. As Proposition 5 Is p1 b Π 1 E, c states, Fir plays the stationary duopoly 1 + b + E optiu, E, p = 1 q, p c b q1 4 c 1 + b =, c +b+q1 and + E q1 will be as large as possible such that: and p = c + b + q1 q1 c b q = c b q1 0 q1 c b This work was supported by a INESC Porto fellowship in the setting of the Optiization Interunit Line. > c b thus q1 = E 1. Note that q > 0, since q = c b E 1 c 1 b = c b c 1 > 0 c 4 4 < b+c 1. An iportant fact is that p = c +b+e1 > c, since this is equivalent to E 1 < c b which is true. On the other hand q E depends on the instance of our = c b E 1 15 proble, so Fir ay have to play the extree point of the duopoly optiu. 1. Suppose E b E 1. Fir 1 plays q 1 = E 1, p 1 < c and Fir plays q = c b E 1, p = c +b+e It is easy to see that Fir does not have advantage in choosing other strategy. Let us see if Fir 1 will change its behavior to p 1 = c 1+c +b E 1. In that case, p 4 1 ust be higher than p : p 1 = c 1 + c + b E 1 4 > c + b + E 1 = p E 1 > c 1 c b 3 and c 1 c b 3 c 1 b, which depends on the instance of our proble. 1

a Let E 1 c 1 c b. Then, Fir 1 does not have stiulus in changing its behavior 3 unilaterally. Here, the Nash equilibriu is given by: Strategies 0. Fir 1: see Equation 3.5 b Let E 1 > c 1 c b 3 Fir : see Equation 3.6 p 1 = c 1+c +b E 1 > c +b+e 1 = p 4. We have: Π 1 p1 b q, p 1, q, p which is larger than the profit = 1 16 c + b c 1 E 1 Π 1 E 1, p 1 < c, q, p = E 1 c + b + E 1 c 1 if E 1 c 1 b c. Therefore, Fir 1 has advantage in changing its strategy. 3 However, like we already did in the Section 3.3.3, Fir can pick q > c b E 1 sufficiently large such that Fir 1 does not have advantage in changing its strategy. This case is copletely analogous to the one treated in Section 3.3.3, let K 1 = E1 + c + 4c 1 be 1. If E 1 b+ K 1 we have the Nash equilibriu: Strategies 1. Fir 1: see Equation 3.5 Fir : see Equation 3.4. Suppose E < c b E 1. Here Fir s duopoly optiu is an extree point, q = E, p = E + E 1 + b and q 1 = E 1. Notice that p = E + E 1 + b > c, since both firs are playing saller quantities than the last case the arket clearing price increases when the arket clearing quantity decreases. Clearly, Fir will not change its strategy, but Fir 1 ay have incentive in choosing a higher price p 1 > p and that requires: p 1 = c 1 + b + E > p = E 1 + E + b E 1 > c 1 b E which depends on the instance of our proble. a Let E 1 c 1 b E. Then, Fir 1 will not change its behavior. The Nash equilibriu is given by: Strategies. Fir 1: see Equation 3.9 Fir : see Equation 3.10

E 1 < c1 b c1+b and Fir decides P d = p > p 1 E b E1 E < c b E1 E 1 c1 c b 3 See the NE 3.5 to 3.6 E < c1 b+ K1 E 1 > c1 c b 3 E 1 b+ K1 E 1 > c1 b E No NE under these conditions E 1 c1 b E See the NE 3.9 to 3.10 No NE under these conditions See the NE 3.5 and 3.4 Strategies 1. b Let E 1 > c 1 b E. Is higher than Figure 3.8: Fir decides P d = p > p 1. Figure 3.8: Fir decides P d = p > p 1. Fir 1: see Equation 3.9 Fir : see Equation 3.10 b Let E 1 > c1 b E. Is p1 b Π 1 E, c 1 + b + E, E, p higher than p1 b Π 1 E, c 1 + b + E, E, p This is equivalent to solve: This is equivalent to solving: Π 1 E 1, p 1 < c, E, E 1 + E + b? Π 1 E 1, p 1 < c, E, E 1 + E + b? 1 4 c 1 + b + E < E 1 + E + b c 1 E 1 1 E1 + c 1 b E E 1 1 4 c 1 b + E < 0 4 c 1 + b + E < E 1 + E + b c 1 E 1 E 1 c 1 b E, so Fir 1 has advantage in changing its strategy. E1 + c 1 b E E 1 1 4 c 1 + b + E < 0 Fro the above we reach the decision tree of Figure 3.8. 4. Conclusions E 1 c 1 b E, 5. Acknowledgents so Fir 1 has advantage in changing its strategy. This work was supported by a INESC Porto fellowship in the setting of the Optiization Interunit Line. Fro the above we reach the decision tree of Figure 3.8. 17 3.4.. Fir 1 decides P d = p 1 > p Fir 1 wants to play the stationary point q1, p 1 = requires: c1 b q, c 1+b+q, which { c c 1 b c 1 + b + q thus q ε E c 1 b = E otherwise the instance of our proble. > c q < c c 1 b. Let us note that: c 1 b q E 1 depends on 3

1. Suppose E c 1 b, then q = c c 1 b ε. a Let E 1 1 b q = c 1 c ε. Hence, Fir 1 can play q 1, p 1 and Fir can play c c 1 b ε, c. Will Fir 1 decrease the price p 1 < c? This eans: Π 1 E 1, c 1, q, c = c c 1 E 1 li ε 0 Π 1 q 1, p 1, q, c c c 1 E 1 1 c 1 c E 1 1 c. Since E 1 1 b q = c 1 c ε, Fir 1 will change its behavior. b Let E 1 < c 1 c then q 1 = E 1, p 1 = E 1 + q + b and { c b q = E 1 ε E b E 1 E otherwise. If E b E 1, Fir will change its strategy as with the present one, its profit is alost zero and increasing p fro c to p = c +b+e 1 > c E 1 < c b 1 which holds by assuption, Fir s profit is higher: c 4 + b + E 1 > 0. Hence, let E < c b E 1 that iplies q = E. In this case, it is easy to see that Fir 1 does not have advantage in changing its strategy: Π 1 E 1, E 1 + E + b, E, c Π 1 E 1, p 1 < c, E, c E 1 + E + b c 1 E 1 c 1 E 1 E c b E 1 which holds. Now, we are going to see under which conditions Fir does not have incentive to change its behavior. First of all, if Fir changes its strategy, it will be with p = c +b+e 1, requiring p > p 1 = E 1 + E + b E > c b E 1 which depends on our instance. If E > c b E 1, Fir chooses this new strategy, see AppendixF. If E c b E 1, we have the NE: Strategies 3. Fir 1: see Equation 3.11 Fir : see Equation 3.1 4

. Suppose E < c c 1 b, which iplies q = E. a Let E 1 1 b E. So, Fir 1 can play q 1, p 1 = q1 1 b E, c 1+b+E. Fir 1 does not have advantage in decreasing the price to p 1 < c when E < c 1 b+ E 1 c 1 c, see AppendixG. Finally, we have to check if Fir has advantage in increasing the price p > p 1. In that case, p = c +b+q1 which requires: p = c + b + q 1 > p 1 = c 1 + b + E q 1 < c 1 c + E. Note that c 1 c + E c 1 b E E c c 1 b. In this way, if E 3 c c 1 b q 3 1 1 c + E and we have the NE: Strategies 4. Fir 1: see Equation 3.16 Fir : see Equation 3.17 Let E > c c 1 b c 1 c +E 3 > c 1 b E, thus q 1 need to be q1 < c 1 c +E p = c +b+q1 > p 1 = c 1+b+E. Will this new proposal price for Fir increases its profit? If E 1 > c b+ K, the answer is no see AppendixI and thus we have the following NE: Strategies 5. Fir 1: s NE c b + ] K 1 = q 1, E 1, c 1 + b + E 3.5 Fir : see Equation 3.17 b Let E 1 < c 1 b E, then q 1 = E 1, p 1 = E 1 + q + b and { c b q = E 1 ε E b E 1 E otherwise. If E b E 1 then E 1 b E. Since E 1 < c 1 b E, then c b E < c 1 b E E > c c 1 b and we previously assued E < c c 1 b. Thus E < c b E 1, iplies q = E. Fir 1 does not change its strategy, since: Π 1 E 1, E 1 + E + b, E, c Π 1 E 1, p 1 < c, E, c E + E 1 + b c 1 E 1 Π 1 = c c 1 E 1 E c b E 1 5

E 1 < c1 b c1+b and Fir 1 decides P d = p 1 > p E c 1 b E 1 < c 1 c E 1 1 c No NE under E < c b E 1 E b E 1 No NE these conditions E b E 1 No NE under these conditions See the NE 3.11 and 3.1 E < c b E 1 under these conditions Figure 3.9: Fir decides P d = p 1 > p. Figure 3.9: Fir 1 decides P d = p 1 > p. 4. Conclusions and the last inequality holds. 5. Acknowledgents Fir does not change its strategy note that c +b+e 1 > E 1 + E + b This work E > was c bsupported E 1 by a INESC Porto fellowship in the setting of the Optiization when E = c b E 1, since Interunit Line. Π E 1, E 1 + E + b, E, c AppendixA. Fir changes its strategy Assue that E 1 1 b, c1+b and > c c1 b 3, and that Fir 1 is playing q1 = c1 b q, p 1 = c1+b+q Π. E 1, E 1 + E + b, p b By Equation 3.13, since E > c c1 b 3 then q > c c1 b E 1, c + b + E 1 3. Let us prove thatfir will change its strategy q, p = c to q = p b q 1, p = b+c1+c q 4 : E + E 1 + b c E 1 Π q, p, q1, p 1 < Π q, p, q1, 4 E 1 + b c p 1 q c 1 + b + q c < 1 16 b + c 1 c q E = c b E 1 9 q + 3q c 1 + b c + 1 c 1 + b c < 0 Thus, for E c b E 1, we have the NE: q c b c Strategies 6. 1 3 for this reason, Fir has benefit in odifying Fir 1: itssee strategy. Equation 3.11 AppendixB. Fir 1 does not change its strategy Fir : see Equation 3.1 Assue that E 1 1 b, c 1+b, E c c1 b 3 and E 1 b, and that Fir is Proceeding playing q as= before, E, p = we c. have Let us the prove decision in which tree conditions of Figures Fir 1 does not change its strategy q 1 = c1 b E, p 1 = c1+b+e to q1 = c ε b, p 1 = c ε 3.9 and 3.10. : In AppendixI there are the trees with all the possible equilibria in pure strategies. Π 1 q1, p 1, q, p li Π 1 q 1, p 1, q ε 0, p 6 0

Fir 1 decides P d = p 1 > p E c 1 b E 1 < c 1 c E 1 1 c No NE under E < c b E 1 E b E 1 No NE these conditions E b E 1 No NE under these conditions See the NE 3.11 and 3.1 E < c b E 1 under these conditions Figure 3.9: Fir decides d 1. Figure 3.9: Fir 1 decides P d = p 1 > p. 4. Conclusions 5. Acknowledgents E 1 < c1 b c1+b and This work was supported by a INESC Porto fellowship in the setting of the Optiization Interunit Line. Fir 1 decides P d = p 1 > p AppendixA. Fir changes its strategy Assue that E 1 1 b, c1+b and E > c c1 b 3, and that Fir 1 is playing E By Equation 3.13, 1 < since c 1 b E E > c c1 b then q > c c1 b E 1 1 b E its strategy q, p = c to E b E 1 No NE under these conditions See q the NE 3.11 and 3 q = p b 3 q 1, p = b+c1+c q 4 Π q, p, q1, p 1 < Π q, p, q1, p 1 E < c b E 1 E < c c 1 b E > c 1 b+ E 1 c 1 c c 1 + b + q c < 1 16 b + c 1 c q No NE under these conditions 3.1 9 q + 3q c 1 + b c + 1 c 1 + b c E c c 1 b < 0 3 q c b c 1 3 for this reason, Fir has benefit in odifying its strategy. q 1 = c1 b q. Let us prove that Fir will change : See the NE 3.16 to 3.17 E 1 < c b+ K E c 1 b+ E 1 c 1 c E > c c 1 b 3 E 1 b+ K, p 1 = c1+b+q. AppendixB. Fir 1 does not change its strategy No NE under these See the NE 3.5 and Assue that E 1 1 b, c 1+b, E c c1 b 3 and E 1 b, and that Fir is playing q = E, p = c. Let Figure us prove in which conditions Fir 1 does not change its strategy q 1 = c1 b E, p 1 = c1+b+e 3.10: decides to q1 = c ε b, p P d = 1 = c ε p 1 > p. Figure 3.10: Fir 1 decides P d = : p 1 > p. Π 1 q 1, p 1, q, p li ε 0 Π 1 q 1, p 1, q, p 0 0 conditions 3.17 7

4. Discussion and conclusions In the Iberian duopoly arket odel, the deand and the production costs are linear. As Proposition 5 suggests, there are five types of Nash equilibria. Instances where Fir participates in the arket with infinitesial quantity ε will be considered as a onopoly for Fir 1. When Fir 1 onopolizes the arket in an equilibriu, the selected prices ay be c 1 +b, E 1 + b or c ε, depending on the efficiency/copetitiveness of Fir. For a high arginal cost c, Fir 1 bids the onopoly optiu: p 1 = c 1+b or p 1 = E 1 + b. Furtherore, for c closer to c 1, if Fir s capacity E is large enough, Fir 1 onopolizes bidding p 1 = c ε; otherwise, for a liited capacity E 1, Fir 1 ay prefer to bid a higher price, sharing the arket with Fir. For c even closer to c 1, the arket clearing price in an equilibriu ay be decided by either one of the firs. Fir 1 decides P d, which eans P d = p 1 > p, if the capacity of Fir is liited. The case of Fir deciding P d, eaning P d = p > p 1, is analogous. In soe copetitive situations, there are two NE: one case with P d = p 1 > p and another with P d = p > p 1. By siulating rando instances it was observed that Fir 1 has a high profit in the equilibriu P d = p > p 1, while Fir benefits when P d = p 1 > p and there is no rational way of deciding aong the. We have discretized the space of strategies S 1 and S to copute NE in ixed strategies in these cases. It was observed that a cobination of these two equilibria leads to a new NE in ixed strategies. However, the new equilibriu does not benefit either of the firs coparatively to the pure NE. In conclusion, this work copletely classified the NE that ay occur in a duopoly day-ahead arket. This helps understand the sensitivity of the outcoes to the instances paraeters and the diversity of equilibria that ay arise. Furtherore, it illustrates the rational strategies of each fir. 5. Acknowledgents This work was supported by a INESC TEC fellowship in the setting of the Optiization Interunit Line. References Anderson, E. J., Xu, H., 004. Nash equilibria in electricity arkets with discrete prices. Matheatical Methods of Operations Research 60, 15 38, 10.1007/s001860400364. URL http://dx.doi.org/10.1007/s001860400364 Baldick, R., 9 006-nov. 1 006. Coputing the Electricity Market Equilibriu: Uses of arket equilibriu odels. In: Power Systes Conference and Exposition, 006. PSCE 06. 006 IEEE PES. pp. 66 73. Barforoushi, T., Moghadda, M., Javidi, M., Sheikh-El-Eslai, M., nov. 010. Evaluation of Regulatory Ipacts on Dynaic Behavior of Investents in Electricity Markets: A 8