LEARNING OLIGOPOLISTIC COMPETITION IN ELECTRICITY AUCTIONS

Computational Intelligence, Volume 23, Number 2, 2007

LEARNING OLIGOPOLISTIC COMPETITIONIN ELECTRICITY AUCTIONS

ERIC GUERCI, STEFANO IVALDI, MARCO RABERTO, AND SILVANO CINCOTTI

DIBE-CINEF, University of Genoa, Genoa, Italy

This paper addresses the problem of auction markets efficiency within the context of recently liberalizedelectricity markets. Two different auction mechanisms, i.e., the uniform and the discriminatory price setting rules,have been employed worldwide in designing electricity markets. In this paper, we study the relative efficiencyof the two auction mechanisms in the framework of the learning-in-games approach. The behavior of electricitysuppliers are modeled by means of an adaptive learning algorithm and the demand is assumed to be constant andinelastic, according to a common hypothesis in electricity market modeling. Computational experiments resultsare interpreted according game theoretical solutions, i.e., Nash equilibria and Pareto optima. Different economicscenarios corresponding to a duopoly and a tripoly competition with different level of demand are considered.Results show that in the proposed conditions, sellers learn to play competitive strategies, which correspond to Nashequilibria. Finally, this study establishes that, in the presented computational setting and economic scenarios, thediscriminatory auction mechanism results more efficient than the uniform auction one.

Key words: electricity market, agent-based computational economics, game theory, adaptive learning,oligopolistic theory, design economics.

1. INTRODUCTION

In the last decade, wholesale power markets have been characterized by a progressiveliberalization around the world. The deregulation of the electricity market is assumed to beable to bring competition to the previously monopolistic market. A more competitive marketsetting should give incentives for the production of electricity by means of more efficient gen-erating units. These are wholesalers or generating companies viewed as portfolio generatorswith different market shares and power plant technologies. Sellers are usually in a reducednumber and the repeated interaction among them can produce tacit collusive behavior. Thus,oligopolistic scenarios may arise. Conversely, the demand side is mainly characterized bybeing inelastic with respect to price, thus exhibiting significantly regular seasonal patternsdue to within-day and within-week differences in consumption. Further seasonal cycles arepresent in the medium and long term. In this context, the choice of the market mechanismplays a crucial role to increase market efficiency. This paper addresses auction policy designvia a computational approach. The feasibility should guarantee a powerful instrument to testcomplex auction mechanisms where many sellers with nonlinear cost functions compete inthe market. Several restructuring designs for the electric power industry have been alreadyproposed and implemented for real markets. The main goal is to increase the overall marketefficiency. Currently, the sale and purchase of electric power is performed by means of bilat-eral contracts or through organized markets, i.e., power exchanges (PEs). Power exchangesare characterized by a complex trading procedure, which involves different subsequent marketsessions. The first session is called day ahead market (DAM) because it occurs the day beforedispatching electricity and is commonly considered as the spot market. Successive sessionsregard adjustment and ancillary services markets. It is worth remarking that, in the presentpaper, only the DAM session is considered. In this context, trading is organized according to asealed-bid clearinghouse double-auction mechanism, which collects sale and purchase offersof electricity and determines the market clearing price. The goal of this first market session isto establish the hourly market price of electricity and to provide an indication for the hourly

Address correspondence to Silvano Cincotti at DIBE-CINEF, University of Genoa, Via Opera Pia 11a, 16145 Genoa, Italy;e-mail: [email protected]

C© 2007 Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK.

198 COMPUTATIONAL INTELLIGENCE

unit commitment plan for the next day. The DAM simultaneously and separately collects allsubmitted bids (offers to buy) and asks (offers to sell) for every hour of the following day.Twenty-four clearinghouses operate independently to set market prices for the 24 hours ofthe next day, matching buyer and seller offers. Subsequent market sessions (also online) op-erate to guarantee the quality of services, such as security requirements and no transmissionconstraints. The main reason for this complex market mechanism is that electricity is nota typical commodity as it cannot be stored. Thus generation and consumption have to bebalanced in real time.

Specific market mechanisms must be studied and determined properly before their ap-pliance. In the last decade, an intense activity of research has been carried on studying andmodeling deregulated electricity market. In particular, attention has been focused on the mar-ket power opportunities arising in this economic environment. Auction theory is the standardtheoretical framework to study, to develop and to propose commodity market mechanisms(Friedman 1993; Klemperer 2000). Game theory has provided an extremely useful instru-ment to study and to derive properties of economic “games” (von der Fehr and Harbord 1993;Baldick 2002), but different approaches have been also proposed in the literature: experimen-tal tests (Denton et al. 2001), statistical analysis (Guerci et al. 2005), computer simulations(Bunn and Oliveira 2001; Bower and Bunn 2001; Nicolaisen, Petrov, and Tesfatsion 2001).Within the latter context, the theory of learning-in-games may be an interesting and inspiringcomputational approach, for studying market inefficiencies (Vega-Redondo 2003; Fudenbergand Levine 1999). This methodology is useful in the context of infinitely repeated games.According to this general framework, this paper addresses the issue of modeling a sealed-bidclearinghouse double-auction mechanism via a computational approach, assuming boundedrationality for seller decision-making process.

The paper is organized as follows: Section 2 presents the theoretical setting of the com-putational experiments. Section 3 describes the auction models studied in this work: dis-criminatory and uniform. Both mechanisms are studied with the specific aim to comparemarket inefficiencies arising from repeated interaction among players. Section 4 illustrateshow the decision-making process of the seller has been modeled in the framework of learningalgorithms. Section 5 presents and discusses the results.

2. THEORETICAL FRAMEWORK

2.1. Economic Setting

The goal of this study is twofold. The first purpose is to investigate the capability ofsellers of electric power in organized auctions to learn to play stable strategies in the long-run. This study is performed by means of computational experiments within the frameworkof the learning-in-games approach where agents employ only private past information andhave no beliefs about the environment. In this respect, computational results are interpretedwithin the standard game theory according to the assumption of agents endowed with purerationality and complete information, i.e., employing two well-known solution concepts ofNash equilibrium and Pareto optimum. The second aim is to address the problem of theimpact of auction design on electricity market efficiency. This economical issue has beenrecently debated in the literature, (see, e.g., Nicolaisen et al. 2001; Fabra 2003). This work isfocused on both the uniform and the discriminatory auction mechanism, thus to study theirrelative efficiency.

The computational analysis performed in this study consists in simulating a large numberof auction rounds where players submit single offers at each step. The players have no initial

OLIGOPOLISTIC COMPETITION IN ELECTRICITY AUCTIONS 199

information about the nature of the game and about the opponents. They learn from experiencehow to select the optimal strategy by iteratively exploring the strategy space and competingagainst the opponents. For this purpose, the Marimon and McGrattan adaptive evolutionarylearning algorithm has been adopted (Marimon and McGrattan 1995). The computationalexperiments are performed under constant economic conditions, i.e., the auction mechanism,the parameters of the learning algorithms, the demand, the number of agents, and theirproduction technology are held constant.

Inspired by the original work of on der Fehr and Harbord (1993), two distinct duopolisticscenarios have been considered in this paper. The former case is a “High-Demand” situa-tion, henceforth HD2, where overall demand Qd is greater than the capacity of the greatestproducer, i.e., Qd > max{Qs

1,Qs2}. The latter case is a “Low-Demand” situation, hence-

forth LD2, where overall demand is less or equal to the capacity of the smallest seller, i.e.,Qd ≤ min{Qs

1,Qs2}. A general formulation of LD scenarios for n-suppliers is described by

the following condition:

Qsi ≤ Qd ≤

n∑j=1j �=i

Qsj , (1)

that must hold for every supplier i = 1, . . . , n.A further “Low-Demand” scenario, henceforth LD3, has been studied where the overall

productive capacity of all sellers is now divided into three competing sellers. Indeed, theLD3 economic scenario is an intermediate case between the LD2 and HD2 ones. Splittingof productive capacity determines a new competitive setting where the overall demand isstill less than the total productive capacity, i.e., trade is not guaranteed for all sellers, but,differently from the LD2 case, two sellers are required to satisfy the overall demand. Such ascenario should weaken the opportunity to exercise market power.

Consequently, the LD3 scenario may result interesting because it addresses an hypo-thetical antitrust action of splitting productive capacity to weaken the exercise of marketpower.

Irrespective to the specific situation, the long-run behavior of the computational ex-periments has been analyzed with respect of some solution concepts for strategic gamescommonly employed in game theory, i.e., the notion of Nash equilibrium and of Paretooptimum. Descriptions of these solution concepts are addressed in Appendix A. For everyeconomic scenarios and for both auction mechanisms, payoff matrices are evaluated underthe n-player normal-form game framework. Nash equilibria in pure strategies (see AppendixA1) have been identified and considered as the competitive solutions for the strategic game,according to economic theory. Whereas, Pareto optima (see Appendix A2) are proposed astacit collusive solutions of the game. Pareto outcomes well describe the opportunity thatthe players have to find an agreement, which leads to the solutions that no individual canbe made better off without another being made worse off. In the context of market games,this group’s decision corresponds to a collusive behavior, because an agreement suppose nocompetition. In particular, it is worth noting that in the repeated games framework, even ifno communication among the players is possible, repeated interactions among the agents canlead them to learn to tacitly agree on a noncompetitive solutions. Thus, in this paper, Paretooptima have been considered as the tacit collusive solutions.

Furthermore, in the context of finite repeated games scenario with pure rationality andcomplete information, the whole auction round sequence should represent a single game.Thus, the extended-form game should be employed to analyze it. Therefore, Nash equilib-ria and Pareto optima should be evaluated with respect of the cumulative payoff. The ap-proach, adopted in this paper, considers bounded rational agents, unable to make intertemporal


considerations. They are able to exploit only private past information and to play next auctionround as if it were the last one. Hence, the one-shot game structure, i.e., the game representedby a single auction round, is considered and relative Nash and Pareto solutions adopted tocompare the learning behavior results.

2.2. Auction Models

In this work, a sealed-bid clearinghouse double-auction is considered. According to thismechanism, an artificial power exchange has been implemented. Buyers and sellers submitoffers, i.e., each purchase or sale offer is a couple of values, (pd

i , qdi ) or (ps

i , qsi ), corresponding

to a limit price, pdi or ps

i , and a quantity of energy, qdi or qs

i . The model, adopted in this paper,considers only simple offer mechanisms, while more complex offer schemes, e.g., multipleoffers, are not currently included. The auctioneer collects all the sale and purchase offers andbuilds the empirical demand Qd(p) and supply Qs(p) curves, sorting buyer (qd

i ) and seller(qs

j ) quantity offers according to a price merit criterion, i.e.,

Qd(p) =∑

i | pdi ≥p

qdi , Qs(p) =

∑j | ps

j ≤p

qsj .

Matching mechanisms between supply and demand can be either “discriminatory” or “uni-form.” The former sets individual prices for each matched buyer–seller pair, whereas thelatter sets a unique market–clearing price, so-called system marginal price, determined at theintersection of demand and supply curves. Demand is assumed to be inelastic with respect toprice and it is modeled through a representative buyer offering a constant and inelastic de-mand at every auction round. According to this assumption, in this work the market clearingprice is always set at a price equal to the seller’s offer. This choice guarantees to calculatereasonable market prices for the specific model of inelastic demand adopted in this paper. Inthe following, a complete description for both matching procedures is proposed.

2.2.1. Uniform Double-Auction (UA). In an uniform double-auction (hereinafter UA)(see Figure 1a), the auctioneer builds the supply and demand curves and determines a uniquemarket clearing price at curves’ intersection. This spot price corresponds to the ask pricemade by the last production capacity whose entry into the system was required to satisfythe demand for electric power. Bids are accepted if their price is greater or equal to theclearing price whereas asks are accepted if their price is equal or lower than the clearingprice. Because aggregate electric supply and demand curves are discrete stepped curves,their crossing point may give rise to indeterminacy in the assignment of electric power. Thisrequires the application of a distribution criterion to ration the exceeding purchase or saleoffers of electricity. In this paper, focused on the decision-making process of two or threecompetitive sellers facing an inelastic demand, quantity rationing is significant only to thesupply side. A rationing scheme is then required to equate demand and supply. Rationingis necessary when Qd <

∑i qs

i . In this respect, the standard procedure, proposed for realmarket, leads to rationing the quantity only for offers with a price equal to the market clearingprice. In this paper, we adopt such strategy. In particular, being Qrd the residual demand atthe market clearing price, if there is only one seller offering at that price, her transaction isrationed to Qrd . Otherwise, a quantity assignment problem arises when more than one seller,let’s say n, offer at the clearing price. In this case, the rationing rule consists in subdividingthe n sellers into two sets, A and B. The Set A is composed by the n A producers whosequantity offers qs

i exceed the value Qrd/n, whereas the Set B collects the remaining sellers.


0 2 4 6 830

32

34

36

38

40

42

Quantities (MWh)

(a)

Pri

ces

(Eu

ro/M

Wh

)

0 2 4 6 830

32

34

36

38

40

42

Quantities (MWh)

(b)

Pri

ces

(Eu

ro/M

Wh

)DemandSupply

DemandSupply

FIGURE 1. Example of a clearinghouse double-auction. In both Figures the dotted line represents the supplycurve, while the dash and dotted line the demand curve. Figure (a) describes the uniform auction mechanism,where a unique price (p = 36 Euro/MWh) is determined and all the sale bids having a price below this will beaccepted and paid at this market clearing price. Conversely, Figure (b) describes the discriminatory auction wherethree different prices are established for accepted sale bids. In both cases refused seller bids correspond to thesale bids having a price greater than the crossing-point price.

A rationing applies only for sellers in Set A. The quantity traded by each seller in the SetA, qs

i for i ∈ A, is then given by qsi = (Qrd − ∑

j∈B qsj )/n A.

2.2.2. Discriminatory Double-Auction (DA). In the single-auction mechanism, thediscriminatory rule (see Figure 1b) corresponds to a simple pay as bid procedure, that is,transaction occurs at the price of the corresponded accepted bid. In the double-auction setting,the discriminatory rule has to take into account also the supply side. In the work by Nicolaisenet al. (2001), the following discriminatory double-auction (hereinafter DA) is adopted. Atthe beginning of the clearing procedure, the highest ask-price is matched with the lowestbid price. A transaction then occurs at a price equal to the midpoint between ask and bidprices and at a quantity equal to the minimum between ask and bid quantities. The remainingask or bid quantity is then matched with the second highest bid price or lowest ask price,respectively. This procedure is then iterated until there are ask prices equal or lower than bidprices, that is, the crossing point of the demand and supply curves is reached. Remainingoffers are discarded. If two or more sellers or buyers offer at the same price, an indeterminacyarises in the matching procedure. This situation can be solved defining an appropriate priorityorder, which also addresses the problem of quantity rationing because of the required balancebetween matched demand and supply.


In the present study, because of the assumption of inelastic demand, i.e., undetermined bidprices, the choice has been to set the transaction price at the accepted ask price. Furthermore,due to the inelasticity of demand, there is no need to introduce a priority order. However, iftwo or three sellers ask at the same prices where demand and supply curves intersect, theindeterminacy for quantity assignment is solved according to the rationing scheme previouslyadopted for the uniform auction mechanism.

2.3. Learning Algorithm

Real electricity markets are usually populated by a reduced number of sellers. Repeatedinteractions among them through the auction mechanism might cause the occurrence of im-plicit collusive behavior (Nicolaisen et al. 2001). Sellers can try to behave opportunistically,i.e., increasing their profits, and sell at prices strongly different from their true marginalcosts. Generally speaking, analytic tools in single-round game have been largely used toinvestigate the nature of market inefficiencies, but many interesting properties, such as tacitcollusive behavior, require to consider repeated games. Two main approaches can be adopted:evolutionary game theory (Samuelson 1997) or learning-in-games theory (Fudenberg andLevine 1999). In this paper, the latter approach is adopted and seller’s decision-making pro-cess are modeled by an homogenous individual learning behavior. In particular, the Marimonand McGrattan, henceforth MM, adaptive evolutionary learning algorithm is adopted as notbelief-based, i.e., no exact behavioral rules are provided with the algorithm and it simplyperforms a stochastic search among the strategy space to identify the most profitable strat-egy. Sellers repeatedly act according to two contrasting behaviors: they are characterizedby a certain probability to exploit strategies that perform better in the past, but at the sametime they still keep on exploring for better strategies. Another important aspect is that it isgame-structure independent, i.e., it does not take into consideration any aspect of the game oropponent’s plays. It is conceived to require minimal information for players’ knowledge. Sell-ers gain knowledge only from their own actual and past selected actions and their associatedrealized profit.

The ith seller’s strategy space Asi is a discrete bidimensional space defined by couples

asi = (ps

i , qsi ) for admissible sale orders. As

i is constrained by maximum price P∗ alongthe price-axis and by the maximum productive capacity Qs

i along the quantity-axis, i.e.,As

i := {(psi , qs

i ) | qsi ≤ Qs

i and psi ≤ P ∗}. The ith seller is also endowed with the following

quantities or functions:

� Linear cost functions with constant marginal costs cm,i ≥ 0 are assumed for every pro-ducing technologies.

� A vector of probabilities σi,t (asi ) defined at auction round t on every pure strategy as

i ∈ Asi ,

henceforth mixed strategies. At the beginning, mixed strategies are characterized by anuniform distribution.

� A function Si,t (asi ) defined on pure strategy as

i that can be interpreted as a measure ofperformance for every strategy exploiting experience. This function is called strength forthe MM algorithm. At the beginning, value’s function are initialized according to Si,1(as

i ) =g · (P ∗ − cm,i ) · Qs

i , where g is an appropriate positive scaling parameter, which needs tobe estimated.

At auction round t, the ith seller selects a strategy asi (t) = ( ps

i (t), qsi (t)) as a random

draw form the distribution determined by current mixed strategies. Thus, she realizes apositive profit �i,t (as

i ) = qsi (t) · ( ps

i (t) − cm,i ) if i-th seller bid is accepted, otherwise a nullprofit, i.e., �i,t (as

i ) = 0. psi (t) and qs

i (t)) are the price paid and the quantity effectively traded,


respectively, i.e., either the quantity bid qsi (t) = qs

i (t) or a rationed amount of qsi (t). The

specific auction mechanism determines different values for (qsi (t), ps

i (t)). Realized profitdepends only on opponents bids and on her strategy as

i (t). The mathematical formulation forthe MM algorithm is described in Appendix B.

3. COMPUTATIONAL EXPERIMENTS AND RESULTS

3.1. Framework

The economic analysis leaded in this work focuses on six different economic cases.Three economic scenarios (HD2, LD2, LD3), mainly identified by the level of demand, areconsidered. Each scenario is examined under two different auction mechanisms (uniform ordiscriminatory).

In the following, a detailed description of the common features characterizing eachexperiment is reported. The demand is modeled by means of a representative buyer. At eachauction round, she asks a constant quantity of energy Qd , which is inelastic with respectto price. The supply-side of the market is provided by two or three players, depending onthe specific economic case considered. Sellers are endowed with a maximum productivecapacity Qs and they are characterized by linear cost functions with different marginal costscm,i ≥ 0 and null fixed costs. At each round, the i-th seller (player) selects a strategy as

i , i.e.,a price–quantity pair (ps

i , qsi ) in the strategy space Ai := {(ps

i , qsi ) | qs

i ≤ Qsi and ps

i ≤ P ∗},where P∗ is an upper bound for the price. Table 1 reports the values of the time-invariantparameters employed for the MM adaptive evolutionary learning algorithm. These have beendetermined through direct search, i.e., the three values were calibrated trying to reach singlepeaked probabilities distribution over the strategy space by the final auction round in eachrun, according to Nicolaisen et al. (2001). Finally, a unitary quantization for the values ofboth ps

i and qsi has been chosen. The units are arbitrary. Sellers are considered as producers

with linear cost functions. This implies that sellers, according to their profit-maximizingbehavior, should surely bid their full capacity to increase their expected payoff. Thus, theproper strategy space for the two players should be characterized only by the price dimension.However, the framework, adopted in this paper, considers a discrete bidimensional strategyspace, characterized by both the price and quantity axes thus to verify if the employedlearning-in-game approach can generalize over higher strategy space by playing strategiescharacterized by maximum quantity bids. Indeed, game theoretical and computational resultsshow that, in the uniform auction settings with low demand, there are vector of strategies,characterized by quantity supply offers lower than maximum capacity, which result to beNash equilibria played in the long-run. An example is given in Section 3.2.

The equilibrium analysis performed in this study focuses on the long-run behavior ofthe learning processes. The convergence of the learning dynamics for each seller states thedetermination of a stable competitive/collusive solution of the single experiments. A total

TABLE 1. Values of the Parameters for the Marimonand McGrattan (MM) Learning Algorithm

MM Algorithm

ρ ε g

0.03 0.0003 0.4


of 10,000 computational experiments have been performed, each characterized by 10,000auction rounds. Computational experiments are analyzed according to three concepts: theoccurrence of Nash equilibria and Pareto optima, and the value of profits earned by sellers.All these observables are evaluated for each auction round averaging across the 10,000experiments. Nash equilibria and Pareto optima are monitored throughout the experiments, tounderstand which behavior prevails in the long run: competition or tacit collusion. Profits areconsidered to understand how the properties of the price-clearing rule of the specific auctionmechanism affects the overall market efficiency. It is worth remarking that, in an auctionmechanism, competitive solutions do not assure that the bid price equals the marginal costfor every seller.

3.2. Low Demand Case with Two Sellers (LD2)

In the LD2 case, the initial endowments are the following. Sellers’ productive capacityis the same and equal to Qs

1 = Qs2 = 6 and their marginal costs are equal to cm,1 = 4 and

cm,2 = 7, respectively. The units are arbitrary. Demand is equal to Qd = 6. Finally, the upperprice bound P∗ has been set equal to 10. Due to unitary quantization, these settings producea strategy space of 77 pure strategies as

i for each seller and a set of 5,929, i.e., 77×77, vectorsof strategies (as

1, as2).

These settings refer to both auction mechanisms. Figure 2 shows the frequencies of Nash(top part) and Pareto (down part) solutions played at each auction round for both the uniformauction (UA) and discriminatory auction (DA) mechanisms, respectively. Frequencies areevaluated as ensemble averages over all 10,000 experiments.

It is worth remarking that sellers are not endowed with collusion opportunities, e.g.,strategy communication or price signaling. Thus, in the long run, they are expected to playcompetitive solutions, i.e., Nash equilibria. In the uniform auction case (see top of Figure 2),the frequency of Nash equilibria grows rapidly toward the value of 0.8. It is worth noting that,in this case, the majority of Nash equilibria results to be also Pareto optima, the latter set beingfar greater (see Table 2). This fact implies that Nash equilibria, i.e., competitive equilibria,are also optimal solutions for the two players, thus strengthening the probability to play thosesolutions for self-interested players. Conversely, in the discriminatory auction game, the setsof Nash equilibria and Pareto optima are disjoint. In this respect, the bottom part of Figure 2clearly shows that the two curves tend to present a divergent behavior and that, in the DAcase, the two sellers learn to play the competitive solutions, as expected, while Pareto optimaare no longer played in the long run. Thus, Figure 2 clearly highlights the different gamestructures for the two auction mechanisms and points out that the DA mechanism is moreeffective if compared to the UA one to reduce the market power of sellers, thus improvingthe overall efficiency of the market.

It is worth noting that, at the beginning of the learning process, the frequency of Paretooptima grows at a faster rate than that of Nash equilibria, both in the UA and DA cases.This depends on the fact that, during the first rounds of each experiment, sellers have notyet learned to behave strategically and are encouraged by their rewarding scheme to increasetheir probabilities to offer larger quantities at high prices. Therefore, in the first part of thelearning process, vector of strategies corresponding to Pareto optima are more frequentlyplayed. This fact accelerates the learning of competitive solutions when Nash equilibria arealso Pareto optima, as in the UA case. Conversely, in the DA case, where Pareto optima nevercorrespond to competitive solutions, Nash equilibria are learnt at a lower rate and Paretooptima, as expected, are no longer played in the long run.


0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

0.2

0.4

0.6

0.8

1UA case

Auction rounds

Fre

quen

cy

NashPareto

0 1,000 2,000 3,000 4,000 5,000 6,000 7000, 8,000 9,000 10,0000

0.2

0.4

0.6

0.8

1DA case

Auction rounds

Fre

quen

cy

NashPareto

FIGURE 2. LD2 case. Frequencies of Nash equilibria and Pareto optima in the uniform auction case (toppart) and the discriminatory auction case (bottom part). Frequencies have been evaluated as ensemble averagesover 10,000 experiments.

Figure 3 presents the evolution of profits (payoffs) realized by the two sellers for boththe UA (top part) and DA (bottom part). These plots show that in the uniform auctionmechanism the profit realized by the two sellers are greater than in the DA case. This factcan be explained according to the results presented in Figure 2. Indeed, both the UA andthe DA game experiments point out competitive outcomes in the long run. However, thedifferent structure of the two games emerges in the profits because, in the uniform auctioncase, competitive solutions are also Pareto optima, thus providing higher payoffs for bothsellers.

Tables 2 and 3 report the vectors of strategies, which result to be more often played atthe final auction round in the UA and DA mechanisms, respectively. Vectors are ranked indescending order according to their frequency. Table 2 lists the 36 vectors of strategies outof 5,929 with frequencies equal or greater than 1%. It is worth noting that this reduced set ofvectors covers more than 70% of the total set of combinations of strategies. Furthermore, Ta-ble 2 shows that in the UA case the vectors of strategies more frequently chosen are all Paretooptima and, except for three marginal cases, these vectors are also Nash solutions. Thesefindings confirm previous analysis about the structure of the UA game. Conversely, in the DAgame, as Nash equilibria and Pareto optima are disjoint sets, one of the two solution conceptsshould prevail in the long run competition. In this respect, Table 3 highlights that in the DAgame with the MM learning algorithm competitive solutions, i.e., Nash equilibria, emerge asthe four most frequently played combinations of strategies. These Nash equilibria correspond


TABLE 2. LD2 UA Case

ps1 qs

1 ps2 qs

2 Payoff 1 Payoff 2 Nash Pareto Frequency

7 5 10 6 30.00 3.00 X X 0.0326 5 10 5 30.00 3.00 X X 0.0307 5 10 5 30.00 3.00 X X 0.0296 5 10 6 30.00 3.00 X X 0.0285 5 10 6 30.00 3.00 X X 0.0266 5 10 4 30.00 3.00 X X 0.0264 5 10 6 30.00 3.00 X X 0.0255 5 10 4 30.00 3.00 X X 0.0247 5 10 4 30.00 3.00 X X 0.0243 5 10 6 30.00 3.00 X X 0.0224 5 10 5 30.00 3.00 X X 0.0225 5 10 5 30.00 3.00 X X 0.0213 5 10 5 30.00 3.00 X X 0.0217 5 10 3 30.00 3.00 X X 0.0212 5 10 6 30.00 3.00 X X 0.0210 5 10 6 30.00 3.00 X X 0.0201 5 10 6 30.00 3.00 X X 0.0201 5 10 5 30.00 3.00 X X 0.0202 5 10 5 30.00 3.00 X X 0.0203 5 10 4 30.00 3.00 X X 0.0194 5 10 4 30.00 3.00 X X 0.0192 5 10 4 30.00 3.00 X X 0.0195 5 10 3 30.00 3.00 X X 0.0190 5 10 5 30.00 3.00 X X 0.0181 5 10 4 30.00 3.00 X X 0.0174 5 10 3 30.00 3.00 X X 0.0176 5 10 3 30.00 3.00 X X 0.0172 5 10 3 30.00 3.00 X X 0.0153 5 10 3 30.00 3.00 X X 0.0150 5 10 4 30.00 3.00 X X 0.0151 5 10 3 30.00 3.00 X X 0.0148 5 10 6 30.00 3.00 X 0.0146 5 10 2 30.00 3.00 X X 0.0120 5 10 3 30.00 3.00 X X 0.0118 5 10 4 30.00 3.00 X 0.0118 5 10 5 30.00 3.00 X 0.0107 5 10 2 30.00 3.00 X X 0.010

Total 0.724

The table reports the frequencies of vectors of strategies ((ps1, qs

1), (ps2, qs

2)) played at the final auction round.Frequencies are ranked in descending order. Only vectors of strategies with frequencies equal or greater than 1%are listed. The symbol “X” is applied if the corresponding vector of strategies is a Nash equilibria and/or a Paretooptima. Units are arbitrary.


0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

5

10

15

20

25

30UA case

Auction rounds

Pro

fits

Most efficient sellerLeast efficient seller

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

5

10

15

20

25

30DA case

Auction rounds

Pro

fits


FIGURE 3. LD2 case. Profits for the two sellers in the uniform auction case (top part) and the discriminatoryauction case (bottom part). Profits have been evaluated as ensemble averages over 10,000 experiments.

TABLE 3. LD2 DA Case

ps1 qs

1 ps2 qs


7 6 8 6 18.00 0.00 X 0.2857 6 8 5 18.00 0.00 X 0.2077 6 8 4 18.00 0.00 X 0.1247 6 8 3 18.00 0.00 X 0.0367 6 9 6 18.00 0.00 0.0317 6 9 5 18.00 0.00 0.0277 6 9 4 18.00 0.00 0.0237 6 9 3 18.00 0.00 0.014

Total 0.747


1), (ps2, qs



to vectors of strategies where the most efficient seller offers her maximum capacity at themarginal cost of the opponent. Table 3 reports the eight vectors of strategies out of 5,929 withfrequencies equal or greater than 1%. These correspond to the four previously mentionedNash equilibria and to other four combinations of strategies, which, however, are close to Nashsolutions. Indeed, these four noncompetitive solutions differ from the four Nash equilibriaonly with respect to the price set by the least efficient seller. These are not Nash equilibriabecause the first seller has an incentive to deviate from such strategy (ps

1 = 7, qs1 = 6) to the

strategy (ps1 = 8, qs

1 = 6) increasing her payoff from 18 to 24.Generally speaking, a reasonable economic conjecture is that producers characterized

by linear cost functions should offer their maximum capacity. The logical reason behind thischoice, in the framework of noncooperative games, is that agents have incentives to increasetheir offered quantities, to maximize their expected payoffs. Bertrand model of oligopoly andwork by Nicolaisen et al. (2001) exactly adopt such assumptions. It is worth remarking thatthis paper points out that this behavioral choice is not always justified for market-games withlinear production costs. In particular, it has been shown that within the uniform auction gameNash equilibria in pure strategies exist for quantity offers lower than the maximum capacity.Table 2 clearly highlights the existence of several Nash equilibria for quantity offers lowerthan the maximum capacity contemporary for both sellers, e.g., ((ps

1, qs1), (ps

2, qs2)) equal to

((7, 5), (10, 5)). These equilibria characterize competitive solutions where one seller acceptsto share part of the trade with the competitor, considering that the best strategy for theopponents is to offer the residual demand at maximum price and that, according to theuniform auction mechanism, the trade is cleared at the maximum price.

3.3. High Demand Case with Two Sellers (HD2)

In the HD2 case, the initial endowments are the following. Seller productive capacitiesare identical and equal to Qs

1 = Qs2 = 6 and their marginal costs are equal to cm,1 = 4 and

cm,2 = 7, respectively, as in the LD2 case. Demand is set to Qd = 11. Finally, the upper pricebound P∗ has been set equal to 10. Due to unitary quantization, these settings produce astrategy space of 77 pure strategies as

i for each seller and a set of 5,929, i.e., 77×77, vectorsof strategies (as

1, as2).

Figure 4 presents computational results in the HD2 case for both auction mechanisms.The top and the bottom parts of Figure 4 shows the frequencies of Nash and Pareto solutionsplayed at each auction round for the uniform and discriminatory auction, respectively. As inthe LD2 case, frequencies are evaluated as ensemble averages over all 10,000 experiments.

In the HD2 UA game the majority of Nash equilibria results to be also Pareto optima, asin LD2 UA case. Therefore, the HD2 UA game (see top of Figure 4), is characterized by avery fast convergence toward Nash and Pareto solutions whose frequencies are even closerto 1 than in the LD2 UA case. Conversely to the LD2 DA game, the HD2 DA game presentsoverlapping of Nash and Pareto sets. This fact is clearly evident in the bottom part of Figure 4,where the two curves related to the two solutions exhibit similar shapes. Furthermore, as in theHD2 UA case, they reach high-frequency values, close to 1, within the 10,000 auction rounds.Indeed, the convergence rate is lower than in the uniform auction mechanism. This findingmay be related to the different numbers of Nash equilibria that characterize the two auctiongames. It is worth remarking that the DA game is characterized by only one Nash solution,whereas the UA game is characterized by several Nash equilibria. Considering that in thefirst rounds of the learning process the probability distribution of selecting pure strategiesis still close to the initial uniform distribution, the probability of a random occurrence of aNash equilibrium is higher in the UA case than in the DA case.

It is worth noting that the long run level of frequency values to play Nash equilibria is lowerthan 1 in both auction mechanisms. In the DA case, this small gap, which amounts to nearly


0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

0.2

0.4

0.6

0.8

1UA case

Auction rounds

Fre

quen

cy

NashPareto

0 1,000 2,000 3,000 4000 5,000 6,000 7,000 8,000 9,000 10,0000

0.2

0.4

0.6

0.8

1DA case

Auction rounds

Fre

quen

cy

NashPareto

FIGURE 4. HD2 case. Frequencies of Nash equilibria and Pareto optima in the uniform auction case (toppart) and the discriminatory auction case (bottom part). Frequencies have been evaluated as ensemble averagesover 10,000 experiments.

0.05, is explained by paper considerations on the value of the experimentation parameter εemployed in the MM learning algorithm. ε corresponds to the minimum probability value thatcan be assigned to any pure strategy out of the 77 available to each seller. Stated that only one ofthese 77 strategies for each seller (as,∗

i i = 1, 2) gives the Nash solution highlighted in Table 5,the probability that one seller plays this strategy can not exceed the value 1 − 76ε. Therefore,the probability for the occurrence of the Nash equilibrium is given by the joint probabilitythat both sellers play such as strategy, as,∗

1 and as,∗2 , respectively, i.e., (1 − 76ε)2 � 0.95. This

value is close to the long run frequency level of the learning dynamics in the DA case. Similarconsiderations and results could be applied to explain the long run frequency level in the UAcase. However, in the UA framework, the analysis is more complex due to the higher numberof Nash equilibria than the DA case.

Figure 5 reports the evolution of the profits for the two sellers in the UA case (top part)and in the DA case (bottom part), respectively. In the HD2 scenario, the difference betweenthe two auction mechanisms results smaller than in the LD2 case. It is worth noting that, inthe HD2 scenario, profits, i.e., payoffs, realized by each seller, are similar in the two auctionmechanisms. The small difference in terms of profit in the HD2 case is certainly due to thedifferent price-assignment rule, which assures that in the UA mechanism the most efficientseller trades at the price of the least efficient seller price offer, whereas in the DA case themost efficient seller trades at her price offer.

Tables 4 and 5 report the vectors of strategies, which result to be more often played atthe final auction round in the UA and DA mechanisms, respectively. Vectors are ranked in


0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

5

10

15

20

25

30

35

40UA case

Auction rounds

Pro

fits


0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

5

10

15

20

25

30

35

40DA case

Auction rounds

Pro

fits


FIGURE 5. HD2 case. Profits for the two sellers in the uniform auction case (top part) and the discriminatoryauction case (bottom part). Profits have been evaluated as ensemble averages over 10,000 experiments.

descending order according to their frequency. Table 4 lists the 20 vectors of strategies outof 5,929 with frequencies equal or greater than 1%. This reduced set of vectors covers nearlythe 95% of the total set of combinations of strategies. Furthermore, Table 4 shows that inthe UA case the vectors of strategies more frequently chosen are Nash solutions, which arealso Pareto optima. It is worth noting that Nash equilibria can be subdivided in two setsaccording to the price offer applied by the two sellers. The first set is characterized by alow price offered by the most efficient seller and the maximum price offered by the leastefficient one. Conversely, in the second set, the maximum price offer is submitted by themost efficient player and the least efficient one offers price smaller than the maximum value.Table 4 highlights that a refinement of Nash equilibria has been realized by the learningprocess and, in particular, that the first set of Nash equilibria is most frequently played. Table5 shows that in the DA game there exists one vector of strategies, which results to be by farthe most frequently played. This vector is: ((ps

1, qs1), (ps

2, qs2)) equal to ((10, 6), (10, 6)) and

results to be both a Nash equilibrium and a Pareto optimum.

3.4. Low Demand with Three Sellers (LD3)

The LD3 case is an extension of the LD2 economic scenario. In particular, a thirdproducer/seller is assumed to enter into the market as consequence of an antitrust actiondrawn to reduce market power by distributing the total productive capacity from the originaltwo to three sellers. Thus, the demand is considered identical to the LD2 case, i.e., Qd = 6.Conversely, the new sellers’ endowments are Qs

1 = Qs2 = Qs

3 = 4 and their marginal costs


TABLE 4. LD2 UA Case

ps1 qs

1 ps2 qs


9 6 10 6 36.00 15.00 X X 0.24110 6 9 6 30.00 18.00 X X 0.1128 6 10 6 36.00 15.00 X X 0.0967 6 10 6 36.00 15.00 X X 0.0616 6 10 6 36.00 15.00 X X 0.04810 6 8 6 30.00 18.00 X X 0.0475 6 10 6 36.00 15.00 X X 0.0424 6 10 6 36.00 15.00 X X 0.0403 6 10 6 36.00 15.00 X X 0.0361 6 10 6 36.00 15.00 X X 0.0362 6 10 6 36.00 15.00 X X 0.0350 6 10 6 36.00 15.00 X X 0.03410 6 7 6 30.00 18.00 X X 0.02510 6 6 6 30.00 18.00 X X 0.02010 6 5 6 30.00 18.00 X X 0.01510 6 3 6 30.00 18.00 X X 0.01310 6 4 6 30.00 18.00 X X 0.01310 6 2 6 30.00 18.00 X X 0.01310 6 1 6 30.00 18.00 X X 0.01310 6 0 6 30.00 18.00 X X 0.012

Total 0.952


1), (ps2, qs


TABLE 5. HD2 DA Case

ps1 qs

1 ps2 qs


10 6 10 6 33.00 16.50 X X 0.93410 5 10 6 30.00 18.00 X 0.015

Total 0.949


1), (ps2, qs


are equal to cm,1 = 4, cm,2 = 5.5 and cm,3 = 7, respectively. The upper price bound P∗ hasbeen maintained equal to 10. Due to unitary quantization, these settings produce a strategyspace of 55 pure strategies as

i for each seller and a set of 166,375, i.e., 55 × 55 × 55, vectorsof strategies (as

1, as2, as

3).Figure 6 shows the computational results for the LD3 case. Both auction mechanisms

have been considered. The top and the bottom parts of Figure 6 show the frequencies of Nashand Pareto solutions played at each auction round for the uniform and discriminatory auction,


0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

0.2

0.4

0.6

0.8

1

Auction rounds

Fre

quen

cy

UA case

NashPareto

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

0.2

0.4

0.6

0.8

1

Auction rounds

Fre

quen

cy

DA case

NashPareto

FIGURE 6. LD3 case. Frequencies of Nash equilibria and Pareto optima in the uniform auction case (toppart) and the discriminatory auction case (bottom part). Frequencies have been evaluated as ensemble averagesover 10,000 experiments.

respectively. As in the previous economic cases, frequencies are evaluated as ensemble av-erages over all 10,000 experiments. It is worth noting that in the LD3 case, both UA and DAgames are characterized by Nash equilibria that are not Pareto optima, see Tables 6, 7, and8. This condition produces a significant difference on the LD3 scenario in the UA case withrespect to the LD2 setting. The antitrust action has changed the game-structure and now thePareto optima are no longer the competitive solutions.

As expected, in both auction mechanisms, the learning dynamics progressively convergestoward the Nash solutions, which result the most frequently played solutions in the long run.This finding is also confirmed by Tables 6, 7, and 8 that report the vectors of strategies mostoften played at the final auction round in the UA and DA mechanisms, respectively. Vectorsare ranked in descendent order according to their frequency. It is worth remarking that inthe LD3 case, the Set of vectors of strategies includes more than 160,000 elements. Eachelement of such a huge set corresponds to a possible event occurring in an auction round,i.e., price and quantity offers of every seller, i.e., (as

1, as2, as

3), with asi = (ps

i , qsi ). For the sake

of compactness, in the LD3-UA scenario, subset of different vectors of strategies have beengrouped in Tables 6 and 7 according to the criteria that different events are considered similarif they have equal payoffs and exhibit the same game-theoretical solution.

It is worth noting that, at the beginning of the learning process, the frequency of Paretooptima grows at a faster rate than those of Nash equilibria, both in the UA and DA cases. Thisfact can be explained according to considerations similar to those discussed in the LD2 case.The difference with respect to the LD2 setting is that, as previously stated, the Pareto optima


TABLE 6. LD3 UA Case (Part 1)

ps1 qs

1 ps2 qs

2 ps3 qs

3 Payoff 1 Payoff 2 Payoff 3 Nash Pareto Frequency

0:6 4 7 2:4 8 2:4 12.00 3.00 0.00 X 0.2947 4 0:5 4 8 2:4 6.00 6.00 0.00 X 0.0427 3 0:5 4 8 2 6.00 6.00 0.00 X 0.0120:6 4 7 4 8 1 12.00 3.00 0.00 0.0077 4 0, 1, 3:5 4 8 1 6.00 6.00 0.00 X 0.0077 3 1:5 4 8 3, 4 6.00 6.00 0.00 X 0.0120:6 4 8 2, 4 8 3 16.00 2.50 1.00 0.0127 3, 4 6 4 8 2:4 6.00 6.00 0.00 0.0081:7 4 8 3 8 2 16.00 2.50 1.00 0.0052:6 4 8 4 8 4 16.00 2.50 1.00 0.0050, 2:6 4 10 3 8 1 24.00 4.50 3.00 X 0.0052:6 4 8 3 8 3 16.00 2.50 1.00 0.0050, 1, 3:6 4 10 2 8 1 24.00 4.50 3.00 X 0.0050, 2, 4:6 4 7 3 8 1 12.00 3.00 0.00 0.0050, 1, 4:6, 8 4 9 4 9 4 20.00 3.50 2.00 0.0040:3, 5:7 3 7 2 10 3 18.00 9.00 3.00 X 0.0041:6 4 10 4 8 1 24.00 4.50 3.00 X 0.0041:3, 5, 6 4 8 3 8 4 16.00 2.50 1.00 0.0040, 3:6 4 9 3 9 4 20.00 3.50 2.00 0.0042, 5:7 3 5:7 2 10 2 18.00 9.00 3.00 X 0.0041, 2, 4, 6 4 8 2 8 2 16.00 2.50 1.00 0.0041, 3:6 4 10 3 7 1 24.00 4.50 3.00 X 0.0031, 3, 5, 6 4 10 1 8 1 24.00 4.50 3.00 X 0.0033:7 3 6 7 2 10 4 18.00 9.00 3.00 X 0.0031, 2, 4:6 4 9 4 9 3 20.00 3.50 2.00 0.0030, 2, 4:6 4 8 4 8 2 16.00 2.50 1.00 0.0030:2, 4, 6 4 10 4 7 1 24.00 4.50 3.00 X 0.0030, 4:6 4 9 3 9 3 20.00 3.50 2.00 0.0032, 3, 5, 6 4 8 2 8 4 16.00 2.50 1.00 0.0032, 4:6 4 9 3 9 2 20.00 3.50 2.00 0.0032, 3, 6 4 9 4 9 2 20.00 3.50 2.00 0.0020, 2, 4, 5 4 9 3 8 1 20.00 3.50 2.00 0.0022, 3, 6 4 9 2 9 2 20.00 3.50 2.00 0.0024, 6 4 10 3 6 1 24.00 4.50 3.00 X 0.0021, 4, 6 4 9 2 9 4 20.00 3.50 2.00 0.0024:6 4 10 1 6 1 24.00 4.50 3.00 X 0.0025, 6 4 10 1 0, 7 1 24.00 4.50 3.00 X 0.0022, 6 4 9 4 8 1 20.00 3.50 2.00 0.0013, 4 3 10 3 7 2 18.00 4.50 6.00 X 0.0015, 6 4 8 3 7 1 16.00 2.50 1.00 0.0011, 3 4 8 3 8 1 16.00 2.50 1.00 0.0015, 6 4 10 2 7 1 24.00 4.50 3.00 X 0.0016 4 8 4 7 1 16.00 2.50 1.00 0.0011, 6 4 9 2 9 3 20.00 3.50 2.00 0.0013, 4 3 10 3 6 2 18.00 4.50 6.00 X 0.001

(continued)


TABLE 6. (Continued)

ps1 qs

1 ps2 qs

2 ps3 qs


7 4 5 6 3 8 2 9.00 4.50 0.00 0.0012, 4 3 10 4 7 2 18.00 4.50 6.00 X 0.0015, 7 3 7 2 9 3 15.00 7.00 2.00 0.0011, 7 3 7 2 9 2 15.00 7.00 2.00 0.0013, 6 3 10 2 7 2 18.00 4.50 6.00 X 0.001


1), (ps2, qs

2), (ps3, qs

3)) played at the final auctionround. Frequencies are ranked in descending order. Only vectors of strategies with frequencies equal or greaterthan 0.05% are listed. The symbol “X” is applied if the corresponding vector of strategies is a Nash equilibriaand/or a Pareto optima. Units are arbitrary. The notation n1 : n2 refers to all integers number from n1 to n2.

are no longer played in the long run, even in the case of the uniform auction mechanism.Being Nash solutions and Pareto optima disjoint sets, only competitive solutions prevail inthe long run.

The LD3 computational experiment is proposed to test effectiveness of possible antitrustpolicies aiming to reduce market power of producers/sellers in an electricity market. As aconsequence, a proper answer to this question requires comparison of profits for any seller inthe long run of the two economic scenarios, i.e., LD2 and LD3 cases. Under this respect, theaggregate profit realized by all sellers in the LD3 case is nearly 36% lower than in the LD2scenario for the UA case, whereas for the DA mechanism the profit reduction is about 27%.This allows one to conclude that the redistribution of total productive capacity to a largernumber of producers is a proper antitrust policy. Moreover, effectiveness of such an antitrustpolicy is larger for a UA mechanism rather than for a DA one, as the latter is relatively moremarket efficient. Indeed, as regarding relative market efficiency of UA and DA mechanisms,the LD3 economic scenario confirms previous findings, i.e., the discriminatory pricing ruleincreases the market efficiency as clearly pointed out in Figure 7. As shown, profits for anyof the three sellers in the UA case (see top part of Figure 7) are significantly larger than inthe DA case (see bottom part of Figure 7). Therefore, even if the LD3-UA game has becomemore competitive than the LD2-UA game due to the antitrust action, the uniform auctionpricing mechanism still results to be less efficient than the discriminatory rule.

4. CONCLUSIONS

This paper has investigated the relative efficiency of two common auction mechanismsusually employed in organized electricity markets. This question has been addressed by alearning-in-games approach. In particular, computational experiments, characterized by anoligopolistic scenario, have been performed. The proposed economic framework consideredtwo or three producers/sellers of electricity that learn to compete in a market environmentcharacterized by a constant and inelastic demand. Sellers are characterized by linear costfunctions and make price and quantity decisions based on an adaptive learning algorithm.Computational results have been interpreted according to classical game theoretical solutions,i.e., Nash equilibria and Pareto optima. Three main economic scenarios are studied. The firsttwo cases correspond to a duopolistic competition between sellers with different marginalcosts facing a low and a high demand level. Furthermore, a third economic case reproduces


TABLE 7. LD3 UA Case (Part 2)

ps1 qs

1 ps2 qs

2 ps3 qs


3 4 8 4 5 1 16.00 2.50 1.00 0.0016 4 7 4 9 4 12.00 3.00 0.00 0.0015 4 10 3 4 1 24.00 4.50 3.00 X 0.0017 4 7 2 8 3 12.00 3.00 0.00 0.0013 4 10 1 5 1 24.00 4.50 3.00 X 0.0016 4 10 2 5 1 24.00 4.50 3.00 X 0.0017 4 5 4 9 4 6.00 6.00 0.00 0.0012 4 10 4 5 1 24.00 4.50 3.00 X 0.0015 4 9 3 4 1 20.00 3.50 2.00 0.0014 4 9 2 7 1 20.00 3.50 2.00 0.0015 4 10 2 6 1 24.00 4.50 3.00 X 0.0016 4 10 2 2 1 24.00 4.50 3.00 X 0.0014 3 10 2 5 2 18.00 4.50 6.00 X 0.0010 4 8 4 9 4 16.00 5.00 0.00 0.0016 4 8 4 9 3 16.00 5.00 0.00 0.0012 4 8 4 8 1 16.00 2.50 1.00 0.0016 4 7 4 7 1 12.00 1.50 0.00 0.0014 4 10 4 6 1 24.00 4.50 3.00 X 0.0016 4 8 4 6 1 16.00 2.50 1.00 0.0016 4 9 4 4 1 20.00 3.50 2.00 0.0016 4 10 4 2 1 24.00 4.50 3.00 X 0.0016 4 8 3 9 4 16.00 5.00 0.00 0.0016 4 7 3 9 3 12.00 3.00 0.00 0.0016 4 8 3 5 1 16.00 2.50 1.00 0.0015 4 8 3 4 1 16.00 2.50 1.00 0.0015 4 10 3 2 1 24.00 4.50 3.00 X 0.0016 4 8 3 0 1 16.00 2.50 1.00 0.0011 4 9 2 6 1 20.00 3.50 2.00 0.0016 4 9 2 5 1 20.00 3.50 2.00 0.0015 4 10 1 1 1 24.00 4.50 3.00 X 0.0016 3 10 4 6 2 18.00 4.50 6.00 X 0.0017 3 2 2 10 1 18.00 9.00 3.00 X 0.001

Total (Part 1 + Part 2) 0.525


1), (ps2, qs

2), (ps3, qs

3)) played at the final auctionround. Frequencies are ranked in descending order. Only vectors of strategies with frequencies equal or greaterthan 0.05% are listed. The symbol “X” is applied if the corresponding vector of strategies is a Nash equilibriaand/or a Pareto optima. Units are arbitrary. The notation n1 : n2 refers to all integers number from n1 to n2.

the effects of an antitrust action, i.e., the total productive capacity of the duopolistic lowdemand scenario is now shared among three producers/sellers.

Results pointed out that the discriminatory auction mechanism is able to increase marketefficiency and to reduce market power with respect to the uniform auction pricing rule. Thisevidence emerges clearly in all the three economic scenarios and is properly appreciated bycomparing the profit curves in different cases. Indeed, the payoffs realized by sellers in the


TABLE 8. LD3 DA Case

ps1 qs

1 ps2 qs

2 ps3 qs


7 4 7 4 8 4 9.00 4.50 0.00 X 0.2857 4 7 4 8 3 9.00 4.50 0.00 X 0.2227 4 7 4 8 2 9.00 4.50 0.00 X 0.1067 4 7 3 8 4 9.00 4.50 0.00 X 0.0617 4 7 3 8 3 9.00 4.50 0.00 X 0.0457 3 7 4 8 4 9.00 4.50 0.00 X 0.0447 3 7 4 8 3 9.00 4.50 0.00 X 0.0357 4 7 4 9 4 9.00 4.50 0.00 0.0207 4 7 4 9 3 9.00 4.50 0.00 0.0187 4 7 4 9 2 9.00 4.50 0.00 0.0187 4 7 3 8 2 9.00 4.50 0.00 X 0.0157 4 7 4 9 1 9.00 4.50 0.00 0.0127 4 7 4 8 1 9.00 4.50 0.00 X 0.0087 3 7 4 8 2 9.00 4.50 0.00 X 0.0077 3 7 3 8 4 9.00 4.50 0.00 X 0.0077 3 7 3 8 3 9.00 4.50 0.00 X 0.0067 4 7 4 10 3 9.00 4.50 0.00 0.0057 4 7 4 10 1 9.00 4.50 0.00 0.0047 4 7 4 10 2 9.00 4.50 0.00 0.0047 4 7 4 10 4 9.00 4.50 0.00 0.0037 4 7 3 9 4 9.00 4.50 0.00 0.002

Total 0.927


1), (ps2, qs

2), (ps3, qs

3)) played at the final auctionround. Frequencies are ranked in descending order. Only vectors of strategies with frequencies equal or greaterthan 0.1% are listed. The symbol “X” is applied if the corresponding vector of strategies is a Nash equilibriaand/or a Pareto optima. Units are arbitrary.

discriminatory auction setting are always lower than in the uniform auction case. A reasonableexplanation of this finding is related to the price rule of the uniform auction, i.e., transactionsoccur at the price fixed by the least efficient producer. Furthermore, another interestingresult of this papers stems from the third experiment on antitrust action. Computationalexperiments clearly show that splitting the overall productive capacity among an increasednumber of producers reduces their market power, thus decreasing their aggregate profits.

Moreover, it is worth remarking that the employed adaptive learning algorithm alwaysconverges toward competitive solutions identified as Nash solutions. This repeated evidencesupports the proposed learning-in-games approach. Indeed, the long run solutions convergeto the classical game-theoretical competitive solutions. Different rates and levels of conver-gence may occur depending on the specific game-structure. In particular, the convergenceis higher and faster if the majority of Nash solutions are also Pareto optima. Finally, theresults show that, under certain economic conditions, competitive solutions, i.e., Nash equi-libria, may exist where self-interested producers with linear cost functions do not offer attheir maximum capacity. This finding concerns the low demand situation with the uniformauction mechanism. In this, each seller is able to satisfy the entire demand. However, dueto the uniform auction mechanism, the most efficient producer has an incentive to offer a


0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

5

10

15

20

25

30

35

40UA case

Auction rounds

Pro

fits

Most efficient seller2nd sellerLeast efficient seller

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0000

5

10

15

20

25

30

35

40DA case

Auction rounds

Pro

fits

Most efficient seller2nd sellerLeast efficient seller

FIGURE 7. LD3 case. Profits for the three sellers in the uniform auction case (top part) and the discriminatoryauction case (bottom part). Profits have been evaluated as ensemble averages over 10,000 experiments.

quantity lower than its maximum capacity to permit trading also by the least efficient seller.The rationale of this choice is that the best strategy for the least efficient seller is to offer theresidual demand at the maximum price and that, due to the uniform pricing rule, transactionsby the two producers are cleared at the maximum price.

As a conclusion, agent-based modeling has been shown to be a suitable approach toinvestigate the nature of auction mechanisms to test market performance. This computationalapproach provides an interesting and versatile instrument for performing feasible marketdesign. New auction mechanisms and rationing rule can be implemented and validated withinthe proposed agent-based computational approach. Moreover, future research will considerproducers with nonlinear cost functions to better reproduce realistic technological constraints.Finally, alternative reinforcement learning algorithms may be adopted. In particular, theQ-learning algorithm seems an interesting and promising approach to model the learningbehavior of intelligent software agents in a repeated game scenario where intertemporalconsiderations may play an important role.

ACKNOWLEDGMENTS

The authors wish to thank the anonymous reviewers for their helpful suggestions, whichsignificantly improved the readability and correctness of the paper. This work has been


partially supported by the University of Genoa, by the Italian Ministry of Education, Univer-sity and Research (MIUR) under grants FIRB 2001 and COFIN 2004 and by the EuropeanUnion under NEST PATHFINDER STREP Project COMPLEXMARKETS.

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VON DER FEHR, N., and D. HARBORD. 1993. Spot market competition in the UK electricity industry. EconomicJournal, 103:531–546.

APPENDIX A: SOLUTION CONCEPTS FOR STRATEGIC GAMES

Let us consider an n-player game. A profile of pure strategies x = (a1, . . . , an) deter-mines an outcome of the game, which corresponds to n payoff values (�1(x), . . . , �n(x))one for each player. Therefore, each ith player (i = 1, . . . , n) evaluates its payoff�i (x) = �i (ai , a−i ), which depends on the chosen strategy ai and on the vector of purestrategies a−i played by all its opponents.


A.1. Nash Equilibrium

A specific vector of strategies x∗ = (a∗i , a∗

−i ) is a Nash equilibrium (Nash 1950; Kreps1990; Vega-Redondo 2003) if the following conditions are satisfied:

�i

(a∗

i , a∗−i

) ≥ �i

(ai , a∗

−i

), ∀ i (A.1)

In other terms, the previous formula states that x∗ is a Nash equilibrium if no player hasincentive to unilaterally change her action.

A.2. Pareto Optimum

A.2.1. Strong Pareto Optimum. A specific vector of strategies x∗ is not a strong paretooptimum (SPO) (Kreps 1990) if there exists another strategy combination x that satisfies thefollowing conditions:

�i (x) ≥ �i (x∗), ∀ i (A.2)

�i (x) > �i (x∗), ∃ i (A.3)

Previous formula can also be expressed saying that a strong Pareto optimal allocation (x∗) isone for which there cannot be any other feasible allocation (say x) such that the allocation(x) is strictly preferred by at least one player, and weakly preferred by everyone else.

A.2.2. Weak Pareto Optimum. A specific vector of strategies x∗ is not a weak Paretooptimum (WPO) (Kreps 1990) if there exists an other strategy combination x that satisfiedthe following conditions:

�i (x) > �i (x∗), ∀ i (A.4)

Previous formula can also be expressed saying that a weak Pareto optimal allocation (x∗) isone where there is no other feasible reallocation that would be strictly preferred by all agents.

APPENDIX B: MARIMON AND MCGRATTAN LEARNING ALGORITHM

Marimon and McGrattan aimed to provide a behavioral foundation for equilibrium the-ory in the theoretical context of bounded rationality (Marimon and McGrattan 1995). Theyintroduced a useful and general classification for adaptive learning algorithms, where playershave not perfect knowledge of the consequences of their actions and they need to learn theeconomic environment to determine their “best” strategy. Inside this category of learningalgorithms, they propose a distinction among three subclasses according to agents’ infor-mation of the past history play and opponent’s plays. This paper is focused on the thirdclass of algorithms proposed by their original classification, which is referred to adaptiveevolutionary learning subclass. In this framework, players have minimal information aboutthe evolution of the game. They keep track only on their own realized payoffs and on thenumber of pure strategies played by themselves on the recent past and they do not considerthe strategic consequences of their actions. They introduce three properties in the algorithmto replicate some human learning characteristics: adaptation, experimentation, and inertia.Adaptation stands for the tendency to exploit strategies, which performed better in the past,those strategies more likely will be played in the future. Experimentation corresponds to thefact that mixed strategies will always keep positive probabilities over every pure strategy. Thisimplies that all pure strategies will always have a minimum probability of being played at


every time. Inertia is a mechanism that allows players to keep constant their mixed strategiesover a certain period without updating them to better test the evolving environment. Thismechanism is conceived to have no correlation among players.

The mathematical formulation of the algorithm is the following: each seller assigns astrength to every strategy and keep memory of its value for updating according to the realizedprofits, i.e.,

Si,t

(as

i

) =

Si,t−1(as

i

) − 1

ηi,t−1(as

i

) · [(Si,t−1

(as

i

) − �i,t−1(as

i

))]if as

i = asi

Si,t−1(as

i

)if as

i �= asi

where ηi,t (asi ) is the number of times that strategy as

i was played within the period of inertiaof the ith player, whose updating value is

ηi,t

(as

i

) ={

ηi,t−1(as

i

) + 1 if asi = as

i

ηi,t−1(as

i

)if as

i �= asi .

The inertia at auction round t is determined according to the parameter ρ, which establishesthe probability of the ith player to update her mixed strategy σi,t (as

i ) at auction round t. Theupdating formula is

σi,t (asi ) =

σi,t−1(as

i

) · exp(Si,t−1

(as

i

))∑σi,t−1

(as

i

)exp

(Si,t−1

(as

i

)) with probability ρ

σi,t−1(as

i

)with probability 1−ρ.

An important feature of this algorithm is to have always a positive probability for everystrategy. This mechanism is called experimentation and is described by

σi,t

(as

i

) =

ε if σi,t

(as

i

) ≤ ε

σi,t

(as

i

)∑σi,t

(as

i

) (1 − ε) otherwise

where ε = ε · | {σi,t (asi ) ≤ ε} | , ε ∈ (0, 1). ε corresponds to the minimum probability value

that can be assigned to any pure strategy.

LEARNING OLIGOPOLISTIC COMPETITION IN ELECTRICITY AUCTIONS

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