An exact solution method for binary equilibrium problems with compensation and the power market uplift problem

Transcription

1 An exact soluton method for bnary equlbrum problems wth compensaton and the power market uplft problem Danel Huppmann a,b, Sauleh Sddqu a,c Preprnt of manuscrpt publshed n the European Journal of Operatonal Research DOI: /j.ejor We propose a novel method to fnd Nash equlbra n games wth bnary decson varables by ncludng compensaton payments and ncentve-compatblty constrants from non-cooperatve game theory drectly nto an optmzaton framework n leu of usng frst order condtons of a lnearzaton, or relaxaton of ntegralty condtons. The reformulaton offers a new approach to obtan and nterpret dual varables to bnary constrants usng the beneft or loss from devaton rather than margnal relaxatons. The method endogenzes the trade-off between overall socetal effcency and compensaton payments necessary to algn ncentves of ndvdual players. We provde exstence results and condtons under whch ths problem can be solved as a mxed-bnary lnear program. We apply the soluton approach to a stylzed nodal power-market equlbrum problem wth bnary on-off decsons. Ths llustratve example shows that our approach yelds an exact soluton to the bnary Nash game wth compensaton. We compare dfferent mplementatons of actual market rules wthn our model, n partcular constrants ensurng non-negatve profts no-loss rule and restrctons on the compensaton payments to non-dspatched generators. We dscuss the resultng equlbra n terms of overall welfare, effcency, and allocatonal equty. Keywords: bnary Nash game, non-cooperatve equlbrum, compensaton, ncentve compatblty, electrcty market, power market, uplft payments JEL Codes: C72, C61, L13, L94 MSC Codes: 90C11, 90C46, 91B26 a b d Department of Cvl Engneerng & Center for Systems Scence and Engneerng, The Johns Hopkns Unversty Internatonal Insttute for Appled Systems Analyss IIASA Department of Appled Mathematcs & Statstcs, The Johns Hopkns Unversty The authors would lke to thank Ibrahm Abada, Benjamn F. Hobbs, J. Davd Fuller, Danel Robnson, Carlos Ruz, Tue Vssng Jensen, Ercson Davs as well as several anonymous revewers for valuable comments and dscussons. 1

2 1 Introducton There are many real-world settngs where several players nteract n a non-cooperatve game wth bnary decsons, such as electrcty markets on-off decson for a power plant, transportaton and faclty locaton models Caunhye et al., 2012, engneerng Rao, 1996, as well as agrculture and land-use plannng Tóth et al., Modellng Nash equlbra between players whch face both bnary and contnuous decsons s a challengng problem Scarf, Economsts and game theorsts usually apply brute-force methods by explorng all possble combnatons and check every soluton for devaton ncentves of each player. When market-clearng prces to support a pure-strategy Nash equlbrum n the Walrasan sense do not exst, economsts suggest to use mult-part prcng Hotellng, 1938 or devate from margnal-cost prcng to a second-best market outcome, such that no player should lose money from partcpatng Baumol and Bradford, However, a canoncal approach to fnd pure-strategy Nash equlbra n bnary games does not exst. In many large-scale practcal applcatons, explorng the entre soluton space s not realstcally possble. A common approach n such cases s to lnearze the bnary decsons; the Nash equlbrum can then be computed by solvng the system of frst-order optmalty condtons, a.k.a. equlbrum modelng usng mxed complementarty problems or varatonal nequaltes, f certan assumptons on convexty of the lnearzed problem hold. Recent work seeks a trade-off between relaxaton of the complementarty slackness condtons or the ntegralty of dscrete constrants to obtan statonary ponts that are presumed to be equlbra of the orgnal problem Gabrel et al., 2012, 2013; Fuller and Çeleb, In ths work, we focus on applcatons where a relaxaton of optmalty condtons or contnuous relaxaton of the bnary decson varable lnearzaton s ether not practcal or yelds ncorrect results. Instead, we derve frst-order optmalty condtons of the contnuous varables for both states of each bnary varable and nclude those n an overall equlbrum problem smultaneously. Our method then selects the state of the bnary varable and correspondng contnuous varable whch provdes the best response for each ndvdual player. Due to the nature of a bnary game, there are many nstances where no set of strateges and no prce vector exsts that supports a Nash equlbrum n pure strateges;.e., there s no outcome where the pay-offs to each stakeholder are such that no player has a proftable devaton. Ths s due to the non-convexty ntroduced by the bnary decson varables and ndvsbltes O Nell et al., We ntroduce the noton of a quas-equlbrum to descrbe stuatons where no equlbrum exsts, but where a market operator or regulator can assgn compensaton payments n order to obtan an ncentve-compatble outcome. These payments algn the ncentves of ndvdual players wth the objectves of the overall system, such as cost mnmzaton or welfare maxmzaton. A regulator may also choose to ntervene when an equlbrum exsts but ts outcome s nferor to the soluton that a benevolent planner mght acheve. That s, the market operator may seek to mnmze the devaton from the system optmum.e., all decsons by one planner caused by the non-cooperatve game among a number of decson makers, each seekng to optmze competng objectves. Our soluton approach allows to endogenously consder the trade-off between regulatory nterventon to mprove market effcency, and the dstortons caused by these nterventons. Electrcty markets are the real-world applcaton of bnary games whch have receved the most attenton n the mathematcal optmzaton lterature O Nell et al., 2013; Lu and Hobbs, 2013; Wogrn et al., 2013; Lu and Ferrs, 2013; Phlpott et al., 2013; Bjørndal and Jörnsten, 2008; Hu and Ralph, 2007; Phlpott and Schultz, 2006; O Nell et al., A challengng problem arses from the on-off decson of power plants, whch usually ncur substantal start-up or shut-down costs and, f operatonal, 2

3 face mnmum-generaton constrants. Because power markets are usually based on margnal-cost, short-term prcng, the commtment costs.e., start-up costs are not necessarly covered by resultng market prces. As a consequence, many electrcty systems have rules that generators must be made whole or have to be n the money ;.e., they receve uplft payments to make sure that they do not lose money from partcpatng n the market. Ths s commonly referred to as a no-loss rule. However, ths may not be requred from a game-theoretc pont of vew, and thereby lead to hgher-than-necessary compensaton payments. At the same tme, there mght exst regulatons that only power plants that are actually generatng electrcty can receve compensaton the ratonale beng that t may create perverse ncentves for market partcpants to be pad to not do somethng. We wll dscuss and llustrate n a numercal example how such market rules can actually overly restrct operatonal effcency and thereby reduce welfare. The outlne of ths paper s as follows: n the next secton, we summarze current approaches to solve bnary Nash games and place our contrbuton n the context of methods appled to solve such problems n the power sector. In Secton 3, we propose an exact soluton method to solve bnary equlbrum problems. The obtaned multobjectve program explctly ncorporates the trade-off between overall effcency and compensaton payments n cases where no equlbrum exsts. Secton 4 apples our method to a power market example from the lterature to llustrate ts advantages and flexblty to ncorporate dstnct market rules regardng uplft payments. Secton 5 concludes wth a dscusson on methods, other possble applcatons, and future work. 1 2 Current approaches to solve bnary games In ths secton, we motvate our method by descrbng how current soluton methods for bnary games obtan equlbra, and we dentfy where our formulaton can mprove ths process. Whle there exst brute-force methods Avs et al., 2010; Audet et al., 2006; Von Stengel, 2002 that solve for an equlbrum consderng all possble combnatons of the bnary varables and check ex-post for devaton ncentves, we want to concentrate on mathematcal programmng technques for obtanng equlbra. For large-scale applcatons such as those consdered n ths work, computatonal effcency proves a hurdle n these brute-force methods. Solvng a large number of equlbrum problems s not very elegant and suffers from a curse of dmensonalty, because the number of equlbrum problems to be solved s 2 k, where k s the number of bnary varables. Therefore, mathematcans and Operatons Researchers are constantly lookng for ways to apply advances n Varatonal Inequaltes and Integer Programmng to develop faster methods to solve such problems. 2.1 Optmzaton and equlbrum modelng Game theory and equlbrum problems have been an ntegral part of the hstory of mathematcal programmng. Frst-order optmalty Karush-Kuhn-Tucker, KKT condtons, derved from each ndvdual player s optmzaton problem, can be solved smultaneously by stackng them to form an equlbrum problem. Interpretatons from dual varables to constrants n a game theory analyss provde essental nformaton n equlbrum problems and are often nterpreted as prces or margnal benefts for 1 The Appendx provdes computatonal results for a numercal test case usng a larger data set than the stylzed example n Secton 4. The GAMS codes for the stylzed example, the numercal test case, as well as an addtonal example for a resource market applcaton wth multple bnary nvestment decsons n producton and ppelne capacty for several player are avalable for download at under a Creatve Commons Attrbuton 4.0 Internatonal Lcense. 3

4 ndvdual players Facchne and Pang, 2003; Ferrs and Pang, 1997; Murphy et al., However, ths relatonshp between optmalty condtons and equlbrum problems fals once a game ncludes bnary decson varables. The reason s that optmalty condtons cannot be drectly derved for bnary optmzaton problems. Thus, appled researchers am to solve such optmzaton problems n other ways. A method based on a trade-off between relaxng the ntegralty and the complementarty constrants s developed by Gabrel et al Whle relaxng ntegralty has been employed as a way to solve nteger programs, relaxng complementarty essentally the optmalty condtons was the novel dea of ther contrbuton. A smlar problem s tackled by Fuller and Çeleb 2017; they propose a mnmum dsequlbrum model, defnng dsequlbrum as the dfference between the pay-off n the socally optmal outcome and the ndvdually optmal decson, summed over all players. That s, they seek to mnmze the aggregated opportunty costs for all market partcpants from followng the nstructons of a socal planner. The authors relate the MD model both to the results obtaned by a socal planner and to the model proposed by Gabrel et al One alternatve recent method to tackle bnary equlbrum problems focuses on solvng ntegral Nash-Cournot games Todd, 2014 and provdes an effcent algorthm to obtan equlbra. Ths method works very well for a specfc nteger game wth no constrants, but the algorthm s not applcable to the broad class of bnary-constraned games consdered n ths paper. 2.2 Dual varables n bnary programs As mentoned above, dual varables n constraned convex optmzaton contan useful nformaton both for computatonal purposes and nterpretaton of the problem under consderaton. However, n mathematcal programs wth bnary or dscrete constrants, the nterpretaton of dual varables as margnal relaxaton s not vald because of the non-convex and dsjont feasble regon. Ths s related to the dffculty of determnng the value functon of the orgnal problem Guzelsoy and Ralphs, To overcome ths caveat and obtan dual varables n such cases, the followng approach s often used cf. O Nell et al., Consder the general constraned problem: f x, y mn x,y s.t. g x, y 0, where x { 0, 1 }n, y R m 1 To obtan dual varables to the constrants gx, y, ths problem s commonly solved n a two-step procedure: frst, the orgnal problem 1 s solved usng nteger programmng technques; then, the bnary varables x are lnearzed,.e., the orgnal problem s replaced by the followng: f x, y mn x,y s.t. g x, y 0, where x [ 0, 1 ]n R n +, y R m 1 lnear Fnally, constrants are added to fx these varables at the level determned to be optmal, x, n the frst step: f x, y mn x,y s.t. g x, y 0 λ 2 x = x µ, where x, y R n+m Solvng the reformulated problem 2 allows to nterpret the dual varables λ, µ n the sense of multplers or shadow values; offerng these prces as contracts to market 4

5 partcpants yelds a Nash equlbrum. The dual varables µ are not part of the orgnal problem, they are obtaned from the lnearzaton and can be thought of as a prce [... ] representng the ntegral actvty for each agent O Nell et al., 2005, p These duals are also mportant for nteger programs, so that most numercal solvers automatcally report these values when solvng mxed-nteger programs. However, one must be careful when usng ths approach n practcal applcatons, as these duals cannot be readly nterpreted as margnal relaxatons of the orgnal bnary model that s, the margnal value λ of the lnearzed fxed program cannot be nterpreted as dual to the constrant of the orgnal, mxed-nteger program problem 1. Ths s, however, what many power markets are currently dong n practce: they use the dual varable to the energy balance constrant as locatonal margnal prce and clear the market based on these pay-offs. The dual prces of the bnary actvtes µ are neglected. Instead, market operators assgn compensaton payments to make whole ndvdual generators after the fact. 2.3 Uplfts, compensaton, and equlbra n power markets There already exsts a substantal breadth of Operatons Research lterature wth regard to electrcty markets and prcng n non-convex problems, and bnary games are a prevalent concern n ths area. The current practce n many centrally dspatched power markets s that, frst, the welfare-optmal dspatch s computed by the Independent System Operator ISO and locatonal margnal prces LMP n the network are determned usng the two-step approach outlned above. Compensaton to ndvdual players are then calculated after market-clearance to ensure that no market partcpant ncurs fnancal losses based on these prces. These are often called uplft, make-whole payments or bd cost recovery, though actual mplementatons and rules dffer across markets. System operators usually have non-confscatory compensaton rules Soshans, Ths means that they do not assgn penaltes for devaton, but only dsburse postve compensaton payments. In that respect, current market operaton devates from contracts T proposed by O Nell et al. 2005, whch are derved from all duals λ, µ. Instead, standard compensaton payments are based on the pay-offs from LMPs the dual varable or vector λ only, n partcular the duals to the nodal energy balance constrant. It s mportant to note that these two are not equvalent. Ths approach does not actually guarantee that the ncentves of all players are algned n the resultng market outcome, because the nature of the non-cooperatve bnary game between market partcpants s sde-stepped. Generators that are not dspatched by the ISO may have an ncentve to enter the market, f they earned postve profts gven observed market prces, or to devate from the announced schedule. Some markets allow self-schedulng, whch gves generators the opton to determne ther dspatch ndvdually rather than surrenderng ther generaton decson to the ISO cf. Soshans et al., An alternatve to the current approach s the mnmum uplft or convex hull prcng method, whch reles on a convex approxmaton of the lower bound of the aggregate cost functon to derve prces and the mnmal uplfts to support the market outcome Schro et al., 2015; Grbk et al., 2007; Hogan and Rng, Ths method acknowledges that compensaton s requred to deter generators from followng proftable devatons from the dspatch chosen by the ISO. Alas, usng the convex hull relaxes the ntegralty of the underlyng problem, and therefore also does not solve for the exact soluton to the non-cooperatve game between generators. An mportant problem of the two-step approach arses from the fact that the budget for necessary compensaton payments s not consdered when determnng the dspatch, but only computed ex-post. Ths neglects the potental trade-off between effcent market operaton and mnmzng the budget requred for compensaton payments, 5

6 whch s usually funded from fees or leves on market partcpants. These fees may n turn cause dstortons n the market. It s easy to conceve of stuatons where acceptng a slght reducton n market effcency.e., lower welfare, hgher costs for dspatch allows to sgnfcantly reduce the compensaton payments requred. The llustratve example n Secton 4 shows just such a stuaton. The method developed n ths work tackles these caveats of current approaches and proposes an exact soluton method for games n bnary varables. Our method offers an mportant practcal advantage: t allows to drectly balance effcent market operaton based on an exact method for fndng solutons to bnary equlbrum problems, on the one hand, wth the amount of compensaton payments to ensure that these outcomes are stable aganst devaton by ndvdual players, on the other. 2.4 Margnal relaxaton vs. the loss from a bnary devaton There s a further caveat of usng the duals of problem 2 for algorthms and economc nterpretaton of results: ths approach ntroduces the dual vector µ as the margnal relaxaton of the constrant that fxes x at ts optmal value. However, t s more approprate to ask not about a margnal relaxaton, but a swtch from one possble value of the bnary varable to the other. We ntroduce the swtch value κ as the beneft or loss ncurred by swtchng from one soluton to the bnary problem fx, y to the optmal value of the objectve functon gven that the bnary varable takes the other value, x = 1 x. Here, y s chosen so as to mnmze fx, y,.e., y = arg mn y fx, y, and y s determned equvalently. Then, κ can be determned by computng: κ = f x, y + f x, y. If κ s strctly postve, swtchng n the bnary varable from x to x ncurs a loss of κ; hence, x s the optmal decson. If κ = 0, the objectve values are dentcal and the player s ndfferent between the two optons. When x {0, 1} n s a bnary vector rather than a one-dmensonal varable, the swtch value can be computed by comparng the objectve value for a possble realzaton x to the outcome for all other permutatons S{0, 1} n of the bnary vector and choosng the most benefcal mnmal alternatve: κx = f x, y + mn f x, y x S{0,1} n \x As before, f κ s strctly postve, ths mples that x s optmal, and κ = 0 means that there s at least one alternatve n the bnary decson vector wth the same objectve value. Ths formulaton stll requres comparng the objectve values of 2 n alternatves and solvng for the optmal level of the contnuous varables y n each case. Hence, ths approach may not seem lke an mprovement. The bg advantage wll become apparent n settngs where multple players I = {1,..., p} nteract and one solves for an equlbrum between them. A brute-force approach would requre to solve all permutaton across players and ther optons n bnary varables 2 pn. Buldng on the approach dentfed above, ths can be transformed to a mult-objectve optmzaton problem wth p 2 n optons. We wll dscuss the analytcal propertes n subsequent sectons and present a numercal analyss usng a larger-scale dataset n the Appendx. In the method proposed below, we use ths noton of a swtch value κ to choose between equlbra n such games wth bnary decsons. Ths varable also serves as a selecton mechansm n such cases where no bnary equlbrum exsts; t can then be used as a soluton strategy to fnd an approprate quas-equlbrum. Ths approach 6

7 holds promse wth regard to algorthmc advances of bnary and nteger programmng, as well as allow a better representaton of real-world problems n economcs, engneerng, and beyond. 3 An exact soluton for bnary equlbrum problems We now turn to our exact soluton method to solve an equlbrum problem wth bnary varables. The core dea for our approach s as follows: for each player, we derve the frst-order optmalty condtons wth respect to the contnuous decson varables for each state of the bnary varable. In addton, we formulate an explct ncentvecompatblty constrant to ensure that each player chooses the state of the bnary varable that s most benefcal to her. For ease of notaton and formulatng a concse and smple exposton of our approach, we drop the ndex on the bnary varable and descrbe the method n the case where each player has exactly one bnary decson varable, whle the number of contnuous decson varables and constrants s arbtrary. Nevertheless, the approach works for any problem wth a fnte number of bnary decson varables. To llustrate ths feature, the electrcty market example presented n the followng secton has multple bnary decson varables per player. The game s defned by a set of players I = {1,..., p}, where each player seeks to mnmze an objectve functon f. In the followng formulaton, each player has a vector of contnuous decson varables y R m, bnary decson varable x {0, 1} and a set of k constrants g : R m {0, 1} R k wth a vector of length k of assocated dual varables λ. As elaborated n the prevous secton, these dual varables are only meanngful for a fxed x. The feasble regon of each player s denoted by K = { x, y g x, y 0 }. Each player s optmzaton problem reads as follows: mn f x, y, y x 3a x {0,1},y R m s.t. g x, y 0 λ 3b The vector y = y j j I\{} s the collecton of all rvals decsons n contnuous varables, and thus s of dmenson m p 1. The set of feasble strateges by the rvals s K = j I\{} yjx j. Because the contnuous varables of the rvals depend on ther bnary decsons, K s usually parwse dsjont and non-convex. The formulaton mplctly assumes that each player s pay-off s only affected by the contnuous decson varables of her rvals, but not drectly affected by ther bnary varables. Ths s a smplfcaton only for notatonal convenence and can easly be relaxed. A Nash equlbrum to ths game s a set of strateges such that each player chooses an optmal strategy gven the acton by the rvals. Ths s equvalent to the noton that no player has an ncentve to unlaterally change her decson upon observng the decsons of the rvals; there exsts no proftable devaton. Ths s formally defned below; we dstngush between devaton ncentves n the bnary and the contnuous varables to facltate the exposton. Defnton 1 Nash equlbrum n a bnary game. We defne the bnary game as a set of players I, each seekng to solve an optmzaton problem as gven by problem 3. A Nash equlbrum to ths game s a vector x, y K such that y s the optmal decson.e., best response by player gven x and y x, f x, y, y x f x, y, y x y { y g x, y 0 } I 4 7

8 and such that there s no proftable devaton wth regard to the bnary varable, f x, y, y x f x, y, y x I, 5 where x s the alternatve value of x,.e., x = 1 x, and y s a best response of player under the assumpton that x = x,.e., f x, y, y x f x, y, y x { y y g x }, y 0 I. 6 Because exstence or unqueness of equlbra cannot be guaranteed n bnary games, we need to devse a method to select among several outcomes, or to arrve at a desred pont whch s almost an equlbrum. For ths purpose, we ntroduce a market operator, as a coordnaton agent and equlbrum selecton mechansm. Ths entty s modeled as the upper-level player wthn a herarchcal, two-stage setup, where the lower-level constrants represent the bnary equlbrum problem. She gudes the players towards a desrable outcome and assgns compensaton payments f necessary. We formally ntroduce the term quas-equlbrum for solutons to the bnary game that are not Nash equlbra accordng to Defnton 1, but where ncentve-compatblty can be ensured wth approprate compensaton payments. Defnton 2 Quas-equlbrum n a bnary game wth compensaton. We defne the bnary game wth compensaton as a set of players I, each seekng to solve an optmzaton problem as gven by problem 3. A bnary quas-equlbrum to ths game s a vector x, y K and a compensaton vector ζ R + such that for each player: 1. y s the optmal feasble decson.e., best response by player gven x and y x, f x, y, y x f x, y, y x y {y g x }, y 0 I, 7 2. no player can mprove her own pay-off by devatng from x by more than the compensaton payment ζ ;.e., the compensaton s at least as great as the beneft from devaton wth regard to the bnary varable. Hence, there s no proftable devaton wth regard to the bnary varable gven the compensaton payment, f x, y, y x ζ f x, y, y x I 8 where x and y are defned as n Defnton 1, 3. and the compensaton payments are mnmal,.e., f a compensaton payment s requred for a player, then the ncentve-compatblty condton 8 holds wth equalty. That s, ζ = mn ζ R + ζ s.t. f x, y, y x ζ f x, y, y x I. 9 Note that when ζ = 0, the bnary quas-equlbrum s also a Nash equlbrum n a game wthout compensaton. In the defnton of the quas-equlbrum, we drectly ncorporate the noton that the compensaton payments should be mnmal. 8

9 Ths s helpful because t elmnates those ncentve-compatble solutons where the market operator over-compensates some players, and t allows to focus on a smaller set of canddate solutons Determnng each player s best response In the defntons above, we have smply stated that the contnuous decson varables y are optmal for player gven the bnary varable and the rvals decsons. In order to effcently compute ths best response of each player, we use frst-order optmalty condtons wth regard to the contnuous decson varables. Hence, we need to make sure these condtons are necessary and suffcent so that we can capture the entre equlbrum set. An assumpton on compactness s also needed for the selecton of certan parameters of our method. A1 Assume that for each player I, problem 3 s such that the frst-order optmalty KKT condtons are necessary and suffcent wth respect to the varables y, and the feasble regon defned by the constrants g x, y s compact and non-empty, for a fxed realzaton of x and for any fxed feasble strategy by the rvals y K. As an example, the KKT condtons are necessary and suffcent for problem 3 f f x,, y are convex and g x, affne for any fxed value x {0, 1} and any fxed vector y K. Let the vector x, y denote the best response for each player wthn the overall problem, gven the decson vector y x by all rvals, and let ỹx denote the best response of player for a fxed x = x {0, 1}. Then, the objectve value f x, ỹ x, y x s the best pay-off that a player can do gven x and the rvals strateges. Under Assumpton A1, f the value of x s fxed at x, the best response ỹ x can be found by solvng the respectve frst-order optmalty condtons: 0 = y f x, ỹ x, y x + λ x y g x, ỹ x, ỹ x free 10a 0 g x, ỹ x λ x 0 10b Player wll choose the bnary varable x such that ts objectve value s mnmal gven the decsons of the rvals y x. Mathematcally, the best response of player regardng her bnary varable x can be wrtten as follows: f 1, ỹ 1, y x < f 0, ỹ 0, y x x = 1 11a f 1, ỹ 1, y x > f 0, ỹ 0, y x x = 0 11b f 1, ỹ 1, y x = f 0, ỹ 0, y x x = {0, 1} 11c The logc of condtons 11 s smlar to the noton of ncentve compatblty n game theory,.e., there exsts no proftable devaton gven the decsons of all rvals. Hence, a vector x, y x that satsfes the ncentve-compatblty constrants n Defnton 1 for each player consttutes a Nash equlbrum. If the ncentve-compatblty condton s not satsfed for any feasble strategy, t may be necessary to fnancally compensate a player to ensure that she doesn t devate, as stated n Defnton 2. A drect mplementaton of the mplct f-then -logc requres addtonal bnary varables and thereby consderably ncreases numercal complexty. We overcome ths drawback by proposng a mathematcally equvalent formulaton usng the orgnal 2 For games where the ndvdual players optmzaton problems are non-convex wth contnuous varables, Pang and Scutar 2013 ntroduce the noton of a quas-nash equlbrum to descrbe solutons that are statonary ponts derved from relaxed constrant qualfcatons. In the defnton used here, we are lookng at a dstnct concept of an equlbrum. 9

10 bnary varables of the players. The resultng overall program wll be shown n problem 14; but frst, we wll dscuss the reformulaton and ntroduce the equlbrum selecton mechansm n more detal. 3.2 An effcent formulaton of ncentve compatblty We ntroduce four non-negatve varables κ 1, κ 0, ζ 1, ζ 0 for each player, and a suffcently large scalar or vector of scalars K. The vector κ x s the swtch value ntroduced n Secton 2.4; t can be nterpreted as the loss the player would ncur by swtchng from her optmal value of the bnary varable to the alternatve. The vector ζ x denotes compensaton payments to guarantee ncentve compatblty; n cases where the market operator requres a player to act aganst her own objectves, ths payment ensures that the player does not have a proftable devaton. The scalar K must be large enough so that t does not nadvertently constran the varables κ 1, κ 0, ζ 1, ζ 0. Snce these are dfferences n objectve functon values, ths mples that K must be larger than the sze of the range of f. By Assumpton A1, each f s contnuous over a compact feasble regon, thus acheves both ts maxmum and mnmum wthn the feasble regon. An effcent technque to choose K s to lnearze the bnary varables n the ndvdual optmzaton problems and mnmze and maxmze over f to fnd the largest dfference possble. Note that the role of K here and n the subsequent sectons s to enforce the dsjuncton between two choces of the bnary varable x. It s the dsjunctve constrants formulaton ntroduced by Fortuny-Amat and McCarl The vectors κ x and ζ x are not dual varables n the orgnal sense, but they do contan smlar nformaton regardng the soluton. Hence, they are analogous n nterpretaton to a dual but n terms of a bnary devaton, not n the sense of a margnal relaxaton. Alas, the term shadow prce often used n economcs as synonymous for dual varables could also be appled here. We can now replace the ncentve compatblty condtons equatons 11 by a more effcent formulaton: 1 1 f 1, ỹ, y + κ ζ 1 κ 0 + ζ 0 0 = f 0, ỹ, y 12a κ 1 + ζ 1 x K 12b κ 0 + ζ 0 1 x K 12c κ 1, κ 0, ζ 1, ζ 0 R + The market operator selects a soluton such that the frst-order optmalty condtons and the ncentve-compatblty constrants are satsfed for all players. In lne wth the defnton of the quas-equlbrum as the mnmal compensaton payment for each player, the varables κ x and ζ x cannot both be strctly greater than zero at a soluton; ths wll be shown after we formally ntroduce the market operator. Let us llustrate and dscuss the nterpretaton of the varables κ x and ζ x n more detal. The queston s whether the soluton for the overall equlbrum problem chosen by the market operator s algned wth the best response of each player. By ths, we mean whether a player s ndvdually optmal decson concdes wth the quasequlbrum chosen by the market operator. There are fve possble outcomes regardng the ncentve algnment of an ndvdual player and the market operator; the cases are llustrated n Table 1. In cases I and II, the ncentves are algned, as the player would ncur losses a strctly worse pay-off by devatng from the outcome decded by the market operator. The respectve swtch value κ x s strctly postve. In cases III and IV, the soluton chosen by the market operator s not n lne wth the player s ndvdual best response; only by dsbursng compensaton payments can the market 10

11 operator convnce the player not to devate to the ndvdually optmal decson. As a consequence, the respectve compensaton payment ζ x s strctly postve, and a quas-equlbrum s realzed. In the last case no. V, the player s ndfferent between her optons, so the market operator s not restrcted n selectng ether outcome. The player does not have a postve swtch value n ether drecton, and no compensaton s requred. 3.3 Translatng each player s best response nto the overall game From equatons 10, we have obtaned two optmal decson vectors, ỹ x, for each player for both values that the varable x can take. We now need to translate whch of these two decson varables s realzed n the quas-equlbrum and seen by the rvals n ther own optmzaton problem: ỹ 0 x K y ỹ 0 + x K 13a ỹ 1 1 x K y ỹ x K 13b The logc of constrants 13 s straghtforward: the decson vector y, as t s consdered by the rvals and the market operator n ther optmzaton problems, must be equal to the optmal decson ỹ x for whchever value of x s the soluton n the quas-equlbrum,.e., x = 0 y = ỹ 0 and x = 1 y = ỹ 1. The parameter K must be chosen sutably large so as not to constran the contnuous decson varables. Ths mples that K must be larger than the sze of the doman of y. As argued before, each y s contnuous and, by Assumpton A1, over a compact feasble regon. A sutable value for K can be found by lnearzng the bnary varables n the ndvdual optmzaton problems and mnmze and maxmze over y to fnd the largest dfference possble. We can now combne the ncentve-compatblty constrants 12 wth the equlbrum condtons 10 for the contnuous decson varables nto one set of constrants. The non-lnearty of the complementarty condtons 10 can be readly reformulated applyng dsjunctve constrants Fortuny-Amat and McCarl, 1981 or usng SOS type 1 varables Sddqu and Gabrel, A mult-objectve program subject to bnary quas-equlbra So far, we have only replaced a number of equlbrum problems for each possble combnaton of bnary varables by a set of nteger constrants that exactly represent all bnary quas-equlbra. Next, we can apply mult-objectve programmng to drect the game towards desred solutons. To ths end, we ntroduce the market operator, and we assume that she seeks to mnmze an objectve functon consstng of two terms: a functon F, whch only depends on the actual market outcome effcency of the ndvdually equlbrum x chosen ncentves case optmal x by market operator κ 1 κ 0 ζ 1 ζ 0 algned I 1 1 > yes II > yes III > 0 0 no IV > 0 no V ndfferent 1 / yes Table 1: Incentve algnment between a player s ndvdually optmal decson her best response and the quas-equlbrum chosen by the market operator 11

12 soluton; cost-mnmzaton or welfare-maxmzaton may be a natural choce for ths term and a functon G, whch serves as a regularzer. Parameterzng ths functon approprately allows to weght between the dfferent terms; n economc applcatons, t can be nterpreted as a penalty term that seeks to mnmze the compensaton payments requred to ensure ncentve compatblty of the market soluton. The overall problem s a mathematcal program subject to a bnary equlbrum problem, where x = {0, 1} are the bnary varables n the lower-level problem. mn x,y,ỹ x, λ x κ x,ζ x x, F y s.t. y f 1, ỹ 1, y y f 0, ỹ 0, y ζ x + G + λ1 + λ0 14a T y g 1, ỹ 1 = 0 14b 0 g 1, ỹ 1 1 λ 0 14c T y g 0, ỹ 0 = 0 14d 0 g 0, ỹ 0 0 λ 0 14e f 1, y 1, y + κ 1 ζ 1 κ 0 + ζ 0 = f 0, y 0, y 14f κ 1 + ζ 1 x K 14g κ 0 + ζ 0 1 x K 14h ỹ 0 x K y ỹ 0 + x K 14 ỹ 1 1 x K y ỹ x K 14j x {0, 1}, y, ỹ x R 3m, λ x,κ x, ζ x R 2k+4 + It s mportant to note that the bnary varable of each player has an addtonal role n ths formulaton: t also controls whch of the potental states wth regard to the contnuous varables are actve and vsble to the rvals constrants 14 and 14j. Furthermore, t ensures that the correct swtch values and compensaton payments are actve constrants 14g and 14h, n lne wth Table 1. Theorem 1 Exact solutons of the bnary Nash game. Under Assumpton A1, any vector x, y s a soluton to the bnary game n Defnton 1 f and only f there exsts a vector ỹ x, λ x, κ x, wth ζx = 0 I, such that x, y, ỹ x, λ x, κ x s a feasble pont to problem 14. Proof. Frst, assume x, y, ỹ x, λ x, κ x wth ζx = 0 I s a feasble pont to problem 14. Then, by Assumpton A1, we know that the pont x, y s an optmal soluton for each player gven fxed values of x and y I. Ths satsfes the frst part of Defnton 1. Furthermore, we know that ζ x = 0 I, and κ x wll be selected accordng to the constrants of problem 14. By these constrants, we know that f x, y, y x f x, y, y x I, where x s the alternatve value of x.e., x = 1 x and y s a best response of player,.e., f x, y, y x f x, y, y x y { y g x, y } 0 I. Ths satsfes the second part of Defnton 1 and thus we have shown that x, y s a soluton to the bnary game defned by Defnton 1. Now, we assume that x, y s a soluton to the bnary game defned by Defnton 1. Choose K large enough so that t s greater than the dfference between any upper and lower bounds on y I and greater than the dfference between the mnmum and maxmum 12

13 value of f I. Such a value exsts snce by Assumpton A1; the feasble set s compact and f s contnuous, so the maxmum and mnmum must exst. Then, by Defnton 1, for any fxed value of x and y, y s an optmal soluton to the ndvdual player s optmzaton problem. Thus, you can fnd ỹ x, λ x such that x, y, ỹ x, λ x satsfes the constrants 14b 14e. Take ζ x = 0 I, and choose κ x accordng to constrant 14f. Thus, for any soluton to the bnary game n Defnton 1 gven by x, y, we have shown that there there exsts a vector ỹ x, λ x, κ x, wth ζx = 0 I, such that x, y, ỹ x, λ x, κ x s a feasble pont to problem 14. The next theorem shows that the method can also be appled to obtan a quasequlbrum. Note that we need an assumpton on the objectve functon before we can prove that our method can obtan a quas-equlbrum. A2 Assume that F and G are convex quadratc or lnear functons for every fxed bnary varable x and that G / ζ > 0 I. Theorem 2 Exact solutons of the bnary Nash game wth compensaton. Under Assumptons A1 and A2, f there exsts a vector x, y, ỹ x, λ x, κ x, ζ x that s an optmal soluton to problem 14, then the vector x, y wth compensaton ζ s a soluton to the bnary game as stated n Defnton 2. Followng the term ntroduced n Defnton 2, we refer to ths as a bnary quas-equlbrum. Furthermore, under Assumptons A1 and A2, f x, y s a soluton to the bnary game wth compensaton ζ as stated n Defnton 2, then there exsts a vector ỹ x, λ x, κ x, ζ x, such that x, y, ỹ x, λ x, κ x, ζ x s a feasble pont to problem 14. Proof. Frst, assume x, y, ỹ x, λ x, κ x, ζ x s an optmal soluton to problem 14. Then, by A1, we know that the pont x, y s an optmal soluton for each player gven fxed values of x and y. Ths satsfes the frst part of Defnton 2. Furthermore, we know that κ x, ζ x I wll be selected accordng to the constrants of problem 14. By these constrants, we know that f x, y, y x ζ f x, y, y x I, where x s the alternatve value of x.e., x = 1 x and y s a best response of player wth fxed x,.e., f x, y, y x f x, y, y x { y y g x, y } 0 I. Ths satsfes the second part of Defnton 2. By Assumpton A2, we know that G / ζ > 0 I and, hence, for any optmal soluton, for each player, ζ s mnmal. Ths satsfes the thrd part of Defnton 2 and hence we have shown that f x, y, ỹ x, λ x, κ x, ζ x s an optmal soluton to problem 14, then x, y wth compensaton payments ζ s a soluton to the bnary game defned by Defnton 2. Now, we assume that x, y wth compensaton payments ζ s a quas-equlbrum to the bnary game wth compensaton defned by Defnton 2. Choose K large enough so that t s greater than the dfference between any upper and lower bounds on y I and greater than the dfference between the mnmum and maxmum value of f I. Such a value exsts snce by Assumpton A1, the feasble set s compact and f s contnuous, so the maxmum and mnmum must exst. Then, by Defnton 2, for any fxed value of x and y, y s an optmal soluton to the ndvdual player s optmzaton problem. Thus, you can fnd ỹ x, λ x such that x, y, ỹ x, λ x satsfes the frst two constrants n problem 14. Calculate ζ x from ζ I and choose κ x accordng to the thrd constrant n problem 14. Thus, for any soluton to the bnary game n Defnton 2 gven by x, y and compensaton payment ζ, we have shown that there there exsts a 13

14 vector ỹ x, λ x, κ x, ζ x, such that x, y, ỹ x, λ x, κ x, ζ x s a feasble pont to problem 14. Corollary 1. Under Assumptons A1 and A2, any vector x, y s a bnary quas-equlbrum for the Nash game n bnary varables wth compensaton ζ as defned n Defnton 2 f there exsts a vector ỹ x, λ x, κ x, ζ x, such that x, y, ỹ x, λ x, κ x, ζ x s a feasble soluton to problem 14 and ζx s mnmal as defned n Defnton 2. Proof. By the arguments n Theorem 2, for any pont x, y, ỹ x, λ x, κ x, ζ x that s feasble to problem 14, the vectors x, y and ζ satsfy the frst two condtons of Defnton 2. If, n addton, Condton 3 of Defnton 2 s satsfed,.e., ζ s mnmal for each player I, then x, y s a bnary quas-equlbrum wth compensaton ζ. If a vector s a global mnmum to the objectve functon 14a, ths s the bnary quas-equlbrum wth the lowest objectve functon value F + G. The followng lemma and theorem provde condtons for the exstence of a bnary quas-equlbrum that can be supported by approprate compensaton payments. Lemma 1 Exstence of a Nash equlbrum n a game wth fxed bnary varables. If for a fxed vector x, the objectve functon f x, y, y x of every player I s contnuous wth regard to y and y x, and quas-convex n y, and the feasble regon defned by the constrants g x, y s compact, convex and non-empty, then the resultng contnuous game has a soluton. Proof. The exstence follows from Kakutan s fxed-pont theorem Glcksberg, Relaxatons of these condtons for the exstence of a Nash equlbrum n contnuous decson varables are also dscussed n the lterature Facchne and Pang, 2003; Nshmura and Fredman, The exstence result for Nash equlbra n contnuous games gven fxed bnary varables n Lemma 1 can be combned wth Theorem 3 to provde reasonable condtons for the exstence of bnary quas-equlbra. Theorem 3 Exstence of a bnary quas-equlbrum. Under Assumpton A1 and A2, f for any fxed vector x, the resultng contnuous game has a soluton, then a correspondng bnary quas-equlbrum exsts for the Nash game n bnary varables. Proof. For any y that s a Nash equlbrum gven the fxed vector x, we can fnd a vector ỹ x, λ x such that x, y, ỹ x, λ x s a feasble pont to constrants 14b 14e. Recall that x = {x, 1 x }. Choose K as n the proof for Theorem 2. We can then fnd values the vector for κ x, ζ x such that x, ỹ x, λ x, κ x, ζ x satsfy constrants 14f 14j, and where ether κx = 0 or ζ x = 0 for every player I. By equaton 14f, ths mples that ζ x s mnmal. By Corollary 1, x, y wth compensaton payments ζ s a bnary quas-equlbrum. Theorem 3 mples that, f a Nash equlbrum soluton to the contnuous game exsts for any fxed realzaton of the bnary varables, then ths soluton can be supported as a quas-equlbrum wth approprate compensaton payments. The reformulated mult-objectve problem subject to a bnary quas-equlbrum 14 s a mxed-nteger program. However, the ncentve-compatblty constrant condton 14f can stll cause numercal dffcultes, because the players objectve functons 14

15 are often not lnear and not even convex n terms of all varables, even when they are lnear from the pont of vew of the player tself. We wll now ntroduce a specal case, whch allows to reduce problem 14 to a lnear or quadratc convex mxed-nteger program wth lnear constrants. A3 Assume that each lower-level player s objectve functon f x, y, y x can be separated nto two functons, where the frst part s lnear wth respect to x and y, and the second part s lnear only wth respect to y, f x, y, y x = f x x, y x + f y y y x, and the partal dervatve of the objectve functon wth regard to the contnuous varable y, y f y x y y x, s lnear n all varables. Furthermore, assume that all constrants g x, y are affne and can therefore be wrtten as: g x, y 0 ax + A y b where a, b are vectors and A a matrx of sutable dmensons. Ths assumpton mples that the functons f y y y x need not be lnear wth respect to all varables, only wth regard to the player s own contnuous decson varable y. One consequence of Assumpton A3 s the caveat that the followng theorems are not drectly applcable to Nash-Cournot equlbrum models, where a player s aware of ts own mpact on the fnal demand prce. These models are usually formulated such that a player faces an objectve functon of the form max y py y, where py s the nverse demand curve; ths volates the lnearty requrement for the reformulaton. However, we only need Assumpton A3 to prove Theorem 4 below, whch allows us to wrte the problem as a mxed-nteger quadratc program, and obtan global optmalty results. Whenever Assumpton A3 does not hold, as n general game-theoretc settngs, the reformulaton ntroduced above as well as Theorems 1 and 2 are stll applcable, but we cannot mathematcally prove global optmalty of a numercal soluton. If we can show that problem 14 can be solved to optmalty wthout Assumpton A3, our method wll obtan globally optmal results to the bnary equlbrum problem. However, ths wll requre novel research nto general game-theoretc settngs as well as nonlnear, mxed-nteger optmzaton problems, both of whch we plan to address n future work. We proceed wth the theoretcal results below to justfy the general settng of the power market uplft problem, whch s the topc of ths paper. Theorem 4 Exact reformulaton as a mxed-nteger lnear/quadratc program wth lnear constrants. Under Assumptons A1, A2 and A3, the mult-objectve program subject to a bnary quas-equlbrum problem 14 can be reformulated as a quadratc nteger program wth lnear constrants. Theorems 1 and 2 reman vald. Proof. By Assumptons A1, A2 and A3, the objectve functon s lnear or convex quadratc, and the frst-order condtons and the players constrants are lnear. The complementarty condtons 14c and 14e can be reformulated usng dsjunctve constrants Fortuny-Amat and McCarl, In the next steps, we wll show how the ncentve-compatblty constrant 14f can be reformulated as a lnear constrant. Frst, by assumpton A3, the functon f y x y y x s lnear wth respect to y x. We know that all lnear functons can be wrtten as the product of ther partal dervatve and the varable wth whch the partal dervatve was taken. More specfcally, ths can be wrtten as: f y x y y x = y f y x y y x T x y. 15

16 By frst-order optmalty, y f y x y y x λ x = T y g x, y x. Then, applyng the defnton of g n Assumpton A3, y g x, y x = A, and usng the complementarty condton of constrants 14c and 14e, a x + A y b T λ = 0, the reformulaton proceeds as follows: f x, y, y x = f x x, y x + f y x y y x = f x x, y x + y f y x y y x T x y = f x x, y x λ x T y g x, y x T x y = f x x, y x λ x T T x A y = f x x, y x λ x T b a x Therefore, problem 14 can be reformulated as a lnear/quadratc convex nteger program wth lnear constrants. Wth ths theorem, we show that our approach can be appled to a large number of problem classes and stll be solved as a mxed-nteger lnear program. These nclude operatonal constrants such as capacty bounds or mnmum generaton requrements, and market forms ncludng lnear nverse demand functons. 3.5 Comparng the bnary equlbrum to a socal-planner model As a benchmark for comparng the bnary-equlbrum method to commonly used approaches, we apply the followng method: frst, solve the welfare-maxmzng problem assumng a central planner, then derve prces from the lnearzed model usng the O Nell et al method, and fnally compute compensaton payments for each player that has a proftable devaton, to ensure that every market partcpant s at last ndfferent and the outcome s stable n the Nash sense;.e., no proftable devaton exsts gven rvals actons and market prces. Ths method s, n a way, a lexcographc soluton approach to the overall problem; mathematcally, t can be seen as a herarchcal mn-mn problem. For llustraton, we formulate ths problem along the notaton used n the dervaton of the bnary equlbrum method: mn ζ s.t. ζ G + mn x,y x, F y s.t. g x, y 0 I f x, y, y x ζ f x, y {, y x x, y arg mn f x, y, y x g x, y } 0 x,y I Throughout the followng dscusson, we wll denote the optmal soluton of the nner welfare optmzaton of problem 15 as x, y, whle ζ s the mnmal compensaton payment to guarantee ncentve compatblty for all players. Smlar as n the defnton of the bnary game ntroduced earler, the objectve functon of player evaluated at x, y s the best a player can do by devatng cf. Defntons 1 and 2. In the context of ths example, devaton s to be understood as not followng the decson of the benevolent socal planner. For comparson, we restate the mult-objectve program subject to a bnary quas

17 equlbrum problem 14 n smplfed form: ζ x, mn G + F y ζ y f x, ỹ x, y + λx T y g x, ỹ x 0 g x, ỹ x s.t. = 0 λ x 0 f 1, y 1, y + κ 1 ζ 1 κ 0 + ζ 0 = f 0, y 0, y translaton constrants 14g 14j 14 I In the followng dscusson, the optmal values of each player s bnary and contnuous decson varables n equlbrum and the compensaton payments are denoted by x, y, ζ. The key dfference between the above mathematcal programs s that problem 15 solves an optmzaton whle problem 14 solves for a noncooperatve equlbrum among all players. Thus, problem 15 s constraned by general power market constrants whle problem 14 has addtonal equlbrum constrants. Clearly, problem 14 s a restrcted verson of problem 15 whenever there are no compensaton payments or when the objectve functon does not nclude a penalty term G ζ. The followng two theorems formalze the dea that the socal planner problem s a less constraned verson of the bnary equlbrum problem. Frst, we show that f the socally optmal outcome x, y does not requre any compensaton payments, ths soluton weakly domnates the bnary equlbrum outcome. Second, a smlar argument can be made f proftable devatons exst n the socally optmal outcome, but the socal planner does not assgn a penalty for compensaton n ts objectve functon,.e., G = 0. Ths s the case f such payments are assumed to ncur no loss to overall welfare. Theorem 5. If any optmal soluton obtaned by the socal planner problem 15 does not requre compensaton payments to any player I, then the objectve value acheved by the socal planner s at least as small as soluton of the bnary quasequlbrum model,.e., ζ = 0 f x, y, y x f x, y, y x x F, y x F, y. I Proof. Assume ζ = 0 at any optmal pont for problem 15. Whenever ζ = 0 n problem 15, the feasble regon of problem 14 s a subset of the feasble regon for problem 15. Moreover, snce G ζ s an ncreasng functon of ζ, whenever ζ = 0, G ζ s at ts mnmum. Thus, the optmal objectve functon value of problem 15 forms a lower-bound to any optmal objectve functon value of problem 14. These two facts combne to show that whenever ζ = 0 at any optmal pont for problem 15, F x, y F x, y. The reasonng for Theorem 5 s qute straghtforward: f compensaton payments are not requred, the regularzer G ζ does not add anythng to the objectve value of problem 15 and the ncentve-compatblty constrants are not relevant. Then, the bnary-equlbrum model problem 14 s a more constraned verson of the nner problem of the socal-welfare maxmzng planner: the objectve functon F x, y and the constrants g x, y are present n both programs, but the bnary-equlbrum reformulaton adds further constrants. 17