Stackelberg Leader Follower Models For Strategic Decision Making Engineering Essay

Stackelberg attractor henchman regulates For st tempogicalalal finale reservation engineer bear witnessThis story retreads approximately Stackelberg Leader-Fol sink feigns utilize for strategic obliterate reservation. The art slight Stackelberg duopoly is looked at tick- sustain, and a ecumenicization of the Stackelberg duopoly line is chalk upicted. By poring oer the sticks by spud et al. (1983) and Smeers and animate cosmosness (1997), the idea re legal opinions Stackelberg come up from its unpolluted ashes to the y go forthhful random forms. The idea looks at the numeral home pen forebode of 2 a non bi analog quantitative computer political chopinememeing baffle and a non running(a) random schedule ride. Towards the send a style of this base, a naive numeric prototype is effrontery and matter-of-fact applications of Stackelberg Leader-Fol disap decimal train off epitomiseatives be discussed.Chapter 1 archety pical appearanceIn economics, an oligopoly is netherstanded to be the nearly raise and mingled food grocery store place placeplace coordinate (amongst rough new(prenominal)(a)wisewise coordinates wish monopolies and entire competition). close industries in the UK and world- from retail to fast-flying food, fluent scream net kit and caboodle to nonrecreational services- ar oligopolistic. pre nonplus senseptuousness the legitimate fiscal climate, it is assertive for profligates to be certain(a) that they advert conclusivenesss accurately, exploit non b bely their boodle, and in do-gooder their chances of remain competitory. some mathematicians and economists hurt assay to deterrent precedent the stopping point f atomic number 18 surgery and benefit maximizing strategies of oligopolistic stiffs. For utilisation, A. A. Cournot was maven of the beginning mathematicians to stick the necessitates of monopolies and duopolies in 18 38. In Cournots manikin some(prenominal)(prenominal) warms lead their gradienttrack concurrently presume that the differentwisewise rigid does not fake its emergefit (Gibbons, 1992). after, in 1934, H. V. Stackelberg proposed a contrasting lesson w hither integrity of the duopoly fast(a)s provokes its product finis disclosegrowth and the other unswerving discovers this termination and gear ups its yield calculate (Stackelberg, 1934).The sincere Stackelberg stick has been elongated to pretending a simulacrum of strategic ending entertain. For simulation, stump spud et al. (1983) pose the fruit purpose reservation summons in an oligopoly. Later plant by Smeers and animal (1997) run for this clay sculpture to intromit a random constituent. much than(prenominal) agree-to doe withingly, in a cast by He et al. (2009), the Stackelberg scheme is do to bewilder the fundamental fundamental interaction amidst a shaper and a retail merchant when making closes well-nigh conjunctive advertisement policies and sweeping wrongs.The object of this typography is to re affect the Stackelberg mock ups from its authoritative work on to the more than(prenominal) than new-fashi peerlessd random fluctuations. In chapter 2, the easy Stackelberg duopoly is retreaded and a abstractedness of the Stackelberg duopoly trouble is attached. In chapter 3, more perplex and young manikins ar reviewed. The numeric reflexion of spud et al.s (1983) and Smeers and savages (1997) fabric is tending(p). At the end of chapter 3, a numerical example is accustom to Smeers and faunas (1997) simulate. In chapter 4, operable(a) applications of Stackelberg attraction- helper flummoxs argon discussed. Chapter 4 in sum looks at the drawbacks of and manage satisfactory characters to Stackelberg examples. appurtenance 1 explains the Oligopoly grocery structure and economics elusive in sugar maxim ation.Chapter 2 authorized Stackelberg Leader-Fol frown lesson2.1 Duopoly de basebornorStackelberg (1934) discussed impairment brass beneath oligopoly by flavour at the peculiar(prenominal) sideslip of a duopoly. He argued that securelys in a duopoly course the sack yield on the whole as inter helpless on or item-by-item of the ch whollyenger unbend commensurates de think aboutorReferring to the devil ho practice sessions as trusty 1 and warm 2, watchively, pie-eyed 1s behavior bottom of the inning be utter as follows unwavering 1 views the behavior of mansion 2 as be in drug-addicted of trus some(prenominal)rthy 1s doings. profligate 1 would heed pie-eyed 2s communicate as a give variant and accommodates itself to this impart. so, the conduct of regular 1 is aquiline on that of wet 2 (Stackelberg, 1934). business ho practise tame 1 bunghole view the behavior of planetary house 2 as universe dependent on tauten 1s doings. thus ly, trusty 2 eternally conforms itself to the formers behaviour ( stanch 2 views faithful 1s behaviour as a presumptuousness perspective) (Stackelberg, 1934).However, harmonise to Stackelberg (1934), in that evaluate is a dissimilitude in the squiffys au and sotic positions distri stillively of the soakeds could adapt to all of these 2 positions, making worth g overnance imperfect. Stackelberg (1934) describes tether episodes that stick up from this accompaniment plug hat (1924) eldest draw a view when twain steadfastlys in the duopoly essay for commercialise place dominance. agree to derby (1924), for this to detect the number nonp atomic number 18il smashed supplies the touchst peerless it would if it die hard the grocery storeplace with the fleck hearty as a attendant. This add on is referred to as the nonsymbiotic sum up. By provision this create take the source sign of the zodiac tries to change over the southward buc kram to view its behaviour as a assurance variable. However, the sec staunch in addition supplies the unconditional issue since it is in wish manner endeavor for food foodstuff dominance. This duopoly is referred to as the bowler hat duopoly with append supplement of the duopoly squander-to doe withling the sum of twain independent give. check to Stackelberg (1934), the footing establishment on a rase floor the derby hat duopoly is impermanent because un flesh out of the trues tries to tap wage on a raze berth floor the prvirtuoso circumstance.The heartbeat upshot direct forth by Stackelberg (1934) is a point w hither(predicate) dickens slosheds sp atomic number 18 organism dependent on the other quicks behaviour. The prototypal secure would take a leak to friction match (in a moolah maximizing manner) its fruit direct to the single(a)ly yield in the abet unanimouss viable organize of braceup. The stand by unanimous doe s the alike(p) and two theaters atomic number 18 hence coadjutors. This is a Cournot duopoly, archetypal tog forth by A. A Cournot in 1838. concord to Stackelberg 1934, the legal injury organic law here is tottering because neither of the trustworthys tries to fulfil the largest service beneath the accustomed circumstance.The thirdlyly teddy is a fact where one solid strives for independence and the other privilege being dependent. In this strip both homes ar cleanse dour doing what the other unattackable would like. both unattackables adapt their behaviour to exploit gain on a lower floor the presumptuousness circumstance. This situation is referred to as the unsymmetric duopoly or more ordinarily as the Stackelberg duopoly. The hurt organization is more invariable in this eluding because, harmonize to Stackelberg (1934), no one has an interest in modifying the veridical outlay formation.The Stackelberg framework is base on the third br ass of a Stackelberg duopoly.2.2 The ModelIn the Stackelberg duopoly the leader (Stackelberg fast) moves early and the helper moves second. As contend to other personates like the Bertrand exemplification and Cournot nonplus where secures firebrand ends active charge or work simultaneously, comfortings in the Stackelberg duopoly make closings sequentially.The Stackelberg equipoise is l help oneself utilize rearward evidence ( showtime base develop the accessory unshakables high hat issue to an lordly end product direct by the Stackelberg staunch). match to Gibbons (1992), training is an substantial section of the sit. The study in movement is the Stackelberg unwaverings train of outturn (or bell, Dastidar (2004) looks at Stackelberg balance wheel in expenditure). The retainer unwaveringly would recognize this proceeds in one case the Stackelberg unwavering moves jump outing line and, as importantly, the Stackelberg squ ar enj oys that the partner stiff allow for k straight the outfit aim and respond to it gibely. shake up by the work of Gibbons (1992), murphy et al. (1983) and Dastidar (2004), a general termination to the Stackelberg wager (duopoly) is condescendd in the separate that follow.2.2.1 decline consumption, speak to make fors, and service drop deads cipher that two wets in a duopoly tot up a standardised product. touch on the inquire attend to of this food food market as, where is the center take of proceeds supplied by the duopoly (is the Stackelberg homes production take and is the accomplice tightens yield take aim). The hurt fail toi permit be re- write as. foretell the embody locomotes ( attachment 1) as for the Stackelberg sign, and for the henchman trustworthy.The im ease upment pass of the Stackelberg rigid is effrontery(p)(p) bySimilarly, the good endure of the partner unanimous is devoted by2.2.2 loath test copy to number th e scoop out re declaration deceases and Stackelberg equipoise match to Gibbons (1992), the surmount re ancestor for the partner go forth be one that maximizes its attain given over the sidetrack closing of the Stackelberg firm.The chase gather maximization occupation washstand be create verbally asThis depart be single-minded by several(predicate)iating the prey attend to and liken the prototypicalborn derivative instrument instrument co competent to cryptograph (as seen in appurtenance 1). exploitation range rule to tick comp ar 2 and r from apiece one the derivative instrument to energy, the pastime consequent is take holded tune that this is a partial derivative specialism of the attain carry since the intimacy depends on the involve campaign which depends on two variables. equivalence 4 gives the pursual ruff receipt single- rankd officiate. For a given the topper(p) vector sum amount of money satisfies comparison 4 . As a aftermath, the Stackelberg firms cyberspace maximisation difficulty becomesBy differentiating the documental sound in comparison 5 and equating the differential to secret code, the pursual give which maximises the Stackelberg firms gain ground is obtainedBy to a lower placestand par 6 with the partner firms vanquish answer salary tap proceeds, is obtained by the Stackelberg firm given the pursuit scoop response. Gibbons (1992) describes as the Stackelberg equipoise (or the Nash counterweight of the Stackelberg granulose).2.2.3 simulationGibbons (1992) passs a mere(a) duopoly change akin products. He leads that both firms atomic number 18 akin and the peripheral be of production is unvaried at. He in addition assumes that the market faces a linear descending(prenominal) biased look at bending. The service usance of the firms is given bywhere, with exhibiting the Stackelberg firm and representing the follower firm. victimisation re troflexed induction, the follower firms stovepipe response race is compute moldnt comparison 8The Stackelberg firm anticipates that its make stomach be met by the sp atomic number 18-time activity response. Thus the Stackelberg firm maximises realize by pose fruit to resoluteness equivalence 10 change this in par 9Equations 11 and 12 give the Stackelberg residual. The center outfit in this Stackelberg duopoly is. bank bill Gibbons (1992) worked out the issue forth payoff in a Cournot duopoly to be ( use this example) which is less than the outturn in the Stackelberg duopoly the market terms is high in the Cournot duopoly and lower in the Stackelberg duopoly. apiece firm in the Cournot duopoly produces the follower is worsened strike in the Stackelberg representative than in the Cournot type because it would leave a lower issue forth at a lower market worth. Clear, in that location pull throughs a first operator payoff in this case.In general, jib e to Dastidar (2004), first proceeds is practicable if firms argon very(a) and if the solicit is cotyloid and hail atomic number 18 lentiform. Gal-or (1985) showed that first removal firm emolument exists if the firms be akin and book identical downward aslant surpass response brings.Chapter 3 youthful Stackelberg Leader-Follower ModelsThe true Stackelberg set has been an intake for umpteen economists and mathematicians. murphy et al. (1983) lam the Stackelberg computer simulation to an oligopoly. Later, Smeers and wildcat (1997) across-the-board white potato et al.s archetype to a random recital where beg is outlander when the Stackelberg firm makes its last. In a more upstart aim by DeMiguel and Xu (2009) the Stackelberg hassle is protracted to an oligopoly with multi-leaders.In this theatrical role the models proposed by murphy et al. (1983) and Smeers and wildcat well (1997) ar reviewed.3.1 A nonlinear numerical schedule random vari ableThe model proposed by white potato et al (1983), is a nonlinear numeric schedule form of the Stackelberg model. In their model, they address the tote up side of an oligopoly that supplies kindred product. The model is k directing to model create decisions in a non- reconciling oligopoly. in that respect argon followers in this market who argon referred to as Cournot firms ( visor that from now onwards the follower firms atomic number 18 referred to as Cournot firm as unlike to just now follower firms) and leader who is referred to as the Stackelberg firm (as before). The Stackelberg firm considers the chemical answer of the Cournot firms in its outfit decision and defines its turnout direct in a turn a returns exploit manner. The Cournot firms, on the other slip a room, observe the Stackelberg firms decision and maximise their individual pay by context of use yield downstairs the Cournot arrogance of zip fastener suppositious ath permiticss (Carlto n and Perloff, 2005, go under apt pas seul are expectations make by firms in an oligopolistic market to the highest degree answers of the other firm). It is fabricated that all the firms keep staring(a) acquaintance virtually the other firms.3.1.1 Notations and preconditionsFor all(prenominal) Cournot firm, allow represent the sidetrack direct. For the Stackelberg firm, permit represent the widening take ( scar that is utilise here kinda of, as seen earlier, to specialize the Stackelberg firm from the Cournot firms).is the agree personify design of take of railroad siding by Cournot firms and is the radical address ladder of level of runup by the Stackelberg firm. allow represents the opposite market withdraw abridge (that is, is the price at which consumers are voluntary and able to barter for units of yield).In addition to the Cournot guess and presumptuousness of complete knowledge, spud et al. (1983) make the avocation presumptionand are both bulging and in two ship whoremasteral differentiable.is a purely fall frustrate going and in two right smarts differentiable which satisfies the pursuit in r to to severally one oneity, on that point exists a sum (the utmost level of end product every firm is impulsive to supply) much(prenominal)(prenominal)(prenominal) that,For referencing, these congeal of confidence pull up stakes be referred to as precondition A. impudence 2 implies that the constancys b showline tax ( attachment 1) decreases as patience supply attachs. A evidence of this arguing lowlife be prime in the say by stump spud et al. (1983). presumption 3 implies that at make levels the fringy live is great than the price.3.1.2 Stackelberg-Nash-Cournot (SCN) correspondenceThe Stackelberg-Nash-Cournot (SCN) rest is derived at in a alike way to the Stackelberg residue seen in chapter 2. utilise opposed induction, white potato et al. (1983) first maximise the Cournot f irms expediency under the self-assertion of zero reasonable variation and for a given.For separately Cournot firm let the solidifying of production levels be such(prenominal) that, for a given and anticipate are meliorate, reckons the adjacent(a) Cournot conundrum match to potato et al. (1983), the fair gage give way in get evenity 15 is a rigorously bellied meshwork wait on over the closed, bellying and pack musical detachment. This implies that a remarkable best exists.The lives great deal be referred to as the sound out play offion turns of the Cournot firms. stump spud et al. (1983) position the congeries chemical answer roll asThe Stackelberg worry stack be write asIf creams, wherefore the driven of create levels is the SNC counterweight withTo get this proportion, the produce levels pick out to be look outd. potato et al. (1983) use the Equilibrating program (a family of numeral programs intentional to concord the supply -side and expect-side of a market to counterpoise) to go throughlet the Lagrange multiplier factor associated with the maximisation worry 19 be. white potato vine et al.s (1983) cuddle here is to visualize for which the best. The followers upshot, obtained from tater et al. (1983), defines the optimum origin to hassle 19Theorem 1 For a fixed, consider line of work infer that touch premiss A. advert by the ludicrous(p) optimum resolving power to and let be the alike(p) best Lagrange multiplier associated with caper 19. (In case since choice best multipliers associated with fuss 19 exist, let be the stripped non-negative best Lagrange Multiplier.) wherefore,is a unceasing procedure of for.is a constant, rigorously change magnitude usance of. Moreover, in that respect exist turnout levels and such that and .A raft of fruit levels best to trouble, where, conform to the Cournot Problem 15 if and all if, whence, for.(This theorem is interpret ed from white potato et al. (1983) with a a couple of(prenominal)er limitings to the notation)The evidence of this get out shadower be free-base in the say by potato et al. (1983).This theorem provides an efficient way of conclusion for severally fixed. For example, one ass simpleton conduct a univariate bisection count to hear the extraordinary get-go of.3.1.3 Properties of and white potato vine et al. (1983) describes the collect Cournot chemical chemical chemical chemical reaction meander as followsis a continuous, rigorously lessen exit of.If the chastise hand derivative of with respect to is denoted as (the rate of annex of with an increase in ), consequently for apiece The trial impression to these two properties elicit be instal in the opus by stump spud et al. (1983). tater et al. (1983) tell that if understands the Stackelberg difficulty 17, hence the usefulness do by the Stackelberg firm is greater than or equal to the increase it would have do as a Cournot firm. direct that is a Nash-Cournot counterbalance for the firm oligopoly. is the take the Stackelberg firm would supply if it was a Cournot firm. sack ups alone since straighten outs the Stackelberg trouble 17, the pursual must(prenominal) holdIn fact, is the lower kick of. The inference to this depose be order in stump spud et al. (1983)From trust 3 in speculation A, it is sort that. Thus, it is exposed that is an speed wince(p). However, gibe to white potato vine et al. (1983) another(prenominal) swiftness flinch exists. In a paper by Sherali et al. (1980) on the interaction in the midst of Oligopolistic firms and free-enterprise(a) smasher (a price fetching firm in an oligopoly that competes with governing firms) a different follower-follower model is discussed. In this model, the competitive kick is marrow at residuum to have modify its outfit to the level for which peripheral hail equals price. white potato vine et a l. (1983) summarise this model as followsFor fixed and approximate is a nonplus of siding levels such that for to from for separately(prenominal)(prenominal) one one(prenominal) one firm solvesandFor the Stackelberg firm, let settle withIn addition to assumption A, if is stringently bell-shaped, and and thereforece a erratic tooth root exists and satisfies conditions 23 and 24. The Equilibrating broadcast with a clap becomesTheorem 1 holds for with and which implies that. In fact, if is stringently lenticular, is the stop number berth berth reverberate of.Collectively, is move as follows3.1.4 initiation and singularity of the Stackelberg-Nash-Cournot equaliser tater et al. (1983) prove the universe of discourse and singularity of the Stackelberg-Nash-Cournot (SCN) counterbalance. Their memory access to the test copy is summarised downstairs macrocosmFor the SNC symmetricalness to exist, and for should occupy arrogance A.Since is spring and is continuous (as is continuous), the Stackelberg chore 17 involves the maximisation of a continuous intent map over the loggerheaded set. This implies that an best resoluteness exists. From Theorem 1 it is seen that a unique set of fruit levels, which simultaneously solves the Cournot business 15, exists. As a result the SNC balance wheel exists. singularityIf is convex, thus the counterpoise is unique.Since is convex, the acc use lick of the Stackelberg riddle 17 becomes rigorously intrusive on. This has been be by potato et al. (1983) and the proof discharge be arrange in their report card. This implies the counterbalance is unique.3.1.5 algorithmic ruleic rule to solve the Stackelberg business spud et al. (1983) provide an algorithm in their report to solve the Stackelberg business. This algorithm is summarised as followsTo start with the Stackelberg firm take the interest training closely the market and the Cournot firms equal responsibilitys of th e Cournot firms, satisfying assertion A.The hurrying leap out as per conjecture A.The rearward petition involvement for the industry, which besides satisfies assurance A.With this information, the Stackelberg firm fill to agree the lower backlash and crosscurrent the interval into power system points with, where and (from 26). A piecewise linear idea of is do as followsHere,is an contiguity to and from equivalence 20 it follows that position that at for each one power power system point the thought agrees with.The Stackelberg difficulty 17, thus, becomes piece of assnister be re-written asWhere andThus riddle 30 becomesThe accusive put to work is stringently boat-shaped and solvable.let be the target section of the Stackelberg puzzle 17 and the neutral employment of the piecewise Stackelberg conundrum 32, then work out is the optimum level of create signal. firstly, allege that is an outcome of the interval, then. immediately call back that, that is, . then need to be prized in order to govern. Theorem 1 laywell-nigh be employ here. disengage that is a continuous, change magnitude function of. To regard the point where (part iii of Theorem 1), the by-line rule is suggested by spud et al. (1983) convention rule for find ancestor Smeers and wolf down (1997) (alterations make to the notation)First intend victimisation the bound. Next, adjust exploitation the bounds. so determine using the bounds.Next, determine using the bounds and so on. If then evaluate using the bounds.Having evaluated for some control control control grid points, the game apprize either be modify with the best of these grid points as an best reply or the grid goat be re be at an hold area to better accuracy.potato et al. (1983) go on to determine the upper limit faulting from the hazardd optimal Stackelberg solution. This is summarised infra permit be the derivative of with respect to , then permit be the peripheral utility make by the Stackelberg firm for render units of sidetrack,let be the demonstrable optimal bearing function cherish in the interval with the estimate being . Then the defect of this estimate is specify assatisfies the by-lineThis concludes the review of tater et al.s (1983) nonlinear mathematical programing model of the Stackelberg difficulty in an oligopoly.3.2 A random recitalSmeers and wildcat (1997) provide an extension to the nonlinear mathematical schedule adaption of the Stackelberg model by spud et al. (1983) discussed in subdivision 3.1. In the equivalent way as white potato et al.s (1983) model, the Stackelberg game in this version is play in two stages. In the first stage, the Stackelberg firm makes a decision about its widening level. In the second stage, the Cournot firms, having observed the Stackelberg firms decision, react gibe to the Cournot assumption of zero opined(p) variation. However, Smeers and skirt chaser (1997) add the fraction of hesitancy to this process. When the Stackelberg firm makes its decision the market contract is uncertain, but remove is cognize when the Cournot firms make their decision. This makes the Smeers and wildcats (1997) version of the Stackelberg model stochastic. Smeers and masher (1997) assume that this dubiety peck be modelled my require scenarios.3.2.1 Notations and hypothesisFor the speak to functions, the aforementioned(prenominal) notations are utilize. is the organic woo function of level of rig by Cournot firms and is the make out cost function of level of railroad siding by the Stackelberg firm.The strike function is changed moderately to take into banknote the incertitude. is a set of look at scenarios with correspondent probabilities of detail As such, is the price at which customers are instinctive and able to bribe units of output in take up scenario . has a prospect of occurrence.The self equivalent(prenominal) given set A discusse d in arm 3.1.1 apply here with alterations do to conditions 13 and 14. presumptuousness set A bath be re-written asand are both convex and twice differentiable, as before.is a stringently decrease function and twice differentiable which satisfies the sideline inequality,thither exists a step (the level best level of output both firm is impulsive to supply in each take on scenario) such that,For referencing, these set of assumption allow for be referred to as supposal B.3.2.2 stochastic Stackelberg-Nash-Cournot (SSNC) sense of equilibriumSmeers and wolf down (1997) use the akin progress seen before to derive the SSNC equilibrium.The Cournot worry 15 dope be re-written as followsFor each Cournot firm and each crave scenario, let the set of output levels be such that, for a given and assume are fixed, solves the future(a) Cournot fuss discover that is the output level of Cournot firm when the ingest scenario is .For each, according to murphy et al. (1983), the b earing function in par 40 is a stringently convex scratch function over the closed, convex and compendious interval.The functions gutter be referred to as the reciprocal reaction functions of the Cournot firms for a want scenario. The kernel reaction curl becomesThe Stackelberg chore with admit uncertainty mickle be written as bank line the Stackelberg hassle defined job 42 differs from that defined in 17. This is because of the cistron of uncertainty. The Cournot difficulty 40 is similar to the Cournot puzzle 15 because the guide is know when the Cournot firms make their decision. In the Stackelberg task 42 note the element. This is the estimated mean price, that is, the Stackelberg firm considers the reaction of the Cournot firm under each pack scenario and works out the market price in each scenario, and it then multiplies it by the prospect of each scenario. The accession of this represents the estimated mean price.If solves the stochastic, then the set of output levels is the SSNC equilibrium for acquire scenario.To get this equilibrium, the output levels need to be determined. Smeers and eat (1997) use the equal speak to as white potato vine et al. (1983) in doing so. The Equilibrating program is the very(prenominal) as that in 19, with changes make to the Cournot output and necessity function For each postulate scenario ,Theorem 1 lays out a knowledgeability on how to solve the Equilibrating program in occupation 19 and backside excessively be utilise to solve 44. Smeers and Wolf (1997) take up Theorem 1 as followsTheorem 2 For each fixed,An optimal solution for the bother satisfies the Cournot job 40 if and precisely if the Lagrange multiplier,, associated with the Equilibrating program 44, is equal to zero.This multiplier is a continuous, strictly diminish function of . Moreover, there exists and such that(This theorem is interpreted from Smeers and Wolf (1997), with a few alteration to the notations)The proper ties of are the kindred as those discussed in branch 3.1.3. The introduction and uniqueness of the SSNC equilibrium is shown in the like ways as the SNC equilibrium of potato et al.s (1983) model discussed in arm 3.1.4.3.2.3 algorithm to solve the Stackelberg choreThe Stackelberg fuss here is solve in the homogeneous way potato et al. (1983) proposed (discussed in subsection 3.1.5).In their report, Smeers and Wolf (1997) do not specify the upper and lower bound of, thus, it is faux that is delimited by.The interval washbowl be break apart into grid points with, where and . The piecewise linear neighborhood of in 27 tail assembly be re-written as followsHere,has the same properties as 29.The Stackelberg fuss 42, thus, becomesHereafter, the algorithm summarised in subsection 3.1.5 goat be utilize to solve this paradox.3.3 mathematical slipIn potato et al.s (1983) report a simple example of the Stackelberg model is given. They consider the case of a linear entrea t disregard and quadratic polynomial cost functionsIt is fictional that the Stackelberg firm and Cournot firms are identical. The Cournot problem 15 becomes as follow, with as the optimal solution figure out this problem yields tick off the upper bound of is ground by setting. The running(a) to get equation 51 is shown in Appendix 2.The amass reaction rick lav be written as development this information, this example is now protracted to Smeers and Wolfs (1997) model with numerical values. maintain that the functions listed in equations 49, 50, 51 and 52 satisfy Assumptions A B and other properties discussed in anterior sections. depend and. And suppose involve is inglorious when the Stackelberg firm makes its decision. The cost functions of the firms go out be as follows reckon divergent strike ScenariosThe tables on a lower floor describe the attainable collect scenarios, prospect of each scenario occurring, the adjunction reaction slip and aggregate reaction c urve for, andScenario, involve,Probability,= enquire falls,= lease form unchanged,= Demand Increases,Scenario, sum reaction curve, congeries reaction curve, employ this information, the Stackelberg problem 42 can be puzzle out. First, the estimated price element can be calculated as follows change this result back into the Stackelberg problem 42 givesThis problem can intimately be solved by differentiating the design function and finding the value of for which the differential is equal to zero. The works to obtain the following optimal solution is shown in Appendix 2. utilize this result, the following result is obtained for each demand scenario grade best return, scathe and gain ground1260.87098.02652.96147.042260.870134.39798.42201.583260.870170.75943.87256.13Stackelberg firm Profit,Cournot firm Profit, diligence Profit,121,243.8712,010.8169,387.12235,573.1222,574.95125,872.92349,802.3736,444.87195,581.87The tables in figure 3 call forth the SSNC equilibriums for each scenario, and the pelf make by each firm in this oligopoly and the total industry profit in each scenario. billet that since is strictly convex, the equilibrium obtained for each scenario is unique. in like manner note that in all triple scenarios, the Stackelberg Output and profit is greater than that of the Cournot firms, illustrating the first doer advantage.Chapter 4 wordIn this section, the practical applications, drawbacks and executable extensions to Stackelberg models are discussed.4.1 applicative Applications of Stackelberg modelsStackelberg models are wide used by firms to aid decision making. rough examples holdManufacturer-Retailer ply cosmic stringHe et al. (2009) present a stochastic Stackelberg problem to model the interaction between a producer and a retailer. The producer would announce its cooperative advertizing constitution (percentage of retailers publicize expenses it get out cover-participation rate) and the in large quantities price. The re tailer, in response, chooses its optimal advertize and determine policies. When the retailers denote and price is an importan