Stackelberg Leader Follower Models For Strategic Decision Making Engineering Essay

Stackelberg Leader Follower Models For Strategic finality Making Engineering EssayThis paper re crop f all in all outs some Stackelberg Leader-Follower moldings used for strategic decision qualification. The simple Stackelberg duopoly is looked at off confine printing of solely, and a generalisation of the Stackelberg duopoly trouble is apt(p). By studying the feigns by tater et al. (1983) and Smeers and animal (1997), the paper revaluations Stackelberg simulate from its classical form to the recent random recitations. The paper looks at the numeral formulation of both(prenominal) a non additive numerical weapons platformming molding and a nonlinear stochastic programming regulate. Towards the end of this paper, a simple numeric example is r all(prenominal)n and applicative applications of Stackelberg Leader-Follower models ar discussed.Chapter 1 IntroductionIn economics, an oligopoly is considered to be the most interesting and complex marketplace structur e (amongst an other(prenominal) structures like monopolies and perfect competition). Most industries in the UK and world- from retailing to fast food, mobile phone networks to professional services- are oligopolistic. Given the current financial climate, it is tyrannical for houses to be sure that they gear up decisions accurately, maximising not only their shekels, but also their chances of remaining competitive. some mathematicians and economists reach attempted to model the decision making process and gelt maximizing strategies of oligopolistic planetary houses. For example, A. A. Cournot was one of the first mathematicians to model the de spuriousours of monopolies and duopolies in 1838. In Cournots model both loyals choose their sidetrack simultaneously assuming that the other secure does not alter its end product (Gibbons, 1992). Later, in 1934, H. V. Stackelberg proposed a different model where one of the duopoly sozzleds contacts its fruit decision first an d the other unanimous observes this decision and sets its output train (Stackelberg, 1934).The classical Stackelberg model has been prolonged to model a variety of strategic decision making. For example, stump spud et al. (1983) model the output decision making process in an oligopoly. Later works by Smeers and Wolf (1997) extend this model to include a stochastic element. More interestingly, in a model by He et al. (2009), the Stackelberg theory is used to model the interaction between a manufacturer and a retailer when making decisions just about cooperative advertize policies and wholesale bells.The accusive of this paper is to review the Stackelberg models from its classic form to the more recent stochastic versions. In chapter 2, the simple Stackelberg duopoly is reviewed and a generalisation of the Stackelberg duopoly enigma is given. In chapter 3, more complicated and recent models are reviewed. The mathematical formulation of white potato vine et al.s (1983) and S meers and Wolfs (1997) model is given. At the end of chapter 3, a numerical example is applied to Smeers and Wolfs (1997) model. In chapter 4, practical applications of Stackelberg leader- participator models are discussed. Chapter 4 also looks at the drawbacks of and possible extensions to Stackelberg models. Appendix 1 explains the Oligopoly market structure and economics involved in service maximisation.Chapter 2 Classical Stackelberg Leader-Follower Model2.1 Duopoly BehaviourStackelberg (1934) discussed outlay fundamental law under oligopoly by looking at the special carapace of a duopoly. He argued that smasheds in a duopoly evict behave either as dependent on or independent of the rival inviol equals behaviorReferring to the two firms as firm 1 and firm 2, respectively, firm 1s behaviour ass be speak as followsFirm 1 views the behaviour of firm 2 as being independent of firm 1s behaviour. Firm 1 would regard firm 2s planning as a given variable and adapts itself to this supply. Thus, the behaviour of firm 1 is dependent on that of firm 2 (Stackelberg, 1934).Firm 1 can view the behaviour of firm 2 as being dependent on firm 1s behaviour. Thus, firm 2 always adapts itself to the formers behaviour (firm 2 views firm 1s behaviour as a given situation) (Stackelberg, 1934).However, according to Stackelberg (1934), thither is a difference in the firms actual positions severally of the firms could adapt to either of these two positions, making scathe constitution imperfect. Stackelberg (1934) describes three cases that arise from this situationBowler (1924) first described a situation when both firms in the duopoly strive for market dominance. accord to Bowler (1924), for this to happen the first firm supplies the quantity it would if it reign the market with the second firm as a follower. This supply is referred to as the independent supply. By preparation this output level the first firm tries to convince the second firm to view its behaviour a s a given variable. However, the second firm also supplies the independent supply since it is also striving for market dominance. This duopoly is referred to as the Bowler duopoly with positive supply of the duopoly equalling the sum of two independent supply. According to Stackelberg (1934), the set administration under the Bowler duopoly is unstable because neither of the firms tries to maximise boodle under the given circumstance.The second case described by Stackelberg (1934) is a situation where both firms favour being dependent on the other firms behaviour. The first firm would have to match (in a realize maximising manner) its output level to the each output in the second firms feasible set of output. The second firm does the comparable and both firms are thus followers. This is a Cournot duopoly, first described by A. A Cournot in 1838. According to Stackelberg 1934, the price formation here is unstable because neither of the firms tries to achieve the largest profit u nder the given circumstance.The third case is a situation where one firm strives for independence and the other favour being dependent. In this case both firms are better off doing what the other firm would like. Both firms adapt their behaviour to maximising profit under the given circumstance. This situation is referred to as the asymmetric duopoly or more commonly as the Stackelberg duopoly. The price formation is more stable in this case because, according to Stackelberg (1934), no one has an interest in modifying the actual price formation.The Stackelberg model is based on the third case of a Stackelberg duopoly.2.2 The ModelIn the Stackelberg duopoly the leader (Stackelberg firm) moves first and the follower moves second. As opposed to other models like the Bertrand model and Cournot model where firms make decisions about price or output simultaneously, firms in the Stackelberg duopoly make decisions sequentially.The Stackelberg sense of balance is boundd using backward in ductive reasoning (first determine the follower firms best response to an arbitrary output level by the Stackelberg firm). According to Gibbons (1992), information is an important element of the model. The information in question is the Stackelberg firms level of output (or price, Dastidar (2004) looks at Stackelberg equilibrium in price). The follower firm would know this output once the Stackelberg firm moves first and, as importantly, the Stackelberg firm knows that the follower firm entrust know the output level and respond to it accordingly.Inspired by the work of Gibbons (1992), Murphy et al. (1983) and Dastidar (2004), a general firmness of purpose to the Stackelberg game (duopoly) is derived in the parts that follow.2.2.1 Price suffice, follow mathematical crops, and profit accountabilitysSuppose that two firms in a duopoly supply a homogeneous product.Denote the gather up function of this market as, where is the total level of output supplied by the duopoly (is the Stackelberg firms output level and is the follower firms output level). The price function can be re-written as.Denote the cost functions (Appendix 1) as for the Stackelberg firm, and for the follower firm.The profit function of the Stackelberg firm is given bySimilarly, the profit function of the follower firm is given by2.2.2 Backward induction to derive the best response functions and Stackelberg equilibriumAccording to Gibbons (1992), the best response for the follower exit be one that maximises its profit given the output decision of the Stackelberg firm.The followers profit maximisation conundrum can be written asThis can be cleard by differentiating the design function and liken the derived function to zero point (as seen in Appendix 1). use chain rule to differentiate equation 2 and vista the differential to zero, the hobby result is obtained poster that this is a partial differentiation of the profit function since the function depends on the pauperization func tion which depends on two variables. Equation 4 gives the followers best response function. For a given the best response quantity satisfies equation 4. As a result, the Stackelberg firms profit maximisation enigma becomesBy differentiating the quarry function in equation 5 and equating the differential to zero, the pursual result which maximises the Stackelberg firms profit is obtainedBy solving equation 6 with the follower firms best response profit maximising output, is obtained by the Stackelberg firm given the followers best response. Gibbons (1992) describes as the Stackelberg equilibrium (or the Nash equilibrium of the Stackelberg game).2.2.3 ExampleGibbons (1992) considers a simple duopoly merc dedicateising homogeneous products. He assumes that both firms are akin and the bare(a) cost of production is constant at. He also assumes that the market faces a linear downward sloping demand thin. The profit function of the firms is given bywhere, with representing the Stacke lberg firm and representing the follower firm.Using backward induction, the follower firms best response function is deliberateSolving equation 8The Stackelberg firm anticipates that its output will be met by the followers response. Thus the Stackelberg firm maximises profit by setting output toSolving equation 10Substituting this in equation 9Equations 11 and 12 give the Stackelberg equilibrium. The total output in this Stackelberg duopoly is.Note Gibbons (1992) worked out the total output in a Cournot duopoly to be (using this example) which is less than the output in the Stackelberg duopoly the market price is higher in the Cournot duopoly and lower in the Stackelberg duopoly. to each one firm in the Cournot duopoly produces the follower is worse off in the Stackelberg model than in the Cournot model because it would supply a lower quantity at a lower market price. Clear, there exists a first mover advantage in this case.In general, according to Dastidar (2004), first advantage is possible if firms are identical and if the demand is concave and be are convex. Gal-or (1985) showed that first mover advantage exists if the firms are identical and have identical downward sloping best response functions.Chapter 3 Recent Stackelberg Leader-Follower ModelsThe classical Stackelberg model has been an uptake for m each economists and mathematicians. Murphy et al. (1983) extend the Stackelberg model to an oligopoly. Later, Smeers and Wolf (1997) extended Murphy et al.s model to a stochastic version where demand is unknown when the Stackelberg firm makes its decision. In a more recent brood by DeMiguel and Xu (2009) the Stackelberg business is extended to an oligopoly with multi-leaders.In this section the models proposed by Murphy et al. (1983) and Smeers and Wolf (1997) are reviewed.3.1 A nonlinear Mathematical course of instructionming random variableThe model proposed by Murphy et al (1983), is a nonlinear mathematical programming version of the Stackelberg model. In their model, they consider the supply side of an oligopoly that supplies homogeneous product. The model is designed to model output decisions in a non-cooperative oligopoly.There are followers in this market who are referred to as Cournot firms (note that from now onwards the follower firms are referred to as Cournot firm as opposed to just follower firms) and leader who is referred to as the Stackelberg firm (as in advance). The Stackelberg firm considers the reaction of the Cournot firms in its output decision and sets its output level in a profit maximising manner. The Cournot firms, on the other hand, observe the Stackelberg firms decision and maximise their case-by-case profits by setting output under the Cournot assumption of zero probable variations (Carlton and Perloff, 2005, touch on conjectural variation are expectations made by firms in an oligopolistic market about reactions of the other firm). It is assumed that all the firms have comp allowe knowledge ab out the other firms.3.1.1 Notations and assumptionsFor each Cournot firm, allow represent the output level. For the Stackelberg firm, let represent the output level (note that is used here instead of, as seen earlier, to distinguish the Stackelberg firm from the Cournot firms).is the total cost function of level of output by Cournot firms and is the total cost function of level of output by the Stackelberg firm.let represents the inverse market demand curve (that is, is the price at which consumers are unforced and able to purchase units of output).In supplyition to the Cournot assumption and assumption of complete knowledge, Murphy et al. (1983) make the following assumptionand are both convex and doubly differentiable.is a rigorously decreasing function and twice differentiable which satisfies the following inequality,There exists a quantity (the maximum level of output any firm is willing to supply) such(prenominal)(prenominal) that,For referencing, these set of assumption wi ll be referred to as assertion A.Assumption 2 implies that the industrys marginal revenue (Appendix 1) decreases as industry supply increases. A trial impression of this severalisement can be found in the report by Murphy et al. (1983).Assumption 3 implies that at output levels the marginal cost is greater than the price.3.1.2 Stackelberg-Nash-Cournot (SCN) equilibriumThe Stackelberg-Nash-Cournot (SCN) equilibrium is derived at in a similar way to the Stackelberg equilibrium seen in chapter 2.Using backward induction, Murphy et al. (1983) first maximise the Cournot firms profit under the assumption of zero conjectural variation and for a given.For each Cournot firm let the set of output levels be such that, for a given and assuming are fixed, solves the following Cournot problemAccording to Murphy et al. (1983), the objective function in equation 15 is a strictly convex profit function over the closed, convex and squelch legal separation. This implies that a unique optimum exis ts.The functions can be referred to as the joint reaction functions of the Cournot firms. Murphy et al. (1983) define the aggregate reaction curve asThe Stackelberg problem can be written asIf solves, so the set of output levels is the SNC equilibrium withTo get this equilibrium, the output levels need to be determined. Murphy et al. (1983) use the Equilibrating program (a family of mathematical programs designed to reconcile the supply-side and demand-side of a market to equilibrium) to determineLet the Lagrange multiplier associated with the maximisation problem 19 be. Murphy et al.s (1983) approach here is to determine for which the optimum. The following result, obtained from Murphy et al. (1983), defines the optimal solution to problem 19Theorem 1 For a fixed, consider Problem mull over that satisfy Assumption A. Denote by the unique optimal solution to and let be the corresponding optimal Lagrange Multiplier associated with problem 19. (In case since alternative optimal mu ltipliers associated with problem 19 exist, let be the minimum non-negative optimal Lagrange Multiplier.) Then,is a continuous function of for.is a continuous, strictly decreasing function of. Moreover, there exist output levels and such that and .A set of output levels optimal to Problem, where, satisfy the Cournot Problem 15 if and only if, whence, for.(This theorem is taken from Murphy et al. (1983) with a few alterations to the notation)The proof of this result can be found in the report by Murphy et al. (1983).This theorem provides an efficient way of finding for each fixed. For example, one can simple conduct a univariate bisection search to find the unique root of.3.1.3 Properties of andMurphy et al. (1983) describes the aggregate Cournot reaction curve as followsis a continuous, strictly decreasing function of.If the right hand derivative of with respect to is denoted as (the rate of increase of with an increase in ), then for each The proof to these two properties can be fo und in the report by Murphy et al. (1983).Murphy et al. (1983) arouse that if solves the Stackelberg problem 17, then the profit made by the Stackelberg firm is greater than or equal to the profit it would have made as a Cournot firm. Suppose that is a Nash-Cournot equilibrium for the firm oligopoly. is the output the Stackelberg firm would supply if it was a Cournot firm. solvesBut since solves the Stackelberg problem 17, the following must holdIn fact, is the lower set of. The proof to this can be found in Murphy et al. (1983)From assumption 3 in Assumption A, it is clear that. Thus, it is clear that is an amphetamine bound. However, according to Murphy et al. (1983) another upper bound exists. In a paper by Sherali et al. (1980) on the Interaction between Oligopolistic firms and Competitive Fringe (a price taking firm in an oligopoly that competes with dominant firms) a different follower-follower model is discussed. In this model, the competitive fringe is content at equilibri um to have adjusted its output to the level for which marginal cost equals price. Murphy et al. (1983) summarise this model as followsFor fixed and suppose is a set of output levels such that for each firm solvesandFor the Stackelberg firm, let satisfyIn addition to Assumption A, if is strictly convex, then a unique solution exists and satisfies conditions 23 and 24. The Equilibrating Program with a fringe becomesTheorem 1 holds for with and which implies that. In fact, if is strictly convex, is the upper bound of.Collectively, is bounded as follows3.1.4 universe and uniqueness of the Stackelberg-Nash-Cournot equilibriumMurphy et al. (1983) prove the existence and uniqueness of the Stackelberg-Nash-Cournot (SCN) equilibrium. Their approach to the proof is summarised below conceptionFor the SNC equilibrium to exist, and for should satisfy Assumption A.Since is bounded and is continuous (as is continuous), the Stackelberg problem 17 involves the maximisation of a continuous objective function over the compact set. This implies that an optimal solution exists. From Theorem 1 it is seen that a unique set of output levels, which simultaneously solves the Cournot problem 15, exists. As a result the SNC equilibrium exists.UniquenessIf is convex, then the equilibrium is unique.Since is convex, the objective function of the Stackelberg problem 17 becomes strictly concave on. This has been proven by Murphy et al. (1983) and the proof can be found in their report. This implies the equilibrium is unique.3.1.5 Algorithm to solve the Stackelberg problemMurphy et al. (1983) provide an algorithm in their report to solve the Stackelberg problem. This algorithm is summarised as followsTo start with the Stackelberg firm needs the following information about the market and the Cournot firmsCost functions of the Cournot firms, satisfying Assumption A.The upper bound as per Assumption A.The inverse demand function for the industry, which also satisfies Assumption A.With this infor mation, the Stackelberg firm need to determine the lower bound and split the interval into control grid points with, where and (from 26). A piecewise linear approximation of is made as followsHere,is an approximation to and from equation 20 it follows thatNote that at each grid point the approximation agrees with.The Stackelberg problem 17, thus, becomescan be re-written asWhere andThus problem 30 becomesThe objective function is strictly concave and solvable.Let be the objective function of the Stackelberg problem 17 and the objective function of the piecewise Stackelberg problem 32, thenSuppose is the optimum level of output. First, suppose that is an endpoint of the interval, then. Now suppose that, that is, . Then needs to be judged in order to determine. Theorem 1 can be used here. find that is a continuous, decreasing function of. To find the point where (part iii of Theorem 1), the following method is suggested by Murphy et al. (1983)Figure Method for determiningSource Sm eers and Wolf (1997) (alterations made to the notation)First determine using the bounds. Next, determine using the bounds. Then determine using the bounds.Next, determine using the bounds and so on. If then evaluate using the bounds.Having evaluated for some grid points, the game can either be terminated with the best of these grid points as an optimal solution or the grid can be re be at an appropriate contribution to improve accuracy.Murphy et al. (1983) go on to determine the maximum error from the estimated optimal Stackelberg solution. This is summarised belowLet be the derivative of with respect to , thenLet be the marginal profit made by the Stackelberg firm for supplying units of output,Let be the actual optimal objective function value in the interval with the estimate being . Then the error of this estimate is defined assatisfies the followingThis concludes the review of Murphy et al.s (1983) nonlinear mathematical programing model of the Stackelberg problem in an oligopo ly.3.2 A Stochastic VersionSmeers and Wolf (1997) provide an extension to the nonlinear mathematical programming version of the Stackelberg model by Murphy et al. (1983) discussed in branch 3.1. In the same way as Murphy et al.s (1983) model, the Stackelberg game in this version is played in two stages. In the first stage, the Stackelberg firm makes a decision about its output level. In the second stage, the Cournot firms, having observed the Stackelberg firms decision, react according to the Cournot assumption of zero conjectural variation. However, Smeers and Wolf (1997) add the element of uncertainty to this process. When the Stackelberg firm makes its decision the market demand is uncertain, but demand is known when the Cournot firms make their decision. This makes the Smeers and Wolfs (1997) version of the Stackelberg model stochastic. Smeers and Wolf (1997) assume that this uncertainty can be modelled my demand scenarios.3.2.1 Notations and AssumptionFor the costs functions, the same notations are used. is the total cost function of level of output by Cournot firms and is the total cost function of level of output by the Stackelberg firm.The demand function is changed slightly to take into account the uncertainty. is a set of demand scenarios with corresponding probabilities of occurrence As such, is the price at which customers are willing and able to purchase units of output in demand scenario . has a luck of occurrence.The same Assumption set A discussed in subsection 3.1.1 apply here with alterations made to conditions 13 and 14. Assumption set A can be re-written asand are both convex and twice differentiable, as before.is a strictly decreasing function and twice differentiable which satisfies the following inequality,There exists a quantity (the maximum level of output any firm is willing to supply in each demand scenario) such that,For referencing, these set of assumption will be referred to as Assumption B.3.2.2 Stochastic Stackelberg-Nash-Courn ot (SSNC) equilibriumSmeers and Wolf (1997) use the same approach seen before to derive the SSNC equilibrium.The Cournot problem 15can be re-written as followsFor each Cournot firm and each demand scenario, let the set of output levels be such that, for a given and assuming are fixed, solves the following Cournot problemNote that is the output level of Cournot firm when the demand scenario is .For each, according to Murphy et al. (1983), the objective function in equation 40 is a strictly convex profit function over the closed, convex and compact interval.The functions can be referred to as the joint reaction functions of the Cournot firms for a demand scenario. The aggregate reaction curve becomesThe Stackelberg problem with demand uncertainty can be written asNote the Stackelberg problem defined problem 42 differs from that defined in 17. This is because of the element of uncertainty. The Cournot problem 40 is similar to the Cournot problem 15 because the demand is known when the Cournot firms make their decision. In the Stackelberg problem 42 note the element. This is the estimated mean price, that is, the Stackelberg firm considers the reaction of the Cournot firm under each demand scenario and works out the market price in each scenario, and it then multiplies it by the probability of each scenario. The summation of this represents the estimated mean price.If solves the stochastic, then the set of output levels is the SSNC equilibrium for demand scenario.To get this equilibrium, the output levels need to be determined. Smeers and Wolf (1997) use the same approach as Murphy et al. (1983) in doing so. The Equilibrating program is the same as that in 19, with changes made to the Cournot output and demand function For each demand scenario ,Theorem 1 lays out a universe on how to solve the Equilibrating program in problem 19 and can also be used to solve 44. Smeers and Wolf (1997) Summarise Theorem 1 as followsTheorem 2 For each fixed,An optimal solution for the problem satisfies the Cournot problem 40 if and only if the Lagrange multiplier,, associated with the Equilibrating program 44, is equal to zero.This multiplier is a continuous, strictly decreasing function of . Moreover, there exists and such that(This theorem is taken from Smeers and Wolf (1997), with a few alteration to the notations)The properties of are the same as those discussed in subsection 3.1.3. The existence and uniqueness of the SSNC equilibrium is shown in the same ways as the SNC equilibrium of Murphy et al.s (1983) model discussed in subsection 3.1.4.3.2.3 Algorithm to solve the Stackelberg problemThe Stackelberg problem here is solved in the same way Murphy 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 assumed that is bounded by.The interval can be split into grid points with, where and . The piecewise linear approximation of in 27 can be re-written as follo wsHere,has the same properties as 29.The Stackelberg problem 42, thus, becomesHereafter, the algorithm summarised in subsection 3.1.5 can be used to solve this problem.3.3 Numerical ExampleIn Murphy et al.s (1983) report a simple example of the Stackelberg model is given. They consider the case of a linear demand curve and quadratic cost functionsIt is assumed that the Stackelberg firm and Cournot firms are identical. The Cournot problem 15 becomes as follow, with as the optimal solutionSolving this problem yieldsNote the upper bound of is found by setting. The working to get equation 51 is shown in Appendix 2.The aggregate reaction curve can be written asUsing this information, this example is now extended to Smeers and Wolfs (1997) model with numerical values.Note that the functions listed in equations 49, 50, 51 and 52 satisfy Assumptions A B and other properties discussed in previous sections.Suppose and. And suppose demand is unknown when the Stackelberg firm makes its decisio n. The cost functions of the firms will be as followsFigure Different Demand ScenariosThe tables below describe the possible demand scenarios, probability of each scenario occurring, the joint reaction curve and aggregate reaction curve for, andScenario,Demand,Probability,= Demand falls,= Demand stiff unchanged,= Demand Increases,Scenario,Joint reaction curve,Aggregate reaction curve,Using this information, the Stackelberg problem 42 can be solved. First, the estimated price element can be calculated as followsSubstituting this result back into the Stackelberg problem 42 givesThis problem can easily be solved by differentiating the objective function and finding the value of for which the differential is equal to zero. The working to obtain the following optimal solution is shown in Appendix 2.Using this result, the following result is obtained for each demand scenarioFigure Optimal Output, Price and Profit1260.87098.02652.96147.042260.870134.39798.42201.583260.870170.75943.87256 .13Stackelberg firm Profit,Cournot firm Profit,Industry Profit,121,243.8712,010.8169,387.12235,573.1222,574.95125,872.92349,802.3736,444.87195,581.87The tables in figure 3 state the SSNC equilibriums for each scenario, and the profits made by each firm in this oligopoly and the total industry profit in each scenario. Note that since is strictly convex, the equilibrium obtained for each scenario is unique. Also note that in all three scenarios, the Stackelberg Output and profit is greater than that of the Cournot firms, illustrating the first mover advantage.Chapter 4 DiscussionIn this section, the practical applications, drawbacks and possible extensions to Stackelberg models are discussed.4.1 Practical Applications of Stackelberg modelsStackelberg models are widely used by firms to aid decision making. virtually examples includeManufacturer-Retailer Supply ChainHe et al. (2009) present a stochastic Stackelberg problem to model the interaction between a manufacturer and a retailer. The manufacturer would announce its cooperative ad policy (percentage of retailers advertising expenses it will cover-participation rate) and the wholesale price. The retailer, in response, chooses its optimal advertising and pricing policies. When the retailers advertising and pricing is an importan