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This paper reviews some Stackelberg Leader-Follower models used for strategic decision making. The simple Stackelberg duopoly is looked at first, and a generalisation of the Stackelberg duopoly problem is given. By studying the models by Murphy et al. (1983) and Smeers and Wolf (1997), the paper reviews Stackelberg model from its classical form to the recent stochastic versions. The paper looks at the mathematical formulation of both a nonlinear mathematical programming model and a nonlinear stochastic programming model. Towards the end of this paper, a simple numeric example is given and practical applications of Stackelberg Leader-Follower models are discussed.
Chapter 1: Introduction
In economics, an oligopoly is considered to be the most interesting and complex market structure (amongst other 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 imperative for firms to be sure that they make decisions accurately, maximising not only their profit, but also their chances of remaining competitive. Many mathematicians and economists have attempted to model the decision making process and profit maximizing strategies of oligopolistic firms. For example, A. A. Cournot was one of the first mathematicians to model the behaviours of monopolies and duopolies in 1838. In Cournot’s model both firms choose their output simultaneously assuming that the other firm does not alter its output (Gibbons, 1992). Later, in 1934, H. V. Stackelberg proposed a different model where one of the duopoly firms makes its output decision first and the other firm observes this decision and sets its output level (Stackelberg, 1934).
The classical Stackelberg model has been extended to model a variety of strategic decision making. For example, Murphy 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 about cooperative advertising policies and wholesale prices.
The objective 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 problem is given. In chapter 3, more complicated and recent models are reviewed. The mathematical formulation of Murphy et al.’s (1983) and Smeers and Wolf’s (1997) model is given. At the end of chapter 3, a numerical example is applied to Smeers and Wolf’s (1997) model. In chapter 4, practical applications of Stackelberg leader-follower 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 profit maximisation.
Chapter 2: Classical Stackelberg Leader-Follower Model
2.1 Duopoly Behaviour
Stackelberg (1934) discussed price formation under oligopoly by looking at the special case of a duopoly. He argued that firms in a duopoly can behave either as dependent on or independent of the rival firm’s behaviour:
Referring to the two firms as firm 1 and firm 2, respectively, firm 1’s behaviour can be generalised as follows:
Firm 1 views the behaviour of firm 2 as being independent of firm 1’s behaviour. Firm 1 would regard firm 2’s supply 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 1’s behaviour. Thus, firm 2 always adapts itself to the former’s behaviour (firm 2 views firm 1’s behaviour as a given situation) (Stackelberg, 1934).
However, according to Stackelberg (1934), there is a difference in the firms’ actual positions; each of the firms could adapt to either of these two positions, making price formation imperfect. Stackelberg (1934) describes three cases that arise from this situation:
Bowler (1924) first described a situation when both firms in the duopoly strive for market dominance. According to Bowler (1924), for this to happen the first firm supplies the quantity it would if it dominated the market with the second firm as a follower. This supply is referred to as the “independent supply”. By supplying this output level the first firm tries to convince the second firm to view its behaviour as 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 total supply of the duopoly equalling the sum of two “independent supply”. According to Stackelberg (1934), the price formation under the Bowler duopoly is unstable because neither of the firms tries to maximise profit “under the given circumstance”.
The second case described by Stackelberg (1934) is a situation where both firms favour being dependent on the other firm’s behaviour. The first firm would have to match (in a profit maximising manner) its output level to the each output in the second firm’s feasible set of output. The second firm does the same 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 “under 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 Model
In 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 equilibrium is determined using backwards induction (first determine the follower firm’s 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 firm’s 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 will 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 solution to the Stackelberg game (duopoly) is derived in the parts that follow.
2.2.1 Price function, cost functions, and profit functions
Suppose that two firms in a duopoly supply a homogeneous product.
Denote the demand function of this market as, where is the total level of output supplied by the duopoly (is the Stackelberg firm’s output level and is the follower firm’s 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 by:
Similarly, the profit function of the follower firm is given by:
2.2.2 Backward induction to derive the best response functions and Stackelberg equilibrium
According to Gibbons (1992), the best response for the follower will be one that maximises its profit given the output decision of the Stackelberg firm.
The follower’s profit maximisation problem can be written as:
This can be solved by differentiating the objective function and equating the differential to zero (as seen in Appendix 1).
Using chain rule to differentiate equation  and setting the differential to zero, the following result is obtained:
Note that this is a partial differentiation of the profit function since the function depends on the demand function which depends on two variables. Equation  gives the follower’s best response function. For a given the best response quantity satisfies equation . As a result, the Stackelberg firm’s profit maximisation problem becomes:
By differentiating the objective function in equation  and equating the differential to zero, the following result which maximises the Stackelberg firm’s profit is obtained:
By solving equation  with the follower firm’s best response profit maximising output, is obtained by the Stackelberg firm given the follower’s best response. Gibbons (1992) describes as the Stackelberg equilibrium (or the Nash equilibrium of the Stackelberg game).
Gibbons (1992) considers a simple duopoly selling homogeneous products. He assumes that both firms are identical and the marginal cost of production is constant at. He also assumes that the market faces a linear downward sloping demand curve. The profit function of the firms is given by:
where, with representing the Stackelberg firm and representing the follower firm.
Using backward induction, the follower firm’s best response function is calculated:
Solving equation :
The Stackelberg firm anticipates that its output will be met by the follower’s response. Thus the Stackelberg firm maximises profit by setting output to:
Solving equation :
Substituting this in equation :
Equations  and  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. Each 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 costs 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 Models
The classical Stackelberg model has been an inspiration for many 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 report by DeMiguel and Xu (2009) the Stackelberg problem 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 Programming Version
The 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 before). 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 firm’s decision and maximise their individual profits by setting output under the Cournot assumption of zero conjectural variations (Carlton and Perloff, 2005, define 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 complete knowledge about the other firms.
3.1.1 Notations and assumptions
For each Cournot firm, let 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 willing and able to purchase units of output).
In addition to the Cournot assumption and assumption of complete knowledge, Murphy et al. (1983) make the following assumption:
and are both convex and twice differentiable.
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) such that,
For referencing, these set of assumption will be referred to as Assumption A.
Assumption 2 implies that the industry’s marginal revenue (Appendix 1) decreases as industry supply increases. A proof of this statement 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) equilibrium
The 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 problem:
According to Murphy et al. (1983), the objective function in equation  is a strictly convex profit function over the closed, convex and compact interval. This implies that a unique optimum exists.
The functions can be referred to as the joint reaction functions of the Cournot firms. Murphy et al. (1983) define the aggregate reaction curve as:
The Stackelberg problem can be written as:
If solves, then the set of output levels is the SNC equilibrium with
To 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 determine:
Let the Lagrange multiplier associated with the maximisation problem  be. Murphy et al.’s (1983) approach here is to determine for which the optimal. The following result, obtained from Murphy et al. (1983), defines the optimal solution to problem :
Theorem 1: For a fixed, consider Problem suppose that satisfy Assumption A. Denote by the unique optimal solution to and let be the corresponding optimal Lagrange Multiplier associated with problem . (In case since alternative optimal multipliers associated with problem  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  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 and
Murphy et al. (1983) describes the aggregate Cournot reaction curve as follows:
is 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 found in the report by Murphy et al. (1983).
Murphy et al. (1983) state that if solves the Stackelberg problem , 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. solves:
But since solves the Stackelberg problem , the following must hold:
In fact, is the lower bound 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 upper 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 equilibrium to have adjusted its output to the level for which marginal cost equals price. Murphy et al. (1983) summarise this model as follows:
For fixed and suppose is a set of output levels such that for each firm solves:
For the “Stackelberg firm”, let satisfy:
In addition to Assumption A, if is strictly convex, then a unique solution exists and satisfies conditions  and . The Equilibrating Program with a fringe becomes:
Theorem 1 holds for with and which implies that. In fact, if is strictly convex, is the upper bound of.
Collectively, is bounded as follows:
3.1.4 Existence and uniqueness of the Stackelberg-Nash-Cournot equilibrium
Murphy et al. (1983) prove the existence and uniqueness of the Stackelberg-Nash-Cournot (SCN) equilibrium. Their approach to the proof is summarised below:
For the SNC equilibrium to exist, and for should satisfy Assumption A.
Since is bounded and is continuous (as is continuous), the Stackelberg problem  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 , exists. As a result the SNC equilibrium exists.
If is convex, then the equilibrium is unique.
Since is convex, the objective function of the Stackelberg problem  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 problem
Murphy et al. (1983) provide an algorithm in their report to solve the Stackelberg problem. This algorithm is summarised as follows:
To start with the Stackelberg firm needs the following information about the market and the Cournot firms:
Cost 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 information, the Stackelberg firm need to determine the lower bound and split the interval into grid points with, where and (from ). A piecewise linear approximation of is made as follows:
is an approximation to and from equation  it follows that:
Note that at each grid point the approximation agrees with.
The Stackelberg problem , thus, becomes:
can be re-written as:
Thus problem  becomes:
The objective function is strictly concave and solvable.
Let be the objective function of the Stackelberg problem  and the objective function of the piecewise Stackelberg problem , then:
Suppose 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 evaluated in order to determine. Theorem 1 can be used here. Recall 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 determining
Source: Smeers 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 redefined at an appropriate region to improve accuracy.
Murphy et al. (1983) go on to determine the maximum error from the estimated optimal Stackelberg solution. This is summarised below:
Let be the derivative of with respect to , then:
Let 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 as:
satisfies the following:
This concludes the review of Murphy et al.’s (1983) nonlinear mathematical programing model of the Stackelberg problem in an oligopoly.
3.2 A Stochastic Version
Smeers and Wolf (1997) provide an extension to the nonlinear mathematical programming version of the Stackelberg model by Murphy et al. (1983) discussed in subsection 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 firm’s 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 Wolf’s (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 Assumption
For 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 probability of occurrence.
The same Assumption set A discussed in subsection 3.1.1 apply here with alterations made to conditions  and . Assumption set A can be re-written as:
and 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-Cournot (SSNC) equilibrium
Smeers and Wolf (1997) use the same approach seen before to derive the SSNC equilibrium.
The Cournot problem can be re-written as follows:
For 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 problem:
Note 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  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 becomes:
The Stackelberg problem with demand uncertainty can be written as:
Note the Stackelberg problem defined problem  differs from that defined in . This is because of the element of uncertainty. The Cournot problem  is similar to the Cournot problem  because the demand is known when the Cournot firms make their decision. In the Stackelberg problem  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 , with changes made to the Cournot output and demand function: For each demand scenario ,
Theorem 1 lays out a foundation on how to solve the Equilibrating program in problem  and can also be used to solve . Smeers and Wolf (1997) Summarise Theorem 1 as follows:
Theorem 2: For each fixed,
An optimal solution for the problem satisfies the Cournot problem  if and only if the Lagrange multiplier,, associated with the Equilibrating program , 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 problem
The 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  can be re-written as follows:
has the same properties as .
The Stackelberg problem , thus, becomes:
Hereafter, the algorithm summarised in subsection 3.1.5 can be used to solve this problem.
3.3 Numerical Example
In 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 functions:
It is assumed that the Stackelberg firm and Cournot firms are identical. The Cournot problem  becomes as follow, with as the optimal solution:
Solving this problem yields:
Note the upper bound of is found by setting. The working to get equation  is shown in Appendix 2.
The aggregate reaction curve can be written as:
Using this information, this example is now extended to Smeers and Wolf’s (1997) model with numerical values.
Note that the functions listed in equations , ,  and  satisfy Assumptions A & B and other properties discussed in previous sections.
Suppose and. And suppose demand is unknown when the Stackelberg firm makes its decision. The cost functions of the firms will be as follows:
Figure : Different Demand Scenarios
The tables below describe the possible demand scenarios, probability of each scenario occurring, the joint reaction curve and aggregate reaction curve for, and:
= Demand falls,
= Demand remains unchanged,
= Demand Increases,
Joint reaction curve,
Aggregate reaction curve,
Using this information, the Stackelberg problem  can be solved. First, the estimated price element can be calculated as follows:
Substituting this result back into the Stackelberg problem  gives:
This 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 scenario:
Figure : Optimal Output, Price and Profit
Stackelberg firm Profit,
Cournot firm Profit,
The 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: Discussion
In this section, the practical applications, drawbacks and possible extensions to Stackelberg models are discussed.
4.1 Practical Applications of Stackelberg models
Stackelberg models are widely used by firms to aid decision making. Some examples include:
Manufacturer-Retailer Supply Chain
He et al. (2009) present a stochastic Stackelberg problem to model the interaction between a manufacturer and a retailer. The manufacturer would announce its cooperative advertising policy (percentage of retailer’s advertising expenses it will cover-participation rate) and the wholesale price. The retailer, in response, chooses its optimal advertising and pricing policies. When the retailer’s advertising and pricing is an importan
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