Mean Field Model for an Advertising Competition in a Duopoly

by   René Carmona, et al.
Princeton University

In this study, we analyze an advertising competition in a duopoly. We consider two different notions of equilibrium. We model the companies in the duopoly as major players, and the consumers as minor players. In our first game model we identify Nash Equilibria (NE) between all the players. Next we frame the model to lead to the search for Multi-Leader-Follower Nash Equilibria (MLF-NE). This approach is reminiscent of Stackelberg games in the sense that the major players design their advertisement policies assuming that the minor players are rational and settle in a Nash Equilibrium among themselves. This rationality assumption reduces the competition between the major players to a 2-player game. After solving these two models for the notions of equilibrium, we analyze the similarities and differences of the two different sets of equilibria.



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1 Problem Statement and Literature Review

In this paper, we analyze an advertising competition in a duopoly with special attention paid to consumer behavior. We consider a model with large companies which we regard as major players, and a large number of consumers whom we treat as minor players. Company , produces product , and product differentiation is horizontal which means that even if the quality of the products are the same, they are differentiated in the consumers’ perception. Therefore, at the same price, some consumers prefer Product 1, while others prefer the other product. The model is designed as a static (one-shot) game.

As expected in a duopoly, one of the goals of the companies is to increase their sales. According to [Bass et al., 2005], a company in a duopoly can increase its sales either by increasing its market share or if there is a market expansion. We assume that there is no market expansion; in other words, the total sales of the products stay constant. This is reasonable since we work with a static model instead of a dynamic one. Since we are assuming that the market size is constant, one of the goals of the companies is to increase their market share. [Doyle, 1968] states that market share can be increased either by decreasing the price, or increasing the advertising. He also mentions that in a market with few companies, competition is through non-price ways.

[Mankiw, 2012] states that companies in oligopoly with differentiated consumer products such as soft drink, perfume or breakfast cereal have incentive to invest in advertising to make the consumers less price elastic. One of the goals of advertisement is to convince consumers that companies’ products are more differentiated than they really are. Therefore, advertisements are more often than not persuasive instead of informative, trying to create a brand name and foster brand loyalty. On the other hand, consumers may perceive advertisement as a signal of quality, and this may make them likely to prefer the highly advertised product. Therefore, persuasive advertisements affect consumer’s preference by boosting the product’s perceived value.

According to [Clarke, 1973], advertisement of a company does not only affect their bottom line, it also affects the opposing company. When a company advertises more, it increases its own sales and decreases the opposing firm’s sales. Therefore, a company would like to increase its relative advertising which is the ratio of their own advertisement efficiency to the total advertisement efficiency. This is an instance of negative externality as increasing one of the product’s advertisement efficiency leads to a decrease in the opposing company’s demand.

It is particularly hard to find Nash equilibria in games with large numbers of players. However, by assuming a form of symmetry among the players’ behaviors, and letting the number of players go to infinity while the influence of each individual player fades, we can make use of the recently developed theory of Mean Field Games (MFGs). Mean Field Game models were introduced by [Lasry and Lions, 2007], and independently by [Huang et al., 2003, Huang et al., 2004, Huang et al., 2006].

The history of the subject and the development of the probabilistic approach to the solution of Mean Field Games introduced in [Carmona and Delarue, 2013] and further information can be found in the two volume book of [Carmona and Delarue, 2018]. While MFGs are relevant in plenty of practical situations, in many real life applications there exists a player that affects the system disproportionally, for example a government or a regulator. In these cases, the addition of a major player may be required. Mean Field Games with major and minor players were introduced by [Nourian and Caines, 2013] and analyzed by [Carmona and Zhu, 2016], [Carmona and Wang, 2017], and [Bensoussan et al., 2016]. In these types of games, the minor players’ decisions are affected by the aggregate of the other minor players as well as the decision of the major player. On the other hand, the decision of the latter is only affected by it own costs and rewards, and aggregate statistics from the population of the minor players. The originality of our contribution is to consider the competition between two major players affecting the field of minor players in a way akin to what was considered in the literature we just cited.

In this paper, we analyze two different equilibrium notions for advertisement and product consumption levels in a duopoly. In the first case, a Nash equilibrium between both major players and the consumers is analyzed. In the second one, the major players compete in a 2-player game assuming that the consumers are rational, and anticipating their purported behaviors. They choose their advertising policies assuming that the consumers will react to their choices and settle in a Nash equilibrium among themselves. We call this equilibrium “Multi-Leader-Follower Nash Equilibrium”. So for this second equilibrium notion, the major players compete among each others, but vis-a-vis the consumers, they behave as in a Stackelberg game by taking actions assuming that the minor players will react rationally. Even though leaders and followers do not act contemporaneously in the original Stackelberg game model introduced by Heinrich Stackelberg in 1934, this will be the case in the first of our models. Note also that a model of a Multi-Leader-Follower game for followers was analyzed in [Hu and Fukushima, 2015], but the mean field limit and the subsequent Mean Field Game formulation for the minor players were not considered.

We call the first model setting where we search for a Nash equilibrium among all major and minor players “MFG Formulation for a Nash Equilibrium (NE)”, and the second setting where we search for an equilibrium when the minor players are settling in a Nash equilibrium among themselves while reacting to the major players who are playing a 2-player game “MFG Formulation for a Multi-Leader-Follower Nash Equilibrium (MLF-NE)”.

With this model, we conclude that for companies, it is inefficient to use Nash Equilibrium advertising strategies instead of using Multi-Leader-Follower Nash Equilibrium strategies. The reason for this is that companies overly advertise and consequently incur high costs, if they are not able to understand how the consumers are going to react to their strategies (NE setup). Therefore, it is recommended that they should understand the consumers’ behavior and use the MLF-NE strategies. Further, we also deduce that a company in an adverse position initially (i.e. having a lower market share at the beginning) may end up as a market leader, if the companies are able to analyze consumers’ reaction and in other words, use MLF-NE strategies. However, if the companies are using NE strategies while advertising, the market leader protects its position.

Advertising behavior of one major player with a large number of minor players is analyzed in [Salhab et al., 2016]. However, in that paper, the model is dynamic and there is no competition among major players. Competition in terms of price and quantity in an Oligopoly by using Mean Field Games was analyzed by Chan and Sircar [Chan and Sircar, 2015]. In this model, consumers are not included as players and a large number of firms are set as players; moreover, the competition between them is not in terms of advertising. Therefore, our model is the first model that analyzes advertising competition in a duopoly under the Mean Field Games paradigm with multiple major and minor players.

The paper is structured as follows. First, we introduce the model with consumers and articulate the equilibrium notions in Section 2. Then we give the mean field game formulation in Section 3. We state amd prove our existence and uniqueness results for both equilibrium notions in Section 4. Finally we compare the properties of these two equilibrium notions through numerical experiments in Section 5.

2 N-Player Model

2.1 Minor Players

We first consider the case of a finite number of consumers and we assume they behave in a symmetric manner. A generic consumer (minor player) is denoted as minor where .

Each consumer controls their preference rate for Product 1 which is denoted as . In particular, if , then consumer buys Product 1 only, whereas, if , they buy Product 2 only. Whenever , their consumption of Product 1 is of their total consumption. Like for the type of a player in a Bayesian game, we assume that the initial value of the control of consumer is random and has a-priori distribution where

denotes the set of probability measures on

. Each player knows their initial preference, but does not know the others. We shall assume that the actual control will be a feedback function of its initial value.

Given the major players’ advertisement efficiencies and the empirical distribution of the other minor players’ controls, each consumer decides on their own control according to their goals and costs. Because of our symmetry assumption, we assume that all the consumers have the same objectives. Firstly, they want to be faithful to their initial preferences, so they do not want to change their initial choices by much. However, consumers care about the choices of others and they do not want to deviate from the average, so they want to buy the more commonly preferred product. Finally, they want to increase their total utility from the products. With all these conditions in mind, we define the optimization problem of consumer as follows:


where and . The rationale for the choice of the above objective function can be explained as follows:

The first term represents the unwillingness of a consumer to change preference. This may be caused from brand loyalty or not being prone to change. The second term represents the fact that a typical consumer does not want to deviate from the average: denotes the mean of the controls of other consumers, and can be interpreted as the market share of Product 1 when the number of players is large. Here and represent the relative importance given to these cost terms. In what follows we use for simplicity. The last part is the maximization of the utility of a consumer from the consumed products. We use the utility function already used by [Hattori and Higashida, 2012], and previously by [Singh and Vives, 1984] and [Garella and Petrakis, 2008]:


Here, represents the substitutability degree of the two products: as it becomes closer to 1, consumers become more price elastic. Since we want consumers to be perfectly price inelastic, we assume . We also assume that the true qualities of Product 1 and 2 are the same, and we denote their common value by . The number denotes the perceived incremental quality as a result of the advertisement of Product . Here we assume:


with and . This intuitively means that the utility from Product increases with the advertisement efficiency of Product and the advertisement of the opposing firm does not have an effect on consumer’s perceived quality for the Product . For the sake of simplicity, we are defining and as follows:


2.2 Major Players

The two major players are the competitive companies in the duopoly. Major players and produce Product 1 and Product 2 which are not differentiated in quality but are horizontally differentiated in the perception of the consumers. This can be understood as the famous example of the Pepsi and Coke advertisement competition: even if people fail blind-folded test, they continue to like Coke over Pepsi, or the opposite.

According to [Doyle, 1968], companies in a duopoly tend to compete with non-price means. Therefore in our model, companies are not controlling the price. They may compete in terms of loyalty schemes, quality differentiation or advertisement. Since advertising is one of the main forms of competition in a duopoly, in this model both major players control their advertisement efficiency which we take as the square root of the amount they spend on advertisement and which we denote by for major player , . Here, advertisement is persuasive and it does not have predefined targets. In other words, it affects every consumer in the same way and hopefully, positively.

Major players have similar goals and costs. Firstly, they want to maximize their market share, and secondly, they want to advertise relatively more than the opposing company to be better known by consumers. Finally, they want to minimize their cost of advertisement. We define their optimization problems as follows:

For major player 1:


For major player 2:


where . Here is the rationale for these choices:

The first part of the cost function is for increasing their own market share. As previously stated, market share of Product 1 is taken as the mean of the control of minor players and denoted here as . Assuming that both companies own the entire market and that there is no market expansion, market share of Product 2 is given by . Here, we use ideas from the classical dynamic Lanchester Model used by [Fruchter and Kalish, 1997] and [Fruchter, 1999]. Lanchester Combat Model is a competitive extension for the Vidale-Wolfe Model proposed by [Vidale and Wolfe, 1957] and used by [Bass et al., 2005]. Different modifications of this model are also used by [Erickson, 1995], and [Prasad and Sethi, 2003, Prasad and Sethi, 2004]. According to the Lanchester model, over time the dynamics of market share are given respectively for Product 1 and 2 by:


where denotes the market share of Product 1 and denotes the market share of Product 2. Here, denotes advertisement efficiency of major player , where is the positive efficiency constant and is the square root of advertisement amount of major player . Intuitively, each company takes a part of the opposing company’s market share which is proportional to their advertisement efficiency, and at the same time, each of them is losing a part of their own market share proportionally to the opposing company’s advertisement efficiency. For the sake of simplicity, in the remaining of this paper, we take and and are called advertisement efficiencies of major player 1 and 2 respectively. The above equation gives the dynamics of the market share for Product 1 over time. However, since our model is designed as a static game, we assume that companies are focusing on increasing their market share instantly. Since they are minimizing, they are taking the negative signed versions of above change rates.

The second part of the cost function comes from the desire to be known more widely by increasing their relative advertisement efficiency. For example, major player 1 tries to increase the ratio , where is a constant. Since the cost function is minimized we use again negative sign for this part. Here the addition of the constant is to enable the analysis of the cases where a company does not advertise, namely or .

The last contribution to the cost is intended to minimize advertisement spending. Here gives the cost per unit of advertisement and gives the advertisement amount of major player . For the sake of simplicity, we assume that both companies have the same unit advertisement cost. Therefore major player tries to minimize .

2.3 Equilibrium Notions

As explained in the introduction, we analyze two different types of equilibrim.

Definition 2.1 (Nash Equilibrium).

With the same notations as in the previous definition, a strategy profile is called a Nash Equilibrium if:

  • [label=]

  • For any fixed , for all , we have:

  • For all , we have:

  • For all , we have:

    where and .

Definition 2.2 (Multi-Leader-Follower Nash Equilibrium).

Assume there exist N many minor players and 2 major players. Let and ,
, are strategy profiles and cost functions for N minor players and 2 major players, respectively. Then a strategy profile is called a Multi-Leader-Follower Nash Equilibrium if:

  • [label=]

  • For any fixed , for all , we have:

  • For all , we have:

  • For all , we have:

    where and .

3 Mean Field Game Formulation

The present formulations correspond to the asymptotic regime whereby the number of minor players goes to . Since players are identical, we focus on a representative minor player.

Remark 3.1.

In the limit , the representative player becomes infinitesimal; therefore, can be taken as . Hereafter, is used for the mean of the control of other minor players in the infinite number of player game instead of and it is equal to the mean of the controls of the all minor players, .

When we analyze the mean field game regime, representative minor player’s cost function can be written as:


where is the feedback control function used by the representative minor player to update their initial preference rate . Recall that we use the notation for the mean of the control of the minor players. Hereinafter, the mean of the initial preference rate is denoted as or .

The major players cost functions remain the same as in the case of finite. Only for consistency in the notation, is changed to . We define the equilibrium notions in the mean field game model as follows:

Definition 3.2 (Nash Equilibrium in the Mean Field Game with Multiple Major Players).

A strategy and a mean field tuple form a Nash Equilibrium in the Mean Field Game regime with Multiple Major Players if for any , we have:

Definition 3.3 (Multi Leader Follower Nash Equilibrium in the Mean Field Game).

A strategy and a mean field tuple form a Multi Leader Follower Nash Equilibrium in the Mean Field Game regime if for any , , we have:

4 Main Theoretical Results

4.1 Nash Equilibrium in the Mean Field Game with Major Players

First, we focus on finding the Nash Equilibrium between major players and minor players. Here, all players are giving their best responses given other players’ controls. We approach the model as follows:

  1. First we fix the mean field and solve 2-player game of Major Players to find their best responses given the mean field and the other major player’s control:

    • [label=]

    • For major player 1, find

    • For major player 2, find

  2. We solve the 2-equation system of and to find the equilibrium controls and of major players in the 2-player game given the mean field of minor players.

  3. Then we fix mean field , and :

    • [label=]

    • By considering the limit , solve the following mean field game problem for Minor Player where is the initial control of minor players which is random, in other words:

      • Find s.t:

    • Fixed Point Argument:

      • Find

  4. Solve the following 3-player system:

Remark: In step 2, instead of solving 2-player game and finding and , we can continue directly to step 3. In this case, we would have the following 3-equation system at the end:

Proposition 4.1.

The final equation system is given as:

Proof of Proposition 4.1.

The proof is consisted of 3 parts that are given above.

  1. Solution of 2-Major Player Game with Given Mean of Minor Player Control. In this part, with the given mean of the minor players’ controls, , we are analytically solving 2-player game of major players. For this reason, first we need to find best responses of major players, and , that minimizes their cost functions.

    Remark: For finding the controls that minimize the cost functions of major players, first order derivatives can be calculated. Although, since we deal with a constrained optimization, this minimizer may be out of the domain that the function is tried to be minimized. In this case, the minimizer would be on the boundary, this refers to case 2 in Figure 1.

    Moreover, when the cost functions of major players, (4.1) and (4.2) are checked, it can be seen that they are strictly convex in and , respectively since it is assumed that . This means that we have unique minimizers.

    (a) Case 1: Minimizer is in
    (b) Case 2: Minimizer is out of
    Figure 1: Different Cases for the Minimizers of Strictly Convex Functions

    With above remark in our minds, first order conditions are calculated and minimizers are found as: