Excerpt

## Table of Contents

List of Figures

List of Tables

1 Introduction

1.1 Motivation and Problem Definition

1.2 Research Question and Objectives

1.3 Course of Investigation

2 Theoretical Background

2.1 The Prisoners‘ Dilemma

2.2 The Application of the Prisoners’ Dilemma in Pricing

2.2.1 Example: T-Shirt Retailer

2.2.2 Example: Two Restaurants

2.2.3 Price Wars as a Consequence of Price Cutting

2.2.4 The American Airline Industry as an Example

2.3 Solving the Prisoners’ Dilemma According to Theory

2.3.1 Rewards Approach

2.3.2 Repeated Games

2.3.2.1 Finite Repetition

2.3.2.2 Infinite Repetition

2.3.2.3 General Theory

3 The German Fitness Industry

3.1 General Development Within the Industry

3.2 Market Characteristics of the Chain Segment

3.2.1 Porter’s Five Forces

3.2.2 Defining the Target Firms

3.2.2.1 McFit

3.2.2.2 FitX

4 Application of the Prisoners’ Dilemma to the Fitness Industry

4.1 Rebuilding the Prisoners’ Dilemma

4.1.1 Assumptions for the Model

4.1.2 Mutual Cooperation

4.1.3 Unilateral Defection of FitX

4.1.4 Unilateral Defection of McFit

4.1.5 Mutual Defection

4.2 The Applicability of the Prisoners’ Dilemma

4.3 Solving the Pricing Dilemma of FitX and McFit

4.3.1 Rewards Approach as a Solution

4.3.2 Repetition as a Solution

4.3.2.1 Does Defecting Pay Off for FitX

4.3.2.2 Does Defecting Pay Off for McFit

4.3.3 Outcome and Implication

5 Conclusion

5.1 Comparison with the Real Rivalry Between FitX and McFit

5.2 Limitations and Further Research

6 References

## List of Figures

**Figure 1.** Gym members in Germany from 2003 to 2017

**Figure 2.** Distribution across the different gym types

**Figure 3.** Price structure 2017

## List of Tables

**Table 1.** Payoff Prisoners’ Dilemma

**Table 2.** Profits T-Shirt Retailers

**Table 3.** Profits of Restaurants

**Table 4.** Profits of Restaurants

**Table 5.** General Payoffs

**Table 6.** Top-Three Fitness Clubs in Germany (31. December 2017)

**Table 7.** Monthly Revenues of Gyms

## 1 Introduction

### 1.1 Motivation and Problem Definition

The fitness industry in Germany developed remarkably in the last one and a half decades. Between 2003 and 2017, the number of active members rose from 4.38 million to 10.61 million making it the most popular sport in Germany concerning members (Capelan et al., 2018, p. 10; Deloitte & DSSV, 2018). The main reason for the enormous increase in members is the rising health awareness among the German population. Society further agrees that becoming active and working out has a positive impact on well-being, health, and mobility especially for older adults (Capelan et al., 2018, p. 10).

Increasing demand fosters rivalry within the industry as all competing firms want to benefit from the high demand and bind potential customers to long-term contracts. The high interest on the consumer side can result in price wars as the rewards immediately gained from cutting the price are high (Haltiwanger & Harrington, 1991; Rotemberg & Saloner, 1986). Even though price cutting may be beneficial in the short run, the long-run consequences of price wars are decreasing profits within the industry (Morrison & Winston, 1996).

The fight on prices seems especially present in the discounter chain segment as gyms are homogenous, and differentiation amongst the competitors in this segment is somewhat limited. Thus, competing on price could be the instrument chosen by firms engaging in the discounter fitness sector to attract new members, increase market share, and raise short-term income while simultaneously putting long-term profitability at risk.

The fitness industry itself, as well as the dealings with the omnipresence of the risk of price wars in the German discounter fitness industry, have not been analyzed before. Consequently, the industry is particularly interesting to study as there is only limited knowledge so far within the academic world.

### 1.2 Research Question and Objectives

The author sees parallels between the discounter fitness industry, their pricing decision and one of the fundamental game-theoretical models, the prisoners’ dilemma, by simplifying the strategic choices and reducing the number of competitors in the market down to the two major firms.

This paper deals with the question whether the framework of the prisoners’ dilemma applies to the context of pricing in the German discounter fitness industry. A deeper understanding of the theory of the prisoners’ dilemma is required to answer this question. Consequently, the author wants to point out the most critical aspects of the framework, as it can be widely applied and does not necessarily need to have an economic context.

Furthermore, the author wants to inform the reader about the fitness industry in Germany. The main reason is that the industry itself is a market with high potential and experts see an even further growing demand for fitness. However, to the author's knowledge, the fitness industry and especially the discounter fitness industry in Germany has not been part of academic analyses.

Lastly, the author aims to find a solution to the dilemma that the fitness companies are trapped in by applying the general theory applicable to the prisoners’ dilemma to the context of the fitness industry and compares this with the real strategies used in the industry.

### 1.3 Course of Investigation

In the subsequent section, the reader receives information about the research context. Chapter 2 introduces the prisoners’ dilemma and provides theoretical background. More concrete, the basic dilemma is explained, applied to price strategies, and potential consequences are pointed out. The chapter concludes with possible solutions on how the prisoners’ dilemma can be dissolved.

Chapter 3 provides an industry analysis of the German fitness market. The general development within the industry is presented. Furthermore, the author lays the focus on the discounter chain sector and explains why this sector is particularly interesting. The rivalry within the industry as well as the attractiveness is determined using the extension of Porter’s (1979) five forces framework. This chapter ends with the introduction of the two leading firms operating and rivaling in the discounter chain sector on members.

Chapter 4 is based on the theory introduced in chapter 2 and applies the presented knowledge to the pricing strategies which can be used by the rivals competing in the discounter chain sector. To analyze whether the pricing decisions in the market draw back to the game-theoretic case of the prisoners’ dilemma, the author rebuilds the dilemma using data from the industry and presents the consequences on revenues given the strategic choices of the competing firms. Concluding, the author applies theoretical solutions to the real-life example to possibly solve the dilemma and further compares the results with real processes going on in the industry.

The author finalizes with summarizing the findings as well as elaborating on the limitations and avenues for further research.

## 2 Theoretical Background

### 2.1 The Prisoners‘ Dilemma

The prisoners’ dilemma is one of the most known examples of game theory. According to Kuhn et al. (1996), Al Tucker came up with the prisoners’ dilemma in 1950 while he was on leave at Stanford. A psychology professor approached him and asked whether he could give a seminar on his work concerning game theory. Tucker used the prisoners’ dilemma as an example of the “attendant paradoxes of non-socially-desirable equilibria” (Kuhn et al., 1996, p. 155) and inspired dozens of research papers and books (pp. 154-155).

Dixit, Skeath, and Reiley (2015) describe the prisoners’ dilemma as a situation in which two people are accused of having committed a murder. Since the prosecutors lack sufficient evidence and could only charge them three years for kidnapping, they rely on the help of the accused to confess the major crime. Now each of the prisoners has two options. One can either admit or deny the murder. Since the police interrogate both prisoners in separate rooms, they cannot communicate with each other. Depending on the choices each of them makes, they will either be convicted of murder or kidnapping only. If both prisoners admit, they will be sent to jail for ten years each. If only one captive concedes while the other denies, the one confessing will get a short sentence of one year for cooperating, while the other will face 25 years in jail. If both decide to stay silent and deny, each of them will be punished with three years in prison (Dixit et al., 2015). Table 1 summarizes the choices and outcomes for this game.

Abbildung in dieser Leseprobe nicht enthalten

**Table 1 .** Payoff Prisoners’ Dilemma. Adapted from Dixit et al. (2015, p. 99).

The value in the top right corner is the time in prison for player 2 and the amount in the bottom left of each cell is the punishment allocated to Player 1.

Dixit et al. (2015) continue that both players find themselves in a two-person, simultaneous-move game with the choice of either confessing or denying the crime. Simultaneous-move means that both players have to make their decisions at the same time and not one after the other knowing the previous decisions. As both of them act as individuals with the aim to optimize their own situation, they both try to minimize their punishment (pp. 99-101).

Player 1 will pick based on what he thinks Player 2 is going to do. If he assumes that Player 2 will confess, then it is also better for Player 1 to admit as ten years are better than 25 years. If Player 1 considers that Player 2 will deny, then it is still better for Player 1 to confess as this would mean a reduced punishment of one year instead of three years. Thus, in this game, it is always better for Player 1 to confess, regardless of Player 2’s choice. In game theory, a strategic decision that is regularly the best response irrespective of the selection of the other player is called a dominant strategy. So in this example, confessing is the dominant strategy for Player 1 as confessing is continuously better than denying for any alternative chosen by Player 2 (Dixit et al., 2015, pp. 99-101).

As rational players always seek the best individual outcome, they will always choose the strategy that strictly dominates the other strategies. Thus, Player 1 will decide to confess.

Since the outcome for Player 2 is symmetric, the dominant strategy for Player 2 is also to confess, regardless of Player 1’s choice as it is always better to admit than to deny.

Consequently, the predicted outcome of this game is (confess, confess). This solution is also the Nash equilibrium. A Nash equilibrium is a solution in which no player can improve the own outcome by switching the strategy and keeping the choices of other players constant (Dixit et al., 2015, p. 95). If both players have a dominant strategy, the solution is also the Nash equilibrium and no player has an incentive to unilateral change the decision.

When looking at the outcomes in Table 1, the solution (confess, confess) will lead to a custodial sentence of ten years each. Compared with the result for (deny, deny), both players are worse off using their dominant strategy than if they would both be using their non-dominant strategy of denying. Their dominant action will lead to ten years in prison while they both would only go to jail for only three year if they would cooperate with each other and choose to deny the crime. However, looking at the optimal solution of (deny, deny) the situation is an unstable equilibrium as both players have an incentive to cheat on the other and change their strategic choice as this would reduce their own punishment. As both players would try to cheat on the other, they would both decide to confess and thus, go back to the Nash equilibrium (confess, confess) (Dixit et al., 2015).

To conclude, a prisoners’ dilemma needs to fulfill three criteria. First, each of the two players has two different strategies. They can either cooperate, in this example denying the murder, or defect from cooperation meaning to cheat on the other, in this example confessing. Second, both players have dominant strategies. And lastly, the dominant approaches lead to a worse equilibrium than the non-superior strategies (Dixit et al., 2015, p. 101).

### 2.2 The Application of the Prisoners’ Dilemma in Pricing

According to Aumann (1987), the concept of strategic equilibrium is the most frequently used game theoretic solution in economics. He continues that it can be applied to so many different situations such as in oligopolistic markets, market entrance and exit, and in bargaining situations. However, Aumann (1987) also mentions that it can even be used in non-economic cases such as political votes, international politics, and biology.

The previous does not only count for the equilibrium concept itself but also applies to the prisoners’ dilemma. The theory behind it can be discovered in several different contexts. Some examples mentioned by Dixit et al. (2015) are political votes or biology. However, the prisoners’ dilemma is mostly applicable to economics.

#### 2.2.1 Example: T-Shirt Retailer

To apply the prisoners’ dilemma to the economy, Dixit and Nalebuff (2008) use two firms as an example and compare their pricing strategies to the conflict of interest depicted in the prisoners’ dilemma. The following chapter is based on Dixit and Nalebuff’s (2008) story of two rival firms that engage as mail-order firms selling clothes (pp. 66-71).

Every fall they have to decide on what prices to charge for the winter collection. Those prices will be printed in catalogs and mailed to the customers. This means it takes a while and prices have to be set simultaneously, meaning without knowing what the competitor charges for the same product. Both firms see that they have a shared pool of potential customers and that those are price sensitive.

Dixit and Nalebuff (2008) state that both firms offer the same t-shirt which has total costs per unit of $20, including production, transport, storage and order fulfillment. They also estimate, that if both firms charged a price of $80 each, each would sell 1,200 t-shirts, leading to profits of each. This is also the price, which maximizes the overall gain and thus would be charged if both firms would collude. The firms also estimate that reducing the price by $10 while the other firm still charges the higher price will lead to an increase of 1,000 units in sales volume for the price-cutting firm. 800 out of those new customers come directly from the competitor while the other 200 are new customers. Looking at the new outcomes, the price-cutter would make profits of . The other firm would make earnings of (Dixit & Nalebuff, 2008, pp. 66-71).

Dixit and Nalebuff (2008) continue that if both firms decide to reduce the price by $10, then the existing customers stay put, and each firm would only get 200 new customers, leading to sales of 1,400 per firm. Concerning profits, this would result in each. These profits are also depicted in Table 2. Consequently, each firm has to make a strategic choice of either charging the high price or cutting and charging the low price.

Abbildung in dieser Leseprobe nicht enthalten

**Table 2 .** Profits T-Shirt Retailers. Adapted from Dixit and Nalebuff (2008, p. 69).

As firms are not colluding and making choices that are best for them individually, each firm would pick the best response to any given action by the other firm. For example, if Rainbows End charges the high price, B. B. Lean should respond by choosing the lower price. This will lead to higher profits as $110,000 is preferred over $72,000.

Dixit and Nalebuff (2008) proceed that if Rainbow’s End decides to cut the price and charge only $70, then B. B. Lean should also charge the lower price as $70,000 is still better than $24,000. It becomes evident that B. B. Lean’s choice does not depend on Rainbow’s End’s decision. Thus, assumptions about the choice of the other firm do not matter as it is always better to charge the lower price. Therefore, cutting the rate is the dominant strategy for B. B. Lean (pp. 66-71).

Vice versa, Rainbow’s End behaves similarly as the payoffs are symmetrical. Consequently, the dominant strategies for both are to charge the lower price, leading to the Nash equilibrium of ($70, $70) and profits of $70,000 per firm.

However, now looking at the possible outcomes, one can observe that both firms would be better off if they would have conspired and would not have used the dominant strategy.

Dixit and Nalebuff (2008) notice that by cutting the prices and engaging in a price war, both firms contributed to decreasing profits within the industry.

#### 2.2.2 Example: Two Restaurants

Another example for the dilemma of right pricing decisions is introduced by Dixit et al. (2015). They use two restaurants in a small town to demonstrate the prisoners’ dilemma. The following example is taken from Dixit et al. (2015, pp. 134-139).

To keep it simple, Dixit et al. (2015) assume Xavier’s Tapas Bar and Yvonne’s Bistro only steward one dish, and the cost of serving one customer is $8. Xavier’s price is Px, Yvonne’s price is Py, and their number of customers are respectively and (measured in hundreds per month).

Xavier’s demand is given by the function:

Abbildung in dieser Leseprobe nicht enthalten

Since both have homogeneous products, Yvonne’s demand is:

Abbildung in dieser Leseprobe nicht enthalten

The equations imply that if one restaurant reduces its price by $1, it will gain 200 customers, while the other will lose 100. Hence, 100 customers switch from one restaurant to the other, and 100 new consumers will start going to the restaurant as it becomes more affordable for them to eat in the restaurant. Dixit et al. (2015) continue by setting up the profit functions.

Xavier’s profit function for a month is:

Abbildung in dieser Leseprobe nicht enthalten

And vice versa, Yvonne’s profit function is:

Abbildung in dieser Leseprobe nicht enthalten

The equation above shows that Xavier’s profit does not only depend on his choice of price but also on Yvonne’s. The best response function needs to be calculated to find the optimal solution.

Here, the author of this paper deviates from the solution presented by Dixit et al. (2015) to introduce a generally applicable solution. To find Xavier’s best response function, the author uses the approach of taking the partial derivative.

Xavier’s profit function can be rearranged to:

Abbildung in dieser Leseprobe nicht enthalten

Now, the partial derivative with respect to is calculated:

Abbildung in dieser Leseprobe nicht enthalten

The partial derivative is set to be equal to zero and solved for:

Abbildung in dieser Leseprobe nicht enthalten

The equation above is Xavier’s best response function and depends on the price Yvonne charges.

Respectively, Yvonne’s profit-maximizing price can be calculated similarly as all relevant aspects are identical:

Abbildung in dieser Leseprobe nicht enthalten

To find the equilibrium price, in Xavier’s function can be substituted by Yvonne’s best response function.

The equation can be simplified to and due to symmetry . Thus, in equilibrium each restaurant serves the meals for $20 each to 2400 customers, summing up to profits of $28,800 per month per restaurant.

However, this is not the optimal solution for both restaurants. To calculate the price if both firms would collude, Dixit et al. (2015) set up a new profit function with both restaurants charging the same price P:

Abbildung in dieser Leseprobe nicht enthalten

P represents the price the firms charge and Q represents the quantity supplied by each firm. Deriving the equation with respect to P and setting it equal to zero leads to , which can be rearranged to . Thus, the jointly best outcome for the restaurants would be achieved by a price of $26 because it will lead to profits of $32,400 for each restaurant.

For simplicity, Dixit et al. (2015) assume that the two prices $20 and $26 are the only options for the restaurants in a strategic game. These two choices are depicted in Table 3. The missing outcomes can be calculated by plugging in the prices of $20 and $26 into the original profit functions, leading to profits of $21,600 for the firm charging more and $36,000 for the one charging only $20 per meal.

Abbildung in dieser Leseprobe nicht enthalten

**Table 3 .** Profits of Restaurants. Adapted from Dixit, Skeath, and Reiley (2015, p. 380).

By analyzing the different strategic choices, one notices that both firms are trapped in a prisoners’ dilemma, as they both have the dominant strategy to cut prices, but by doing so the joint outcome ($20, $20) results in lower profits for both than if they both would have chosen the non-dominant strategy of keeping the price high (Dixit et al., 2015).

#### 2.2.3 Price Wars as a Consequence of Price Cutting

Reducing the price to undercut competitors, increase market share, and optimize the own position within the market can lead to retaliation of competitors. To be in a competitive situation, they will adapt their strategy so that their price meets the new price or even undercuts it. This behavior can lead to price wars. Busse (2002) defines the term price war in the following way:

A price war is a period in which the firms in an industry or a market set prices that are significantly below the usually prevailing prices, generally implying a change in strategy within a set of oligopolists. In game-theoretic term, this can be modelled as a temporary period of noncooperative behavior among players whose normal course is a (tacitly) collusive equilibrium. (p. 299)

Models of price wars can be divided into three different classes, giving different explanations on why price wars occur (Busse, 2002, p. 299).

**[...]**

- Quote paper
- Moritz Moerke (Author), 2018, Pricing Strategies in the German Discounter Fitness Industry. The Prisoners’ Dilemma, Munich, GRIN Verlag, https://www.grin.com/document/943445

Publish now - it's free

Comments