How efficient is dynamic pricing for sport events? Designing a dynamic pricing model for Bayern Munich.

Kemper, Christoph ; Breuer, Christoph

Introduction

Football clubs usually generate revenues from three different sources: broadcasting, commercial activities, and ticket sales (Buhler & Nufer; 2010). In contrast to clubs in the English Premier League, teams in the Bundesliga are unable to capitalize on ticket sales (Nufer & Fischer, 2013). Until now ticket prices in the Bundesliga have been the lowest among Europe's top five leagues despite its clubs having the highest stadium attendance in European football (Nufer & Fischer, 2013). Yet, revenues from ticket sales are the most controllable stream of income for football clubs, and charging a higher price for tickets can drive revenues without any upfront investment (Nufer & Fischer, 2013). Hence, the optimization of ticket revenues represents a major opportunity for sport clubs of the Bundesliga to cope with increasing financial obligations like rising player salaries, transfer fees, or stadium maintenance costs (Bundesliga Report, 2014; Drayer, Shapiro, & Lee, 2012). While in recent years all Bundesliga clubs have implemented some form of variable ticket pricing, the new developed pricing strategy of dynamic ticket pricing has not been applied thus far by football clubs in Germany.

After the San Francisco Giants introduced dynamic pricing to the sports industry in 2009 the subject attracted a good deal of interest. Dynamic pricing refers to the concept of adapting ticket prices depending on the actual demand; this concept entails that prices may change from day to day or even from minute to minute (Drayer et al., 2012). Although the San Francisco Giants experimented with the new pricing approach for only 5% of their stadium's capacity in 2009, the financial results were so impressive that in the following season they priced all stadium seats dynamically, which resulted in a ticket revenue increase of 7% (Kahn, 2011). Consequently, in the 2011 season three other Major League Baseball (MLB) teams implemented dynamic pricing (Paul & Weinbach, 2013), and in the 2012 season 30 organizations from MLB, National Basketball Association (NBA), National Hockey League (NHL), Major League Soccer (MLS), and National Association for Stock Car Auto Racing (NASCAR) were working together with Qque, the company that developed the dynamic pricing software (Dunne, 2012).

Due to the success of dynamic pricing in American sports leagues, this pricing approach has become a topic for scientific research as well. Drayer et al. (2012) examined the applicability of a dynamic pricing strategy for the sports industry from a managerial point of view and concluded that dynamic pricing may be regarded as an appropriate pricing strategy for sports events. Nufer and Fischer (2013) came to the same conclusion based on an analysis of European football leagues and stated that it can be only a matter of time before dynamic pricing will be applied by major football clubs in Europe. The number of scientific publications on the subject matter of dynamic pricing in sports, however, is low; Drayer et al. (2012) pointed out that more research in this new field of research is necessary.

While previous research has focused exclusively on analyzing already-existing dynamic pricing models in American baseball (Paul & Weinbach, 2013; Shapiro & Drayer, 2012, 2014), this is the first study to pursue the objective of designing a dynamic pricing model for a sports club in a country where clubs currently do not apply such a pricing approach. Moreover, no published study has ever applied the mathematical theory of dynamic pricing to the context of sports clubs. Hence, this study adds to the existing scientific literature. It does so by combining the theoretical principles of dynamic pricing with empirical methods that assist in estimating demand-functions, with the aim of crafting a dynamic pricing model for a Bundesliga club, using the example of Bayern Munich. Following this approach, the current study is actually the first to simulate the financial effects of a dynamic pricing approach for a sports club prior to its implementation.

In this paper, first, the basic theoretical principles of dynamic pricing are explained. Second, the literature on sport demand, dynamic pricing in general, and on dynamic pricing with a particular focus on the sports industry is reviewed. Because Bayern Munich is used as an example, the current pricing structure of that football club is presented in the third step, which serves as an orientation for the dynamic pricing model crafted later in the paper. The next step illustrates the data collection process, followed by a description of the empirical data and a presentation of the estimated demand functions. These statistical models are used to specify a dynamic pricing model, and a concrete scenario is evaluated. After that, a simulation of the effects is illustrated and the simulation's results are discussed. A conclusion summarizes the paper and points out limitations and further research demand.

Theoretical Framework

The current paper is based on mathematical models of dynamic pricing, which operate from the perspective of neoclassical demand theory. Here, a consumer is subject to a budget constraint and is assumed to choose a consumption bundle that maximizes utility based on the customer's preferences. Thus, the price is negatively correlated with the demanded quantity (Andreff & Szymanski, 2009). This relation is supported by numerous studies on sport demand (e.g., Borland & MacDonald, 2003; Villar & Guerrero, 2009).

So far there does not exist a standard definition of dynamic pricing. Generally, though, dynamic pricing appears to be understood as a strategy of price setting by which the seller sets a price that is not negotiable and that varies dynamically over time (Gonsch, Klein, & Steinhardt, 2009). Gallego and van Ryzin (1994) take a theoretical-methodical approach in terms of "... an initial inventory of items and a finite horizon over which sales are allowed," which gives rise to the ".tactical problem of dynamically pricing the items to maximize the total expected revenue" (p. 999). In an overview paper, Bitran and Caldentey (2003) describe dynamic pricing as ". the problem faced by a seller who owns a fixed and perishable set of resources that are sold to a price sensitive population of buyers. In this framework, where capacity is fixed, the seller is mainly interested in finding an optimal pricing strategy that maximizes the revenue collected over the selling horizon" (p. 203).

Closely related to the concept of dynamic pricing is revenue management (also known as yield management). However, a general hierarchical classification is not possible (Gonsch et al., 2009). Whereas some authors argue that dynamic pricing and revenue management are alternative concepts of equal value (Boyd & Bilegan, 2003), others, such as Bitran and Caldentey (2003), subsume revenue management under the more general concept of dynamic pricing; authors such as Gonsch et al. (2009), Talluri and van Ryzin (2004), and Tscheulin and Lindenmeier (2003) regard dynamic pricing as an additional version of the generalized concept of revenue management. In this respect the last-mentioned authors distinguish between quantity-based revenue management, which corresponds to the classic form of revenue management, and pricebased revenue management, which corresponds to dynamic pricing. From that distinction also emerges the point of view of this paper. Quantity-based revenue management was originally developed by the airline industry and focused primarily on capacity management. The idea was to establish fencing criteria like cancelation policies or minimum stays in order to segment the market and to capitalize on the different preferences of the customers. The objective of the revenue management system was then to allocate the resources to the product variants and to decide whether to accept or reject the incoming product inquiries (Gonsch et al., 2009), that is, to sell the right seat to the right customer at the right price to maximize yield (Kimes, 1989). Price-based revenue management, on the other hand, focuses on the dynamic variation of the price in order to steer demand and not on the allocation of capacities. Low cost carriers often apply such a pricing approach. They even abandon all fencing criteria in order to offer only one price, which is varied over the selling period (Gonsch et al., 2009). Nevertheless, from a methodical perspective a clear distinction between quantity-based revenue management and price-based revenue management is difficult because in certain circumstances the different pricing approaches can be converted into each other (Gonsch et al., 2009).

However, the overall issue of dynamic pricing refers to the maximization of a monetary variable (e.g., revenues or profit). A frequently considered dynamic pricing model, which is also used as the theoretical foundation of this paper, is the Bernoulli model of dynamic pricing. The model assumes that a monopolist sells one single product with a fixed number of items C over a limited period of time to an indefinite population of time homogenous and myopic (1) customers. The model assumes that in every period t only one customer wants to purchase one single item. When the selling period is over the product becomes worthless and cannot be sold any more. Furthermore, it is assumed that the selling period can be divided into discrete segments, starting with period T and ending with period 1. At the beginning of each period the seller can set a price pt and the customer purchases the ticket, if his individual willingness to pay is equal or higher than the seller's price offer. Hence, there exist only two possible outcomes (purchase or no purchase) and a Bernoulli approach can be formulated. The customer's purchase probability [d.sub.t]([p.sub.t]) is modeled as a continuous random variable. The mathematical term can be specified by the following Bellman-Equation (Gonsch et al. 2009):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

for all [less than or equal to] c [less than or equal to] C and t = T,..., 1.

with boundary conditions:

V(c,0) = 0 for all c [greater than or equal to] 0 and (2)

V(0,t) = 0 for all t=T,..., 1 (3)

The equation maximizes the expected revenues, and consists of two summands:

* The revenues that can be yielded directly in a certain period of time based on the chosen price pt. This means that with the probability [d.sub.t]([p.sub.t]) a purchase takes place in period t, resulting in revenues pt. As a consequence, the capacity c is reduced by one and for the remaining periods t-1 the expected revenues account for V(c-1, t-1).

* The second summand models the case that no purchase takes place. This occurs with probability 1- [d.sub.t]([p.sub.t]). Therefore, no revenues are yielded in that period and the capacity is not reduced. In this case, the remaining expected revenues account for V(c, t-1).

The boundary conditions (2) and (3) ensure that no additional revenues are yielded at the end of the selling period (2) or in the case that all items are sold (3).

The objective of the dynamic pricing problem is to determine a price policy that maximizes the expected revenues V(c,t) over all periods. In this respect all possible states (c,t) have to be evaluated. The evaluation can be realized by applying a roll-back-approach (Gonsch et al., 2009). However, the main issue concerning the specification of a dynamic pricing model is to determine the relevant demand function [d.sub.t]([p.sub.t]). In this matter, the current paper estimates the demand function [d.sub.t]([p.sub.t]) empirically by the evaluation of auctions on ebay.de. Based on pricing theory, auctions are considered to be the most suitable and valid method to measure willingness to pay, because prices can fluctuate freely and the seller of the product, not the buyer, determines the price (Lusk & Shogren, 2007; Miller et al., 2011; Volkner, 2006a, 2006b). In the primary ticket market, however, prices are set by the sports clubs and are not subject to an auction format. Consequently, the evaluation of ebay auctions seems to be the only possible way to apply the findings of pricing theory research and to estimate the fans' willingness to pay most accurately. In this respect, Drayer, Stotlar, and Irwin (2008) argued that the existence of ticket brokers and the reselling of tickets is evidence that tickets in the primary market are priced inefficiently. Therefore, Drayer and Shapiro (2009) concluded that the knowledge of the secondary market price provides valuable information for price setting in the primary market and Drayer (2011) as well as Drayer and Martin (2010) argued that the secondary market should be used in order to enhance the knowledge of the actual value of tickets.

Review of Related Literature

The overview of the literature is organized in three steps. In a first step, literature on sport demand and the role of ticket prices is reviewed. Second, an overview of the development in the field of dynamic pricing in general is presented. In a third step, the literature on dynamic pricing with a focus on the sport industry is presented.

Sport Demand

Sport demand has been a much discussed topic in the field of sport economics (overviews can be found in Borland & MacDonald, 2003; Cairns, Jennett, & Sloane, 1986; Downward & Dawson, 2000; Villar & Guerrero, 2009). Studies suggest that the ticket price has a negative effect on stadium attendance and that sport managers usually set ticket prices in the inelastic portion of the demand curve, which was interpreted in the way that sports clubs do not act as revenue optimizers (Andreff & Szymanski, 2009; Borland & MacDonald, 2003; Villar & Guerrero, 2009). However, further studies like Coates and Humphreys (2007) as well as Krautmann and Berri (2007) explain this strategy by concluding that sport managers do not merely optimize ticket revenues but rather optimize overall revenues, which also include ancillary revenues like concessions and parking.

Dynamic Pricing

Whereas in the past it was possible to set prices based exclusively on allocable costs (cost-plus-pricing), today it is necessary to orientate price strategies in relation to competitors (market-based-pricing) and particularly in relation to the customer (value pricing) (Phillips, 2005). Meanwhile a company's capability to adapt prices to changes in surrounding conditions over short periods of time is regarded as a critical factor of running a business successfully in the service industry (Kretsch, 1995). Therefore, in the last two decades dynamic pricing has been established as one of the most methodically sophisticated interdisciplinary research topics. Published papers stem primarily from operations research, marketing, economics, and e-commerce (Gonsch et al., 2009). The standard textbook in the field of dynamic pricing was written by Talluri and van Ryzin (2004); overviews of the current state-of-the-art can be found in Bitran and Caldentey (2003), Chan, Shen, Simchi-Levi, and Swann (2004), Elmaghraby and Keskinocak (2003), Gonsch et al. (2009), and Tscheulin and Lindenmeier (2003).

Kimes (1989) outlined the conceptual framework when a dynamic pricing approach is appropriate for a firm and identified six prerequisites characteristics of an industry: the ability to segment markets, the sale of perishable inventory, advance sale of the product, low marginal sales costs, high marginal production costs, and fluctuating demand. Kimes, Chase, Choi, Lee, and Ngonzi (1998) added predictable demand as a seventh criterion.

Gonsch et al. (2009) established a catalog of eight criteria in order to classify dynamic pricing models. First, they distinguish between stochastic and deterministic models. Deterministic models are often used as easier-to-manage alternative models for the more complex stochastic models that explicitly incorporate uncertainty. Second, the authors consider time-continuous and time-discrete models. While time-discrete models allow the adjustment of prices only from one period to the next, time-continuous models allow price changes every time. Third, they distinguish between price-continuous and price-discrete models. Price-continuous models use a mathematical function to specify all prices within a certain range; in contrast, price-discrete models are limited to a fixed number of price points. The fourth criterion concerns the question of when to consider a single- or a multi-product model. In the fifth criterion, models with a fixed capacity and models that allow capacity to be increased are distinguished. The sixth criterion addresses customer behavior; strategic and non-strategic (myopic) customer behavior is differentiated as well as time-homogeneous and time-inhomogeneous behavior. Seventh, monopolistic and oligopolistic models are distinguished. The eighth criterion concerns the forecast of the model parameters: it discriminates between models with known parameters and models with not-identified parameters, which uses demand learning procedures. A detailed discussion of the eight criteria is provided in Gonsch et al. (2009).

Kincaid and Darling (1963) published the first paper on dynamic pricing. Due to the Airline Deregulation Act of 1978, the airline industry appears as being initially responsible for the development of revenue management and dynamic pricing strategies (Tallury & van Ryzin, 2004). In the following decades other industries such as hotel, car rental, and tourism adopted these new pricing strategies and added specific characteristics concerning the mathematical modeling of the dynamic pricing approaches (Klein & Steinhardt, 2008). Some of the latest studies have even applied the principles of dynamic pricing to restaurants (Heo & Lee, 2011), cruise lines (Maddah, Moussawi-Haidai, El-Taha, & Rida, 2010), golf courses (Kimes & Schruben, 2002), spas (Kimes & Singh, 2009), and theme parks (Heo & Lee, 2009).

Dynamic Pricing in Sports

While there exist numerous studies on dynamic pricing in the field of the airline or the hotel industry, the number of scientific publications on dynamic pricing in sport is limited. Drayer et al. (2012) presented a general overview on dynamic pricing in sports by discussing general managerial issues concerning the specification of a dynamic ticket pricing system in the field of professional sports. The authors examined the applicability of a dynamic pricing strategy for the sport industry based on the criteria established by Kimes (1989) and Kimes et al. (1998) and concluded that the sport industry is an appropriate platform to implement the technics of dynamic pricing. However, they pointed out that a more comprehensive analysis of this new pricing approach required a lot of additional research. Nevertheless, they also emphasized the striking revenue potential. Nufer and Fischer (2013) analyzed European football leagues and drew the same conclusion. They stated that based on the experience with dynamic pricing in America it can only be a matter of time before dynamic pricing will be applied by a major football club in Europe.

Apart from these two rather theoretical papers, three empirical studies were contributed by Shapiro and Drayer (2012), Paul and Weinbach (2013), and Shapiro and Drayer (2014). All three focused on teams in the MLB and analyzed real ticket pricing data from sports franchises that had already been applying dynamic pricing; the shared objective of these three studies was to identify determinants of ticket price setting.

Shapiro and Drayer (2012) evaluated ticket prices for 12 selected games of the San Francisco Giants during the 2010 season. They found that time was a significant factor for all three evaluated tier categories: as the game drew nearer stadium ticket prices gradually increased. Furthermore, they compared dynamic ticket prices with prices of the secondary market and reported that customers in the secondary market were willing to pay nearly 50% more for tickets. The authors concluded that the pricing system of the San Francisco Giants was not yet capable to fully take advantage of the customers' willingness to pay.

An extension of the study of Shapiro and Drayer (2012) was conducted by Paul and Weinbach (2013), who analyzed a data set of ticket prices for the San Francisco Giants, St. Louis Cardinals, Chicago White Sox, and Houston Astros--the four baseball teams that had used a dynamic pricing approach in the 2011 season. This study specified a regression model for each club and found that the following independent variables notably influenced the ticket price: day of the week, month of the year, the home team's winning percentage, opponent, promotions, starting pitcher, and weather conditions. The authors identified as key variables the opponent, the day of week, and the home team's winning percentage. They observed that promotions and the starting pitcher had only limited effects on the ticket price, and pointed out, though, that results varied widely depending on the team. Above all, the results were mixed concerning the impact of weather conditions.

The third empirical study in this field was presented by Shapiro and Drayer (2014), who evaluated ticket prices through dynamic pricing for 12 selected games of the San Francisco Giants' 2010 season and included 10 of 29 potential variables in their final model. The authors reported significant effects for ticket-related variables (seat location), team performance (whether the opponent had reached the playoffs in the previous year, the home team's winning percentage for the past 10 games), individual performance (whether Tim Lincecum was scheduled to pitch, starting pitcher's earned run average, number of all-stars on opponent's roster), time-related variables (the game's start time, part of the season, and number of days before the game), and game-related variables (whether the opponent is from the same division).

Summarizing the existing literature about dynamic pricing in sports it can be pointed out that previous studies approached this topic merely from a managerial point of view or tried to identify determinants of already-existent dynamic pricing models in the MLB. The current paper, however, seems to be the first to apply the mathematical theory of dynamic pricing to the context of sports clubs. Furthermore, the presented study seems to be first to address this topic in the context of European soccer and to simulate the revenue potential of a dynamic pricing model for a sports club in a country where this pricing approach has not been applied thus far, using the example of Bayern Munich.

The Current Pricing Structure of Bayern Munich

The ticket pricing structure of Bayern Munich for the 2013-14 season is presented in Figure 1. Bayern Munich applies a simple form of variable ticket pricing and differentiates their ticket prices based on only two dimensions: seat category and price category. It is, therefore, a very structured and clear-cut pricing approach. The seat category is more or less determined by the distance to the field and whether the seat is located on the sideline of the field or behind the goal. In general, seats that are closer to the field are priced higher than seats that are farther away and thus offer an inferior view. Additionally, seats at the side of the field are generally priced higher than seats behind the goal. Overall, there are four different seat categories used to differentiate ticket prices for seated places. A fifth category is applied to price the standing room area, which is located directly behind the goals.

The second dimension concerns the price category and relates to the quality of the opponent. In the 2013-14 season 11 games were categorized as A games whereas only six games were classified as B games. The price difference between A and B games accounts for a difference of 10 [euro] concerning the first and second seat category and 5 [euro] concerning the third and fourth seat category. Regarding the standing room, no markup is charged for A games compared to B games.

General Model Design

Concerning the design of a dynamic pricing model for Bayern Munich, the team's current pricing approach serves as a starting point; the four seat categories and two price categories are applied in that model, too. There are, then, eight separate products, all of which will be priced individually. The stadium's standing room, though, will not be considered in this dynamic pricing model because the fans who attend a game in the stands can be regarded as the most valuable asset of a football club; they are essential for creating atmosphere within the stadium and should not be discouraged by such a pricing approach (Nufer & Fischer, 2013). Hence, the first objective of this paper is to estimate those eight individual demand functions, which will then be put to use in specifying the Bellman-Equation of the dynamic pricing model.

Data Collection

To estimate the necessary eight demand functions for games of Bayern Munich the current paper uses realized transactions on ebay.de. In this respect, only auctions were considered. Buy it now offerings with a listed price were not taken into account. Ebay.de was choosen because this is the only reselling platform in Germany for Bayern Munich tickets that allows an auction-based format, which was important for our research design and to measure the fans' willignesss to pay. The other major reselling platform in Germany, viagogo.de, as well as the official secondary ticket market on the homepage of Bayern Munich only offer a listed price format. The data covers the whole second half of the 2013-14 Bundesliga season and was collected between January 1, 2014, and May 5, 2014. In that period eight games took place in the Allianz Arena, the stadium of Bayern Munich. Six games--those against Bayer 04 Leverkusen, Borussia Dortmund, Eintracht Frankfurt, FC Schalke 04, SV Werder Bremen, and VfB Stuttgart--were classified as A games. B games included the matches against 1899 Hoffenheim and Sport-Club Freiburg. Hence, a broad variety of opponents could be analyzed. Unfortunately, eBay.de was not willing to provide us with the necessary information, so we had to collect the data manually. Every day the auction platform ebay.de was browsed by searching for tickets of Bayern Munich for individual games. For example, one search string was Fussball Ticket Bayern Munchen Borussia Dortmund. VIP tickets were not taken into account because those tickets included additional services like food, beverages, and parking. Tickets for selected customer groups such as students, pensioners, and disabled people were also not considered due to the reduced price and the possibility to upgrade the ticket on match day in the stadium. In total the data sets consisted of 1,385 transactions for the respective price and seat categories. Figure 2 provides an overview of the allocations of the transactions to the price and seat categories.

Games assigned as A games included 1,202 transactions, whereas only 183 transactions could be analyzed for B games. The highest number of transactions was registered for A3 games with 428 transactions. In contrast to A3 games, the lowest number of transactions was registered for B3 games with only 23. Generally, the number of recorded transactions for B games is much lower than for A games. This is true for all seat categories. As a consequence the estimation of demand functions for A games relies on a much broader basis than for B games, which should be taken into account while interpreting the results.

Results

The results are presented in four steps. First, general descriptive results concerning the willingness to pay are presented. Second, the statistical models are illustrated to estimate the demand functions. Third, the results concerning the evaluation of equation (1) for a specific scenario are shown (i.e., the optimal ticket price as a function of the remaining capacity and periods are calculated). Fourth, the results of a Monte Carlo simulation are presented in order to evaluate the effects of the dynamic pricing model in comparison to the optimal fixed price approach and to the current ticket price.

Descriptive Results

Figure 3 displays the descriptive results, which gives an overview of the average final amount of money the buyer paid for tickets of Bayern Munich on ebay.de. While analyzing the data a constant time value of money is assumed. Generally, fans are willing to pay amounts much higher than the current ticket price. Surcharges vary between 75% for B3 games and 224% for B4 games. The highest average willingness to pay was registered for A1 games with 178 [euro] compared to 70 [euro] for B3 games, for which the lowest willingness to pay was recorded.

Demand Functions

Based on the papers of Voeth and Schumacher (2003) and Miller et al. (2011) (2) we calculated the cumulative distribution functions of the price points and evaluated four different functional forms in order to find the optimal statistical model for our data. The tested mathematical functions included a linear, a logistic, an exponential, and a logit model. (3) The models were estimated by means of an OLS approach. The adjusted [R.sup.2] of all four models for the eight combinations of price and seat category are displayed in Figure 4.

In general, the [R.sup.2.sub.adj] is relatively high for all models, ranging from 0.773 for the linear model of A2 games to 0.995 for the logistic model of B1 games. However, the logistic model provided the best fit for seven of the eight demand functions. The exponential model provided a better fitting only for B3 games. The optimal model for each combination of price and seat category is highlighted in Figure 4. The econometric models, coefficients, and standard errors of the selected models are displayed in Figure 5.

That the models fit the empirical data can also be seen in Figure 6, which illustrates the empirical data, the estimated models, and the mathematical equation. Examining the requirements of the regression analyses revealed no critical issues. Consequently, the identified statistical models were utilized and included in the dynamic pricing model.

[FIGURE 6 OMITTED]

Optimal Price Points Based on the Dynamic Pricing Model

The estimated demand functions were used to specify the Bellman-Equation of the dynamic pricing model (equation 1-3) and a concrete scenario was evaluated in order to analyze the dynamic pricing model's effects for Bayern Munich. The scenario draws from the example given in Gonsch et al. (2009) and assumes a starting capacity c of 15 tickets and a selling period t of 50 periods. The willingness to pay [d.sub.t]([p.sub.t]) is modeled by the estimated demand functions for each combination of price and seat category. For example, for the seat category 1 and A games, the purchase probability accounts for [d.sub.t]([p.sub.t]) = (1/((1/1.513)+0.136*1.015^[p.sub.t])). Based on the evaluation of equation (1) the optimal price for all combinations of period t and capacity c was calculated applying a rollback approach (Gonsch et al., 2009). The result is provided in Figure 7, which shows the iso-capacity lines for A1 games of Bayern Munich for all 15 capacities, a potential realization of a price path as well as the optimal fixed price. The lowest iso-capacity line represents the price path for a capacity of 15 tickets; the highest iso-capacity line stands for a capacity of one remaining ticket.

[FIGURE 7 OMITTED]

The two monotonous characteristics of dynamic price policies can be made out from Figure 7. First, the iso-capacity lines do not intersect. This indicates the following relationship: the lower the remaining capacity the higher the optimal price point. Second, the iso-capacity lines are declining monotonously. Hence, in phases in which no purchase occurs the price declines continuously from period to period (i.e., the price path follows the iso-capacity line). As the prices decline, the probability increases that a customer will purchase a ticket. When a ticket is sold, the price path switches over to the next iso-capacity line and the price increases abruptly, followed by a continuous price decline over the next periods until the next purchase takes place. In this example, the first ticket is sold in period 43. Hence, the iso-capacity-line of c=15 is followed until period 43, in which the price path switches over to the iso-capacity line of c=14. In period 42 the next ticket is sold and, consequently, the price path changes to the iso-capacity line of c=13.

This process continues until either all tickets are sold or the last period is reached. Furthermore, the optimal fixed price was calculated based on a stochastic model, (4) and the calculation accounts for 194.47 [euro] for A games of Bayern Munich based on 50 periods and a capacity of 15 tickets. As depicted in Figure 7, the price points of the realized price path through dynamic pricing vary around this optimal fixed price. However, depending on the customer's willingness to pay in each period, the realized price path can differ significantly. The most revenues would be generated if all 15 tickets were sold in the first 15 periods, because in each period the realized price path would switch over to the next higher capacity line. For this example the highest possible revenues would add up to 3,780 [euro]. If all 15 tickets were sold in the last 15 periods, though, revenues would only account for 1,805 [euro], compared to 2,917 [euro] if all 15 tickets were sold for the optimal fixed price. Therefore, carrying out a Monte Carlo simulation will assist in evaluating the effects of the dynamic pricing approach in comparison to the optimal fixed price and to the current ticket price of Bayern Munich.

Results of the Monte Carlo Simulation

A Monte Carlo simulation with 10,000 iterations was conducted. That is, in each of the 50 periods of one iteration the willingness to pay was randomly drawn from the corresponding empirically estimated demand function. If the individual willingness to pay was equal to or higher than the seller's price offer, the ticket was sold. Otherwise no purchase took place. The result was the corresponding price path. This process was repeated 10,000 times for each of the eight dynamic pricing models. The results of the simulation are illustrated in Figure 8, which shows the expected revenues based on the current ticket price, the optimal fixed price, and the dynamic pricing model for the specified scenario, in which 15 tickets are offered over 50 periods.

The simulation indicates that the expected revenues from dynamic pricing are significantly higher than the revenues based on the optimal fixed price for all eight combinations of price and seat category (p <.001, respectively). The differences ranged between 3.9% for B4 games and 5.3% for B3 games. The total revenues from dynamic pricing varied between 1,167 [euro] for B3 games and 2,845 [euro] for A1 games, compared to 1,108 [euro] and 2,718 [euro], respectively, based on the optimal fixed price. Concerning the comparison between revenues from dynamic pricing and those from the current pricing structure of Bayern Munich, the difference was even greater. The difference was greatest for B4 games with 273% compared to 94% for B3 games, which represents the lowest potential revenue enhancement. All differences between revenues from dynamic pricing and from the current price were statistically significant (p <.001).

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Another advantage of a dynamic pricing system discussed in the literature is to enhance the occupancy rate. Therefore, the expected numbers of sold tickets are illustrated in Figure 9.

Although the dynamic pricing approach was not able to sell on average all 15 tickets over 50 periods, like the current pricing structure of Bayern Munich, the difference between these two pricing schemes was quite little, ranging between 0.0% for B3 games and -2.0% for A4 games. The total numbers accounted for 15.0 and 14.7 sold tickets, respectively, with the dynamic pricing system. Nevertheless, except for B3 games the number of sold tickets differed significantly for all combinations of price and seat categories between the current pricing structure and the dynamic pricing approach. Importantly, compared to the optimal fixed price, the dynamic pricing approach was much more efficient. Differences in terms of sold tickets ranged between 2.8% for B4 games and 7.0% for B3 games. The total numbers of sold tickets for the dynamic pricing approach were 14.92 and 15, and for the optimal fixed price approach 14.51 and 14.02. All differences were significant (p <.001).

The last figure concerning the results of the simulation refers to the average ticket price. Figure 10 shows that the differences of the average ticket price between dynamic pricing and the optimal fixed price approach were relatively small, ranging from -2.0% for A4 games to 1.1% for B4 games. Nonetheless, the differences were significant for all combinations of price and seat categories (p <.001) except for B2 games. However, the differences were not always pointing in the same direction. For three combinations of price and seat category the dynamic pricing simulation revealed lower average ticket prices, while for four combinations the average dynamic ticket price was higher than the optimal fixed price. Concerning the comparison of the average dynamic ticket price and the current ticket price, the differences ranged between 94% for B3 games and 275% for B4 games. All differences were statistically significant (p <.001).

[FIGURE 10 OMITTED]

Discussion

The descriptive evaluation of the fans' willingness to pay revealed that stadium attendees could agree to pay significantly higher amounts of money for tickets of Bayern Munich than the current ticket price. The average willingness to pay ranged between 70 [euro] and 178 [euro], depending on seat and price category. Compared to the current ticket prices, an average surcharge between 75% and 224% could be yielded. Nevertheless, the willingness to pay for B games of the third price category, especially, has to be interpreted with caution due to the limited number of observations. The measured willingness to pay seems to be a little too low compared to the results of the willingness to pay for the other combinations of price and seat category. While the average willingness to pay for A games declined from price category one to four, the willingness to pay for B games declined from price category one to three and increased again to price category four. Following this line of reasoning, the willingness to pay for the fourth price category of B games might also be a little too high. The measured price surcharge for this category accounted for 224%, whereas the price surcharges of the other seven combinations of price and seat category ranged between 75% and 158%. However, future studies that rely on a broader data set might solve this issue.

Apart from the data reliability for these two categories, the general results agree with the multitude of studies pointing out that tickets for sports events are generally underpriced (e.g., Borland & MacDonald, 2003; Drayer et al., 2012; Villar & Guerrero, 2009). The current study supports this well-documented observation. Several reasons--such as uncertainty of sales, the value customers place on fair treatment, the idea that consistency in pricing is necessary to attract loyal fans, and the notion that a fuller stadium enhances fan experience--have been mentioned for why sports clubs do not act as revenue or profit optimizers (Courty, 2003). Other reasons refer to the argument that low ticket prices provide the opportunity to yield ancillary revenues for parking and merchandise (Coates & Humphreys, 2007; Drayer, Rascher, & McEvoy, 2012; Fort, 2004). Although these reasons might be understandable, sports clubs are faced with increasing financial obligations such as rising player salaries, transfer fees, or stadium maintenance costs (Drayer et al., 2012). Consequently, sports clubs are searching for new income opportunities. The growth of sponsorships and stadium naming right deals are just two examples of such opportunities (Howard & Crompton, 2005). The optimization of ticket prices can be seen as another step in this development. Therefore, the objective of this paper was not only to evaluate the fans' willingness to pay for tickets of Bayern Munich but also to analyze the revenue potential of a dynamic pricing approach for a sports club in a country where such a pricing approach has thus far not been applied.

In order to measure the fans' willingness to pay and to estimate the necessary demand functions concerning the specification of a dynamic ticket pricing model for Bayern Munich, the current paper made use of transactions on ebay.de. This approach was chosen because, based on pricing theory, auctions can be regarded as the most suitable and valid method to measure willingness to pay (Lusk & Shogren, 2007; Miller et al., 2011; Volkner, 2006a, 2006b). However, it should be mentioned that previous studies suggested that the primary and secondary ticket market might represent different consumer bases (Shapiro & Drayer, 2012). Whereas ticket prices in the primary market seem to increase over time (Shapiro & Drayer, 2012, 2014), studies about ticket prices in the secondary market reported an overall downward price trend (Drayer & Shapiro, 2009; Shapiro & Drayer, 2012; Sweeting, 2012). Nevertheless, Shapiro and Drayer (2012) emphasized that the impact of time concerning the ticket price development in the primary and secondary market is a field for future research since studies like Shapiro and Drayer (2012) as well as Shapiro and Drayer (2014) only examined four or five points in time, respectively. In this respect, Shapiro and Drayer (2012) reported that ticket prices in the primary market increased continuously, whereas ticket prices in the secondary market rose between 20 and five days prior to match day and then decreased significantly on match day itself. Shapiro and Drayer (2012) further found that ticket prices in the secondary market are on average roughly 50% higher compared to ticket prices in the primary market. However, the difference in ticket prices between these two markets decreased significantly towards the day of the match. Hence, this observation relativizes the general oppositional development of ticket prices in the primary and secondary ticket market. These results can be interpreted in the way that a dynamic ticket pricing system in the primary market is not yet capable of tapping the customer's whole willingness to pay (Shapiro & Drayer, 2012). Therefore, the amount of money that was actually paid for tickets in the secondary market might serve as an orientation of the real ticket value (Drayer, 2011; Drayer & Martin, 2010). Consequently, the current study used pricing data of the secondary market in order to specify a dynamic ticket pricing model for Bayern Munich and to evaluate its effects in terms of revenues, number of sold tickets, and price per ticket.

The simulation of the dynamic pricing model revealed several of the advantages that are typically associated with dynamic pricing approaches. First of all, revenues due to ticket sales could be improved significantly as a result of applying a dynamic pricing model. On the one hand, this could be achieved by generally increasing the ticket prices. Due to the high demand for tickets and the extraordinary willingness to pay, the tickets would still be sold. On the other hand, the improvement in terms of revenues is not based solely on the ticket price increase. A crucial aspect of dynamic pricing relies on the adaption of the price depending on demand. That means, in phases of no ticket sale the price decreases continuously, which leads to a higher purchase probability. As a consequence of this flexible price adjustment, more tickets are sold applying a dynamic pricing approach compared to the optimal fixed price approach. However, a dynamic pricing model also increases complexity. Therefore, if a sports club, in this case Bayern Munich, preferred to pursue the currently applied pricing approach of variable ticket pricing, the optimal fixed price, differentiated by seat and price category, would serve as an orientation. The conducted simulation suggests that also in this scenario Bayern Munich could increase ticket revenues substantially. Similarly, Rascher, McEvoy, Nagel, and Brown (2007) analyzed the revenue potential of variable ticket pricing for MLB clubs and concluded that on average about 2.8% additional ticket revenues could have been yielded if variable ticket pricing would have been applied by all clubs in the 1996 season. This estimate, however, is significantly lower than the results of the current study. In this respect, it has to be pointed out that in the Rascher et al. (2007) study only 5% of the respective games were sold out, while in the 2013-14 season all games of Bayern Munich were sold out. These extraordinary high attendance figures support the conclusion that Bayern Munich could yield notably higher ticket revenues. Although this study only analyzed ticket prices of Bayern Munich, an occupancy rate of roughly 92% for all Bundesliga clubs in the considered season (Transfer Market, 2014) might be interpreted as an indication that other clubs could also capitalize more intensively on the fans' willingness to pay. Having in mind that the Bundesliga does not share ticket revenues like American sports leagues, the optimization of this source of income seems to be promising.

Another objective of a dynamic pricing system confirmed by the conducted simulation was that, apart from increasing ticket revenues, the stadium attendance rate could also be optimized. Because ticket revenues and stadium attendance rate are generally considered the two aspects related to the determination of ticket prices (Drayer et al., 2012), a dynamic pricing system makes for an ideal pricing approach for sports clubs. Especially in terms of strategic pricing and the consideration of additional revenue streams like parking and concessions, the optimization of the attendance rate is an important aspect. As the results of the simulation suggest, the dynamic pricing model was almost able to sell all tickets over the selling period. Therefore, a dynamic pricing model could be interpreted as a promising approach of strategic pricing in order to ensure high attendance rates and the related ancillary revenues.

The third result of the simulation was that the average ticket price did not necessarily increase through dynamic pricing as when compared to the optimal fixed price model. When clubs implement a new pricing scheme, sport fans might refuse to accept the new pricing policy due to unfamiliarity (Drayer et al., 2012) and might be concerned with being treated unfairly (Nufer & Fischer, 2013). Consequently, a marketing campaign accompanying the implementation of a dynamic pricing system into a new market should emphasize that a dynamic pricing system does not inevitably result in higher ticket prices. Especially for games with low demand, fans do benefit from substantially reduced ticket prices. Furthermore, a proactive media and communication approach seems reasonable. Studies on revenue management in the airline (Kimes, 1994, 2003) and hotel industries (Choi & Mattila, 2005) pointed out that the perception of price fairness increases proportionally with the amount of information provided. In addition, Nufer and Fischer (2013) concluded that negative responses are likely to fade away as football fans become more familiar with dynamic pricing.

In summary, sports clubs, if they applied a dynamic pricing approach, would profit from higher ticket revenues. But fans would also benefit both from average ticket prices that are comparable to the optimal fixed price and from an enhanced stadium experience due to an increased attendance rate. Even if a club decided not to increase the ticket prices to the optimal level but instead chose to only partially exploit the revenue potential, the additional income from the application of a dynamic pricing system in comparison to a fixed price approach would be significant.

Conclusion and Limitations

Conclusion

The current study is the first one conducted at the intersection of operation research, economics, marketing, and sports and was concerned with designing a dynamic pricing model for Bayern Munich. For the first time the mathematical principles of dynamic pricing were combined with empirical marketing research methods using ebay.de auctions to determine demand functions for football tickets. These estimated demand functions were fundamental for specifying a dynamic pricing model. By means of a Monte Carlo simulation the effects of such a pricing approach in terms of revenues, number of tickets sold, and average ticket price were evaluated.

This paper demonstrates that the stadium attendees' willingness to pay could be significantly higher than the current ticket prices of Bayern Munich, thereby emphasizing the presently existent revenue potential. Based on the calculated demand functions, however, the application of a dynamic pricing approach would yield even more income than the optimal fixed price. Therefore, the study at hand contributes to the existing literature by analyzing a pricing approach that is completely new for sport clubs in Germany. Because the application of dynamic pricing systems has to date been limited to American and British sports clubs, the current paper lays the foundation for a new and very promising field of research both in terms of theoretical perspectives as well as practical implications.

Although this study was specifically designed for Bayern Munich, it might set a good example to other clubs and inspire them to implement new pricing strategies. Especially in times of escalating player salaries, transfer fees, and stadium maintenance costs, sports clubs should regard the current study as encouragement to evaluate whether a dynamic pricing approach would suit their interests and their financial situations.

Limitations

There are at least the following limitations to this pioneering study. First, only the Bayern Munich football club was subject to this study. An evaluation of other football clubs or sports clubs in other leagues might reveal different demand functions, which are the core element of a dynamic pricing model. Thus it seems advisable to repeat this study in a different context. Furthermore, the estimated demand functions for B games rely on a rather small data set. Hence, future research could improve the current study by analyzing a larger set of data and by extending the data collection period to an entire season. Secondly, the designed dynamic pricing model is based on only eight demand functions, one for each combination of price and seat category. However, one reason why dynamic pricing of sports events has been discussed intensely in public is the interest in different kinds of factors, such as current position in the league standings, team performance, or weather conditions, in order to price the tickets. Future research should therefore extend the presented dynamic pricing model by including additional factors that are generally associated with dynamic pricing in sport via estimating individual demand functions for the interesting factor combinations. Apart from the limitations concerning the empirical data collection, the specified mathematical model should be reviewed as well. Future research should, therefore, extend this study by generalizing the model's assumptions and restrictions.

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Endnotes

(1) Here, "myopic" means that the customer does not anticipate price changes in the future, and hence does not show strategic customer behavior.

(2) Both Voeth and Schumacher (2003) and Miller et al. (2011) applied an equivalent approach to determine demand functions, but neither study made any connections to the theory of dynamic pricing, and both used the data only to identify relevant price points.

(3) Functional forms of the tested models for the demand functions

Linear model y = [alpha] + b[chi]

Logistic model y = 1/u + [alpha] x [b.sup.[chi]]

Exponential model y = [alpha] x [e.sup.b x [chi]]

Logit model y = [e.sup.[alpha]] + b[chi]]/1 + [e.sup.[alpha] x b x [chi]]

(4) The optimal fixed price (OFP) was calculated by evaluation the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Klein &Steinhardt, 2008)

Christoph Kemper [1] and Christoph Breuer [1]

[1] German Sport University Cologne

Christoph Kemper is a PhD student in the Department of Sport Economics and Sport Management. His research interests include pricing strategies, willingness to pay, and statistical modelling.

Christoph Breuer, PhD, is a professor in the Department of Sport Economics and Sport Management. His research interests include sport organizational economics and economics of sponsoring.

Figure 1. Ticket pricing structure of Bayern Munich for the 2013-14
season
(http://www.fcbayem.de/media/native/tickets/Preisliste_080n4.pdf)

Seat category        Price    category
               A      B
   1          70 [euro]   60 [euro]
   2          60 [euro]   50 [euro]
   3          45 [euro]   40 [euro]
   4          35 [euro]   30 [euro]
   5 Stands      15 [euro]   15 [euro]

Figure 2. Number of transactions for the
corresponding price and seat category
([SIGMA]=1.385)

Seat         Price       Price
Category   Category A  Category B

1             240          68
2             273          52
3             428          23
4             261          40

Note: Table made from bar graph.

Figure 3. Current ticket prices and average
ticket prices on ebay.de

Price Category A

            Current       Ticket
Seat         Ticket     Price on    Standard
Category      Price      Ebay.de    Deviation   Surcharge

1          70 [euro]   178[euro]                   +154%
2          60 [euro]   136[euro]                   +127%
3          45 [euro]   116[euro]                    158%
4          35 [euro]    88[euro]                   +151%

Price Category B

Seat
Category

1          60 [euro]   130 [euro]                  +117%
2          50 [euro]   98 [euro]                    +96%
3          40 [euro]   70 [euro]                    +75%
4          30 [euro]   97 [euro]                   +224%

Figure 4. [R.sup.2] adj for the estimated models

Price and
Seat
Category  Linear         Logistic     Exponential      Logit

1    A    0.799            0.992         0.989         0.979
     B    0.978            0.995         0.856         0.995
2    A    0.773            0.99          0.970         0.988
     B    0.965            0.976         0.887         0.959
3    A    0.898            0.990         0 882         0.990
     B    0.787            0.934         0.961         0.919
4    A    0.856            0.993         0.980         0.961
     B    0.958            0.983         0.824         0.983

Figure 5. Coefficients and standard errors of the selected models

Price and
Seat
Category  Model            u/SE          a/SE           b/SE

1    A    Logistic      1.513/0.046   0.136/0.010   1.015/0.000
     B    Logistic      1.017/0.013   0.013/0.002   1.034/0.001
2    A    Logistic      1.075/0.010   0.009/0.001   1.038/0.001
     B    Logistic      1.416/0.128   0.071/0.019   1.031/0.002
3    A    Logistic      0.990/0.007   0.013/0.001   1.038/0.001
     B    Exponential       --        3.087/0.286   -0.029/0.001
4    A    Logistic      2.558/0.188   0.294/0.017   1.023/0.001
     B    Logistic      0.995/0.028   0.005/0.002   1.055/0.003