Upgrade auctions in the airline industry : a game-theoretic approach

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Master Thesis

Upgrade Auctions in the Airline

Industry: A Game-Theoretic

Approach

Nicolai Schlage (11088338)

nicolai.schlage@student.uva.nl

Supervisor: Sander Onderstal

Markets and Regulation track, Faculty of Economics and Business

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Statement of Originality

This document is written by Student Nicolai Schlage who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervi-sion of completion of the work, not for the contents.

1 Introduction

The airline industry is traditionally one of the most researched industries in Industrial Organization (Alderighi et al. (2016)). This is due to its clear boundaries to other industries but also to its dynamic nature that often brings up new pricing schemes earlier than in other industries. In this thesis, a new price discrimination mechanism is examined that is implemented for the first time in airline pricing in recent years and is widely spread across Europe by now. Airlines started to auction off free seats in upper classes to customers that have tickets in a lower class. This scheme has now been adopted by over 40 airlines. An overview of the airlines currently using class upgrades can be seen in Table 1. This replaces the long-standing practice of handing out free

Table 1: Airlines with Upgrade Auction

Lufthansa Virgin America Aero Mexico Royal Brunei

Air Canada Hawaiian Airlines Malaysia Air Seychelles

Air China Aer Lingus Philippine Airlines Air Astana

Singapore Airlines Swiss Garuda Indonesia SriLankan Airlines

Etihad Airways Alitalia South African Gulf Air

Cathay Pacific Austrian Ethiopian tigerair

Virgin Australia Airberlin Kenya Airways Silk Air

Air New Zealand Brussels Airlines Tap Portugal Air Niugini

Qantas Icelandair Lot Polish Airlines Czech Airlines

Virgin Atlantic Copa Airlines Air Serbia Aircalin

SAS Avianca Aerolineas Niki

Latam

upgrades to regular customers. This new mechanism can be seen as part of a trend to monetize additional aspects of flying through upselling, as for example noted by Warnock-Smith et al. (2015). This might be due to competition in the flight business increasing for the recent past, especially with the rise of low-cost carriers such as Ryanair.

This thesis proposes two versions of a model for upselling auctions in the air-line industry and discusses solutions for optimal prices. In addition, profit

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maxi-mizing properties of the auction solutions are discussed. The goal of this thesis is to examine how prices for different classes are affected by upgrade auctions and if this is advantageous for the airlines. We hypothesize that the implementation of an upgrade auction system helps airlines to keep lower class prices low, which would also be useful in competition with low-cost carriers. The thesis could also lead to a better understanding of the dynamic pricing structure of airline pricing. To achieve this, we will compare the situations before and after implementing the auction mechanism in a monopoly setting, in a duopoly setting where only one airline has different classes (the other might be a low-cost carrier), and in a duopoly setting where both airlines potentially could implement an auction system. The monopoly setting can be examined in closed-form solutions, while much of the duopoly setting relies on a computer simulation, solving the maxi-mization problem numerically for a range of plausible values for the horizontal and vertical differentiation.

In the next chapter, an overview of the relevant literature follows. Subse-quently the structure of an appropriate model is defined in section 3. In chapters 4–6, we examine analytically and numerically how optimal firm profits and prices change when the auction mechanism is established. For this purpose, we deal with a set-up without uncertainty on the distribution of the valuations, but costs incurred by the auction mechanism. This model is calculated for the monopoly situation and all possible duopoly situations in sections 4–6. Then we turn to a richer model with ex ante uncertainty and look at the monopoly situation again to see if the insights from the simplified model might be robust to other – maybe more realistic – types of modelling of the costs (section 7). Finally in section 8 a conclusion is drawn.

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2 Related Literature

In this chapter, we discuss the literature related to my research question. Literature connected with general competitive properties of the airline market are manifold. First there is general literature about vertical and horizontal dif-ferentiation in competitive Industrial Organization models. This strand also has a small subset of papers that deal with multi-dimensional differentiation models. Additionally, the literature on auctions and especially second price auctions are connected to this thesis.

The airline industry is a market with horizontal differentiation as evidenced by many marketing and branding efforts of many airlines. The different landing times are also an example of a horizontal structure. Additionally, consumer loy-alty programs, like air miles, will induce a preference in customers for a specific airline (see, for example, Sandada and Matibiri (2016)). How strong this effect is can remain unanswered for now. There is also vertical differentiation present in the sense that products are on the market that are of higher value to every customer than other products. This is notably done by selling different classes of services. Classic models to look at the phenomenon of price differentiation are the linear city model by Hotelling (1929) and the Salop (1979) circular city model. In an industry where there are no clear extremes in the parameters of differentiation, the circle model is to be preferred. This also leaves the possi-bility of extending the model to more than two competitors in the future. The object of interest – the upgrade auction – makes it necessary to implement a system of vertical differentiation at the same time. In this case we need two different classes, so there can be an upgrade from one to the other. There is some literature that combines vertical and horizontal aspects of differentiation. Neven and Thisse (1987) were the first to fully integrate these two strands of one-dimensional models. However, they not only looks at optimal prices and quality, but also incorporates the location choices of the firms. Furthermore, firms only

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have one product each. In the present case, it is necessary that firms have more than one product. This fundamentally changes the model. Further papers like Gabszewicz and Wauthy (2011) and Economides (1993) follow a similar setting as Neven and Thisse (1987). Bar-Isaac et al. (2014) even change these location choice type models to allow for locating on the inside of the circle, making it possible to choose the degree of differentiation.

In another strand of literature the Salop model is generalized to higher dimen-sions. Such papers are, for example, Hanson (1999), Vandenbosch and Weinberg (1995), and Weber (2008). This is in a way similar to the present analysis, as the models deal with additional dimensions of product differentiation. But they usually do not mix vertical and horizontal differentiation like we do in this the-sis. Weber (2008) specifically uses many of the elements also needed here, but does not generalize to more than one firm. For example, Parlakt¨urk (2012) and Gallego and Hu (2014) look at the dynamic component of differentiation under competition. In this setting we ignore this dimension altogether, although it is especially very important for airline pricing. Production costs are widely seen as problematic in differentiation models if they are asymmetric (see also Lin and Wu (2015)). The airline industry has the fortunate property that marginal costs are near zero compared to the fixed costs. So in this paper, we ignore costs altogether.

It is generally unclear if bigger or lower differentiation is yields optimal profits for firms, both in horizontal and vertical differentiation. This is highly dependent on the circumstances, as B¨ockem (1994) shows. For example, an outside option distorts the standard finding that maximum differentiation is preferred. We will examine what the impact of horizontal and vertical differentiation in the current situation is.

As for auction design, for simplicity’s sake, we will use a second price sealed bid auction in the model for which the equilibrium in bidding strategies is well

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known. It is furthermore revenue equivalent to the first price sealed bid auction as shown by Vickrey (1961). This means that an optimization done with this auction type is likely also valid for first prize auctions. In this specific auction, every customer who bought a lower-class ticket in the first step bids his residual valuation, (α − 1)v as later explained. After the winners of the auction are defined, each of them pays the residual valuation of the first bidder who did not win and gain additional utility according to their valuation. By employing the second price auction design, we make the problem much easier and avoid additional specification problems.

The need to buy a lower-class ticket first could also be seen as equivalent to an auction with participation costs such as in Samuelson (1985) or Tan and Yilankaya (2006). Here we see it as an independent path. This auction system has similarities to the buy-now-auctions on online platforms such as eBay. Buyers can either buy the item right away or choose to bid. Bose and Daripa (2009) showed that it is optimal to set a “buy-it-now” price below the reservation price to pool higher valuation types. Reynolds and Wooders (2005) showed that the temporary version, where when the auction has begun, nobody can use the buy option any more is indeed optimal. Yu et al. (2006) showed that this dominates selling through a traditional auction mechanism non-trivially. The difference to our model is that this literature generally assumes that there is only one good to sell, whereas in our case the number of goods (available upgrades) depends on the set capacity in the first step and the manifestation of the valuation. Also in this paper, the competition mechanism between the firms interferes to a degree with the auction. To summarise, this kind of upselling auction has not been examined in the relevant literature before as far as we know. Therefore we hope to contribute some new results to the discussion about three dimensional differentiation modelling and upselling auctions.

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3 The Model

In this section the general outline of the two versions of our model for the upselling auction is discussed. In the first subsection, we present the building blocks for a baseline model that combines vertical and horizontal differentiation. In the second subsection, an auction mechanism is defined, so that in the follow-ing chapters these parts can be used to compare the outcome of the model in different competitive settings.

3.1 Baseline Model

Taking the Salop model as a base, we add a vertical dimension where con-sumers vary in valuation for flying. In the basic Salop-Model from Salop (1979) companies and customers are located around a circle. Customers have to in-cur costs linked to the distance to a company if they consume a product from this company. The circular form has the advantage over the Hotelling-model (Hotelling (1929)) that all consumers are located between companies because there is no end to the spectrum. This model is very good to model actual dis-tance, like distance to several supermarkets, or distance in the sense of tastes. In its base form however, it cannot be used to describe vertical discrimination.

The vertical dimension implemented in this thesis transforms the Salop circle into a cylinder. In this dimension, the customers vary over their valuation for flying. Specifically, they have a valuation for flying of v ∼ U (0, ¯v). In the first model in sections 4–6, the distribution of v is ex ante clear to customers and airlines. In the model with uncertainty in section 7, it is ex ante unclear to the airlines and the customers what ¯v is, but they have some information on how it is distributed. To simplify, we assume ¯v ∼ U (¯v, ¯v). In other words, the airline has to set prices and capacities, and the indifferent customer is defined without knowing the actual manifestation of the distribution of customers. It will be shown in the thesis that this is equivalent to the firms systematically

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undervaluing ¯v. Every customer gets v − Plow − T from flying lower class and αv − Pup− T from flying upper class, with α > 1, where T notes the travel costs of each consumer. Plow and Pup are the ticket prices for lower and upper class. These are the decision variables of the airlines.

In addition to the (vertical) valuation, every consumer has a horizontal loca-tion x ∈ [0, 1] on a circle of circumference 1. Accordingly, the maximum demand density for a given v is 1 when every possible customer buys. When making the decision from which airline to buy or if to buy at all, consumers have to incur linear travel costs of tx depending on the horizontal differentiation factor t > 0 and their location x.

With these two characteristics the space [0, ¯v] × [0, 1] is a cylinder and every atomic customer is defined by a combination (v, x). In Figure 1, one can see the transition of the Salop circle to the cylinder model. The dark grey area

Figure 1: Transition of Circle to Cylinder

signifies the customers buying lower class. The light grey area stands for those in the higher class, whereas in the white area are the customers that do not value

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flying enough to buy (on this route). This area has the shape of a bent triangle as will be seen in the thesis. n airlines are positioned in equal distance around the cylinder. There is no location choice in this model to keep it as simple as possible. Depending on the specific competitive situation, there are then several indifferent consumers positioned on the lines separating these areas. Finding their locations yields profit functions for the airlines.

When the airline does not know the specific distribution of the valuations ex ante a capacity choice has to be made. As long as the auction is not implemented, free capacity in the upper class is gifted away for free to customers who would transparently not buy upper class tickets anyway. This is revenue equivalent to keeping the seats free. After some inquiries, this seems to be the practice of most airlines which have not implemented an auction system for upgrades.

3.2 Auction

In this subsection we will define a suitable auction mechanism to use in the analysis. In the consumer surplus for the auction mechanism, there have to be costs, denoted by s, or risk to not win the auction, denoted β, involved for the consumers that enter the auction. If this was not the case, the gains from participating in the auction would be equal to the gains from the upper-class ticket. Then, if prices are not equal, rational customers would always choose the cheaper option, making the other one redundant. So customers would buy upper-class tickets from the start or not participate in the auction at all. Foreseeing this, the airline would not offer the auction.

Chapters 4–6 in this thesis use the simplest possible model, where people who get an upper-class ticket by auction have to bear some non-zero hassle costs. These might be due to the costs involved with bidding, inconvenience or an abstract form of uncertainty. This simple way is chosen to reduce the complexity of the duopoly situations.

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Another way of modelling these costs is by including the uncertainty which is produced by the auction mechanism explicitly. If no direct costs are incurred, the valuation of the customers cannot be known beforehand. Otherwise the customers will anticipate who wins in the auction, making the decision between upper class and lower class with auction upgrade risk-neutral. Consequently, only the pattern of the distribution of valuation is known ex ante for the airline and the customers. But the specific manifestation has a stochastic element to it. The game then develops as in Figure 2. Knowing the expected probabilities of

Figure 2: Auction Game

winning in the auction changes the position of the indifferent customers. Since ex ante it is not clear to the customers if they win the auction or not, the benefits of this option are lower than buying an upper class ticket right away. Therefore the lower class ticket price plus the bid price can also vary compared to the other option. This system of looking at the auction mechanism is a lot more complicated than the first one. That is why it will only be used on the monopoly situation, where we will see if it leads to qualitatively different solutions.

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4 Monopoly Without Uncertainty

In this section we take a closer look on the monopoly situation, where the consumers and the airlines are not uncertain about the distribution of consumers valuation v, which is then not stochastic in this case. In the first subsection we will discuss the baseline monopoly model without auction. In section 4.2, the auction mechanism is introduced in this setting.

4.1 Baseline

In this case there is just one level of vertical differentiation, the two classes. Following from this, there are two indifferent consumers v1 and v2:

αv1− Pup− tx = v1− Plow− tx v1 = Pup− P low α − 1 v2− Plow− tx = 0 v2(x) = Plow + tx

By design, the travel costs have to be “low”, because consumers far away hori-zontally, but vertically in the upper-class area would not consume otherwise. We have to decide where the break of consumers occurs, because the build of the corresponding areas fundamentally depends on that. The chapters five and six on the duopoly situation are supposed to be a related to the monopoly case. To ensure an equivalent structure, we assume v1 ≥ v2(x = 1₂), i.e. the point where all consumers buy tickets in the lower class lies below the upper indifferent consumer or all customers in upper class buy tickets. That is the same situation in that regard as in the duopoly case. To ensure the break lies in the lower class, travel costs have to be sufficiently low. Then the monopolists maximization problem is

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defined as: max Π = Pup ¯ v − v1 ¯ v + Plow v1− v2(x = 1₂) ¯ v + 1 2Plow v2(x = 1₂) − v2(x = 0) ¯ v (1)

, where the last part is the triangular area, where only part of the consumers – on average one half – with the same valuation are located close enough horizontally to buy.

Lemma 1 Profit maximizing prices and profits in the monopoly case are:

P_low∗ = ¯v 2 + 3 8 t α − t 2 P_up∗ = αv¯ 2 − t 8 Π∗ = 9t 2_{− 8αt}2_{− 8αt¯}_{v + 16α}2_v_¯2 32α¯v

Observing the derivatives of the optimal profits we see that: ∂Π ∂α > 0 if 4 3α¯v > t ∂Π ∂ ¯v > 0 if (8α − 9)t 2 + 16α2¯v2 > 0 ∂Π ∂t > 0 if (8α − 9)t + 4α¯v < 0

The first one just requires horizontal travel costs to be low enough compared to the maximum valuation of the consumers. The second derivation definitely holds true if α > 9₈. Otherwise it holds true if t is small enough. For all values of t also fulfilling the requirements of the first derivation, this is the case. In the third derivative it is just the other way around. ∂Π_∂t > 0 is never true for low t’s. So, unsurprisingly, the profit increases with α and ¯v, but falls in t.

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4.2 With Auction

In this subsection we discuss the monopoly situation where the monopolist implements the auction mechanism. In this case, the monopolist and the cus-tomers know ex ante how the valuations will be distributed, i.e. ¯v is not stochas-tic. The utility from bidding into upper class is assumed to be (α − s)v > v with 0 < s < (α − 1).

There are two classes of indifferent customers. The first is indifferent between buying a lower class ticket and then bidding for the upper class and buying upper class from the start:

(α − s)v − Plow− b − tx = αv − Pup− tx v1 =

1

s(Pup− Plow− b)

, where b is the bid of the last customer not getting in. This is the residual valuation of the last one not getting in through bidding b = (α − s − 1)vlast. Because the airline controls capacity, it can freely move this position and thereby change b.

The second indifferent customer is the one between buying lower class and not buying at all:

0 = v − Plow− tx v2(x) = Plow+ tx

The position of the second indifferent consumer depends on the horizontal dis-tance of the customer from the airline. Here and in the following cases we will make the assumption that travel costs are relatively low. That means v2 < vlast for all x.

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Then the maximization problem of the monopolist is: max Π =Pup ¯ v − v1 ¯ v + (Plow+ b) v1− vlast ¯ v + Plow vlast− v2(x = 1₂) ¯ v +1 2Plow v2(x = 1₂) − v2(x = 0) ¯ v (2)

As capacity does not cost anything, the monopolist will want to set it so that it can get the most people in the auction for a given price b. Therefore (α − s − 1)vlast = b. Using this we get:

Lemma 2 The optimal prices and profit are:

P_low∗ = 6¯v(α − s − 1) 8α − 8s − 9 P_up∗ = (8α 2_{− s − 8α(1 + s))¯}_v 2(8α − 8s − 9) b∗ = 2(3 + 2α 2_{+ 5s + 2s}2_{− 4sα − 5α)¯}_v (8α − 8s − 9) Π∗ = (8α 2_{− s − 8α(1 + s))¯}_v 2(8α − 8s − 9)

The first thing to notice is that b∗ > 0 if α > s + 3₂. So α must be sufficiently high or hassle costs sufficiently low, so that a non-zero auction area is optimal. This restriction also makes the analysis of derivatives simpler. Looking at the derivatives of the optimal prices, P_up∗ increases together with α, ¯v, and also with s. t decreases P_up∗ . P_low∗ falls in α. Otherwise it reacts like P_up∗ . The winning bid b has the same pattern. Considering the derivatives of the optimal profit of the monopolist: ∂Π ∂α > 0 if 4 3α¯v > t ∂Π ∂ ¯v > 0 if (8α − 9)t 2_{+ 16α}2_¯_v2 _{> 0} ∂Π ∂t > 0 if (8α − 9)t + 4α¯v < 0 ∂Π ∂s > 0 if (8α − 9)t + 4α¯v < 0

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In the area of α > s + 3₂ the derivative to α, ¯v, and s are positive. That means with the auction mechanism established, the monopolist makes more profit if he raises the hassle costs for the auction by any means possible until α = s + 3₂. These higher hassle costs lead to more vertical differentiation and finally to more profit. The derivative to t is always negative, indicating that the monopolist loses profit the higher the horizontal differentiation potential is.

4.3 Comparison

After analysing both the case without and the case with auction, we can compare the two. Lemma 1 and Lemma 2 imply:

Corollory 1

1. The optimal price for lower-class tickets is higher with the auction mecha-nism than in the baseline case.

2. The optimal price for upper-class tickets is also higher with the auction in place than in the baseline case.

3. Profits are higher with the auction mechanism than in the baseline case if α ≥ s + 3₂. They are lower otherwise.

Implementing the auction mechanism is sometimes better for the monopolist, sometimes worse. If α is high, the auction optimizing profits leads to higher profits with the auction mechanism than without. This is in line with the intu-ition. Higher α – the valuation of consumers of flying upper instead of lower class – means that the upper part of the valuation spectrum is more valuable. The auction presents an opportunity to price discriminate in that segment because the customers winning the auction get a part of the additional valuation from the upper class, effectively creating an in-between class.

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5 Duopoly With Only One Airline Having

Dif-ferent Classes Without Uncertainty

Next we introduce competitive dynamics to see if this changes anything about the profitability of the auction mechanism. We will also specifically look at the position of the lower indifferent customer. If optimal lower-class prices are significantly lower with the auction mechanism in place this could help the airline in the competition with an airline that does not use auctions. The simplest possible competitive set-up is one airline located on the cylinder at x = 0 and the other airline located at x = 1₂ on the other side of the cylinder. Only the first airline has an upper and a lower class. The second airline only has a lower class and can therefore not implement the auction mechanism. With the same logic subtracting hassle costs from the surplus from flying upper class, we will now check the implications of this competitive dynamics. As shown below, the system gets too complicated to solve in a closed form. Reaction functions can be derived relatively easily, but solving for optimal prices is only possible numerically. To do so, we estimate these prices for a wide range of plausible values for α and t. Throughout the chapter, Plow,1, Pup,1, and b1 stand for the prices and the target bid of the first airline. Similarly Plow,2, Pup,2, and b2 are the equivalent for the second airline. In the following subsection we look at the competitive case without the implementation of the auction mechanism to construct a benchmark. Following that, we will look at the situation where the airline with different classes

5.1 Baseline

There are now more indifferent consumers to check than in the monopoly case, since there are additional consumers between the two airlines:

Lemma 3 v1 = Pup,1−Plow,1 α−1 and v2(x) = Pup,1−Plow,2 α−1 − t 2(α−1) + 2 t α−1x are the

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additional indifferent customers when compared to the monopoly baseline case.

We also need:

Lemma 4 There is exactly one customer (v, x) = (v1, x3) that is indifferent according to category indifferent between upper class of Airline 1, and both lower clases. And there is exactly one customer (v, x) = (v6, x3) that is indifferent according between not buying at all, and both lower classes.

Using these two Lemmas we can visualize the different areas on the cylinder where specific consumers buy specific product as shown in Figure 3. We assume

Figure 3: Asymetric Areas of Consumer Choice

that travel costs are sufficiently low: t < 2(Pup,1− Plow,2). This ensures that the customer with (v, x) = (¯v,1₂) chooses Airline 1, and therefore makes the involved shapes more defined.

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Lemma 5 The maximization problems of the two airlines are:

max Π1 = 2(Pup,1(U pper1) + Plow,1(Low1)) (3)

max Π2 = 2(Plow,2(Low2)) (4)

The reaction functions for optimal pricing are computationally possible to derive, but too long to list here. In Appendix 2, there is Mathematica-Code included suited to display them.

In the following discussion, the optimal solution for prices and profits are estimated numerically. This is done for α in 20 steps of 0.2 from 1.5 to 5.5, and for t from 0.1 to 0.8 in steps of ₃₀1. The following graphs symbolize the output: As one can see in Figure 4, the optimal profit of Airline 1 – the one

Figure 4: Optimal Profit of Airline 1

with different classes – varies greatly with the valuation for the upper class α. The horizontal differentiation factor t on the other hand has next to no influence on the profits. This stands in sharp contrast to the profits of the other airline portrayed in Figure 5. The profit of Airline 2 varies less in total, but is hugely impacted by t. If α is big, it is preferable for Airline 2 to have closer competition, i.e. less horizontal differentiation. The graphs for the solutions for the optimal prices show why the profits react like this. The price for the upper-class tickets

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Figure 5: Optimal Profit of Airline 2

Figure 6: Optimal Price of Upper Class of Airline 1

for Airline 1 mostly follows α. The price for both low classes in Figures 7 and 8 rise in α and t. So the higher both parameters the higher the price also Airline 2 can set. Conversely for high α and low t Airline 2 has more competition from the upper class of Airline 1 which is getting more and more attractive.

5.2 With Auction

Now we introduce the auction mechanism for the airline with two different classes. Following the monopoly situation with auction, the utility of the indif-ferent customer changes in that there is additional utility from the auction if the residual valuation is high enough to win the auction. The difference to the

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Figure 7: Optimal Price of Lower Class of Airline 1

monopoly case is that there are now additional classes of indifferent customers between the two airlines similar to the previous subsection.

Lemma 6 Compared to the monopoly case and the baseline case, there is one additional class of indifferent consumers between the auction area of Airline 1 and the lower class of Airline 2: v3(x) =

Plow,1−Plow,2+b−1₂t

α−s−1 +

2t α−s−1x

From the position of the indifferent customers we know how the areas on the cylinder change because of the auction. This can be seen in Figure 9. We now have one area more, where customers choose to upgrade into higher class via the

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Figure 9: Areas with Auction in Place auction mechanism. U pper1 = 1 2(¯v − v1) − 1 2(v2(x = 1 2) − v1) ∗ ( 1 2− x ∗ ) Auction1 = (v1− vlast) ∗ x4 + 1 2(v1− vlast) ∗ (x ∗_{− x} 4) Low1 = (vlast− v7) ∗ x4 + 1 2(v7− Plow,1) ∗ x4 Low2 = 1 2(v7− Plow,2)( 1 2− x4) + (v1− v7)( 1 2 − x4) − 1 2(x ∗ _{− x} 4)(v1− vlast) + 1 2(v2(x = 1 2) − v1)( 1 2− x ∗ )

The maximization problems of the two airlines are then:

max Π1 = 2(Pup,1(U pper1) + (Plow,1+ b)(Auction1) + Plow,1(Low1)) (5)

max Π2 = 2(Plow,2(Low2)) (6)

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are then: ¯ v ≥ (v2(x = 1 2) ≥ v1 ≥ vlast ≥ v7 ≥ 0 1 2 ≥ x ∗ _{≥ x} 3 ≥ 0

For the sake of the optimization, we set s = α−1₂ . This always satisfies α −s−1 > 0, which is necessary to have customers participates in the auction. s could be set to other values, but that does not cause the following results to be qualitatively different. With these new objective functions and conditions, we can run the optimization again. The profit of Airline 1 still rises mainly in α as shown in

Figure 10: Optimal Profit of Airline 1 with Auction

Figure 10. Higher or lower t only has a small effect on the optimal profits. For Airline 2 in contrast t is very important to the size of its profits like in Figure 11. A medium differentiation again is preferable for it. The new prices for Airline 1 in Figures 13 to 12 mainly behave in the same way as before. The new “price”, the winning bid for the upgrade, rises straight in α. The prices for both lower classes follow a similar pattern as before.

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Figure 11: Optimal Profit of Airline 2 with Auction

Figure 12: Optimal Price of Upper Class of Airline 1

5.3 Comparison

Having the optimal prices and profits for both the situation without auction and the situation with an auction now makes it possible to compare the two. Surprisingly, the profits of both firms drop after the implementation of the auc-tion mechanism. The profit of Airline 2 in Figure 14 is consistently lower than without the auction. This effect is strongest with a low α and a little stronger with a medium horizontal differentiation. The profit of Airline 2 shown in Figure 15 is lower too in most of the cases. Only for high t and high alpha does the profit not change much. For all other situations, Airline 2 makes less profit than

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Figure 14: Difference in Profit of Airline 1

without the auction system. This might be a analyze use of the auction mech-anism. Although Airline 1 suffers by implementing the auction, it also reduces the optimal profit of Airline 2. In a very competitive situation, this could help to push Airline 2 out of the market.

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Figure 15: Difference in Profit of Airline 2

6 Duopoly Where Both Airlines Have Different

Classes Without Uncertainty

Now we turn to the case where both airlines have two different classes. This changes the dynamic because both have the choice to establish an auction mech-anism. Again we first analyze a baseline case, where none of the airlines uses the auction. After that we explore how the auction mechanism changes the dynamics when one implements it and when both implement it.

6.1 Baseline

In the case without auction, the new competitive situation changes the posi-tions of the indifferent customers. The new specification can be looked up in the appendix. Figure 16 shows the areas on the surface of the cylinder. These areas define the two target functions as before. Both airlines have analogous areas rep-resenting their customers on the cylinder. From the areas and the corresponding prices result the profit functions of the two firms. Because the areas on the cylinder are symmetric, both airlines have analogous reaction functions. From this follows that Plow,1 = Plow,2, Pup,1 = Pup,2 must characterize an equilibrium

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Figure 16: Areas without Auction Mechanism

solution. This then yields optimal prices and profits then. In Figure 17 it can be observed that the optimal profit rises strictly in α and t. This suggests that

Figure 17: Optimal Profit of both Airlines

a big horizontal differentiation gives the airlines room for price setting whereas, for a higher α, the parameter for the vertical differentiation, the upper-class is more valuable and therefore a higher price there is possible. This can also be observed in Figure 18. A higher α and a higher t lead also to a higher price

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Figure 18: Optimal Price for Upper Class

for upper-class tickets. For lower-class tickets it is the other way around (Figure 19). A higher α presses Plow down. That is because the upper class gets more important with rising α. This prompts customers to buy upper class tickets and makes the market for the lower class narrower.

Figure 19: Optimal Price for Lower Class

6.2 Both Implement Auction

This is the other symmetric case where assume the chosen prices are not too different. Then for every airline the customers can only be indifferent between a

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specific area of this airline and the bordering classes of this or the other airline. Not possible is the option that consumers are indifferent between areas that are two steps away vertically or that there are customers indifferent to both a higher and a lower area for the other airline. This simplifying assumption is justified from the assumption that because of the symmetry optimal prices will be equal following the equivalent strategies. Indifferent consumers and the resulting optimal symmetric profit can be looked up in the corresponding appendix.

Analysing the derivations of the profit, we see that ∂Π_∂α > 0 when α > 1059₂₆₀ and ∂Π_∂t > 0 for low t and α > 25₁₃+ s. In a graphic, the dependence of the profits on α and t look like in Figure 20. The shape of the profit looks vastly different

Figure 20: Optimal Profit of both Airlines with Auction

from the other symmetric case. Especially the kink in the graph catches the eye. This coincides with the kink in Graph 23 for the optimal size of the bid of the last winning consumer. This is where using the auction more is optimal. Around the same value of α P_low∗ in Figure 22 does not grow in alpha any more. The price for upper-class tickets however is very straightforward only affected by α in this case (Fig 21).

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Figure 21: Optimal Price for Upper Class

6.3 One Airline Uses an Upselling Auction

Now we turn to the case where both airlines have an upper and a lower class, but only one implements the auction mechanism. Assuming the effect pressing the prices for upper class up and for lower class down, the indifferent customer between upper and lower class of the airline that does not implement the auction has a valuation that corresponds to customers in the auction area of the other airline. This implied indifferent consumers lead to a new definition of areas on the cylinder.

The areas on the cylinder then look like in Figure 24. Notice that the auction area now has a more irregular shape. But with the restriction v1 > v2 > vlast, also described in the introduction to this chapter, the size is easily distinguished. Unfortunately we cannot, as we did in the previous cases, use the symmetry of the set up to simplify reaction functions to optimal price solutions. So this case must remain open as the complexity makes it impossible to achieve solutions.

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Figure 22: Optimal Price for Lower Class

6.4 Comparison

In the former subsections we found solutions for the two symmetric cases where both airlines have different classes, and in the second case, they addi-tionally implement the described auction mechanism. The difference caused by implementing the auction can be seen in Figure 25. We see that implementing the auction actually reduces profits after optimizing prizes for the airlines re-gardless of the specifics of the market in the form of α and t. The difference in profits gets bigger with rising horizontal differentiation and lower in rising α, but stays consistently negative.

7 Introducing Uncertainty in the Monopoly

Sit-uation

An alternative model to deal with the necessary costs for going through the auction is modelling them as the risk incurred that the auction is actually not won. This is a robustness check for the results obtained in chapters 4–6. If this

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Figure 23: Optimal Bid

model, which depicts the necessary costs for the auction participants focussing on another aspect of the auction, yields similar results as the other monopoly case in Chapter 3, this is support for the validity of the results. If the distribution of valuations is not known ex ante, but there is enough information to form an expectation about it, there is a risk associated with buying lower class and bidding in the auction for someone who would otherwise by an upper-class ticket directly. The anticipated risk can be used to evaluate new indifferent customers. This second model is only applied to the monopoly situation as it gets much too complicated to introduce competitive dynamics. This would be something to examine in further studies.

7.1 Baseline

When there is no auction mechanism, there are three critical valuations that need to be derived. v1 = Plow is trivially the critical value below which no consumer buys. In other words, density of demand for v < v1 is dlow = 0. Lemma 7 In the area above v ∈ [v1, v2] the density is d = 2v−P_tlow

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Figure 24: Areas When One Introduces Auction

The second area has upper bound v2 which can be found by setting density to 1, i.e. v is high enough so that everybody buys tickets. From this follows:

v2 = 1

2t + Plow

The last critical valuation can be obtained by searching for the indifferent con-sumer between buying lower class and buying upper class:

αv3− Pup− tx = v3− Plow− tx

v3 =

Pup− Plow α − 1

Setting the capacity k for upper class, the monopolist does not know the mani-festation of the distribution of valuations. If we assume that he cannot be certain to capture all potential upper class customers, he can set a probability β = f (k)ˆ that there are free seats. Combining the three valuations and the demand

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den-Figure 25: Difference in Profits

sities, we get the profit maximization problem of the monopolist:

max E[Π] =E[βPup ¯ v − v3 ¯ v + (1 − β)(Pup ¯ v − k ¯ v + Plow k − v3 ¯ v ) + Plow v3− v2 ¯ v + Plow v2− v1 ¯ v 1 2(v2+ v1) − v1 t ] (7)

, where the last fraction is the average density in the lowest bracket. In the other areas the density is one. Since the valuation is uniformly distributed, the expected profit largely depends on the size of the areas between the critical valuations.

¯

v is a stochastic value. To solve the maximization problem, the monopolist has to suppose a certain estimator. Following Weber (2008), we replace ¯v with the non-linear estimator ˆ¯v = E[1_v_¯]−1. This is no longer a stochastic value. Lemma 8 The expected value expression in the profit function can be solved by replacing ¯v with ˆ¯v = E[1_v_¯]−1

Furthermore, the monopolist uses as a decision relevant valuation a systemati-cally smaller value as ˆv < E[¯¯ v]. This is trivial as there is variation in ¯v and ˆ¯v is a function of a convex function. For this case, Jensen’s inequality predicts strict inequality.

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As ˆ¯v is not stochastic, solving max ˆ¯vE[Π] yields the same results. Transform-ing the profit function accordTransform-ingly yields:

max ˆvE[Π] =βP¯ up(ˆ¯v − v3) + (1 − β)(Pup(ˆ¯v − k) + Plow(k − v3)) + Plow(v3− v2) +

1

2tPlow(v2− v1) 2

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Assembling the critical valuations into the function:

max ˆvE[Π] =βP¯ up(ˆv −¯ Pup− Plow α − 1 ) + (1 − β)(Pup(ˆv − k) + P¯ low(k − Pup− Plow α − 1 )) + Plow( Pup− αPlow α − 1 − 1 2t) + 1 2tPlow( 1 2t) 2 (9)

The capacity k has a direct functional relation to the set probability β that there are free places in upper class. Specifically because ¯v is is uniformly distributed, k is:

k = ¯v − v3+ β(¯v − ¯v) (10)

This is demonstrated in Figure 26. The capacity k is set somewhere inbetween the value of (4) for β = 0 and for β = 1. Other set capacities would be wasteful. Inserting (4) in (3) yields: max ˆvE[Π] =βP¯ up(ˆv −¯ Pup− Plow α − 1 ) + (1 − β)(Pup(ˆv − ¯¯ v + Pup− Plow α − 1 − β(¯v − ¯v)) + Plow(¯v + β(¯v − ¯v) − 2 Pup− Plow α − 1 )) + Plow( Pup− αPlow α − 1 − 1 2t) + 1 2tPlow( 1 2t) 2 (11)

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Figure 26: Capacity Setting in Uncertain Valuation environment

Solving this maximization problem yields an optimal β, Plow, and Pup as follows:

β∗ = 3¯v − 4¯v + q 3¯v2− 2¯v2 6(¯v − ¯v) P_low∗ = 1 16(4(¯v − ¯v) − 3t) P_up∗ = 24(α − 1)¯v 2 − 9t( q 3¯v2− 2¯v2_{− ¯}_{v) + 12¯}_v( q 3¯v2− 2¯v2_{− ¯}_v) 48( q 3¯v2− 2¯v2_{− ¯}_v) +4¯v((5 − α)¯v + (2α + 1) q 3¯v2− 2¯v2₎ 48( q 3¯v2− 2¯v2_{− ¯}_v)

The resulting profit is a function too long to display here, but the code for the computation can be referenced in Appendix 2. Simplifying the derivations of the optimal profit Π∗ shows that Π∗ always rises with α. It rises also in t if t > 3₄(¯v + ¯v), i.e. if horizontal travel costs are high enough or the expected

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valuation is low enough.

7.2 With Auction

We now look at the case of the auction mechanism implemented in the model with uncertainty. The valuation of the lower indifferent customer does not change:

v2 = 1

2t + Plow

This is because the lower indifferent customer is per definition too far away from buying upper class. Even β = 1 would not lead to him winning in the upgrade auction. The upper indifferent customer however is changed by the implementation of the auction mechanism:

αv3− Pup− tx = (β(α − 1) + 1)v3− Plow− βblast− tx

Notice that the additional valuation from buying flying upper class and also blast – the winning bid of the last one getting in because of the auction – only have to be paid if there are places free in upper class, i.e. when β > 0. Consequently, we now define the bid paid by the last customer getting in, blast. The premise of the second price auction is that the optimal strategy of everyone is to bid their own (in this case residual) valuation, so

blast = (α − 1)vlast

The position of the last customer getting in through the auction is dependent on the free capacity. After defining the capacity above, the expected free capacity is kauc = β(¯v − ¯v), depending on the set probability that there are free seats.

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Following from this, the position of the last customer winning the auction is:

vlast= v3− kauc= v3− β(¯v − ¯v)

So the size of the bid is depending on the position of the upper indifferent cus-tomer:

blast = (α − 1)(v3− β(¯v − ¯v)) Inserting this into the above indifference equation:

αv3− Pup− tx = (β(α − 1) + 1)v3− Plow− β(α − 1)(v3− β(¯v − ¯v)) − tx ⇐⇒ (α − 1)v3− Pup= −Plow+ β(α − 1)β(¯v − ¯v) ⇐⇒ v3 = Pup− Plow α − 1 + β 2_(¯_{v − ¯}_v)

Using this to define the new relevant areas the new target function of the mo-nopolist is:

max ˆvE[Π] =βP¯ up(ˆ¯v − v3) + (1 − β)(Pup(ˆ¯v − k) + Plow(k − v3)) + βblast(v3− vlast) + Plow(v3− v2) + 1 2tPlow(v2− v1) 2 (12)

The middle part is now additional income in case there are seats free in upper class, which is the case with probability β. The output from this maximization can again not be displayed in closed form.

7.3 Comparison

After finding solutions for both cases, we analyze the difference of both opti-mal profits. The numerical analysis shows that the size of α and t do not effect the difference in profits. However, holding α and t constant, lets us look at the

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effect ¯v and ¯v have on this difference. This is displayed in Figure 27. From the

Figure 27: Difference in Profit

figure we can see that in some configurations the auction has a positive effect on profits. In other combinations, the auction is detrimental. Specifically if ¯v and ¯

v are close to each other and risk is thereby low, the auction is positive for the monopolist. If the variation is high, establishing the auction is detrimental. In the first case, the auction enables the monopolist to differentiate an additional step. But with growing risk, the existence of the auction mechanism raises the position of the upper indifferent customer so much that it is ultimately bad for the monopolist.

8 Conclusion

This thesis examined upselling auctions in the airline industry. We looked at the profitability of such an auction system in a monopoly and a duopoly setting. We present two ways to model upselling auctions. We looked at a model

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where the customers have to bear some hassle costs for participating in the auction. This model was examined in the context of a monopoly and in a duopoly situation. We then went on to a more complicated, yet more realistic, version where we modelled the costs involved with the auction by introducing a non-zero probability that the auction is lost. The question in all those cases was if the auction mechanism is profit-maximizing when compared to a situation without it. We also wanted to examine if the implementation of the mechanism could be justified from a analyze standpoint.

Our analysis shows that in some settings, airlines can increase profits by using upselling auctions. However, the settings that we identified are arguably extreme though. In a simple monopoly model with auction-hassle costs, the usefulness of the mechanism for the airline depends on a high valuation for the upper-class tickets. Transferring the same model to a duopoly situation yields an even worse result for the airlines. The auction mechanism yields lower profits than the situation without it. On the other hand, the other airline in a duopoly setting also has vastly lower optimal profits, therefore the auction has a downward pressure on the whole market. Consequently, an auction could be useful to push rivals out and monopolize a market.

Another possibility to model the costs of an auction is implementing the ex ante uncertainty about consumers valuation directly. This was done in the last section for the monopoly situation. The solution from the model is that the difference in profits is not affected by the parameters, but that the auction mechanism is slightly better when the risk is low. For high risk situations, the auction mechanism would do worse.

The need to do much of the analysis in numerical form has the downside that the solutions are ultimately less general, because they are obtained for specific parameters. We included a range of likely possibilities, but there is the chance that for other specifications the solutions would be different. To account for these

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weaknesses further research into multi-dimensional differentiation modelling is necessary, as this is also an area where very few studies are published.

Appendix 1 - Proofs

A.1 Proof of Lemma 1

¯

v is not stochastic. Therefore, instead of maximizing Π we can maximize ¯vΠ and the optimal prices are unchanged. First order conditions are then:

∂ ¯vΠ ∂Pup = ¯v − 2 α − 1(Pup− Plow) ! = 0 ∂ ¯vΠ ∂Plow = 2 α − 1(Pup− Plow) − 2Plow− t 4 ! = 0

From easy inserting follows:

P_low∗ = ¯v 2 − t 8 P_up∗ = αv¯ 2 − t 8

This solution breaks the constraint v1 ≥ v2. This shows that this constraint is binding at t = 1

2. Using that, we can reformulate the profit function:

max Π = Pup ¯ v − v1 ¯ v + 1 2Plow v1− v2(x = 0) ¯ v

Now only the triangular area for the lower class is left. The non-triangular area is zero. There is no v where all horizontal customers buy in the lower class.

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Maximizing this yields: P_low∗ = ¯v 2 + 3 8 t α − t 2 P_up∗ = αv¯ 2 − t 8 Π∗ = 9t 2_{− 8αt}2_{− 8αt¯}_{v + 16α}2_v_¯2 32α¯v

A.2 Proof of Lemma 2

As before we can substitute Π for ¯vΠ. We have three choice variables. First order conditions are:

∂ ¯vΠ ∂Pup = ¯v − 2 s(Pup− Plow − b) ! = 0 ∂ ¯vΠ ∂Plow = 2 s(Pup− Plow − b) − 2Plow− t 4 ! = 0 ∂ ¯vΠ ∂b = 2 s(Pup− Plow − b) − 2b α − 1 ! = 0

From the first and the second derivative we get:

P_low∗ = v¯

2−

t 8 From the first and the third we get:

b∗ = α − 1 2 ¯v Finally substituting back yields:

P_up∗ = (s + α)v¯

2 −

t 8

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Inserting back into the profit function we get: Π∗ = (s + α)v¯ 4− t 8 + ( t 8) 21 ¯ v

This again breaks the restriction v1 ≥ v2. So fixing the restriction to the highest possible vlast = v2(x = 1₂) yields the new profit function:

max Π =Pup ¯ v − v1 ¯ v + (Plow+ b) v1− vlast ¯ v + 1 2Plow vlast− v2(x = 0) ¯ v , where there is one less area to compute. Optimising then produces:

P_low∗ = 6¯v(α − s − 1) 8α − 8s − 9 P_up∗ = (8α 2_{− s − 8α(1 + s))¯}_v 2(8α − 8s − 9) b∗ = 2(3 + 2α 2_{+ 5s + 2s}2_{− 4sα − 5α)¯}_v (8α − 8s − 9) Π∗ = (8α 2_{− s − 8α(1 + s))¯}_v 2(8α − 8s − 9)

A.3 Proof of Corollary 1

∆ signifies the value for the auction case minus the one without auction. Then: 1. ∆P_low∗ > 0 ⇐⇒ t 8 + 3 8 α − 1 α t + 4α − 4s − 3 2(8α − 8s − 9) > 0

This is definitely positive if 4α−4s−3 > 0, which is always since α−s−1 > 0 per definition (or nobody would participate in the auction).

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2.

∆P_up∗ > 0 ⇐⇒ 8(α − s)t − 9t + 4(α − s)¯v > 0

For the relevant area of α this is true. 3.

∆Π∗ > 0 ⇐⇒ (8α − 9)(8α − 8s − 9)t2 _{+ 8α(8α − 8s − 9)t¯}_{v + 16α(α − s)¯}_v2 _{> 0}

All three parts of the sum are bigger than zero for α ≥ s + 3₂ and s > 0.

A.4 Proof of Lemma 3

1. There are consumers indifferent between upper class of Airline 1 and lower class of the same airline:

αv − tx − Pup,1= v − tx − Plow,1 v1 =

Pup,1− Plow,1 α − 1

2. There are consumers indifferent between upper class of Airline 1 and lower class of Airline 2: αv − tx − Pup,1= v − ( 1 2− x)t − Plow,2 v2(x) = Pup,1− Plow,2 α − 1 − t 2(α − 1) + 2 t α − 1x

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3. There are consumers indifferent between the two lower classes: v − xt − Plow,1 = v − ( 1 2− x)t − Plow,2 xlow = Plow,2− Plow,1 2t + 1 4

4. There are consumers indifferent between not flying and flying lower class with Airline 1:

0 = v − xt − Plow,1 v4(x) = Plow,1+ tx

5. There are consumers indifferent between not flying and flying lower class with Airline 2: 0 = v − (1 2 − x)t − Plow,2 v5(x) = Plow,2+ 1 2t − tx

A.5 Proof of Lemma 4

First part: Inserting x3 into v2 yields exactly v1. Hence there must be a com-bination (x, v) where all three intersect. Because they are linear on the surface of the cylinder x × v with different slopes, there can only be one intersection.

Second part: Similarly setting v4 and v5 equal yields x3. Therefore they have to intersect. Because they have different slopes, they can only intersect once.

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A.6 Proof of Lemma 5

The area sizes can be measured as follows:

U pper1 = 1 2(¯v − v2(x = 1 2)) + (v2(x = 1 2) − v1) ∗ xlow+ 1 2(v2(x = 1 2) − v1) ∗ ( 1 2 − xlow) Low1 = (v1− v6) ∗ xlow + 1 2(v6− Plow,1) ∗ xlow Low2 = 1 2(v2(x = 1 2) − Plow,2)( 1 2 − xlow) + 1 2(v1− v6)( 1 2− xlow) Using this the problems of the two airlines are:

max Π2 = 2(Plow,2(Low2)) (14)

The profit is double the product of prices with areas because the shown pattern is replicated on the other side of the cylinder. Condition for the profits is that all the areas involved are positive in size. From this follows one vertical and one horizontal condition: ¯ v ≥ (v2(x = 1 2) ≥ v1 ≥ v6 ≥ 0 1 2 ≥ x3 ≥ 0

A.7 Proof of Lemma 5

1. There are consumers indifferent between upper class of Airline 1 and lower class of the same airline with bidding into the upper class:

(α − s)v − Plow,1− b − tx = αv − Pup,1− tx v1 =

1

s(Pup,1− Plow,1− b)

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class of Airline 2. For lower class of Airline 2 bidding is not possible, as that airline does not have an upper class:

αv − tx − Pup,1= v − ( 1 2− x)t − Plow,2 v2(x) = Pup,1− Plow,2 α − 1 − t 2(α − 1) + 2 t α − 1x

3. There are consumers indifferent between the two lower classes inside the area where customers win the upgrade auction:

(α − s)v − Plow,1− b − tx = v − ( 1 2 − x)t − Plow,2 v3(x) = Plow,1 − Plow,2+ b − 1₂t α − s − 1 + 2t α − s − 1x

4. There are consumers indifferent between the two lower classes below the area where consumers win the upgrade auction:

v − xt − Plow,1 = v − ( 1 2− x)t − Plow,2 x4 = Plow,2− Plow,1 2t + 1 4

Note that number 3 and 4 were one and the same area before. After

implementing the auction, the upper part is a linear function. Depending on if α − s < 1 the auction increases or decreases the part of consumers choosing Airline 1 over 2.

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The last two indifferent consumers meet at a specific combination (v, x) like before. This is v5(x4) or v7 = ₂1(Plow,1+ Plow,2) + 1₄t. Other points of interest to distinguish the size of the areas are:

1. v1, v2 and v3 have exactly one intersection point (v, x) = (v1, x∗) with x∗ = α−s−1_2ts Pup,1− α−1_2tsPlow,1+_2t1Plow,2− (α−1)_2ts b + 1₄.

2. x4, v5 and v6 have exactly one intersection which is (v7, x4), with v7 = 1

2(Plow,1 + Plow,2) + 1 4t

3. The position of the last one entering via auction is also the intersection of x4 and v3 at (v, x) = (_α−s−1b , x4)

4. Airline 1 holds as long as b > (1₂(Plow,1+ Plow,2) + 1₄t)(α − s − 1) The proof for that:

1. Reordering v3 for x yields:

x = v(α − s − 1) 2t − Plow,1 2t + Plow,2 2t − b 2t− 1 4 Inserted in v2 this yields exactly v1.

2. Airline 1 is the same as in Lemma 3.

3. The position of the last consumer winning in the auction is defined by there residual valuation which is the bid b = (α − s − 1)v. The position is then

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vlast = _α−s−1b . This is also the intersection of x4 and v3 as yielded, when inserting v3 into x4.

A.8 Proof of Lemma 6

In the Salop model the density to both sides combined is d=2|x∗_consumer − xf irm|. Without loss of generality we can set xf irm = 0. The critical x∗ is then the position of the indifferent customer in the lower class:

v − Plow− tx∗ = 0

Therefore density is

d = 2v − Plow t

A.9 Proof of Lemma 7

For the parts where ¯v is just in the denominator, this is trivial. For the two other parts (where c is non-stochastic):

E[v − c¯ ¯ v ] = E[¯v − c] ∗ E[ 1 ¯ v] + Cov(¯v − c, 1 ¯ v) = (E[¯v] − c) ∗ E[1 ¯ v] + (1 − E[¯v] ∗ E[ 1 ¯ v]) = 1 − c ∗ E[1 ¯ v] = v − cˆ¯ ˆ ¯ v

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Appendix 2 - Specifications of Section 5

B.1 Indifferent Consumers, Areas, and Target Function

of Section 5.1

When both airlines implement the auction mechanism the indifferent con-sumers are as follows:

1

2. The same is true of Airline 2:

1

3. There are indifferent consumers between the two upper classes:

αv − xt − Pup,1 = αv − ( 1 2 − x)t − Pup,2 xup = Pup,2− Pup,1 2t + 1 4

4. There are consumers indifferent between the two lower classes:

v − xt − Plow,1 = v − ( 1 2− x)t − Plow,2 xlow = Plow,2− Plow,1 2t + 1 4

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class of Airline 2, if v1 < v2: αv − tx − Pup,1= v − ( 1 2− x)t − Plow,2 v3(x) = Pup,1− Plow,2 α − 1 − t 2(α − 1) + 2 t α − 1x

6. If v1 > v2 the opposite is true:

αv − (1 2 − x)t − Pup,2= v − tx − Plow,1 v3(x) = Pup,2− Plow,1 α − 1 + t 2(α − 1) − 2 t α − 1x If v1 = v2 there is no connection needed and v3 vanishes.

0 = v − xt − Plow,1 v4 = Plow,1+ tx

This implies like in the baseline case, that where v4, v5, and xlow meet, there is the valuation above which everyone buys a ticket. This is (v, x) = (v4(xlow), xlow) with v4(xlow) = v6 = ₂1(Plow,1+ Plow,2) + 1₄t. Following the same strategy as in the other symmetric case, we set Plow,2 = Plow,1, Pup,2= Pup,1, and b2 = b1. With

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this we can solve for profits in equilibrium, which are here both: Π = (20α − 63)t 2 (16(20α − 43)(5t(20α − 63) +p(66717 − 37560α + 5200α2_)t2₎2₎ × (−5t2_(400α2_{+ 25680α − 92061) + 40(20α − 43)}p (5200α2_{− 37560α + 66717)t}2_¯_v + 4t(−459p(5200α2_{− 37560α + 66717)t}2 _{+ 10α(}p_(5200α2_{− 37560α + 66717)t}2_{− 8536¯}_v) + 113262¯v + 15200α2v))¯

B.2 Indifferent Consumers, Areas, and Target Function

of Section 5.2

Indifferent consumers:

2. Airline 1 is in analogy also true for Airline 2:

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4. There are consumers indifferent between the two lower classes: v − xt − Plow,1 = v − ( 1 2− x)t − Plow,2 xlow = Plow,2− Plow,1 2t + 1 4

5. There are consumers indifferent between upper class of Airline 1 and lower class of Airline 2, if v1 < v2: αv − tx − Pup,1= v − ( 1 2− x)t − Plow,2 v3(x) = Pup,1− Plow,2 α − 1 − t 2(α − 1) + 2 t α − 1x

6. If v1 > v2 the opposite is true:

αv − (1 2 − x)t − Pup,2= v − tx − Plow,1 v3(x) = Pup,2− Plow,1 α − 1 + t 2(α − 1) − 2 t α − 1x If v1 = v2 there is no connection needed and v3 vanishes.

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The following areas on the surface are defined by the consumers choices: U pper1 =        (¯v − v1)xup− 1₂(v2− v1)(xup− xlow) v2 ≥ v1 (¯v − v1)xup v2 < v1 U pper2 =        (¯v − v2)(1₂ − xup) − 1₂(v2− v1)(xup− xlow) v2 ≤ v1 (¯v − v2)(1₂ − xup) v2 > v1 Low1 =       

(v1− v6)xlow +1₂(v6− Plow,1)xlow v2 ≥ v1

(v1− v6)xlow +1₂(v6− Plow,1)xlow− 1₂(v2− v1)(xup− xlow) v2 < v1

Low2 =       

(v2− v6)(1₂ − xlow) + 1₂(v6− Plow,2)(1₂ − xlow) v2 ≤ v1 (v2− v6)(1₂ − xlow) + 1₂(v6− Plow,2)(₂1 − xlow) − 1₂(v2− v1)(xup− xlow) v2 > v1

This leads to the target functions:

max Π2 = 2(Plow,1(Low1) + Plow,1(Low1)) (16)

B.3 Indifferent Consumers, Areas, and Target Function

of Section 5.3

Indifferent consumers are:

1

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2. There are consumers indifferent between upper class of Airline 2 and lower class of the same airline:

4. There are consumers indifferent between the two lower classes:

v − xt − Plow,1 = v − ( 1 2− x)t − Plow,2 xlow = Plow,2− Plow,1 2t + 1 4

5. There are consumers indifferent between the auction upgrade path of Air-line 1 and upper class of AirAir-line 2:

(α − s)v − tx − b − Plow,1 = αv − ( 1 2 − x)t − Pup,2 v3(x) = 1 s(Pup,2− Plow,1− b − 2tx + t 2)

6. There are consumers indifferent between the auction upgrade path of Air-line 1 and lower class of AirAir-line 2:

(α − s)v − tx − b − Plow,1 = v − ( 1 2 − x)t − Plow,2 v4(x) = 2tx −₂t + b + Plow,1− Plow,2 α − s − 1

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Code for Monopoly

(*without auction first*) v1= (Pup - Plow) / (alpha - 1); v2= Plow + t / 2;

Profit=

2_{((Pup * (vbar - v1) / vbar) + (Plow * (v1 - v2) / vbar) + (Plow * (v2 - Plow) / (2 * vbar)));} (*the two times is because of the cylindrical form*)

f=

maximiere

Maximize[{Profit, alpha > 1 && vbar > 0 && t > 0 && t < 4 vbar && v1 ≥ v2}, {Pup, Plow}] (*we see the last inequality is binding*)

(*optimal solutions and derivatives*)

gib aus

Print["optimal solutions and derivatives"]

gib aus Print["Pup"] Puplos= f[[2]][[1]][[2]][[2]] vereinfache Simplify[ leite ab D[Puplos, alpha] > 0] Simplify[ leite ab D[Puplos, vbar] > 0] Simplify[ leite ab D[Puplos, t] > 0] gib aus Print["Plow"] Plowlos= f[[2]][[2]][[2]][[2]] vereinfache Simplify[ leite ab D[Plowlos, alpha] > 0] vereinfache Simplify[ leite ab D[Plowlos, vbar] > 0] Simplify[ leite ab D[Plowlos, t] > 0] gib aus Print["Profit"] Profitlos= f[[1]][[1]][[1]][[1]] Simplify[ leite ab D[Profitlos, alpha] > 0] Simplify_[ leite ab D_{[Profitlos, vbar] > 0]} Simplify[ leite ab D[Profitlos, t] > 0]  9 t

2_{-8 alpha t}2_{-8 alpha t vbar+16 alpha}2_vbar2

32 alpha vbar vbar> 0 && 0 < t < 4 vbar && alpha > 1

-∞ True ,

Pup  Indeterminate1 ! (vbar > 0 && 0 < t < 4 vbar && alpha > 1) 8 (-t + 4 alpha vbar) True

,

Plow

Indeterminate _{! (vbar > 0 && 0 < t < 4 vbar && alpha > 1)}

1

8 -t - 3 (-1+alpha) 2_t2

alpha2 + 4 vbar True

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optimal solutions and derivatives Pup 1 8 (-t+ 4 alpha vbar) vbar> 0 alpha> 0 False Plow 1 8 -t- 3 (-1 + alpha) 2_t2 alpha2 + 4 vbar

(-1 + alpha) alpha (-1 + alpha)2t2

alpha2 < 0

True

t t+ 3 (-1 + alpha)2t2

alpha2 < 0

Profit

9 t2_{- 8 alpha t}2_{- 8 alpha t vbar + 16 alpha}2_vbar2

32 alpha vbar 16 alpha4vbar3> 9 alpha2_t2_vbar

alpha vbar2_{(-9 + 8 alpha) t}2_{+ 16 alpha}2_vbar2_{ > 0}

alpha vbar((-9 + 8 alpha) t + 4 alpha vbar) < 0 (*test auction case*)

lösche alle

ClearAll[alpha, vbar, t] v1a= (Pup - Plow - b) / s; v2a= Plow + t / 2;

vlast_{= b / (alpha - s - 1);}

Profita= 2 (Pup * (vbar - v1a) / vbar + (Plow + b) (v1a - vlast) / vbar + Plow* (vlast - v2a) / vbar + Plow (v2a - Plow) / (2 * vbar)); vbar_{= 1;}

f=

maximiere

Maximize[{Profita, alpha > Pup ≥ Plow + b ≥ Plow > 0 && vbar ≥ v1a ≥ vlast ≥ v2a ≥ 0 && alpha_{> 1 && 0 < t < 4 vbar && 0 < s ≤ alpha - 1}, {Pup, Plow, b}]}

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(*This optimization procedure leads to v2a>v1a which is not possible*)

lösche alle

ClearAll[alpha, vbar, t, s, foc1, foc2, foc3] v1a= (Pup - Plow - b) / s;

v2a= Plow + t / 2;

vlast= b / (alpha - s - 1);

Profita= 2 (Pup * (vbar - v1a) / vbar + (Plow + b) (v1a - vlast) / vbar + Plow* (vlast - v2a) / vbar + Plow (v2a - Plow) / (2 * vbar)); foc1=

löse

Solve[

leite ab

D[Profita, Plow]  0, Plow]; foc2=

löse

Solve[

leite ab

D[Profita, Pup]  0, Plow]; foc3= löse Solve[ leite ab D[Profita, b]  0, Plow]; second1= löse

Solve[(Plow /. foc1[[1]])  (Plow /. foc2[[1]]), b]; second2₌

löse

Solve_{[(Plow /. foc2[[1]])  (Plow /. foc3[[1]]), b];} third=

löse

Solve[(b /. second1[[1]])  (b /. second2[[1]]), Pup]; (*This is the maximizing solution for Pup*)

Puplos= third /. {alpha  2, s  1 / 2, vbar  1, t  1 / 2} (*second1/.third (*test checks out/same as second2*)*)

Plowlos_{= foc3 /. second2[[1]] /. third[[1]] /. {alpha  2, s  1 / 2, vbar  1, t  1 / 2}} blos= second2[[1]] /. third[[1]] /. {alpha  2, s  1 / 2, vbar  1, t  1 / 2}

v1a/. {alpha  2, s  1 / 3, vbar  1, t  1 / 2} /. Puplos[[1]] /. Plowlos[[1]] /. blos v2a_{/. {alpha  2, s  1 / 3, vbar  1, t  1 / 2} /. Puplos[[1]] /. Plowlos[[1]] /. blos} vlast/. {alpha  2, s  1 / 3, vbar  1, t  1 / 2} /. Puplos[[1]] /. Plowlos[[1]] /. blos Pup  15_{16 } Plow  _{16 }7 b  1_{4 } 3 4 11 16 3 8

(*like before but v2a=

vlast i.e. the only consumers buying lower are the one in the triangle*)

lösche alle

ClearAll[alpha, vbar, t, s, foc1, foc2, foc3, v1a, vlast, v2a, Plowalos, Pupalos, blos, Profitalos] v1a= (Pup - Plow - b) / s;

vlast= b / (alpha - s - 1); v2a= vlast;

Profita= 2 (Pup * (vbar - v1a) / vbar + (Plow + b) (v1a - vlast) / vbar + Plow* (vlast - v2a) / vbar + Plow (v2a - Plow) / (2 * vbar));

Upgrade auctions in the airline industry : a game-theoretic approach

Master Thesis