A Mathematical Study of Purchasing Airline Tickets

Transcription

1 A Mathematical Study of Purchasing Airline Tickets THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Mathematical Science in the Graduate School of The Ohio State University By Heather Smith, B.S. Graduate Program in Mathematics The Ohio State University 2013 Thesis Committee: Yuan Lou, Advisor Ian Hamilton

2 c Copyright by Heather Smith 2013

3 ABSTRACT We have sought to apply ideas from foraging theory to the purchase of airline tickets from competing websites in an online shopping environment. We explore how consumers might distribute themselves among websites and different tickets dependent on parameters such as price of ticket, number of stops, input rate of tickets into the site, and so on. To facilitate our exploration of consumer behavior, we have developed a system of difference equations and rules as to why consumers may prefer one website over another. By tuning parameters such as quality of a ticket and input rate of tickets, different distributions of consumers are observed. Based on these distributions, we seek to determine the best way - measured by the average quality of ticket purchased - for consumers to distribute themselves among websites and ticket types. ii

4 ACKNOWLEDGMENTS I would like to thank my advisors, Yuan Lou and Ian Hamilton, for their support and guidance through the completion of my thesis as well as my husband, Hudson Smith, for his help. iii

5 VITA Weddington High School B.Sc. in Physics from Erskine College 2011-Present Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major Field: Mathematics Specialization: Mathematical Biology iv

6 TABLE OF CONTENTS Abstract Acknowledgments Vita ii iii iv List of Figures vi List of Tables vii CHAPTER PAGE 1 Introduction The Model Terminology, assumptions, and notation How consumers distribute themselves within a website Quantifying the quality of a ticket Movement between websites Rating Rules Results The Ideal Distribution Several Scenarios Scenario Scenario Scenario Discussion Bibliography Appendix A: MATLAB Code v

7 LIST OF FIGURES FIGURE PAGE 3.1 A rule that leads to the ideal distribution for Scenario 1. Exact fractions for each factor are given in Table A rule that leads to the ideal distribution for Scenario 2. Exact fractions for each factor are given in Table Several rules that lead to the ideal distribution for Scenario 3. Exact fractions for each factor are given in Table vi

8 LIST OF TABLES TABLE PAGE 3.1 A website environment is shown in the first two rows. Row 4 shows the calculated ideal distribution and associated r value. The remaining rows show the population distribution and r value for various consumer rules A website environment is shown in the first two rows. Row 4 shows the calculated ideal distribution and associated r value. The remaining rows show the population distribution and r value for various consumer rules. *This rule did not reach an equilibrium population distribution A website environment is shown in the first two rows. Row 4 shows the calculated ideal distribution and associated r value. The remaining rows show the population distribution and r value for various consumer rules vii

9 CHAPTER 1 INTRODUCTION In studies of animal behavior, there are many models that attempt to describe the way in which animals distribute themselves among habitats in response to the availability of resources. One well-known model is the ideal free distribution [2]. The ideal free distribution describes the way animals distribute themselves when patches differ in the amount and/or quality of resources available. Ideal animals possess complete knowledge of the quality of each patch and also the density of other foragers in that patch. They are also able to move freely among patches without cost of movement, delay, or any other inhibition. In this situation, the best way for animals to distribute themselves is according to input matching, that is, when the ratio of animal densities matches the ratio of resource input among patches. This is a very simplistic model, of course. Animals are not ideal and there are a multitude of other factors, such as differences in the abilities of individual animals, that complicate matters. In response, there have been many other models proposed to describe the behavior of animals in various situations [3]. In our work, we have sought to apply some of the intuition of foraging models to an analogous situation that modern humans may face - purchasing goods in an online shopping environment. While our ideas could be applied to a number of scenarios, we have focused specifically on the purchase of airline tickets from among different websites, such as Expedia, Priceline, individual airline websites, etc. The website may 1

10 be considered the humans s habitat in which they search for a depletable resource - airline tickets - that other people are also searching for. The consumer must search for the best ticket from amongst different websites, like an animal may search for the best food source among different habitats. We seek to explore how consumers might distribute themselves among websites and among different classes of tickets dependent on parameters such as price of ticket, number of stops, input rate of tickets into the site, and so on. Drabas and Wu consider a similar question, focusing on the analysis of consumer choice and using a rather complex model. However, we will use a simpler, more intuitive model in order to understand the basic, underlying dynamics of the system - specifically, what leads to different distributions of consumers and what are the resulting consequences for consumers. [1] To facilitate our exploration of consumer behavior, we have developed a system of difference equations. The solutions to these equations give the number of tickets available in a ticket class as well as the number of consumers in each ticket class. We have also developed rules as to why consumers may decide to purchase a ticket from a different website. By tuning parameters such as quality of tickets and input rate of tickets, different distributions of consumers are observed. Based on these distributions, we will try and answer several questions. Namely, given quality and number of tickets available from various websites, what is the best way for consumers to distribute themselves among the websites and ticket classes? Furthermore, what rules of ticket-purchasing behavior will lead to these distributions? 2

11 CHAPTER 2 THE MODEL 2.1 Terminology, assumptions, and notation In our model, we consider a single, arbitrary plane trip. For example, Columbus to Atlanta on May 5, There are multiple websites from which consumers may purchase a ticket. We will consider two websites in the following discussion. Within each website, there are different ticket classes. A ticket class is defined by the price and number of stops (layovers) associated with that ticket. Time of departure, total travel time, and other ticket features have been ignored for simplicity. For example, a website may offer one ticket for $500 that has one stop and another ticket for $650 that is a direct flight with no stops. These would be considered two different ticket classes. We will consider four ticket classes per website in the following discussion. Realistically, the system we have chosen to explore is more complex. Websites are likely constantly adjusting the price and distribution of tickets dependent on purchasing patterns, fuel prices, and other factors. Consumers are certainly not always rational in their purchasing decisions and, additionally, every consumer differs in how they weight factors when making a ticket purchasing decision. [1] Some consumers may value price more highly and always purchase the cheapest ticket available. Other consumers may value their time most and purchase a ticket with zero layovers regardless of price. As a population, consumers would be a heterogeneous 3

12 mixture of these ideals. However, in order to facilitate a study of the basic behavior of the website - consumer interaction, we have chosen to make several simplifying assumptions. First of all, all consumers in our model purchase a ticket. Secondly, there is a constant birth rate and death rate for consumers (i.e., the consumer population is constant). This is likely somewhat realistic for popular domestic routes with large numbers of flights per day. Additionally, there are always enough tickets for everyone, although not necessarily in the class they desire. This is also probably realistic for well-traveled routes, as airlines can be expected to meet demand for their product. Finally, consumers are homogeneous in our model. That is, they all use the same function when assigning desirability to a certain ticket class. This assumption is clearly unrealistic but allows us to start with a simple, understandable model. This assumption is one of the first that should be relaxed in any future, expanded models. Any other assumptions will be mentioned later in the text. Vector notation will be used throughout the following discussion to characterize tickets classes within each website. Subscript 1 or entry 1 of a vector will always refer to the best, or most desired, ticket within that website. Subscript 2 or entry 2 is the second best, and so on. The variable x will represent the number of tickets available for purchase. Tickets may accumulate up to 5,000 within a ticket class if they are not purchased. For example, x1 = the number of tickets available for purchase in website 1 = [1000,500,3000,5000]. Here x1(1)=1000 means that within website 1, there are 1000 tickets available of the best ticket that the website has to offer. There are 500 of the second-best ticket type available, and so on. The variable p will represent the number of consumers purchasing tickets from a certain ticket class. For example, p1 = the number of people purchasing tickets from website 1 = [1000,500,1000,0]. Here, p1(1) means that in website 1, 1000 consumers are purchasing tickets from the best ticket class of that website. The variable a will represent the input rate of tickets into 4

13 a ticket class per unit time. Like in the preceding examples, it is defined as a vector for each website. In our model, this input rate is constant with time, though this is not realistic as input of tickets likely depends on many factors. 2.2 How consumers distribute themselves within a website As mentioned previously, the consumers are homogeneous in our model. Thus, within a website, they will always fill up the first (best) ticket class preferentially, then the second class, and so on. This idea lends itself to a very simple equation for the number of customers purchasing from each ticket class: i 1 x1 i if x1 i n1 p1 k k=1 p1 i = i 1 i 1 (2.2.1) n1 p1 k if x1 i > n1 p1 k k=1 k=1 This equation describes behavior within website 1. There would be one of these equations for each website in the model, replacing, for instance, p1, x1, and n1 with p2, x2, and n2 for website 2. Here, p1 i represents the number of people purchasing tickets from the i-th ticket class in website 1 where i=1 is the best ticket class, i=2 the second best, and so on. As mentioned before, x1 i is the number of tickets available in ticket class i for website 1. n1 represents the total number of people in website 1 as a whole. Given n1, this piecewise equation distributes people among ticket classes according to how many tickets are available in each class, filling the best classes preferentially. The expression for x1 i is a simple recursive relationship: x1 t+1 i = x1 t i p1 t i + a1 t i (2.2.2) Once again, other websites would use the same equation, replacing x1, p1, and a1 as appropriate. Here, the number of tickets at a given time step is equal to the 5

14 number of tickets previously available, minus the number of consumers in the ticket class (since they all must purchase a ticket) plus input of tickets into the class. If we run a simulation of the above formulas for the simple case of one website only (i.e., no movement between sites), we observe what we ll call the trickle down effect. As many consumers as will fit in the best ticket class purchase from there. From the leftover consumers, as many as will fit in the second best ticket class purchase from there, and so on. For example, let a1=[500,1000,2000,1000] and let there be n1=4000 people in a website. Then, after only a few iterations of code, an equilibrium is reached in which p1=[500,1000,2000,500]. 2.3 Quantifying the quality of a ticket In our model, the quality of a ticket is an important factor in the purchasing decision. Thus, we need a numerical value that will quantify the quality of a ticket. Since our consumers are homogeneous, they will all assign the same quality value to a given ticket. The quality of a ticket is a function of the price of the tickets, d, and the number of stops, s, according to the relationship u(d, s) = e bd cs (2.3.1) where b and c are parameters. In our simulations we have used b= and c=0.5. This function was chosen because it exhibits certain desirable qualities, but it is arbitrary. Notably, the quality of a ticket is one for a ticket that costs zero dollars and has zero layovers. Quality then decreases with increasing price and stops. A well designed survey may be able to discover a true quality function with parameters. Once again, we will use vector notation to denote the quality of tickets within a website. For example, u1 = the quality of tickets in website 1 = [0.2456, , 6

15 0.1193, ]. Recall once again the important fact that the first entry of any vector is always associated with the best ticket available in a given website. 2.4 Movement between websites The real interest in our system lies in considering why people may decide to purchase tickets from a different website and what population distributions arise when they do. In our model, we consider three rules for why people may choose a certain website to purchase a ticket from. These rules will govern which website is seen as better by the population of consumers and thus determine their movement amongst the sites. The first rule is the average quality of a purchased ticket from a website. Here, the consumer considers the overall quality of tickets offered by the websites when they choose where to purchase a ticket. The second rule regarding why consumers may prefer one website over another is the quality of the best ticket from a website. Here, you can imagine consumers who only compare the top tickets from each website and decide based on this. They would be easily swayed by the advertisement of great tickets, even if there were only a very small number of those tickets available. The final rule is the probability of getting a ticket in the best ticket class. Here, the consumer is interested in the quantity of good tickets available and will choose a website that offers a higher number of best tickets. Of course, most consumers probably do not follow one of these three rules exclusively when choosing a website from which to purchase a ticket. Thus, we allow consumers to choose a website based on a combination of these three rules according to the weights α, β, and γ, where α+β+γ=1. The weight α will correspond to the weight of the decision based on the average quality of the website, β will correspond to the weight of the quality of the best ticket and γ the weight of the probability of getting a ticket from class 1. We will refer to a combination of α,β,γ as a rule. Each 7

16 unique combination of weights defines a different rule as to why a consumer may prefer one website over another. Each website can then be assigned a score under a particular rule as follows u1 x1 x1 ξ1 = α n i=1 x1 + βu1 1 + γ 1 n i i=1 p1 i (2.4.1) Here, the numerator of the first term is the dot product of the quality vector of website 1 with the available tickets vector and the denominator is simply the sum of all tickets available in the site, giving the average quality of a ticket purchased from that website. u1 1 represents the quality of the best ticket in website 1. In the final term, x1 1 is the number of tickets available in the best ticket class and the denominator is the total number of consumers in website 1, giving the probability of getting a best ticket. Note that each website has its own score and all scores are between zero and one. If the score of website 1 is higher than website 2, consumers prefer website 1 and will relocate if possible. However, though a consumer may prefer one website over another, he may or may not be able to actually purchase a ticket from that website depending on the availability of tickets there. In order to avoid movements of a large number of consumers at once, only a certain proportion of consumers may switch websites at any iteration of a simulation. The proportion that may switch, σ, is given by the difference between the scores of the websites σ = ξ1 ξ2 (2.4.2) 2.5 Rating Rules In the following discussion, we will refer to the ticket input vectors along with the quality vectors (determined by prices and number of stops) of the websites as the 8

17 website environment. In other words, the website environment is what the consumers encounter when they decide to purchase a ticket - what quality of tickets are offered from the various websites and how many are available? For a particular website environment, we would like to know what rule is best for consumers to follow when distributing themselves among websites. To do this, we need to quantify how good a rule is for consumers. We ll call this the rule rating and denote it by r. The rule rating is simply the average quality of ticket per consumer for the entire population of consumers. For a model with two websites, this would be r = u1 p1+u2 p2 n (2.5.1) where u1 and u2 are the quality vectors for websites 1 and 2, p1 and p2 are the consumer population vectors, and n is the total population of all consumers in the system. The higher the value of r, the better the rule is for consumers as a whole. 9

18 CHAPTER 3 RESULTS 3.1 The Ideal Distribution For any given website environment, it is easy to imagine that the population distribution that will give the highest r value (i.e. highest average ticket quality per consumer) is the distribution that results from the trickle down effect amongst all ticket classes from all websites. In other words, considering all tickets from all websites, as many consumers who are able based on availability should purchase from the best ticket class, as many leftover consumers as are able should purchase from the second best, and so one. This is simply Equation applied to the entire set of ticket options. This is shown below in Tables 1, 2, and 3 for several website environments. 3.2 Several Scenarios We would like to see if, for various website environments, we can match this ideal distribution with a particular rule. To do so, we create several different website environments and first calculate the ideal population distribution and its associated r value. We then tune the parameters α, β, and γ to illustrate various possible distributions and, ultimately, to try to match the ideal distribution. In the tables that follow, we report a website environment, the calculated ideal 10

19 distribution with r value, and then several rules along with the associated population distributions and r values. Within the website environment, a1 and a2 represent the input rates of tickets into the two websites, d1 and d2 give the prices of each ticket class for each website, s1 and s2 give the number of stops associated with each ticket, and u1 and u2 are the resulting quality scores for each ticket in each website. Note that because of the construction of our model and code, we are unable to guarantee that any matches to the ideal distribution are unique rules (as shown explicitly in Table 3.3). For all the scenarios reported below, there are n=8000 total consumers with an initial condition of n1=n2=4000 consumers in each website. Unless otherwise stated, the population distributions given for various rules are steady state distributions (i.e., these distributions emerge after a certain number of iterations of code and then do not change for future iterations) Scenario 1 In the website environment shown in Table 3.1 below, the two websites have equal input rates of tickets for all ticket classes. However, website 2 has better ticket quality overall. The ideal distribution and associated r value were calculated and are shown in row 4 of Table 3.1. In the rows of the table, the population distributions and r values for various rules are reported. In the last row of Table 3.1, a rule which leads to the ideal distribution is shown. Figure 3.1 illustrates a rule that leads to the ideal distribution with a pie chart where the pieces correspond to the fraction of weight placed on a certain decision factor. Note that in this website environment, the rule we found that leads to the ideal distribution weights the average quality of tickets most heavily. However, recall that we cannot guarantee that this rule is the only one that leads to the ideal distribution and there may very well be others. 11

20 Website Environment: Consumer Behavior: Ideal Distribution: Rules and Results: a1=[1000,2000, 2000,1000] a2=[1000,2000, 2000,1000] d1=[850,600, 500,400] d2=[750,500, 400,300] s1=[0,1,2,3] s2=[0,1,2,3] Rule: Rule rating, r Population Distribution - r= p1=[1000,2000, 0,0] α=1, β=0, r= p1=[1000,1000, γ=0 0,0] α=0, β=1, r = p1=[1000,1000, γ=0 0,0] α=0, β=0, r= p1=[1000,2000, γ=1 1000,0] α=0.4, β=0.3, r= p1=[1000,2000, γ= ,0] α=0.5, β=0.25, r= p1=[1000,2000, γ= ,0] α=0.8, β=0.1, r= p1=[1000,1460, γ=0.1 0,0] α=0.6, β=0.2, r= p1=[1000,2000, γ=0.2 60,0] α=0.59, β=0.23, r= p1=[1000,2000, γ=0.18 0,0] u1=[0.1353, , , ] u2=[0.1738, ,0.1353, ] p2=[1000,2000, 2000,0] p2=[1000,2000, 2000,1000] p2=[1000,2000, 2000,1000] p2=[1000,2000, 1000,0] p2=[1000,2000, 1599,0] p2=[1000,2000, 1740,0] p2=[1000,2000, 2000,540] p2=[1000,2000, 1940,0] p2=[1000,2000, 2000,0] Table 3.1: A website environment is shown in the first two rows. Row 4 shows the calculated ideal distribution and associated r value. The remaining rows show the population distribution and r value for various consumer rules. 12

21 Figure 3.1: A rule that leads to the ideal distribution for Scenario 1. Exact fractions for each factor are given in Table Scenario 2 In the website environment shown in Table 3.2, below, ticket quality for each website is similar to what was explored in Table 3.1. Ticket input rates are the same between the two websites except for the input into ticket class 1. While website 1 inputs 1000 best tickets per unit time, website 2 inputs only 5 best tickets per unit time. Note, though, that those 5 tickets from website 2 are of slightly better quality than those from website 1. The ideal distribution and associated r value were calculated and are shown in row 4 of Table 3.2. In the subsequent rows of the table, the population distributions and r values for various rules are reported. In the last row of Table 3.2, a rule which leads to the ideal distribution is shown. The asterisk in row 8 denotes the fact that this rule did not reach an equilibrium population distribution. Rather, the population distribution fluctuated around what is reported in the table. Figure 13

22 3.2 illustrates a rule that leads to the ideal distribution with a pie chart where the pieces correspond to the fraction of weight placed on a certain decision factor. As is shown in Table 3.2, for this website environment there are a number of rules which lead to the same distribution. Note also that in this website environment, the rule we found that leads to the ideal distribution weights the quality of the best ticket most heavily. Website Environment: a1=[1000,2000, 2000,1000] a2=[5,2000, 2000,1000] d1=[875,600, 450,400] d2=[750,500, 480,300] s1=[0,1,2,3] s2=[0,1,2,3] Consumer Behavior: Rule: Rule rating, r Population Distribution Ideal Distribution: - r= p1=[1000,2000, 2000,0] Rules and α=1, β=0, r= p1=[1000,1995, Results: γ=0 0,0] α=0, β=1, r = p1=[1000,1995, γ=0 0,0] α=0, β=0, r= p1=[1000,2000, γ=1 2000,1000] α=0.4, β=0.3, r= p1=[1000,2000, γ= ,1000] α=0.5, β=0.25, r= p1=[1000,2000, γ= ,1000] * α=0.8, β=0.1, r= p1=[1000,2000, γ= ,1016] α=0.6, β=0.2, r= p1=[1000,2000, γ= ,1000] α=0.16, r= p1=[1000,2000, β=0.695, 2000,1] γ=0.145 u1=[0.1353, , , ] u2=[0.1738, ,0.1108, ] p2=[5,2000, 995,0] p2=[5,2000, 2000,1000] p2=[5,2000, 2000,1000] p2=[5,1995, 0,0] p2=[5,1995, 0,0] p2=[5,1995, 0,0] p2=[5,1979, 0,0] p2=[5,1995, 0,0] p2=[5,2000, 994,0] Table 3.2: A website environment is shown in the first two rows. Row 4 shows the calculated ideal distribution and associated r value. The remaining rows show the population distribution and r value for various consumer rules. *This rule did not reach an equilibrium population distribution. 14

23 Figure 3.2: A rule that leads to the ideal distribution for Scenario 2. Exact fractions for each factor are given in Table Scenario 3 In the website environment shown in Table 3.3, below, both the input rate vectors and quality vectors are different between the two websites. Notably, website 1 is inputting only 5 best tickets per unit time while website 2 is inputting 1000 best tickets per unit time. However, the quality of the best ticket in website 1 is very high and significantly better than that of website 2. This illustrates a case in which one website offers an extremely good ticket but only a very limited number of those good tickets are available. The ideal distribution and associated r value were calculated and are shown in row 4 of Table 3.3. In the subsequent rows of the table, the population distributions and r values for various rules are reported. Figure 3.3 illustrates several rules that lead to the ideal distribution with pie charts where the pieces correspond to the fraction of weight placed on a certain decision factor. 15

24 Interestingly, for this website environment, several rules were found that match the ideal distribution (see rows 5, 7, 10 of the table). There are likely others as well. Particularly notable is the fact that two of these rules are pure strategies and one is a mixed strategy. Website Environment: Consumer Behavior: Ideal Distribution: Rules and Results: a1=[5,1000, 3000,3000] a2=[1000,2000, 2000,1000] d1=[200,600, 550,500] d2=[550,500, 450,300] s1=[0,1,2,3] s2=[0,1,2,3] Rule: Rule rating, r Population Distribution - r= p1=[5,1000, 995,0] α=1, β=0, r= p1=[5,1000, γ=0 995,0] α=0, β=1, r = p1=[5,100, γ=0 3000,3000] α=0, β=0, r= p1=[5,1000, γ=1 995,0] α=0.4, β=0.3, r= p1=[5,1000, γ= ,667] α=0.5, β=0.25, r= p1=[5,1000, γ= ,337] α=0.8, β=0.1, r= p1=[5,1000, γ= ,0] α=0.6, β=0.2, r= p1=[5,1000, γ= ,0] u1=[0.6065, , , ] u2=[0.2528, ,0.1194, ] p2=[1000,2000, 2000,1000] p2=[1000,2000, 2000,1000] p2=[995,0, 0,0] p2=[1000,2000, 2000,1000] p2=[1000,2000, 328,0] p2=[1000,2000, 658,0] p2=[1000,2000, 2000,1000] p2=[1000,2000, 1284,0] Table 3.3: A website environment is shown in the first two rows. Row 4 shows the calculated ideal distribution and associated r value. The remaining rows show the population distribution and r value for various consumer rules. 16

25 Figure 3.3: Several rules that lead to the ideal distribution for Scenario 3. fractions for each factor are given in Table 3.3. Exact 17

26 CHAPTER 4 DISCUSSION For a known website environment, we see that it is possible to calculate an ideal distribution and a rule (or rules) which lead to that distribution. However, the ideal distribution varies dependent upon the website environment encountered and, subsequently, any rules that lead to the ideal distribution can be quite different for different website environments. This study of the dynamics of the ticket purchasing system is introductory and there are many directions in which future research could go. First of all, it would be interesting to verify if, for a given website environment, the rules leading to the ideal distribution are evolutionarily stable strategies [4] - that is, no other strategies can invade and do better if the entire population of consumers is following the proposed rule. [3] In other words, if the entire population of airline ticket consumers is following one of the rules leading to an ideal distribution as described above, can another consumer following a different rule purchase a ticket with higher quality? Furthermore, it would be of interest to explore the ticket purchasing environment from the point of view of the websites that are marketing and selling the tickets. For a given rule that consumers are following, what is the best way for a website to distribute its tickets to outperform a competitor? A good measure of the website s performance could be revenue. For example, if consumers are following a pure strategry in which they only consider the quality of the best ticket in making a website 18

27 decision, it would be advantageous to a company to offer a few tickets at rock-bottom price in order to entice consumers without cutting into their revenues. However, if consumers are more interested in the probability of getting a top ticket, it may be better for the company to offer a large number of first class tickets, even if those tickets aren t significantly better than the other tickets the company has to offer. Of course, consumers following a mixed strategy would necessitate a more complicated strategy from the website. Once both the consumer and website strategies have been explored, one could employ concepts from game theory to answer questions such as, is there a Nash equilibrium between the consumer population and the website? Finally, one could explore the analogy between the consumer-airline system and an animal behavior system. For example, the system explored in this paper may be analogous to a bee chosing a flower from among different flowering plants. The flowers offer limited nectar, they differ in quality, and there is a population of bees all seeking a flower. If data could be collected from such a system showing how bees distribute themselves among the different flowers and plants, it could be determined whether our model is accurate in such a situtation. 19

28 BIBLIOGRAPHY [1] Drabas, T., Wu, C. Modeling Passener s Airline Ticket Choice Using Segment Specific Cross Nested Logit Model with Brand Loyalty (2012) [2] Fretwell, S. D. Lucas, H. L., Jr. On territorial behavior and other factors influencing habitat distribution in birds. I. Theoretical Development., Acta Biotheoretica 19 (1970), [3] Hamilton, I. M. Habitat Selection, Encyclopedia of Animal Behavior, 2 (2010), [4] Maynard Smith, J. Price, G.R. The logic of animal conflict, Nature 246 (5427) (1973),

29 CHAPTER 5 APPENDIX A: MATLAB CODE % x r e p r e s e n t s the number o f t i c k e t s in each patch % p r e p r e s e n t s the number o f people in each patch % a r e p r e s e n t s input r a t e s o f t i c k e t s i n t o each c e l l c l e a r a l l c l c p r i c e 1 = [ 2 0 0, 600, 550, ] ; stop1 = [ 0, 1, 2, 3 ] ; p r i c e 2 = [ 5 5 0, 500, 450, ] ; stop2 = [ 0, 1, 2, 3 ] ; u t i l i t y 1 = exp ( p r i c e stop1 ) ; u t i l i t y 2 = exp ( p r i c e stop2 ) ; u t i l i t y 1 s o r t e d = s o r t ( u t i l i t y 1, descend ) u t i l i t y 2 s o r t e d = s o r t ( u t i l i t y 2, descend ) 21

30 % website 1 x1 = [ 5, 1000, 3000, ] ; lx1 = length ( x1 ) ; x1matrix = [ x1 ] ; p1 = [ , 1000, 1000, ] ; lp1 = length ( p1 ) ; p1matrix = [ p1 ] ; n1 = sum( p1 ) ; n1matrix = [ n1 ] ; a1 = [ 5, 1000, 3000, ] ; %website 2 x2 = [ , 2000, 2000, ] ; lx2 = length ( x2 ) ; x2matrix = [ x2 ] ; p2 = [ , 1000, 1000, 1000 ] ; lp2 = length ( p2 ) ; p2matrix = [ p2 ] ; n2 = sum( p2 ) ; n2matrix = [ n2 ] ; a2 =[1000, 2000, 2000, ] ; n = 8000; %t o t a l consumers 22

31 %number o f time s t e p s T = 5000; tspan = [ 0 : 1 : T ] ; number leaving 1 matrix = [ 0 ] ; number leaving 2 matrix = [ 0 ] ; f o r i = 1 : T f o r j = 1 : lp1 %update people in website 1 / t r i c k l e down i f x1 ( j ) <= n1 sum( p1 ( 1 : j 1) ) ; p1 ( j ) = x1 ( j ) ; e l s e p1 ( j ) = n1 sum( p1 ( 1 : j 1) ) ; end end f o r j = 1 : lp2 %update people in website 2 / t r i c k l e down i f x2 ( j ) <= n2 sum( p2 ( 1 : j 1) ) ; p2 ( j ) = x2 ( j ) ; e l s e p2 ( j ) = n2 sum( p2 ( 1 : j 1) ) ; end end 23

32 p1matrix = [ p1matrix ; p1 ] ; %update p matrices p2matrix = [ p2matrix ; p2 ] ; f o r i = 1 : lx1 x1 ( i ) = x1 ( i ) p1 ( i ) + a1 ( i ) ; %update t i c k e t s a v a i l a b l e in website 1 i f x1 ( i ) <= 5000 x1 ( i ) = x1 ( i ) ; e l s e i f x1 ( i ) > 5000 x1 ( i ) = 5000; end end f o r i = 1 : lx2 x2 ( i ) = x2 ( i ) p2 ( i ) + a2 ( i ) ; %update t i c k e t s a v a i l a b l e in website 2 i f x2 ( i ) <= 5000 x2 ( i ) = x2 ( i ) ; e l s e i f x2 ( i ) > 5000 x2 ( i ) = 5000; end end x1matrix = [ x1matrix ; x1 ] ; %update x matrices x2matrix = [ x2matrix ; x2 ] ; 24

33 %a propotion o f people move in each time step, where that proportion i s %the d i f f e r e n c e in the s c o r e f o r website 1 and the s c o r e f o r website 2, %as d e f i n e d below e x p u t i l i t y 1 = sum( u t i l i t y 1 s o r t e d. x1 ) / sum( x1 ) ; % d e f i n e expected u t i l i t y f o r each website e x p u t i l i t y 2 = sum( u t i l i t y 2 s o r t e d. x2 ) / sum( x2 ) ; prob1 = x1 ( 1 ) / sum( p1 ) ; %p r o b a b i l i t y o f g e t t i n g a c l a s s 1 t i c k e t in a website prob2 = x2 ( 1 ) / sum( p2 ) ; y = 0. 2 ; %weights f o r d i f f e r e n t r u l e s. z = 0. 6 ; s c o r e 1 = y u t i l i t y 1 s o r t e d ( 1 ) + z e x p u t i l i t y 1 + (1 y z ) prob1 ; %weighted average o f expected u t i l i t y and prob. o f c l a s s 1 t i c k e t s c o r e 2 = y u t i l i t y 2 s o r t e d ( 1 ) + z e x p u t i l i t y 2 + (1 y z ) prob2 ; 25

34 prop = abs ( s c o r e 1 s c o r e 2 ) ; i f score1 >s c o r e 2 number leaving 1 = 0 ; number leaving 2 = ( prop sum( p2 ( 1 : 4 ) ) ) ; e l s e i f score2 >s c o r e 1 number leaving 1 = ( prop sum( p1 ( 1 : 4 ) ) ) ; %who l e a v e s and who s t a y s f o r each website number leaving 2 = 0 ; e l s e number leaving 1 = 0 ; number leaving 2 = 0 ; end i f number leaving 1 > sum( x2 ) n2 number leaving 1 = sum( x2 ) n2 ; end i f number leaving 2 > sum( x1 ) n1 number leaving 2 = sum( x1 ) n1 ; end n1 = n1 number leaving 1 + number leaving 2 ; n2 = n2 number leaving 2 + number leaving 1 ; 26

35 n1matrix = [ n1matrix, n1 ] ; n2matrix = [ n2matrix, n2 ] ; number leaving 1 matrix = [ number leaving 1 matrix, number leaving 1 ] ; number leaving 2 matrix = [ number leaving 2 matrix, number leaving 2 ] ; end p1matrix (T, : ) %n1matrix %number leaving 1 matrix p2matrix (T, : ) %x1matrix (T, : ) %x2matrix (T, : ) %n2matrix %number leaving 2 matrix a v e r a g e u t i l i t y = (sum( u t i l i t y 1 s o r t e d. p1 ) + sum( u t i l i t y 2 s o r t e d. p2 ) ) / 8000; a v e r a g e u t i l i t y 27