Formal definition of the interactive data and procedure of the

3.3 Design of the Quasi-Experiment

3.3.2 Formal definition of the interactive data and procedure of the

In order to formally define the interactive nature of the experiment, we need to define both the data presented to the respondents and the computations we used to analyze the experiment. We begin by defining a vector of the arrival times based on Table 3.1 in the following way:

arr := [7:57, 7:53, 8:07, 8:27, 8:23, 8:37, 8:25]

During the experiment, the respondent gives feedback during twenty rounds. The feedback given to the respondent in each of these rounds was generated using a number of matrices. These matrices are drawn a single time prior to the quasi- experiment. As a result, the feedback and crowding indicators observed by the respondents are based on the same data matrices. These data matrices are defined as follows:

The crowding matrix U ∈ {0, 1, 2}6×20_{for train options. An entry U}_c,r_{is the crowding}

during round r for choice c and either 0 (low crowding), 1 (moderate crowding) or 2 (high crowding) where 0 is low crowding and 2 is high crowding. The crowding was drawn uniformly and independently at random. Note that the crowding matrix contains no crowding for the car choice, c = 7.

The delay matrix D ∈ Z7×20_{. An entry of the delay matrix D}_c,r_{contains the delay}

of choice c during round r, which comes on top of the travel time communicated to the respondents. Each entry was computed by taking the absolute value of an independent draw from the normal distribution |N(0, 3)| for the train choices. In the Dutch network, more than 90% of the trains arrive “on time”, which means they are delayed by at most five minutes. For the car choices the delay matrix is used to model stochasticity due to traffic lights and minor traffic. The distribution |N(0, 3)| was used in case the disruption matrix DI (which we define next) contains a 1 for the car option and |N(0, 6)| was used otherwise. We refer to the delay matrix that used |N(0, 3)| by D

for the car choices, while we refer to the |N(0, 6)| matrix by D_{. Thus, we assume that}

the standard deviation of delays of the train is three minutes and of the car is either three or six minutes.

3.3 Design of the Quasi-Experiment 51

The disruption matrix DI ∈ {0, 1}7×20_{. An entry of the disruption matrix DI}_c,r_con-

tains 1 if a disruption occurred for choice c during round r. If a respondent faces a disruption this implies an increase of 30 minutes in the travel time. The entries were drawn independently from the Bernoulli distribution B(0.1) for all the train choices and from B(0.3) in case of a car choice, were a disruption represents an accident or a severe traffic jam. The probability was chosen in such a way that in expectation a respondent runs into a disruption twice when travelling by public transport. The probability for disruptions in car based transport are higher as we model route with very high demands which often result in traffic jams, while rail punctuality is typically quite high in the Netherlands.

The random information matrix I, which is only relevant during the second phase. This matrix is generated in the same way as the matrix U. This way, the crowding indicator levels are generated according to the same distribution as the real crowding. In case we want to provide accurate information to a respondent, the crowding matrix

Uis used and in the base case there is a1₃ probability that an entry in U is accurate

and equal to I and a 2

3 probability that it is not.

During the second phase of twenty rounds, respondents were presented with crowding indicators before making a choice, representing the predicted crowding of each train choice. The crowding indicator indicated the expected crowding level consisting of either one, two or three symbols representing a person.

Our three experimental manipulations yield 23₌_{8 respondent groups, denoted}

by a triplet (δj, qj, ρj). The three manipulations are referred to as occurrence of

disruptions, quality of information and reactive crowding.

The first manipulation, occurrence of disruptions, consists of the occurrence of

large disruptions, indicated by δj∈ {0, 1} for respondent j. One group of respondents,

δj=0, were incidentally confronted with a large disruption and the other group only

faced small delays.

The second manipulation, quality of information, consists of the quality of in-

formation during the second phase of the experiment, indicated by qj ∈ {0, 1} for

respondent j. One group, qj=0, received accurate crowding level information and

the other group received random information.

The third manipulation, reactive crowding, consists of the relation between the

crowding level and the prior choice, indicated by ρj ∈ {0, 1}. One group, ρj = 0,

experienced a purely random crowding level and for the other group the crowding level was partly dependent on the previous choice.

Reactive crowding was implemented as follows. If a respondent picked a certain train journey the previous round, the “crowd” will be attracted to this option, making that option more crowded during the current round. This increase in crowding is modeled to be due to a shift of the crowd from travel options close in time. As such, the crowding of the adjacent (with respect to the choice number) travel options is decreased by one. As a result, a respondent who only cares about avoiding the crowding and who is in the ρ = 1 group has an incentive to keep switching from its

50 Time Choice Data for Public Transport Optimization

that disruptions often cause very crowded situations in public transport, so this is important to consider when studying crowding behavior. The quality of information is something which operators have to predict, so in the real world there is always an error margin in these predictions. Including it as a manipulation allows us to compare the effect of errors in the crowding indicators on choice behavior against a best case scenario. The motivation to include reactive crowding as a manipulation is that it allows us to analyze whether increased crowding levels are give a greater incentive to change travel times than purely random crowding levels.

3.3.2 Formal definition of the interactive data and procedure of the

quasi-experiment

arr := [7:57, 7:53, 8:07, 8:27, 8:23, 8:37, 8:25]

The crowding matrix U ∈ {0, 1, 2}6×20_{for train options. An entry U}_c,r_{is the crowding}

The delay matrix D ∈ Z7×20_{. An entry of the delay matrix D}_c,r_{contains the delay}

for the car choices, while we refer to the |N(0, 6)| matrix by D_{. Thus, we assume that}

the standard deviation of delays of the train is three minutes and of the car is either three or six minutes.

3.3 Design of the Quasi-Experiment 51

The disruption matrix DI ∈ {0, 1}7×20_{. An entry of the disruption matrix DI}_c,r_con-

Uis used and in the base case there is a 1₃probability that an entry in U is accurate

and equal to I and a2

3 probability that it is not.

Our three experimental manipulations yield 23₌_{8 respondent groups, denoted}

by a triplet (δj, qj, ρj). The three manipulations are referred to as occurrence of

disruptions, quality of information and reactive crowding.

The first manipulation, occurrence of disruptions, consists of the occurrence of

large disruptions, indicated by δj∈ {0, 1} for respondent j. One group of respondents,

δj=0, were incidentally confronted with a large disruption and the other group only

faced small delays.

The second manipulation, quality of information, consists of the quality of in-

formation during the second phase of the experiment, indicated by qj ∈ {0, 1} for

respondent j. One group, qj=0, received accurate crowding level information and

the other group received random information.

The third manipulation, reactive crowding, consists of the relation between the

crowding level and the prior choice, indicated by ρj ∈ {0, 1}. One group, ρj = 0,

experienced a purely random crowding level and for the other group the crowding level was partly dependent on the previous choice.

52 Time Choice Data for Public Transport Optimization

current travel option. In case the choice during the previous day was either the car or does not exist, the “regular” crowding level, as defined in the matrix U, is used. Given

a base crowding level c described in U, the function tjtransforms this crowding level

cto the manipulated crowding level based on the current chosen travel option o

and the previously chosen option o_{when ρ}

j=1 and is the identity function in case

ρj=0. In case the crowding level is adjusted, we make sure it will be one of the three

crowding levels by means of the min and max functions. It is formally defined as follows: tj(c, o, o) :=        c if o =7 min{2, c + 1} if ρj=1 ∧ o = o max{0, c − 1} if ρj=1 ∧ |o − o|=1 c otherwise (3.1) This function reflects the idea that the crowding level is just a random draw in case

the respondent is in a ρj=0 group, or has selected the car during the previous round.

If the respondent is in the ρj=1 group, the crowding level is modified depending on

the choice of the previous round.

Example 3.1. Suppose that during round 5 the respondent j picked choice 4. During round 6,

by chance the crowding of travel options 3, 4 and 5 is defined to be 1. If ρj=0, i.e. respondent

jhas random crowding, the reported crowding will be 1 for all these choices. However, if ρj=1, i.e. respondent j has reactive crowding, the reported crowding for options 3 and 5 will

be 0, while it will be 2 for travel option 4.

While one can argue that this crowding behavior may not be realistic in actual transport systems, this rule is designed to have two properties which we believe to be important for the experiment: 1) the rule should be deterministic in a sense that two respondents who make the same choices observe the same crowding levels, as this is important for a fair comparison between correspondents; and 2) the crowding should be noticeable by the respondents, as one of the important questions is whether crowding interactions affect the choice behavior of the respondent at all. Even if our crowding behavior would be exaggerated, it is still helpful to determine whether or not there is an effect.

When the quasi-experiment is currently in the second phase, the respondent is shown a crowding indicator before making a choice, as is visualized in Figure 3.2. Thus we need to compute the value of the crowding indicator for each train choice during every round r before presenting the possible travel choices to respondent j.

We compute the function indj(r, o, o)for each chosen travel option o based on the

travel option chosen by respondent j during the previous round, denoted by o_{. This}

computation depends on j’s quality of information qjand is defined as follows.

indj(r, o, o) =

tj(Ur,o, o, o) if qj=0

tj(Ir,o, o, o) otherwise (3.2)

In document Passengers, Crowding and Complexity : Models for passenger oriented public transport (Page 66-68)