Where are the Carriages?

<rant>

So it's around 9am, and you're waiting for a train. The trains in your area are a bit scummy, but whatever, you can't drive, and you've got to get to work/uni somehow. It's pretty busy, what with it being 9am, and when the train finally pulls up.. it's a single carriage.

Now a single car has around 50 seats, and this train is packed tight, with people standing in the aisles and the doorways, being forced to get intimate. And with everyone on board the conductor can barely get through the door. There are clearly enough people on this train to fill two cars, with people still having to stand.

So what gives? You're pretty pissed about having to stand for half an hour, and you've been inadvertently touched by strangers in ways you're not comfortable with. So you get your complaining hat on, and you turn to the internet to unleash the fury.

The train company's website directs you to their Twitter feed, where a poor public relations person is taking a barrage of vitriol from other angry commuters, while doing their best to remain polite and professional.

Several other people have already made your complaint, but all they're getting in return is "Apologies we try to avoid this where possible", or that the short trains were "due to operational reasons". And that's not really a satisfying response. You're not even sure it means anything.

</rant>


So Where Are The Carriages?

Disclaimer: I have no idea how the train company actually operates. I just like to write blogs about applied maths.

Here's the setup - you're in charge of logistics for the train company. You have a fixed number of trains/cars, and, for each day, a list of services and expected numbers of passengers.

How do you apportion the carriages between the services?

Obviously, every service needs at least one car, and services with more passengers need more cars.

So if you had, for example, three services with 50, 100, and 150 passengers respectively, and 6 cars, then it's easy to divvy them up - one for the first, two for the second, and three for the third.

But in general, it won't be possible to divide up the cars exactly like that; so what do you do about remainders?

This is actually similar to the problem of apportioning parliamentary seats between states in the US.

Currently there are 435 seats in the House of Representatives, which need to be shared out between the 50 states according to each state's population - the idea being that each seat should represent roughly the same number of voters.

There are various methods for doing this, but I'm only going to go over two of them.


The Hamilton method

Hamilton's Method is the easiest and most intuitive.

Say you've got only two services - service A has 85 passengers, service B has 115 passengers (200 passengers total). And you happen to have 4 carriages - a total 200 seats. So seats for everyone! Except service A needs 1.7 cars, and service B needs 2.3 cars. And you can't divide a car into two chunks.

But for starters, you can give one car to service A (50 seats), and two to service B (100 seats). So what about the fourth? Regardless of which service you give it to, some people are going to end up having to stand. So what you want to do is minimise that number.

For service A, 35 people need a seat. For service B, 15 people need a seat. So it makes sense to give the last car to service A. 15 people still have to stand, but - short of building more cars - there's really nothing you can do about that.

This is also known as the largest remainder method.

What if the passenger numbers were 75 and 125? Well in that case, I guess it would have to be a judgement call.

Anyway, that's the basic idea of the Hamilton method - divide up the cars as far as you can, then give the remaining car(s) to the service(s) which 'need them most'.

Things get trickier when you're dealing with larger numbers of passengers and cars, but ultimately it's pretty straight-forward.


The Huntington-Hill Method

Where trains differ from apportioning of parliamentary seats, is that a carriage has a fixed capacity, whereas the number of people a seat can represent is free to change.

And this means that you won't encounter some of the 'quirks' that arise from the Hamilton method - such as the Alabama Paradox (where increasing the total number of seats can mean a state losing seats), or the Population Paradox (where increasing a state's population can result in that state losing seats).

[You won't encounter them, but they're worth mentioning 'cause they're pretty cool.]

Huntington-Hill's Method - the one currently used by the House of Representatives - is, generally, a more powerful method than Hamilton's, and isn't susceptible to the various paradoxes. Another benefit is that it can be set up to guarantee that each state will get at least one seat.

It works by assigning seats, one by one, based on each state's 'priority quotient' - itself based on the state's population, and a 'modified divisor' based on the number of seats already allocated to that state.

But one advantage Hamilton's Method has over HHM, is that it will always give each service its ideal number of cars, either rounded up of down to the nearest whole number. So if a service's ideal number is 3.67, then Hamilton's method will assign either 3 or 4 cars to that service.

HHM, on the other hand, can 'violate quota' - that is, it can result in a given service being assigned more or fewer cars than it's ideal number (e.g. the 3.67 service might end up with only 2 cars). But this problem only occurs when the number of cars is fixed prior to apportioning. And sadly, it usually is.


As an aside, it turns out it's impossible to find a 'perfect' method of apportioning - one with neither paradoxes nor quota violating. Incidentally, the mathematical theory surrounding voting is quite fascinating. You know, if you're into that sort of thing.


Unfortunately..

Either method would work perfectly well. The problem is, while rounding up or down to the nearest car may seem trivial, the 50 seats that that one car represents can be a significant gain/loss for a service.

The lack of precise information can cause problems as well. You can only estimate how many passengers a given service will have in advance, and if you under-estimate, people are gonna be pretty pissed.

And on top of that, the numbers have to be recalculated from time to time as passenger numbers change..

Basically, getting the number of cars right can be tricky.


Of course, this all assumes that Northern Rail (or whatever company) has enough carriages to adequately satisfy its needs in the first place. Some how I doubt that's the case.

So ideally, NR needs to work out how many cars it actually needs, and build them. But I can't see that happening. And even if it did happen, it'd probably mean higher fares. And fares are pretty bad as they are.

Really, though, NR could do with a complete makeover just in general, given how shitty it is. Seriously, you should see the difference between them and the London Midland service. Ridiculous.

Incidentally, I also have strong feelings about the buses; don't even get me started. Trams are alright though - one every 10mins, seldom short of seats, sufficient leg-room.. bliss.


tl;dr - Where are the carriages, then? As it turns out, it's probably just that they don't have enough to go around.

This is what we get for privatising the trains...


Oatzy.


[My complaining hat is a trilby. I like to look bad-ass while I'm complaining.]

Tumbling, Part Two: Over the Threshold

I Didn't Vote For This

A short while back, while people still cared about AV versus FPTP voting, I wanted to come up with a voter simulation, so as to get data to compare the merits of the two (and a few other) systems, since real-world data is limited. Sadly, it came to no avail, since I had trouble coming up with a plausible way to simulate the behaviour of British voters.

The basis of how it worked was this - each voter has ten 'vote points'. These points are divided amongst the parties according to how much the voter likes each party.

So if a voter was a strong Conservative supporter, they might give all their points to Conservative. A voter who's more liberal might give 6 points to the Lib Dems and 4 to Labour. Or whatever.

In the case of the simulation, these points were going to be assigned randomly, but according to some Markov Chain type model - based on the typical voting habits of Brits. This, by the way, is where I was having trouble - for large numbers of voters, the result were always the same.

The other part of the model was to (randomly) assign to each voter an 'approval threshold' - that is, if a voter assigns to a party a number of points higher than their personal approval threshold, then they 'approve' of that party, and will place a vote accordingly.

So if, in the liberal example described above (voter B in the diagram), the voter has a threshold of 5, they would place a vote for only the Lib Dems; if, on the other hand, they had a threshold of 3, then (under an AV voting system) they would place a vote for both - order of preference: 1. Lib Dem, 2. Labour.

(In the diagram, all three 'voters' are given the same approval threshold for simplicity. The thresholds are by no means constant and universal.)


I Like It, But Not that Much

So what does this have to do with Tumblr? Well. On Tumblr, you can either ignore a post, like it, or reblog it.

Now, there are some people who will reblog almost anything that catches their fancy. Then there are people, like me, who are very picky and only reblog things they really like.

This is basically analogous to the approval threshold in the voting sim.

Obviously, the quality of a post is subjective (to some extent). But imagine first that for every post you see, you assign to it some rating from 0 to 10. The rating you assign to a given post will be (relatively) unique to you.

Here is how we get around the problem of subjectivity,

We assume that the ratings a large sample of people will assign to a given post will have some regular distribution - most likely a normal distribution - where the average and spread of ratings reflects (to some extent) the quality of that post. Or, at least according to that particular sample of people.

Each user then has two threshold values - a 'like threshold' and a 'reblog threshold'. And as before, if the rating a person assigns to a given post exceeds one of these thresholds, that person will either like or reblog the post (possibly both) accordingly.


Some Asides

It's worth noting that similar modelling can be applied to any piece of information that can be spread - whether it be on Twitter, by email, whatever - where you have some 'sharing threshold' by which you judge whether or not you think a thing is worth sharing.

This may vary wildly from situation to situation. But in most cases outside of Tumblr, you will tend to find that people typically have a higher share threshold - higher standards, are more pick, are more wary of what they share, etc.

Another thing to consider, with Twitter in particular, the originator of a post can have a massive effect on how much said post is retweeted.

For example, some people (fangirls) will retweet someone like Justin Bieber simply by virtue of the fact that it's Justin Bieber. Conversely, some people may not retweet someone like Piers Morgan because they don't like him - even if they would otherwise like something he posted enough to have retweeted it.

Finally, people don't always share things simply because they like them - they may want to make some comment on the post, they may want to mock the post, whatever.

But for simplicity, we're ignoring that all these eccentricities, on the assumption that including them would have no appreciable effect on the model.


Model Affinity

I wrote a quick model of this behaviour (Python, can be viewed/downloaded here).

For this model, the user's thresholds are generated uniformly - that is, all numbers in the ranges have the same probability of being chosen,

For the 'like threshold' you pick a random number between 1 and 11.
For the 'reblog threshold' you pick a random number between the 'like threshold' and 11.

The user's assessment of a post is randomly generated based on a normal distribution - in this case with mean=5, sd=3, constrained to the interval [0,10]. These values were chosen arbitrarily.

For a simulation of 100,000 users, this model returns approximately 25,575 likes and 14,487 reblogs. This gives a 'reblog to like' ratio of 0.56.

This is quite different from the 'R:L ratio' of the Dr Who image, discussed in the previous post - which turns out to be 1.42

But if we (effectively) increase the quality of the post - mean=10, sd=2, same interval - we now get approximately 32,393 likes and 41,872 reblogs, giving a R:L ratio of 1.29

Which reflects reality quite well - that is, the better a post is, the more reblogs it will get (and the higher its R:L ratio). You could also get a similar effect by lowering the approval thresholds of the users - effectively lowering the users' standards.

Note/ You could probably get a better replication of real world data by fine tuning the random variables. The ones used here were chosen fairly arbitrarily.


Personal or Probable

For large numbers of people, there are two ways you could work this into a model of post spreading:

1) On a per person basis

That is, for each user work out whether they will ignore, like, or reblog, as you go. This would lead to richer behaviour on a per-simulation basis, and is better if you want a more detailed, more 'precise' model.

2) Based on the averaged probability

That is, run a simulation for a large number of people (as above), work out the average number of likes/reblogs, and use this as the 'probability of liking/reblogging', constant for all users. So from the first simulation above, the probabilities would be:

p(reblog) ~ 0.14
p(like) ~ 0.26

This would make the behaviour more uniform across multiple simulations, but would be less computationally taxing.

It can be argued that, when you average over a large number of simulations, you will get (almost) identical results for both approaches. So it's ultimately a matter of preference.


In Part 3 of the series, I'll be looking at constructing a model of how posts spread through Tumblr, building on the model described here.


Oatzy.


[addendum] -  Due to a mild error, the section labeled 'Model Affinity' had to be rewritten. If you read this post before this addendum was added, it's probably worth re-reading that section.