Saturday, June 28, 2008

Choosing Your Seat on a Plane

Sorry to go with back-to-back plane-related posts, but I figured I would get all of them out of the way today. In my previous post, I spoke about the optimal way to fill an aircraft, a dilemma for the airline industry. Now, I'd like to present a problem more relevant to the individual traveler. Consider this:

- You are traveling alone (let's say on a business trip) in coach, or any other flight class that deals in rows of three seats per aisle side.

- When buying your ticket, you are presented with a choice of any seat you want. We will assume that you are the first person to get to select your seat - that is, you can have your choice of *any* seat in coach.

- You want to maximize the probability of ending up with an empty seat next to you. We'll assume the flight is near, but not at, capacity.

Which seat do you choose?

I won't present a conclusion here (because I don't have a great one), but will go over some factors that I use in choosing a seat.

Let's say N = the average number of travelers in a party. I will assume that, during times with the greatest number of business travelers, N = 1 -- people traveling alone. I would guess that on weekends and trips to more recreational destinations, N > 1. Just for simplicity's sake, let's assume N = 1 or 3\N (pronounced "3 divides N"; it basically means you can divide N by 3 and not have a remainder - 3, 6, 9, etc.). N > 0, of course.

In the cases where 3/N, the row cancels out, so we can throw out those cases, leaving us with N=1.

OK, I think it's safe to say that the middle seat in any given row is a losing proposition. You have twice the chance of ending up with a person sitting next to you, on either or both sides. Of course, I guess you could have the greatest payout, as well - having the entire row to yourself - but that seems unlikely. So, we'll consider the only logical choices to be window or aisle. Now, which one, and which row?

I think that the front rows have the highest probability of filling up. People like being the first to exit the plane. That is nice and all, but not the object of our game. So, since people have an incentive to choose a seat toward the front of the plane, we will go with the counter-option, the back of the plane.

I'm not sure about this, but I think that some people may choose the seats in the very back of the plan in order to be close to the bathroom, especially for long flights. So, I don't think we should choose the very last rows. So, if R = the number of rows on the plane and r = our row choice, I think we should go with something like:

(R - 3) <= r <= (R-6)

Finally, as for which seat to choose, aisle or window, I don't think it matters. If you choose, the window, someone will probably choose the corresponding aisle seat for your row (and vice versa), making the middle seat a very unappealing option for another player to select.

So, I guess this assumes that all players of the game will avoid choosing a seat adjacent to another player, if possible. We will assume, no, hope, that no one chooses an adjacent seat on purpose with the motive of being the incessant talker!

I used this algorithm myself recently and it worked. On a plane that was ~95% full, I ended up with an empty seat between me and the guy sitting in the window seat (Sweet!). I had seat 29C out of, I think, 35 rows.

So, to summarize, I would guess the answer to this problem is selecting a window or aisle seat where (R - 3) <= r <= (R-6).

Of course, there are other "quality-of-flight" factors that mess up a perfectly good plan, like this:

- the crying baby factor
- the little kid kicking the back of your seat factor
- the aforementioned incessant talker factor

We've made a lot of assumptions here, so if you have a more abstracted theorem, I would certainly be interested in hearing it!


P.S. Yes, I am a major nerd.


BenVitale said...


It would be pretty straightforward to calculate if one knew what 'next to' means. For example, are two aisle seats next to each other?

Chris Mustazza said...


Good point. No, I would not consider two aisle seats next to each other for the purposes of this discussion. The object of the game is basically not to end up shoulder-to-shoulder with anyone. :)

Thanks for the clarification.


BenVitale said...

Hi Chris,

Do we have any relevant data on the subject? any statistical data to actual base any probabilistic judgment on?

BenVitale said...

Hi Chris,

Could you explain (R - 3) <= r <= (R-6)

BenVitale said...

Hi Chris,

Oh! One more thing:
How can we assume N = 1 or 3\N ?

Chris Mustazza said...

Hi Ben,

Unfortunately, I don't have any statistical data on the subject. All of my axioms and assertions are based purely on conjecture, rather than empirical experiments. :)

So, the (R - 3) <= r <= (R-6) is basically an unnecessarily complex way of saying that I think the best choice of row is anywhere between 3 and 6 rows from the back of the plane, not inclusive of the last three rows. So, if there are 30 rows of seats on the plane (R = 30), I think your best bet is to choose between rows 27 (R-3) and 24 (R-6).

Again, this isn't based on anything but my personal experiences - an admittedly small sample size. But, I would venture to guess that range of preferable rows increases with the size of the aircraft - e.g. if R = 50, maybe the range could be like 5 rows. Just a guess.

Well, I am analyzing the case where N = 1 because I think it is the most frequent case for mid-week, business-time flights. I threw in the 3\N case because it is easy. :)

The problem gets harder when you consider the 2\N case.

So, this is far from a fully abstracted mathematical theorem, but just a fun problem that I spent some time considering.

Thanks for your interest in the post! Hope you'll continue to follow the blog.


Brady said...

The problem gets trickier if you aren't first and if you actually have a preference for aisle versus window. Say you prefer the aisle, then you need to know whether an aisle seat next to a fat dude is better than a window next to a vacant seat. Algorithm that! :-)