Wednesday, July 23, 2014

The Cost Benefit Analysis of Fighting a Ticket

If you ever visit NYC, there is a little hidden surprise when you take the six train.

A passenger can remain on the train as it turns around in order to view a stop that has not been in operation since 1954. The old ‘City Hall’ stop is a huge hit amongst tourists, but here is the problem: the turnaround is debatably legal. Despite many signs, blog posts, news reports and people telling me that it is legal, a friend and I recently received tickets for making this journey. The tickets were $50 each, and I was really upset because it is in fact not illegal to take this journey. I fought the case in the Transit Adjudication Bureau (TAB) court and you guessed it, I lost! This post isn’t about whether riding the six train around the turnaround is legal or not (since we know it is legal), but whether the cost of fighting a ticket is worth your time and money.

Time Is Money

Let’s assume you, like me, think you are going to win this ticket. You are so sure, that the probability of winning the ticket is I P(Winning) = 1. Winning in this case means you do not have to pay the ticket, so your total fine cost = 0. However, you do have to sit in the misery (and boy do I mean misery), of the TAB offices for three hours while you dispute your case.

So EVEN if you know you are going to win, it is still going to cost you:

     Loss = 0 + 3*[Hourly Rate].

If you wanted your loss to be less than the cost of your ticket, you could solve the equation and your hourly rate would need to be less that $16 just to fight the ticket knowing you are going to win.

But what if you don’t, then how much would need to make?

How to read this graph: it is only beneficial if your salary is below that black line at that probability. So if you are 75% sure you are going to win the case, your hourly rate would need to be less than $12.

If you are say 50% sure and make minimum wage, it is not worth your time. Of course, I’m assuming you would be working from 9-3 (the only hours of operation for TAB), and that you are using your Paid Time Off as a factor. 

In a strictly cost-benefit analysis, the cost of fighting a ticket is rarely worth it. So if you are a rational human being, just pay the ticket and move on. But if you want to prove a point to the system (like I did), then go ahead and fight it. Right?

Freedom and Democracy Will Prevail

When I fought the ticket I knew I was going to ‘lose’ money (as in time), but what is the cost of Justice and knowing that the court system got it right? I mean sure, the colonists threw a lot of tea in the ocean that day, but that tea represented the price of freedom and democracy, and that really does not have a price.

The sad part is, the TAB court system was not accepting of these ‘point-makers’ or freedom fighters. The room holds about 250 people, and it was packed. It gave me the sense that the judges do not really care about the cases: they are simply doing their jobs. They are only safety nets so that you don’t sue the TAB for being unconstitutional. They have this system in place because they have to, not because they are trying to change the world. Sadly, when you fight a ticket, you are just one person in a sea of humans. The time within the office is very formal. Though you are attempting to fight your case as an injustice, you are treated as if you are begging them to not fine you.

Sometimes We Do Win

Occasionally people decide to hop over subway turnstiles in order to avoid paying a fare. In NYC you do not need a card to exit the system, only to enter it. Some people know that it is actually smarter to risk a ticket than to pay the fare. The maximum ticket a TAB officer can give is $100 - normally the penalty for jumping the turnstile. The cost of a ride is $2.50, so again the simple math is

                                                     Loss = cost*risk
                         100 = 2.5(#times jumping)

As long as you only get caught once every 40 times, it is actually smarter for you to jump the gate. That means if you get caught about once a month, it may be smarter to risk it and jump the turnstile and not pay the MTA machine. (Can you tell I’m bitter yet?)

Why The MTA SHOULD Ticket

The MTA almost surely has a cost benefit to this: what is the amount of tickets they can issue before the government gets involved? If the MTA keeps their ticket processing capacity the same, but issues more tickets, it means less people would be willing to wait longer and longer hours at TAB just to fight a $50-$100 ticket.

                             Hours to wait = #tickets*(Processing time per judge)

The processing time per judge would likely remain the same, but if the number of tickets issued increases, so do the hours of wait time. The graph above is based off my single sample of 3 hours, but what if that time went up to 4 hours? That would mean even if I was 100% sure I would win, it would only be worth my time if I made less than $12.50 per hour. So the MTA should increase the number of tickets they issue, thus increasing the hours to wait at TAB, and then decreasing the probability someone will fight a ticket.

This is a fine line though, because if the MTA sets the ticket-issuing number too high, the government (hopefully) would come in and regulate the system by enforcing a swifter trial or by making the ticketing rules stricter and thus decreasing the number of tickets issued.


If you ever get a ticket, just pay it. Or at the very least, fight the ticket and blog about it. Maybe I actually got something out the ticket (which I’m finally going to pay). I have a story to tell at bars, coffee shops, and most importantly, on my blog. This post is pretty cynical and critical of the MTA, which it naturally should be. I actually don’t mind the subway too much. Sure it smells like pee 90% of the time. It also manages to remain 110 degrees under ground despite it only being 60 degrees outside. Sure there are something like 113 track fires per year, but running the MTA is not easy, so I’ll be kinder next time I take that six train around.

Wednesday, February 19, 2014

T is for time

Three hours have passed. You sit nervously by your phone, waiting for a reply from your crush. You message your best friend: "Hey, when do you think s(he) will get back to me – it’s been three hours. Three hours, [insert best friends name] !!!!!". You start to wonder why your crush hasn’t responded. Is it because s(he) lacks service? Then you remember it is the middle of the day on Tuesday, and that s(he) might be at work. Wait, who has three hours of meetings? Then, like an angel, the phone buzzes (because who uses ringtones anymore), and the name pops up on the screen with the response: "Not much, what is up with you?" The tables have turned.

You could respond right away, but then think to yourself: Hold on, will I look desperate? I don't want him/her knowing that I was waiting for a reply. You start to wonder how long you should take to respond. Is it five minutes, is it 10? Maybe three hours - maybe that will be our interval between texts.

We’ve all done it -- worried about time -- either having too much, or not having enough. The funny thing about time is that we humans created it. Time is a measurement of the interval between two events, but there is no natural occurrence of it. We used it to measure the time between sunrises (also known as days). Time gives us some level of expectation; we have a sense of how long it takes an egg to boil, or how long our daily commutes are; how long a movie should be before people get up and walk out.

Time is a very important thing; that’s why we created it. Imagine life without time. When will George get here? Erm, when his horse gets him here. Time allows us to build the foundations of expectations and creates structure in our society. We need to be to work "on time,",we make dinner plans with a specific time in mind (Meet you there at 7:30). Time is most certainly not a bad thing, but sometimes our interpretation of it is quite off. We value time in different categories: the immediate, the short run, and the long run. How we handle time depends on context.

Nerd Alert : The science of expected time

The Poisson distribution is how us fancy suspender-wearin’ folk deal with time. The concept, at its most simple level is: If we are given the last occurrence of an event, and how often that event occurs, we can calculate the nth time you will see that event. Boring formula below: where lambda is the rate and k being some occurrence of it, say the 5th.

\!f(k; \lambda)= \Pr(X=k)= \frac{\lambda^k e^{-\lambda}}{k!},

So how does this tie back in? Well, if we know the rate that our crush texts at, and we know the last time s(he) sent us a message, we could calculate the next time they will text us. However, we don't know our crush’s text rate, and to be honest, I don't even know my text rate. So we do what every scientist does: We guess at what his/her rate is. Sometimes we apply the “attacked by lion factor'” which states: " If said person does not respond within t, I will assume they were attacked by a lion and thus could not respond."  To many this seems silly, but the next time someone hasn’t responded to a text, write down the reason why you think they didn’t respond. Sometimes, they are funny months later.

Time, in the immediate

These are short time events, like how long should a pot boil? Their effect can be massive, but more often than not, hey are actually not a big deal. Short term time is where we test our impatience. It’s the time we think we will never get back, and it’s also the one we have the most expectations about. I take the L train (what up, Brooklyn) and my favorite thing about the L train is that it tells you how long until the next train. They have these fancy signs at all the stops (P.S. not every subway line has these).

Now normally, these info boards say something like three, five, or seven minutes, or something more reasonable. However, we expect that the train will arrive in seven minutes, and when the train takes nine minutes, we are angry about it. The L train violated the contract we made with it; I pay you $2.50, and you show up at that time above. Being late falls into the immediate time category. I always push the line of "on-time", generally I am a few minutes late. It angers people, because like the L train, we agreed to show up at that time. The immediate time interval angers us so much because since the interval is so short, we expect few things to get in the way of our planned time. The chaos level is much smaller if the interval is smaller. Our confidence in our expectations is much more solid than say, our short run time interval.

Time, in the short run:
It’s Monday. You have five days to figure out what you are doing Saturday night. This medium planning is where we really learn how to plan. We also do not put a lot pressure on ourselves about short run issues. Your friend's flight gets moved, but that’s ok because you have four days to deal with it before it approaches. Short run planning also has lower stakes. If plans fall through, you are just sitting at home watching some reruns on Netflix. The expected rewards are generally less than those of the long run, and generally there is enough time to change course, unlike the immediate time category.

Time, in the long run
So if you read my last post , you will notice my emphasis that in the long run, we need to focus on what is best for us. However, that long run does end. As Rolen pointed out, we are all dead in the long run. Fight Club said it best, "On a long enough timeline, the survival rate of everyone is zero". It’s kind of depressing, but it’s true. In the back our minds, we know our shot clock is ticking.

We have natural shot clocks. For women, it is having children, for men, it is having a full head of hair. We get old and our time is limited. Being single at 18 is vastly different then being single at 35, but it’s because time is no longer on our side. Time is more telling than just the number - time has an end, an end we often fear. Without an endpoint, time does not exist. Time is why YOLO exists, we don't want to waste these precious minutes, so we must carpe diem (apparently saying it in Latin was too hard, so we switched it to YOLO).

The scary part about time in the long run is our expectations and the rewards from those expectations. The small actions we do everyday affect us in the long run, and we are trying to optimize things that are years away from today. Choosing a college is a great example of this; juniors and seniors have to choose what college they want to attend that will change their life forever.  The thought that we are trying to plan for something 10 years down the road makes us crazy. 

People often overthink the time and the long run. Some people think (seriously, they do) like the following:

Min(marriage) = 2 years, Min(per child)*n(desired children) = (2)*(3)=6. Marriage + Children =  8 years. My age + 8 years = X (age that i will have three children).

Then they often wonder if that is too old.  This concept is known as search theory. According to some economist, we worry too much about being lonely today, and not enough emphasis on the long run. We are pressured (for various reasons) to get married within some time frame, but no one worries about the years after that time frame. Take the person that says: “I want to get married before I'm 35." Well, sure that is fair, but if you rush it, do remember that you might spend the next 35+ years with that person, so make sure that person is good.

About Time:

Time dictates our life, and we created it. We get angry at something that we control! We create these expectations, so we can allow more time for things, and the stress will go away. Our interactions with this data type time is like few others. It can be quite unpredictable but makes us dependable on it. 

Sunday, January 19, 2014

In the long run

It’s midnight, you’re at your favorite bar and the girl or guy of your dreams walks through the door. You know you need to talk to them, but immediately wonder how.  You can use that pick up line you planned a million times, you can fumble over some new line you just learned, or you throw a hail Mary and just say “Hi, I’m [insert your name nervously], and I think we should grab drinks.”

To all you young statisticians out there: don’t try any of these. The problem with statistics is, we always think about the long run; never the short run.

In statistics, we use this term “the long run,” stemming from a really nerdy concept called ”The Central Limit Theorem.”  I’ll leave the work of proving the theorem for our readers; suffice to say it basically states that if we do something enough, we will eventually get the actual permanent outcome.  

To some extent, our lives follow the central limit theorem in that our daily routines tell us what we actually do.  Think about your daily wake-up time: you set your alarm for 6:30am, but rarely actually wake up at exactly that time; some mornings you wake up at 6:35am and other mornings 6:25am.  If we took all those various wake-up times and graphed them, though, they would wind up normally distributed with the mean at 6:30am.  I’m stretching the realities of statistics slightly, but this is the basic concept.

The point here is this: our expectations are build around what is going to happen in the long run. But as the nervous person at the bar, there’s no such thing as a long run; because of time constraints this evening, a really short run is all we have.  

Some life events and decisions – like careers, friendships, and pick of our favorite foods – give us plenty of time to play out.  The sample size of these events is large, so we know if our expected value is going to return positive or negative results.  Other things, however, don’t have a long run.  It’s the championship game and your team needs 3 yards for a first down; if not, you lose the game (and your bet!).  In the long run, the average NFL team goes 2 ½ yards on that play (believe me; I’m a statistician).  But you need 3 yards, not 2 ½.  There is no long run; the whole season comes down to this short run.

I was recently at a casino, but needed to get back home for a friend’s birthday party.  Time was limited, so I couldn’t play the long run.  I took my $20, marched up to the roulette table, put it on my birthday (12 and 6), and hoped the odds would be in my favor.  If I had time to sit down and gamble, I may have hit those numbers in the long run – but I needed to leave.  Lucky for me, I wound up winning on the one hand I played!

When we have time to think about things and make informed decision, we should do what the long run dictates; when we don’t have time, just go with it.  When statistical probabilities exist, use them, but don’t fear when they’re absent.  When thinking about where you’re going to live for the next year, think through the probabilities.  If you’re picking where to go for dinner next week, think through the probabilities.  But if you’re at a bar, and the person of your dreams walks in – forget the probabilities.  In the short run, they probably don’t matter anyway.