Archive for the ‘Math’ Category

Investing in Happiness

Tuesday, October 6th, 2009

I think a lot about investing.  I learned in high school that investing while you’re young is one of the best moves you can make from a financial perspective.  The earlier you start investing, the more time your money has to grow.  For a while I thought it made sense to live as frugally as possible, and to save as much as possible, in order to maximize my financial payoff in life.   I had a realization, though, that changed this attitude.

Think about a really good memory. Maybe it’s a memory of your family gathered for thanksgiving dinner, or maybe its of a fun date you went on. Maybe it’s a memory of the time you found five dollars on the sidewalk. Who knows.  Doesn’t thinking about good memories make you feel good inside? I would say that in my experience, remembering a good time that I had is almost as enjoyable as having the good time itself.

This year, I threw a birthday party for myself, and invited a bunch of friends from out of town. I rented a house on a lake for a weekend, and even sprung for a Jet Ski for the day on saturday.  Five of my 8 siblings were able to make it out, as were a substantial number of my friends, from grad school, college, and high school.  I saw some people I literally thought I’d never see again, and we had a great time.  I will probably remember that weekend on the lake for the rest of my life.  When I think of it, I can’t help but smile and feel good inside. It’s not just a fleeting happiness, like the feeling you get from eating a good filet mignon, It’s a feeling of contentment and metaphysical satisfaction with life.

That weekend cost me money; around $1,000 after all was said and done.  A slightly younger me would have argued that spending $1,000 on a birthday party for yourself is a waste of money and irresponsible. An older (and I’d argue wiser) me would respond that the weekend was not just a way of having fun in the present, but an investment in the future because of the value of the memories it creates.

Let’s suppose that I remember my birthday party 4 times a year for the rest of my life.  They say you can’t put a price on memories, but they say a lot of things that aren’t true, so I’m going to put a price on the memory and say that the good feelings I get from remembering that weekend are comparable to the good feelings I’d get if I found $20 on the sidewalk. That means my $1000 investment pays me $80 a year for the rest of my life.  If I live another 50 years (which seems like a reasonable bet), the memory will pay me $4,000 over the course of my lifetime. Not bad. Is it possible to do better?  If I had put that $1,000 into the stock market, it would not be unreasonable to assume that my return would average close to 8% annually.  That works out to … $80 a year.  It’s true that if I invested that $80 back into the market, my annual return would increase beyond $80, but I’d be giving up the momentary satisfaction of having that $80 now.

I’m not arguing against savings – I think it’s important to prepare for large future expenses like retirement and your childrens’ education – but I think there are diminishing returns to having large amounts of money, and that spending even non-trivial amounts of money now in order to create great memories that will last your entire life is also a great idea.

A Mathematical Case For Optimism

Tuesday, September 22nd, 2009

Given the choice, does it make more sense to be  pessimistic or optimistic?  I was thinking about this one night, and, as usual, I decided to explore the question mathematically, using the tools of game theory. Credit goes to Megan for helping me figure this one out.  The basic problem I had was this: Everyone says you should be optimistic, but my objection to this claim has always been “what if the universe is really a terrible place where mostly bad things happen – why should you be optimistic then?” Megan insisted that it made sense to be optimistic, and I thought I could prove her wrong mathematically. I did some thinking, though, and concluded that it actually does make sense to be optimistic, regardless of the nature of the universe.

Let’s use a simple model of reality: events happen, and they’re either good or bad, to varying degrees.

In this model, one of three things is true:

  1. Reality is fundamentally a good place:  the sum of the good experiences is greater than the sum of the bad experiences
  2. Reality is fundamentally a neutral place: the sum of the bad experiences is roughly equal to the sum of the good experiences
  3. Reality is fundamentally a bad place:  the sum of the bad experiences is greater than the sum of the good experiences

    You might object to my model of reality and claim that it’s too simple to be useful.  Just for you, I’ll add another possibility:

  4. Reality is too complex to be characterized by simple good/ bad models.

As you experience life, you’ll try to build a model of the universe, and part of this model is how or bad the universe is.  Ideally your model is perfect, but invariably you’re going to get things wrong.  If you’re going to model how good or bad the universe is, is it better to err on the side of  good or bad? We’ll say that optimism means believing the universe is better than it really is, while pessimism means believing the universe is worse than it really is. I claim that, regardless of the true nature of the universe, an optimistic strategy is the best strategy to pursue.

To prove my claim, I have constructed a ‘truth table’ which describes the outcomes of optimistic and pessimistic strategies in the four cases listed above. In each case, the Optimistic strategy pays off more than the pessimistic strategy.

Nature of the Universe vs. Life Strategy
Nature of Universe Optimism Result
Pessimism Result
Winner
Good Life is good, and you enjoy it to its fullest. Life is good, but you don’t enjoy it to its fullest because you’re worrying about things that are unlikely to happen. Optimism.
Neutral Life is OK, but you put a positive spin on things and as a result enjoy them more. You spend less time worrying about bad things that might happen, and more time anticipating and enjoying the good things. When bad things happen, you get over them quicker because you’re convinced more good things are in store for you. Life is OK, but you put a negative spin on things, and as a result, you enjoy them less. You spend more time worrying about bad things that might happen,  and less time anticipating and enjoying the good things. When bad things happen, you take longer to get over them because you’re convinced they’re going to keep happening to you. Optimism.
Bad Life is tough, but you’re oblivious to the fact. Bad things happen often, but you don’t linger on them. On the rare occasion good things happen, you enjoy them to their fullest. Life is tough, but you think it’s worse than it really is. When bad things happen, you reflect upon how miserable life is. When good things happen, you tell yourself that they won’t last. Optimism.
Undefined Bad things and good things both happen. When the good things happen, you enjoy them. When the bad things happen, you don’t dwell on them because you know they’re temporary. When nothing good or bad is happening, you stay positive and think good things are ahead in your future. Bad things and good things both happen. When the good things happen, you enjoy them. When bad things happen, you dwell on them because you think you have more of the same to suffer through. When nothing good or bad is happening, you’re in a negative mood because you’re overestimating the probability of bad things happening in your future Optimism.

If you find yourself worrying about the future or fretting about the past, just remember – Optimism is always an intelligence choice that will make you happier in the long run.

A Random Fact About Yourself

Monday, January 26th, 2009

I was tagged in a note on facebook, titled ‘25 random facts about myself’. You’re supposed to write “25 random things, facts, habits, or goals about you”, and tag 25 people, as well as the author of the note.  What does it mean to write a random fact about yourself?

Let’s figure out how many facts I could choose from. To simplify things,  let’s suppose my life occurs in one-second ‘frames’, and that my life is constant for the duration of that frame. In other words, lets assume time is divided into fixed-length quanta. As of the time of this posting, I am a little over 23.5 years old. That’s approximately 700 million seconds.   If a ’story’ about myself is a sequence of quanta, there are 2700 = 5.3 x 10210 possible stories about myself I could tell. For comparison, there are about calculated to be 1080 atoms in the universe. Even if I increase the size of the quantum to an hour, and limit stories to be descriptions of no more than 20 quanta, we have 7x1078 possible stories. That’s still a lot of stories.

Interestingly enough, if I used python’s random number generator,  the Mersenne Twister algorithm, I could actually implement a program to randomly select one of those stories, because it has a period of  219937-1. I thought about generating a truly random story, using a sweet bit of python code, but it probably wouldn’t be too interesting. It’d be really difficult to write anything about what I was doing at 4:00 on March 3, 1994, other than ‘I was 8 years old.’