Buoy Don’t Float

So we were given this example problem in class that happens to be a popular TV show game. It is called the “Best Prize” problem, which if you aren’t familiar with it, it goes sort of like this: (mind you, this is off the top of my head)

You receive n random prizes in order. Prizes arrive one at a time. When a prize arrives, you have two options: ACCEPT the prize–and receive no further prizes–or REJECT the prize–and wait for the next prize without being able to return to any prior prize. Each prize, when it arrives, has a random value. And finally, but most importantly, you DO NOT KNOW what the value of the best prize is.

It’s a popular problem simply because people don’t know how to deal with it. Naturally, everyone wants the best prize. But when it comes down to it, most people are too risk averse to actually get the best prize, often settling for something lesser. The problem itself is interesting, but I think it is most interesting when it is applied to real life. But before I get into that, I’ll give you the secret to maximize your chances of winning the best prize.

Naturally you think your odds of winning the best prize is incredibly low when n is large. However, when n approaches infinity, your chances of winning the best prize is greater than 1/4. The real solution is complicated, but a lower bound is easily found. Because the only information you get from each prize is its relative rank compared to the prizes that came before it, you reject the first 1/2 of the prizes. Now you are left with the second half of prizes. The probability that the best prize is in the second half of prizes is 1/2. Now, when a prize is selected, it is better than n/2 prizes. (All the prizes you already skipped) As a result, it has probability of at least 1/2 of being the best prize. To put it all together and summarize, if you let the first half of the prizes go and accept the first prize greater than those prizes, than you will have a probability greater than 1/4 of obtaining the absolute best prize. Not bad, eh? (Keep in mind this is a lower bound. The real optimal solution is somewhere around 37%)

Ok, I’ll give you a second to digest all that.

Now, lets have some fun. Generally, I would say most people want the best prize. If you think about how many people have the mindset of going-for-it-100%-because-you-only-live-once, then my generalization is well-justified. With that said, settling for second best is stupid: we are all gamblers.

First, lets apply the Best Prize Problem to love–or relationships, if you’d prefer not to gamble your love. The biggest problem is that there is no set number of relationships (ie. no n prizes) that is known before hand. However, this is easily negated if you make some assumptions about how many relationships you are able to get. Sounds arrogant and lame, but deep down inside, I know you know how many relationships you can initiate based on your social skills. For a lower bound and the continuation of my argument, let us make the assumption that each person can have at least 4 relationships in their relationship career, thus n is greater than/equal to 4. Right away, it’s clear that you can’t marry your first–marry being the equivalent of accepting a prize. The more controversial thing is that you should end your first two (half) relationships, NO MATTER WHAT, if you want to maximize your chances of finding your true love. However, at the same time, you risk losing your true love as the price for information. But hey, as a go-for-it-all type person, this is a chance you must make, right?

Ok, I need to sleep, but you get it. More applications to come later. Feel free to argue with me.

P.S.
In the end I don’t really believe most of this stuff. But I think it is a good model and you can take away some things from it. I had an argument with Maria and I agree with her: the human factor that is not taken into account, leaving room for too much error. What do you guys think?