How to choose wisely

May 7, 2019

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So many choices, so little time. In today’s world, everyone struggles sometimes to make a choice quickly, in fear of not picking the best one. Often the amount of options is overwhelming and the paradox of choice strikes us multiple times a day. What if there was a method that helped you pick a movie to watch or a spot to park your car?
Luckily, such a method exists and it is easily explained by the following illustration:
In front of you are 100 envelopes, each containing an unknown amount of cash. You are allowed to open as many envelopes as you want and to keep the content of the last envelope you have opened. Obviously, you do not want to stop at an amount too small, but neither do you want to pass on a number larger than the remaining contents. How do you know when to stop?
It turns out that when solving this seemingly difficult problem, we bump into Euler’s number eOptimal stopping theory tells us that the best decision will be made by creating a sample first.
The sample size is equal to the total amount of options divided by e, rounded to the nearest integer. In our envelope game, this equals \lfloor \frac{100}{e} \rceil = 37.

This means we have to open 37 envelopes to create our sample. After that we simply stop at the next envelope containing any amount greater than every sample envelope. This way we maximize the expected value of our final envelope. Ofcourse, since we do not know what amounts the envelopes contain, we cannot for sure stop at the largest sum, but this method makes sure that our choice is best, given our options. It turns out that the probability of making the best choice is exactly equal to \frac{1}{e}.

Measuring preferences

We now know that this method works for the mathematical envelope problem. However, in some cases, ranking the choices from good to bad is not as easy as just ranking numbers. When you want to buy a house, for example, it may be harder to rate the available options. Even in our envelope game, our utility corresponding to the cash amounts might not scale linearly with the amounts to be won.
For example, one may find the difference between winning nothing and winning one million euros to be larger than the difference between winning 4 million and 5 million euros. This concept indeed changes a lot, but it is not difficult to take into regard.

To easily rate an option we use a so called utility function. An example of a utility function for winning a certain amount of cash is U(x) = \sqrt{x}, in which x is the amount of cash and U the corresponding utility. This specific function takes into account the diminishing effect of increasing the cash amount on the utility that comes with it.
Imagine having the choice between winning ten thousand euros with probability 0.5 and twenty thousand euros with probability 0.25. A person with the above utility function would prefer the first option even though the expected payout is the same:

0.5 * 10.000 = 0.25 * 20.000,
but when using the utility function:
0.5 * \sqrt{10.000} > 0.25 * \sqrt{20.000}

Choosing unconsciously

Now for every situation a utility function can be constituted, albeit sometimes very tedious. With this strategy, our method can be applied not only to maximize an expected value in a mathematical problem, but also to make sure the best choice is made in everyday situations. For many simple choices, our mind already uses the sampling technique unconsciously. Whether you are scrolling through netflix to find something to watch or driving around to pick a parking spot, your brain scans through the options, makes a sample of the first few options and then at some point picks the best out of many possibilities. Often this goes without too much effort, but sometimes your brain needs a little help. Hopefully, you now know how to make the right decision and to stop at the right choice!

Dit artikel is geschreven door Pieter Dilg

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