I still remember when I learnt probability in secondary school, during the class, the teacher talked about expected value. It is a very simple concept but can help people to make more rational decision.

Basically it’s just about calculating the probability of every possible scenarios, and for each scenario, you multiply the probability for the occurrence of that scenario and your return. The result is going to be an expected return if you run this game for a long enough time.

A mathematical representation of that will be

And for a long period of time, I thought a rational person should make decision based on the expected value. It is a quantitative, scientific value and will avoid any human bias. As long as the number is positive, it basically said if you play this game long enough, you are going to be profitable.

Until I heard about the St. Petersburg Paradox.

The St. Petersburg Paradox is a classic problem in decision theory that challenges the idea of making decisions based solely on expected value. In the paradox, a game is played where a fair coin is flipped until it lands on tails. The player receives dollars, where n is the number of times the coin was flipped.

The expected value of the game is infinite, as there is a non-zero probability of receiving an arbitrarily large payout. However, most people would not be willing to pay a large amount to play the game, indicating that expected value alone does not fully capture the decision-making process.

It also explains why any sane person would not advise gambler to double down until he wins the game.

That spawns a new discipline in science called decision theory, which takes probability, psychology etc into account. Using the above example, because one won’t have unlimited money to bet into that game, the game can’t last forever. Because of that, one won’t be able to realize the expected value.

We also need to take human psychology into consideration. For example, loss aversion is the tendency for people to feel more negative emotions from losing something of equal value than they feel positive emotions from gaining something of equal value. In this case, if there is a game that you may have half a chance to win, and in such case, you will gain $20000. But if otherwise, you will lose $15000. If you are only allowed to play it once, you probably won’t decide to enter the game as the stake is too high.

There are lots of strategies in decision theory, trying to quantitatively analyze decision making. An interesting example is a strategy that minimize the maximum regret that one can have. This minmax regret approach, originally proposed by Leonard Savage in 1951, is about say you have four options A, B, C, D. They may bring you different return in different scenarios. Because you don’t know which option can bring you the best result before the fact. So among these options, you should choose the option that has minimum difference from the optimal option (only known after the fact).

It’s an interesting theory that I recommend people to learn more. After all, making a better decision in life is way more important than just put your head down and work.