While the definition of expectation value (EV) is useful, it misses a subtlety. It gives us the ability to optimize on one value - in this case to maximize the amount of money in your possession. While many would consider this to be a worthwhile lifetime goal, many more would not. They would describe this as only part of the picture. Life, we are told, presents many possibilities, and not all of them can be compared or evaluated, so a rigorous approach is fundamentally flawed.

All of us make literally thousands of decisions a day. In many cases we don't consider the decisions for long - they are both trivial, and the best decision is obvious, but we still make decisions based on some reasoning.

The concept of utility is that any rational entity always chooses the option with the highest utility. In mathematical terms this is an Total Order - that is given any two options, there is always one that we would choose over another, and that is the one with the highest utility. We may not be able to predict what option we will take until the situation occurs. We might have incorrect information and make what turns out to be the wrong decision. We might not spend enough time considering the options, and later realize that even with the information we had, that it was the wrong decision. But the principle remains - rationality is choosing the highest utility option.

Sometimes however, we appear to make the wrong decision even though we are completely rational. Sometimes we might calculate the probabilities, and quite rationally make a decision and when the coin is tossed, or the die is rolled the events are the poorer ones. Just because the decision is wrong in hindsight does not mean that it was irrational at the time. Even though it offered the highest expectation value does not mean that it resulted in the most gain. Choosing the throw of the dice in the game above may end up with zero return, but it is still the choice which maximized expectation value of money, and hence was still the rational choice, presuming that maximizing our money was the goal.

However maximizing money is usually not the goal. Even the strongest proponents of the greed is good philosophy will agree that money is only good for what it can buy (status, power, sports-cars, mistresses, whatever). We see an apparent contradiction simply by changing the amounts in triple or nothing.

Consider the following game:

The reality is that the difference between your current wealth and $20,000,000 is probably much more than the difference between $20,000,000 and $60,000,000 dollars. The lifestyle that $60,000,000 can buy is not that much different from the lifestyle that $20,000,000 can buy. However the lifestyle that most people have is a lot different from the $20,000,000 lifestyle.

Even though the difference is greater in monetary terms, the difference is less in utility, or lifestyle, or whatever the money can actually usefully buy.

Sometimes it is logical to actually gamble where the gambling gives an apparent negative expectation value (EV<0). This is because the EV is measuring money, not utility.

Another thought experiment:

Even though your expectation value is negative (on average the bet will lose $1350) it is still a logical thing to do. If you play, you have a 48.6% of saving your child. If you don't play, you have none.

Your expectation value measured in dollars is negative if you play the game, but your expectation value measured in utility is (while possible non-quantifiable), clearly positive. It's only money. Your child's life is at stake.

Another example, what would you do if:

For most of us it would depend on the situation.

While an in depth mathematical understanding of calculus and economics are beyond the understanding and interest of most people, the basic concepts of expectation value and fuzzy logic are not. They belong, not just in our universities, but in our schools.