Is homo economicus rational?

Nicolás Bernoulli

In 1713, Nicolás Bernoulli formulated the St. Petersburg paradox, which can be summarized as follows:

Let us consider the following game: a coin is tossed. If it comes up heads, you receive $2. If it comes up tails, it is tossed again. If it comes up heads, you receive $4. If it comes up tails, it is tossed again. And so on. With each toss, the prize is multiplied by 2. How much would you be willing to pay to participate in the game?

The probability of winning $2 is 0.5; the probability of winning $4 is 0.25; the probability of winning $2^k is 2^-k. The expected value is obtained by multiplying each value by its probability and adding them all together. So the expected value of the profit that could be obtained by playing that game is:

Note that all the terms in the sum equal 1, so the expected value of this game is infinite. The paradox is this: almost anyone, when asked to play this game, will only want to pay a small amount, even though they would have the expectation of winning an unlimitedly large amount, albeit with decreasing probability. Some consider this to prove that people are irrational.

Daniel Bernoulli, a cousin of Nicholas, attempted to solve the paradox and published his solution in the Annals of the St. Petersburg Academy of Sciences (hence the name of the paradox). His solution consisted in proposing that this type of problem should not be solved by considering the expected value, but the expected utility, a different function that would depend on the prior assets of each person. This solution was not final, as even now new explanations of the paradox continue to be proposed.

In the 20th century, some economists tried to define what could be considered reasonable behavior of people participating in economic transactions. In particular, John von Neumann and Morgenstern proposed four axioms of choice, which define what a reasonable choice should look like. The person who will choose is supposed to know information about the probabilities of the different outcomes that his or her choice can lead to. These are the axioms:

1. Information must be complete.
2. The choice must be transitive. If option A is preferred to option B, and option B is preferred to option C, then option A must be preferred to option C.
3. Continuity. If option A is preferred to option B, and option B is preferred to option C, then there must be some combination of A and C that is preferred to B.
4. Independence. If A is preferred to B, then that preference must hold for all existing options. For example, A+C (either A or C) must be preferred to B+C (either B or C).

John von Neumann

The problem is that normal people do not behave like this. Transitivity, for example, does not always hold. There are cases where, in a political election between three candidates, a voter may prefer A over B, B over C, and C over A, without ceasing to be reasonable.

Axiom 1, for example, can only be applied when all the objective probabilities are known, as in some games of chance, but in more complex cases it is practically impossible to know all of them, especially if only subjective probabilities are available.

Let us look at a case that contradicts the axiom of continuity: You are told to cross a public park. If you do it, you will receive $1 (case A). If you don’t, you will receive nothing (case B). Obviously, A is economically better than B. Now let us add situation C: hidden in the park there is a sniper who randomly shoots at the people who cross the park. Can option A+C really be better than B? Is there any combination that makes risking being shot to win $1 to be better than staying at home and receiving nothing? If you refuse to cross when you hear about option C, are you acting irrationally? Who would dare to say so?

The last axiom, independence, is the most controversial of all. Let's look at an example, taken like the rest of this article from the book Radical Uncertainty: Decision Making Beyond the Numbers by Mervyn King and John Kay. Would you rather win a million dollars with probability 11%, or win 5 million dollars with probability 10%? Almost everyone prefers the second option, because the difference in probability seems negligible compared to the increase in profit.

Now we make a modification to the game: Would you rather win a million dollars for sure (case A), or a million dollars with probability 89%, 5 million dollars with probability 10%, and nothing with probability 1% (case B)? Almost everyone prefers the first option. However, while case A gives us a million dollars, the expected value of the gain associated with case B is $1.39 million, considerably higher. However, there is a 1% chance of winning nothing. Are those who prefer the first option irrational? Well, they are the majority.

The conclusion drawn by the authors of the book is that many economists insist on applying mathematics to situations where applying common sense would be a better choice.

The same post in Spanish

Thematic Thread on Politics and Economics: Previous Next

Manuel Alfonseca