### The St-Petersburg Paradox, rationality and normativity

Before Bernouilli (1738]1954) introduces the concept of subjective expected utility (SEU), rational choice theory was a theory of expected value (EV). The EV of a possible gamble was construed as the product of the objective (for instance, monetary) value of its outcomes and their its probability. Suppose, then, a gambler is offered the following lottery:

A fair coin is tossed, and if the outcome is heads, the lottery ends and you win 2$. If the outcome is tail, toss the coin again. It the outcome is heads, the lottery ends and you win 4$, etc. If the nth outcome is heads, you win 2n.

Summing the products of probability and value leads to an infinite EV:

(0.5 x 2) + (0.25 x 4) + (0.125 x 8)…. =

1+1+1 …

After 30 tosses, the gambler could win more than 1 billion $. How much would he pay for a ticket? If a rational agent maximizes EV, he must be willing to buy a ticket for this lottery at any finite price, considering that the EV of this prospect if infinite. But, as Hacking said, “few of us would pay even $25 to enter such a game” (Hacking, 1980). When Bernoulli offered scholars in St-Peterburg to play this lottery, nobody was interested in it. He could then have decided that human beings are irrational. Instead, he uses this important discrepancy between theory and facts to amend the theory, and forged the concept of subjective utility: the value of a monetary gain is not a linear function of the monetary value (according to Bernoulli, it’s a logarithmic). The value of 10$ is different, whether you are Bill Gates or a homeless. It is now standard, in economics, to consider that a rational agent maximizes not the expected value, but the subjective expected utility.

Bernoulli’s discussion of the “St-Petersburg paradox” is often considered as one of the first economic experiment. In this case, the facts were not seen as irrelevant to the normative theory, but as a mean toward a better theory of rational decision-making. Hence one of the cornerstones of contemporary economics and normative theory of decision-making, subjective utility, owe its existence to empirical facts.

References

- Bernoulli, D. (1738]1954). Exposition of a New Theory on the Measurement of Risk. Econometrica, 22, 23-36.

- Hacking, I. (1980). Strange Expectations. Philosophy of Science, 47, 562-567.

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