Daniel Bernoulli[1] (1700-1782) is credited with creating the model of maximization of the expected payoff[2] in the modelling of economic decision-making under risk. Numerous experiments have demonstrated that this approach does not fully correspond to agents' economic behaviour.[3] The assumption that the criterion of a rational agent in a situation of uncertainty is maximization of expected profit can also be easily challenged in other ways than on the basis of empirical experience.

One of the most effective ways of challenging the expected payoff maximization model is the "St. Petersburg paradox') where the profitable strategy for each player is one which brings him success in an absolutely negligible percentage of cases and in practically no case leads to a large loss.

The hypothetical game is as follows: A player pays, say, a million dollars to enter a casino (the St Petersburg casino) where only one game is played — a coin is tossed repeatedly until it lands heads. Let us denote this number of tosses by n. The player then receives 2n dollars, i.e. two dollars if he is unlucky and a head appears on the first toss, four dollars if a head appears on the second toss, and so on. The expected payoff in the St Petersburg casino tends to infinity:

The payoff will exceed the million dollar entry fee if the coin lands tails twenty times in a row. The entry fee should be worth paying (in terms of the expected payoff) whatever the amount (even if it is a million dollars). However, no one with any sense will pay more than a hundred (or even, in most cases, more than ten) dollars to enter this hypothetical casino. The intuitive maximum acceptable entry fee for the basic (minimum) win of two dollars ranges between four and eighty dollars.[4]

For an expected-payoff-maximizing player, the model "profitability” of the entry fee — whatever the level — results from an astronomically high payoff in a virtually negligible number of cases. No real decision-taker will be guided by the expected profit in this case: a willingness to accept an astronomically high entry fee would evidently mean loss of the instinct of self-preservation and therefore also of the agent's (economic) viability.

The view from the "opposite side" is equally convincing — it is not profitable to run the St. Petersburg casino at zero or even negative rent if the decision is based on expected profit.[5] Meanwhile, in reality (even in the world of organized crime) there is no business opportunity that comes even remotely close to such an offer (and any economically rational agent would "take it” even at zero rent).

The questions we ask in this context[6] are the following: Does a final, economically rational price of this game exist for an economic agent? And if it does, what is that price? And which model, or which cardinal utility function, should we choose for the utility of money if we want to avoid and model the paradox described above?

The St. Petersburg paradox cannot be explained satisfactorily using "mean-variance utility” models,[7] where a weighted average of the mean and variance is used as the cardinal utility function. Here, again, it turns out (nonsensically from the real-world perspective) that for any non-zero weight of the mean, any entry fee to the St. Petersburg casino — whatever the amount — is acceptable.

One possible approach to explaining the St. Petersburg paradox is von Neumann's theory of expected payoff utility maximization.[8] This model uses the cardinal money utility function, which for a risk-averse agent is strictly concave. It can take the for m of, for example, a power function


The field of psychology offers a different approach, namely the a for ementioned Weber-Fechner law, according to which a real agent decides not according to the intensity of a stimulus, but according to the change in its intensity. The corresponding utility function is logarithmic:

for this function, the expected utility is proportional to α.

In both these cases (power utility function and logarithmic utility function] an appropriately sized parameter[9] (i.e. one offering a "sensible" entry fee to the St Petersburg casino] exists. However, we have no direct economic guide (no economically justifiable direct argument] for determining this parameter, and its "appropriate” ad hoc setting for the St. Petersburg paradox would not be the right one for other decision-making situations.

The approaches described above share the problem that the decision does not depend on the agent's income. Yet wealth, or income, clearly does affect agents' choices in risky situations — a wealthy player will have a "lighter hand", whereas a drowning man will clutch at straws (i.e. possible but unlikely payoffs).

An alternative approach to the expected payoff utility model is to use subjective probabilities differing from the objective values. It turns out that the people usually overestimate the probability of rare events.[10] This increases the value of a lottery above its expected value, every week causing millions of would-be millionaires to pay one dollar for lottery tickets having an expected payoff of 50 cents. To explain the St Petersburg paradox, however, we would need exactly the opposite tendency, i.e. players would have to underweight (or ignore) extremely unlikely (albeit astronomically high) payoffs. This is how "real" visitors to the St. Petersburg casino think when comparing the extremely high entry fee with the expected payoff in (subjectively perceived) actually expectable cases. In so doing, they in fact "lop off” a portion of the expected payoff. We illustrate this division of the expected payoff into two parts in section 2.2 by for mulating a hypothetical Leningrad casino in which the actually likely payoffs are removed and only the extremely unlikely payoffs remain. Even in the Leningrad casino, an expected-profit-maximizing agent is willing pay the entry fee whatever the amount.

We try here to build on the von Neumann-Morgenstern approach and find an economically justifiable, strictly concave cardinal utility function based on the idea of maximizing the probability of economic survival.

  • [1] Nephew of the more famous Jacob Bernoulli [1654-1705], who among other things developed the law of large numbers.
  • [2] See Bernoulli, D.: Specimen theoriae novae de mensura sortis. Originally published in 1738; English translation; Econometrica 22, 1(1954): 23-36. Bernoulli himself claimed that this model was known long be for e his time.
  • [3] See, for example, Mosteller, F., Nogee, P.: An Experimental Measurement of Utility. Journal of Political Economy 59, 5(1951): 371–404, or Kahneman, D., Tversky, A.: Rational Choice and the Framing of Decisions. Journal of Business 59, 4(1986): 251–78.
  • [4] See Maňas, M.: Teorie her a optimální rozhodování. Praha: Státní pedagogické nakladatelství, 1969, p. 121.
  • [5] Negative rent here means that the operator pays nothing to the owner but instead receives a daily subsidy of a million (ten million, a billion, …) dollars from the owner on top of entry fees.
  • [6] Hlaváček, J., Hlaváček, M.: Petrohradský paradox a kardinální funkce užitku. Politická ekonomie 52, 1(2004): 48–60.
  • [7] See, for example, Meyer, J.: Two-Moment Decision Models and Expected Utility Maximization. American Economic Review 77, 3(1987): 422–30.
  • [8] See Neumann, J. von, Morgenstern, O.: Theory of Games and Economic Behavior. Princeton: Princeton University Press: 1953.
  • [9] In the case of the power utility function u(x) = x α the maximum acceptable entry fee lies in the “sensible” range of $4 to $80 for α  {0.68; 0.98}, regardless of initial income. A similar parameter can be “set” for the logarithmic utility function.
  • [10] See Preston, M. G., Baratta, P.: An Experimental Study of the Auction-Value of an Uncertain Outcome. American Journal of Psychology 61, 2(1948): 183-93.
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