The Saint Petersburg Paradox and Its Solution

Abstract

This article describes the main historical facts concerning the Saint Petersburg paradox, the most important solutions proposed thus far, and the results of new experimental evidence and a simulation of the game that shed light on a solution for this paradox. The Saint Petersburg paradox has attracted the attention of important mathematicians and economists since it was first formulated 300 years ago, and it has strongly influenced the development of new concepts in the economic and social sciences. The main conclusion of this study is that the behavior of the individuals playing the game is not paradoxical at all, and the paradox is intrinsic to the game.

1. Introduction

The Saint Petersburg paradox is most likely the oldest and most famous paradox in decision theory and a fundamental contribution to the birth of the modern theory itself. In the simple game of chance proposed, the fact that the sum that individuals are willing to pay to enter the game is small contradicts the rule that the game should be evaluated according to its (infinite) expected value.

In the 300 years since the paradox was introduced, this contradiction and attempts to solve it have attracted the attention of many of the most important mathematicians and economists, and it has strongly influenced the development of new concepts in the economic and social sciences. The literature on the argument is very rich, and it includes, among others, classical contributions such as those of Samuelson (1977), Dutka (1988), Bassett (1987), Aumann (1977), and more recent studies, such as those of Neugebauer (2010), Seidl (2013), Lahiri (2023), Katsikopoulos (2024), Taylor (2024), just to cite a few.

This study reviews the history of this fascinating paradox and the attempts to solve it, and proposes a new approach to the solution. In particular, Section 2 describes the birth of the paradox and its discovery by Nicholas Bernoulli. Section 3 presents the first attempts to solve it, while Section 4 discusses more recently proposed solutions to the problem. Some experimental studies are discussed in Section 5, the new study is illustrated in Section 6, and conclusions are provided in Section 7.

2. The History of the Paradox

The origin of the calculus of probability dates back to the 17th century, when many aristocrats in France spent their time and fortunes on games of hazard. In 1654, one such aristocrat, Chevalier de Méré, asked the mathematicians Pierre de Fermat and Blaise Pascal to help him optimize gambling. Their suggestions were based on the expected value of the winnings (indeed, Pascal can be considered founder of the idea that behavioral decisions are based on the expected value of money).

At the beginning of the 18th century, the mathematician Pierre Reymond de Montmort published a book in which he analyzed some games of chance, and Nicholas Bernoulli (the leading figure in stochastics and games of chance at that time and nephew of the founder of probability theory, James Bernoulli) had a long correspondence with De Montmort. In a letter dated 9 September 1713, Bernoulli commented on the first edition of Montmort’s treatise and suggested a number of combinatorial or expectational problems, which were included in the second edition of the book (Montmort 1713). Among these was the fourth problem:

“A promises to give 1 crown to B if with an ordinary die he gets six points on the first throw, 2 crowns if he gets the six on the second throw, 3 crowns if he gets this point on the third throw, 4 crowns if he gets it on the fourth, and so on; B’s expectation is required.”

The fifth problem was as follows:

“The same is required if A promises B to give the crowns in the progression  $1,2,4,8,16,$  etc. or  $1,3,9,27,$  etc. or  $1,4,9,16,25$  etc. or  $1,8,27,64$  etc. instead of  $1,2,3,4,5$  etc. as before.”

In his response dated 15 November 1713, Montmort expressed his opinion that these problems can be solved using geometric progressions and methods for the summation of series (developed by Jacob Bernoulli, another member of the Bernoulli family). In another letter dated 20 February 1714, Nicolas Bernoulli demonstrated the importance of his discovery: for the fourth problem, he found the sum of the convergent series, but for the first two cases of the fifth problem, the game has an expected value of infinity. Indeed, the results are as follows:

Proposition 1.

The mathematical expectation for individual B in the fourth problem considered above, and in the last two cases of the fifth problem, is finite, while the expectation in the first two cases of the fifth problem is infinite.

Proof. 

In the fourth problem, the mathematical expectation is as follows

$$\begin{array}{ccc}\hfill E\left(X\right)& =& \sum _{i=1}^{\infty }{x}_i{p}_i=1·{\displaystyle \frac{1}{6}}+2·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+3·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+\dots =\hfill \\ & =& {\displaystyle \frac{1}{6}}\sum _{i=1}^{\infty }i{\left({\displaystyle \frac{5}{6}}\right)}^{i-1}={\displaystyle \frac{1}{6}}{\left(1-{\displaystyle \frac{5}{6}}\right)}^{-2}={\displaystyle \frac{1}{6}}·36=6\hfill \end{array}$$

and in the fifth problem, in the last two cases, the series

$$E\left(X\right)=\sum _{i=1}^{\infty }{x}_i{p}_i=1·{\displaystyle \frac{1}{6}}+4·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+9·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+\dots ={\displaystyle \frac{1}{6}}\sum _{i=1}^{\infty }{i}^2{\left({\displaystyle \frac{5}{6}}\right)}^{i-1}$$

and

$$E\left(X\right)=\sum _{i=1}^{\infty }{x}_i{p}_i=1·{\displaystyle \frac{1}{6}}+8·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+27·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+\dots ={\displaystyle \frac{1}{6}}\sum _{i=1}^{\infty }{i}^3{\left({\displaystyle \frac{5}{6}}\right)}^{i-1}$$

both converge; hence, the mathematical expectation is finite. On the other hand, in the fifth problem, in the first two cases, the series

$$E\left(X\right)=\sum _{i=1}^{\infty }{x}_i{p}_i=1·{\displaystyle \frac{1}{6}}+2·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+4·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+\dots ={\displaystyle \frac{1}{6}}\sum _{i=1}^{\infty }{2}^{i-1}{\left({\displaystyle \frac{5}{6}}\right)}^{i-1}$$

and

$$E\left(X\right)=\sum _{i=1}^{\infty }{x}_i{p}_i=1·{\displaystyle \frac{1}{6}}+3·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+9·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{5}{6}}·{\displaystyle \frac{1}{6}}+\dots ={\displaystyle \frac{1}{6}}\sum _{i=1}^{\infty }{3}^{i-1}{\left({\displaystyle \frac{5}{6}}\right)}^{i-1}$$

diverge, meaning that the mathematical expectation is infinite. □

In conclusion, there was a contradiction between the mathematical expectation (that had been accepted in evaluating games of chance), which equals infinity, and the actual sum people were willing to pay in the real world to participate the game.

Montmort expressed skepticism but had to admit that he was unable to solve this problem.

3. Attempts at a Solution

Fifteen years later, Gabriel Cramer, another famous mathematician, read Montmort’s book in which the problem was presented, and in a letter addressed to Bernoulli dated 21 May 1728 (Cramer 1728), he first simplified the problem by replacing the six-sided die with a two-sided coin and interchanging the roles of A and B. Thus, the problems became the following:

“B promises to give 1 crown to A if tossing a coin he gets head on the first throw, 2 crowns if he gets head for the first time on the second throw, 3 crowns if he gets this point on the third throw, 4 crowns if he gets it on the fourth, and so on; A’s expectation is required”

and then

“The same is required if B promises A to give the crowns in the progression  $1,2,4,8,16,$  etc.”.

In this case, the following result holds:

Proposition 2.

In the first problem stated above, the mathematical expectation of individual A is finite, while in the second problem, it is infinite.

Proof. 

Since the first toss of the coin ends the game yielding 1 crown with a probability of ${\displaystyle \frac{1}{2}}$, the second toss of the coin ends the game yielding 2 crowns with a probability of ${\displaystyle \frac{1}{4}}$, and so on, in the first problem, player A’s expectation is given by

$$E\left(X\right)=\sum _{i=1}^{\infty }{x}_i{p}_i=1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+3·{\displaystyle \frac{1}{8}}+\dots =2$$

therefore, it is finite, even if the payoff is unbounded. On the contrary, in the second problem, the expected value is

$$E\left(X\right)=\sum _{i=1}^{\infty }{x}_i{p}_i=1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+4·{\displaystyle \frac{1}{8}}+\dots ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}+\dots +{\displaystyle \frac{1}{2}}+\dots =+\infty$$

i.e., the series is divergent, and the expectation is infinite. □

Therefore, the paradox that arises is in the second case considered above, the expected value of the gamble is infinite, but individuals are willing to pay only a small amount to participate in the game.

To solve the paradox that arises in the second problem, Cramer suggested that people evaluate money not in proportion to its quantity but to its utility. In particular, very large amounts (which, in the game considered, can be won if heads appears for the first time after a very large number of tosses) provide people no more pleasure than amounts of money equal to a certain threshold. More specifically, Cramer assumed that any amount above $2^{24}$ crowns was considered equal in value to $2^{24}$ crowns; he also considered utility function with a quadratic form. In these cases, the following results hold:

Proposition 3.

Assuming that individuals have a utility function of the form

$$u\left(x\right)=\left\{\begin{array}{cc}x& \mathit{for}x\le 2^{24}\\ & \\ 2^{24}& \mathit{for}x>2^{24}\end{array}\right.$$

the expected value of the game is finite, and the sum equivalent to the game is $c=13$. Assuming, instead, that the utility function is

$$u\left(x\right)=\sqrt{x}$$

the expected value of the game is still finite, and the sum equivalent to the game reduces to $c=2.9$.

Proof. 

In the first case, the mathematical expectation of the game is

$$\begin{array}{ccc}\hfill E\left[u\left(X\right)\right]& =& \sum _{i=1}^{\infty }u\left(x_i\right)p_i=\sum _{i=1}^{24}{2}^{i-1}·{\displaystyle \frac{1}{2^i}}+\sum _{i=25}^{\infty }{2}^{24}·{\displaystyle \frac{1}{2^i}}=\hfill \\ & =& 1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+4·{\displaystyle \frac{1}{8}}+\dots +2^{24}·{\displaystyle \frac{1}{2^{25}}}+2^{24}·{\displaystyle \frac{1}{2^{26}}}+2^{24}·{\displaystyle \frac{1}{2^{27}}}+\dots =\hfill \\ & =& {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}+\dots +{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{8}}+\dots ={\displaystyle \frac{1}{2}}·24+\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{8}}+\dots \right)=13\hfill \end{array}$$

then, the sum equivalent to the game is an amount c such that

$$u\left(c\right)=E\left[u\left(X\right)\right]\Rightarrow c=13$$

In the second case, the mathematical expectation is

$$\begin{array}{ccc}\hfill E\left[u\left(X\right)\right]& =& \sum _{i=1}^{\infty }u\left(x_i\right)p_i=\sum _{i=1}^{\infty }\sqrt{2^{i-1}}·{\displaystyle \frac{1}{2^i}}=\hfill \\ & =& \sqrt{1}·{\displaystyle \frac{1}{2}}+\sqrt{2}·{\displaystyle \frac{1}{4}}+\sqrt{4}·{\displaystyle \frac{1}{8}}+\dots +\sqrt{2^{n-1}}·{\displaystyle \frac{1}{2^n}}+\dots =\hfill \\ & =& {\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\sqrt{2}}{2}}+{\displaystyle \frac{\sqrt{4}}{4}}+\dots +{\displaystyle \frac{\sqrt{2^{n-1}}}{2^{n-1}}}+\dots \right)=\hfill \\ & =& {\displaystyle \frac{1}{2}}·{\displaystyle \frac{1}{1-\frac{\sqrt{2}}{2}}}={\displaystyle \frac{1}{2-\sqrt{2}}}\hfill \end{array}$$

and then the sum equivalent to the game is the amount c such that

$$u\left(c\right)=E\left[u\left(X\right)\right]\Rightarrow \sqrt{c}={\displaystyle \frac{1}{2-\sqrt{2}}}\Rightarrow c={\left({\displaystyle \frac{1}{2-\sqrt{2}}}\right)}^2=2.9$$

□

These results are, therefore, small amounts close to the value that people attribute to the game considered.

However, Nicolas Bernoulli did not accept Cramer’s view and proposed a different solution. In his explanation of the game considered, no player would pay 20 crowns because they would regard their chances of winning a large sum impossible. His result was the following:

Proposition 4.

Assuming, in the game of heads and tails introduced above, that the occurrence of the events with probabilities lower than ${\displaystyle \frac{1}{32}}$ is negligible, the expected value of the game is $E\left(X\right)=2.5$.

Proof. 

In this case, the expectation is given by

$$\begin{array}{ccc}\hfill E\left(X\right)& =& \sum _{i=1}^5{x}_i{p}_i=\sum _{i=1}^5{2}^{i-1}·{\displaystyle \frac{1}{2^i}}+0=\hfill \\ & =& 1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+4·{\displaystyle \frac{1}{8}}+8·{\displaystyle \frac{1}{16}}+16·{\displaystyle \frac{1}{32}}=2.5\hfill \end{array}$$

since the occurrence of the events associated with the sum of the probabilities

$$\sum _{i=6}^{\infty }{\displaystyle \frac{1}{2^i}}=\sum _{i=1}^{\infty }{\displaystyle \frac{1}{2^i}}-\sum _{i=1}^5{\displaystyle \frac{1}{2^i}}=1-{\displaystyle \frac{31}{32}}={\displaystyle \frac{1}{32}}$$

can be disregarded. □

In another letter dated 27 October 1728, Nicolas Bernoulli communicated the problem in Cramer’s simplified version to his cousin Daniel Bernoulli, another mathematician at the University of Saint Petersburg. Daniel answered, saying that the paradox had to be found in the small probability that the gamble would last for more than 20 or 30 throws, and individuals would not be willing to pay an infinite sum for a gamble in which there is only an infinitesimally small probability of winning. Then, in 1731, he composed a draft of his famous expected utility theory. The paper was not published until 1738 in the Commentarii of the Saint Petersburg Academy, and from this, the paradox became known as the “Saint Petersburg paradox”. The formulation of the problem presented by Daniel Bernoulli in his 1738 paper (Bernoulli 1738) is as follows:

“Peter tosses a coin and continues to do so until it should land “heads” when it comes to the ground. He agrees to give Paul 1 ducat if he gets “heads” on the very first throw, 2 ducats if he gets it on the second, 4 if on the third, 8 if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. What is the maximum amount Paul should pay for this game?”

The solution proposed was to take into account the diminishing marginal utility of money, distinguishing the mathematical expectation from what was called the moral expectation (in modern terms, the utility expectation) of a chance event upon which a sum of money depended. The mathematical expectation is defined as the sum of the products of the various sums of money that can be gained times their respective probabilities, i.e.,

$$E\left(X\right)=x_1{p}_1+x_2{p}_2+\dots +x_n{p}_n=\sum _{i=1}^n{x}_i{p}_i$$

while the moral expectation is defined as the sum of the products of the advantages (utilities) accruing from the various sums of money that can be gained times their respective probabilities, i.e.,

$$E\left[u\left(X\right)\right]=u\left(x_1\right)p_1+u\left(x_2\right)p_2+\dots +u\left(x_n\right)p_n=\sum _{i=1}^{n}u\left(x_i\right)p_i$$

The result obtained by Daniel Bernoulli is as follows:

Proposition 5.

Assuming that the increase in utility as a consequence of an increase in wealth is inversely proportional to the present wealth, we have

$$u\left(x\right)=klog{\displaystyle \frac{x}{\alpha }}$$

where $\alpha >0$ is the initial wealth, $x>0$ is the present wealth, and $k>0$ is a proportionality factor. As a consequence, for a person with initial wealth α that can win an amount $a_i$ (that is $1,2,4,8,\dots$) with a probability of $p_i$ (that is, ${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{16}},\dots$), where $i=1,2,3,\dots$ and ${\sum }_{i=1}^{\infty }{p}_i=1$, his moral expectation is finite and is given by

$$\begin{array}{ccc}\hfill E\left[u\left(X\right)\right]& =& \sum _{i=1}^{\infty }u\left(x_i\right)p_i=k\sum _{i=1}^{\infty }{\displaystyle \frac{1}{2^i}}log\left({\displaystyle \frac{\alpha +a_i}{\alpha }}\right)=\hfill \\ & =& klog\left[{\left({\displaystyle \frac{\alpha +1}{\alpha }}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{\alpha +2}{\alpha }}\right)}^{{\displaystyle \frac{1}{4}}}·\dots ·{\left({\displaystyle \frac{\alpha +2^{n-1}}{\alpha }}\right)}^{{\displaystyle \frac{1}{2^n}}}·\dots \right]\hfill \end{array}$$

and the sum of money whose utility equals the moral expectation is the amount c such that

$$u\left(c\right)=E\left[u\left(X\right)\right]$$

from which

$$c=\sqrt{\alpha +1}·\sqrt[4]{\alpha +2}·...·\sqrt[2^n]{\alpha +2^{n-1}}·...$$

Proof. 

See Bernoulli (1738). □

From this formula, it follows that if Paul does not have initial wealth (i.e., $\alpha =0$), the worth of the gamble from his perspective is given by

$$c=\sqrt{1}·\sqrt[4]{2}·\dots ·\sqrt[2^n]{2^{n-1}}·\dots =2$$

while if $\alpha =10$, then $c\simeq 3$; if $\alpha =100$, then $c\simeq 4$; and if $\alpha =1000$, then $c\simeq 6$. As outlined by Bernoulli, therefore, Paul’s prospective gain will increase with the size of Paul’s fortune but will never attain an infinite value unless Paul’s wealth simultaneously becomes infinite. In particular, the worth of the gamble to Paul, who possesses few resources, is quite small and increases only slowly with the magnitude of his fortune. In this way, Daniel Bernoulli attempted to solve the paradox.

Nicholas Bernoulli was not completely satisfied with this explanation; in any case, he thought that combining the ideas of Daniel Bernoulli and Gabriel Cramer with his own, according to which it is necessary to approximate a small probability as null, would enable him to find a finite expected value of the Petersburg gamble and determine the exact equivalent value of this gamble.

4. More Recent Attempts

Over the following 200 years, Daniel Bernoulli’s utility theory was discussed and generalized. Finally, in 1934, Karl Menger showed that utility must be bounded from above (Menger 1934). The following result holds:

Proposition 6.

Given the Petersburg game with logarithmic utility function, if Paul receives $e^{2^i}$ ducats if the first head is obtained at the i-th toss, Paul’s expected utility is infinite.

Proof. 

In this case, the expected utility of the game is

$$\begin{array}{ccc}\hfill E\left[u\left(X\right)\right]& =& \sum _{i=1}^{\infty }u\left(x_i\right)p_i=\sum _{i=1}^{\infty }log\left(e^{2^i}\right)·{\displaystyle \frac{1}{2^i}}=\hfill \\ & =& \sum _{i=1}^{\infty }{2}^i·loge·{\displaystyle \frac{1}{2^i}}=\sum _{i=1}^{\infty }1=+\infty \hfill \end{array}$$

i.e., it is infinite, and the paradox returns. □

In practice, considering the Petersburg game, even with a logarithmic utility function, the paradox reappears if the payoffs grow sufficiently fast. Additionally, if a new, unbounded utility function were introduced for this game to return a finite expectation, a new set of more rapidly increasing payoffs can be designed so that the resulting expectation is infinite again. Samuelson called this result the “Menger’s Super-Petersburg Paradox” (Samuelson 1977). By repeatedly adjusting the payoff strategy, the measurement of utility using logarithms can always be made to result in an infinite expectation, and the paradox can be avoided if and only if the utility function is bounded.

Menger’s research also played a crucial role in persuading von Neumann and Morgenstern to develop a formal treatment of utility, and the axiomatic approach to expected utility was incorporated in the second edition of their fundamental book, “Theory of Games and Economic Behavior” (Von Neumann and Morgenstern 1947). In 1971, then, Arrow showed that to avoid the Super-Petersburg paradox, the utility function must be bounded from both above and below (Arrow 1971).

In the three centuries since Nicolas Bernoulli proposed his fourth and fifth problems to De Montmort in 1713, numerous solutions to the Saint Petersburg paradox have been proposed. Some authors such as D’Alembert attacked the bases of probability, while others such as Lubbock, Drinkwater, and De Morgan accepted the view that A was justified in paying an infinite sum to B because his expectation was infinite. Fontaine, a French mathematician, assumed that the banker’s capital is finite and sought the stake that the player should pay to make the game equitable, obtaining in this way some inequalities for the number of games to be played under these constraints.

Other contributions to the discussion of the paradox were made by Buffon (1777). First of all, he developed Fontaine’s idea further, taking into account the banker’s ability to fulfill his obligation, in particular, paying off the player after a long sequence of tails arises before the first head appears. He pointed out that if a head does not appear until after the 29th toss, there would be no sufficient money in the entire French kingdom to pay the player (he also estimated the number of games which would be possible in a player’s lifetime).

The view that the limitation of the banker’s capital implies that only the restricted game is possible in the real world was adopted by many later writers on the Saint Petersburg paradox such as Poisson, Catalan, Czuber, Pringsheim, and Von Mises, and it is perhaps the most frequently offered resolution to the paradox. In this case, the boundedness of the winnings is justified by the finiteness of the bookmaker’s wealth or by the fact that the time spent playing a Saint Petersburg game cannot be infinitely long. However, this perspective also had opponents; for example, Bertrand suggested that whatever the banker owes, he can compensate the player, whether he is solvent or not, by simply writing a note for the debt. Further elaborations were given by Shapley (1977) and Samuelson (1977).

Buffon suggested another possibility relevant to the solution of the paradox, i.e., the fact that any probability below some small positive fraction can be set equal to zero. In particular, he considered a probability of ${\displaystyle \frac{1}{10,000}}$ or less for an event as a probability that can be disregarded (this was the probability that a 56-year old man in good health would die within 24 h). Daniel Bernoulli approved this idea but, acting more conservatively, chose a threshold, called “the moral probability”, of ${\displaystyle \frac{1}{100,000}}$. Fontaine, Poisson, Condorcet, and D’Alembert also accepted the idea of negligible probabilities. Perhaps the most important proponent of negligible probabilities in modern times was the French mathematician Borel, who considered, with respect to the Saint Petersburg paradox, possible results with a probability lower than ${\displaystyle \frac{1}{1,000,000}}$ negligible.

Buffon also tested his arguments; to this end, he conducted the first experiment recorded in statistics to empirically determine the likely outcomes in the Petersburg gamble. A child played out $n=2^{11}=2048$ trials of the Petersburg gamble, all of which ended after nine tosses at most, and the average payoff was $4.9$ ducats. Hence, he concluded that about 5 ducats should be a fair entry fee to the gamble (compared with the $5.5$ units per game predicted by the theoretical argument).

In the 1930s and 1940s, probabilists paid considerable attention to increasingly general formulations of the law of large numbers, and in particular, Feller produced a result that is directly applicable to the Saint Petersburg problem. He considered the total gain in N games and proved that a modified law of large numbers holds for it (Feller 1945). To compare these estimates with the experimental results previously obtained, computer simulations were conducted that involved $22,528$ Saint Petersburg games. The resulting average payout turned out to be $7.23$ units per game, compared with the Feller’s estimate of $7.34$ units per game and the Buffon’s estimate of $5.5$ units per game. In 1985, Martin-Lof obtained a limit theorem for the total gain in a series of Saint Petersburg games, which can be regarded as a refinement of Feller’s result (Martin-Lof 1985).

An alternative resolution to the Petersburg paradox is the so-called “expectancy heuristic” proposed by Treisman (1983). Instead of computing the product of probabilities and payoffs, the expectancy heuristic suggests computing the outcome at the expected gamble length. The following result holds:

Proposition 7.

In the Petersburg game, the expected number of trials for the first head to occur is 2, and the corresponding payoff is 2 ducats.

Proof. 

In this game, the number of trials to obtain the first head is i, with a probability of ${\displaystyle \frac{1}{2^i}}$; therefore, the expected number of trials it would take for the first head to occur is

$$\begin{array}{ccc}\hfill \sum _{i=1}^{\infty }i·p_i& =& \sum _{i=1}^{\infty }i·{\displaystyle \frac{1}{2^i}}=1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+3·{\displaystyle \frac{1}{8}}+\dots +k·{\displaystyle \frac{1}{2^k}}+\dots =\hfill \\ & =& {\displaystyle \frac{1}{2}}+{\displaystyle \frac{2}{4}}+{\displaystyle \frac{3}{8}}+\dots +{\displaystyle \frac{k}{2^k}}+\dots \hfill \end{array}$$

Rearranging the terms

$${\displaystyle \frac{1}{2}}+\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{4}}\right)+\left({\displaystyle \frac{1}{8}}+{\displaystyle \frac{1}{8}}+{\displaystyle \frac{1}{8}}\right)+\dots$$

and regrouping them

$$\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{8}}+\dots \right)+\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{8}}+{\displaystyle \frac{1}{16}}+\dots \right)+\left({\displaystyle \frac{1}{8}}+{\displaystyle \frac{1}{16}}+{\displaystyle \frac{1}{32}}+\dots \right)+\dots$$

where each regrouping is a geometric series whose sums are

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{8}}+\dots & =& {\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+\dots \right)={\displaystyle \frac{1}{2}}·{\displaystyle \frac{1}{1-\frac{1}{2}}}=1\hfill \\ \hfill {\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{8}}+{\displaystyle \frac{1}{16}}+\dots & =& {\displaystyle \frac{1}{4}}\left(1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+\dots \right)={\displaystyle \frac{1}{4}}·{\displaystyle \frac{1}{1-\frac{1}{2}}}={\displaystyle \frac{1}{2}}\hfill \\ \hfill {\displaystyle \frac{1}{8}}+{\displaystyle \frac{1}{16}}+{\displaystyle \frac{1}{32}}+\dots & =& {\displaystyle \frac{1}{8}}\left(1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+\dots \right)={\displaystyle \frac{1}{8}}·{\displaystyle \frac{1}{1-\frac{1}{2}}}={\displaystyle \frac{1}{4}}\hfill \\ & & \dots \hfill \end{array}$$

The initial expression, therefore, corresponds to

$$1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+3·{\displaystyle \frac{1}{8}}+\dots +k·{\displaystyle \frac{1}{2^k}}+\dots =1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}+\dots +{\displaystyle \frac{1}{2^{k-1}}}+\dots ={\displaystyle \frac{1}{1-\frac{1}{2}}}=2$$

This means that the gamble is expected to terminate on the second trial; therefore, the expected length is 2. As a consequence, the corresponding payoff is 2 ducats, which is therefore the value of the Petersburg gamble according to the expectancy heuristic. □

In summary, there have been three types of solutions offered to the paradox. The first, attributed to Poisson and Condorcet, is that the conditions of the game itself imply a contradiction since the game, by promising to pay an infinite sum, implies a condition that can not be fulfilled. In addition, a lack of time is another limitation preventing the gamble from being well-defined since it could be that, once the game has started, the gamble never ends, and in the real world, nobody can toss a coin an infinite number of times.

The second type of solution, attributed to Buffon, is the principle that events whose probabilities are sufficiently small can be regarded as impossible. The probability that a head will not appear until the i-th toss, is very small for i sufficiently large. As a consequence, if the occurrence of that event is regarded as impossible for all n beyond a certain value, the mathematical expectation of returns becomes finite, and the paradox is resolved.

The third type of solution, attributed to Daniel Bernoulli, is to assume that the marginal utility of money decreases so that the expected value of the game, ${\sum }_{i=1}^{\infty }u\left(2^i\right)·\left({\displaystyle \frac{1}{2^i}}\right)$, is finite. Menger objected to this solution, showing that the paradox can be restored by modifying the game such that the payoff grows sufficiently fast; in this situation, this difficulty can be avoided by requiring that the utility function is finite.

An interesting conclusion is offered by Yukalov (2021). In his words, “…The St. Petersburg paradox is probably the oldest paradox in decision theory and has promoted the birth of modern decision theory itself. […] In economics, it has played a particularly important role in pointing out situations in which supposedly rational decisions based on expected gains or even expected increasing utilities are not endorsed by real rational human decision makers. This paradox has opened a flood of attempts to solve it, which turn out all to modify it in one way or another. The most important change involves the introduction of a concave utility function, which captures the idea (due to Cramer) that individuals estimate money in proportion to the usage that they may make of it, and not necessarily in proportion to its quantity. Motivated by the St. Petersburg paradox, the introduction of concave utility functions, which embody risk aversion and decreasing marginal utility of gains, remains the central pillar of modern economic theory. However, the solution in terms of utility functions of Cramer and Bernoulli is not completely satisfactory since a slight change in the formulation makes the paradox reappear. The conclusion is that from the mathematical and logical point of view, the St. Petersburg paradox is impeccable, while the suggested solutions are not mathematical, since they all modify the initial problem, hence the original St. Petersburg paradox remains unresolved.” (Yukalov 2021).

5. Experimental Studies

Despite the age and importance of the problem, only a few experiments on the Petersburg gamble have been documented. As cited in Neugebauer (2010), some authors (for example, Levy and Sarnat (1984), or Vivian (2003)) report, without detailing the procedure followed, that they made inquiries of a group of students, most of which were willing to pay only two or three dollars to play the game, and some even indicated that zero was a reasonable price for entering the gamble.

Bottom et al. (1989) conducted a more complete study, designing an experiment to test several hypotheses, including the expectancy heuristic of Treisman, which accounted for the possibility that small probabilities were neglected. To this end, they terminated the game after a number of tosses and considered only payoffs below a certain amount reasonable, assuming that funds were limited. In this experiment, two subject pools of 139 students and 47 professionals were involved, taking part in a sealed-bid auction. They were asked to write down a sealed bid and to imagine that these bids were actually competing in an auction under different conditions (the first one was the standard Petersburg gamble; in the second one, the payoffs for each outcome were increased by five dollars; in the third one, the payoffs were increased by ten dollars; and in the last one, the payoffs were doubled with respect to the first condition). The conclusion of this research is that the results supported the expectancy heuristic, meaning that the median bids were approximately equal to the expected median payoff.

Another study was conducted by Cox et al. (2009), which was probably the first experiment with real money payoffs for finite Saint Petersburg lotteries. The authors designed an experiment considering nine possible truncations of the Petersburg gamble, including $\left\{1,2,3,\dots ,9\right\}$ tosses of the coin. Thirty subjects were invited to play this game, paying a price that was $0.25$ dollars lower than the expected value of the gamble. Then, one of the nine possible gambles was chosen randomly and actually played. The result of the experiment was that most of the subjects were not willing to play the game; therefore, the conclusion of the study is that most individuals are risk-averse, not risk-neutral (as the expected value theory would require).

A third experimental study on the Petersburg gamble was conducted by Hayden and Platt (2009). In this case, over 200 individuals were involved in the classical Saint Petersburg game and then in a number of variants, including truncated versions and repeated gambles. The results show that the bids observed for the original form of the gamble were much lower than commonly supposed; specifically, they were lower than twice the smallest payoff. Furthermore, bids were only weakly affected by truncating the gamble, while they were strongly affected by repeating the gamble, in line with the law of large numbers. In particular, the subjects’ valuations converged to the median outcome and therefore supported the proposal of Tolman and Foster (1981) that the median valuation is a reasonable choice for the repeated Petersburg gamble.

The most important study considered here is that of Neugebauer (2010), which takes into account the one-shot Petersburg gamble. The underlying assumption is that preferences are represented by the general expected utility functional form

$$u\left(X\right)=\sum _{i=1}^{N}f\left(p_i\right)u\left(x_i\right)$$

Most of the hypotheses proposed over the history of the paradox truncate the Petersburg gamble, including the bounded utility or finite wealth hypotheses, which put an upper bound on the utility of the gamble, and the physical impossibility or the moral impossibility hypotheses, which assume that a small probability is set equal to zero, i.e., $f\left(p\right)=0$ for $p<$p. The objective of the experimental study of Neugebauer (which consisted of different treatments involving hundreds of students in total) was to understand if it is possible to find evidence for such a cut-off level and, if so, at what level this cut-off would occur.

First, the study shows that people seem to neglect small probability events in the Petersburg gamble. In effect, experiment provides some evidence that subjects’ valuations for the longer gamble are not higher than for the shorter game, from which it seems that the probabilities are set equal to zero at or below a certain probability level. It is therefore important to determine the probability level at which the probabilities are truncated, and from the experiment, it seems that probabilities smaller than ${\displaystyle \frac{1}{32}}$ are neglected by experimental subjects. This result is in line with the conjecture of Nicholas Bernoulli that a probability smaller than ${\displaystyle \frac{1}{32}}$ should be set equal to zero in the Petersburg gamble. In fact, his computation was

$$E\left(X\right)=1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+4·{\displaystyle \frac{1}{8}}+8·{\displaystyle \frac{1}{16}}+16·{\displaystyle \frac{1}{32}}+32·0+\dots =2.5$$

Therefore, he did not account for any payoffs above 16 ducats, and this implied an expected payoff of $2.50$ ducats.

The results of the experiment of Neugebauer also show that willingness to pay exhibits an age effect (more senior individuals who participated in the experiment had a higher willingness to pay). Furthermore, subjects’ offers increase significantly with income (in line with the assumption of Daniel Bernoulli, according to which the willingness to pay for the Petersburg gamble should be an increasing function of wealth).

The conclusion of the Neugebauer study is that the Petersburg problem was originally designed to study the level of moral impossibility in games of chance. During the lifetime of Nicholas Bernoulli, there was little interest in studying moral impossibilities in the games of chance and, generally, in applications to the social sciences, but as a consequence of Daniel Bernoulli publishing the expected utility concept as a solution to the Petersburg paradox, some of the most brilliant minds in the recent centuries have contributed to this area of research. In particular, Nicholas Bernoulli derived a level of moral impossibility of ${\displaystyle \frac{1}{32}}$ to determine the fair value of the Petersburg gamble, beyond which each smaller probability is unlikely. The experiments conducted by Neugebauer support this conclusion, and therefore, it seems reasonable to accept that the neglect of small-probability events is a more relevant solution criterion in the Petersburg paradox than its best known alternatives, bounded utility and the limitation of the experimenter’s expected wealth.

6. A New Study: Materials, Methods, Results, and Discussion

The objective of the present research is to fill the gap represented by the fact that despite the importance of the Saint Petersburg paradox, there is no widely accepted explanation for the low values that most people give to this gamble (which is theoretically infinite) and, even more importantly, the lack of empirical data from testing the explanations proposed.

In order to shed light on these points, we have undertaken a new study, proceeding in two directions. On one hand, an extensive experiment involving a large number of individuals invited to play the Saint Petersburg game was conducted, with the goal of understanding their behavior. On the other hand, a computer simulation of the game was created, with the objective of comparing the results to the actual behavior of the individuals.

6.1. The Design of the Experiment

The first part of the study, i.e., the experiment, involved a large number of students at the School of Economics of the University of Turin who were enrolled in different years of study, distinguishing between undergraduate and Master’s degrees and between business-oriented and economics-oriented students. The number of participants is the main strength of this study with respect to previous experimental studies (which were described in Section 5). In previous studies, 200 individuals at most took part in the research, while this study involved almost 1500 students.

Table 1. Groups of students involved in the experiment.

Groups of Students	Number of Students
Undergraduate students—Business track	850
Undergraduate students—Economics track	220
Master students—Business track	350
Master students—Economics track	45
Total	1465

reports the groups of students involved in the experiment and their numbers:

These data are also represented in Figure 1:

In the first step of the experiment, at the end of a semester at university, these students were invited to participate in the survey experiment by answering three questions. These questions asked them to evaluate the version of the game that does not give rise to the paradox, then the “truncated version” and, finally, the original version of the Saint Petersburg game. The answers were collected using the Moodle platform in order to create a database that has then been used to elaborate the results.

The first question, formulated in order to evaluate the version of the game that does not generate the paradox (corresponding to the fourth problem that Nicholas Bernoulli originally proposed to Montmort) was the following:

“You can enter a game in which a coin is flipped until heads occurs for the first time. You win 1 € if heads appears for the first time at the first toss, you win 2 € if heads appears for the first time at the second toss, you win 3 € if heads appears for the first time at the third toss, you win 4 € if heads appears for the first time at the fourth toss, and so on. How much are you willing to pay to participate in this game?”

The second question, formulated in order to evaluate the “truncated version” of the game, was the following:

“You can enter a game in which a coin is flipped until heads occurs for the first time. You win 1 € if heads appears for the first time at the first toss, you win 2 € if heads appears for the first time at the second toss, you win 4 € if heads appears for the first time at the third toss, you win 8 € if heads appears for the first time at the fourth toss, and so on. However, if after 7 tosses heads has not appeared, the game ends and you don’t win anything. How much are you willing to pay to participate in this game?”

The third question, formulated in order to evaluate the original version of the game, was the following:

“You can enter the same game of the previous question, but this time the game ends only when heads appears for the first time (eventually after a large number of tosses). Also in this case, each additional toss is associated with a payoff that is the double of the previous one. How much are you willing to pay to participate in this game?”

The objective of these questions (i.e., the research question of our study) was the intention to investigate if, with respect to the version of the problem without an infinite expected payoff, the behavior of the individuals is consistent with the theory, and if, regarding the version that generates the infinite expected payoff, moving from a reduced (in the truncated version) to potentially very large (in the complete version) number of tosses, the evaluation of the game by the individuals was very different.

6.2. Results and Discussion of the Survey

In the second step of the experiment, the data collected during the first step were elaborated to gain the information necessary to answer the research question (i.e., how consistent is the behavior of the individuals with the theory, and what is the eventual difference between the complete and truncated versions of the game?). The basic results are reported in

Table 2. Results of the experiment.

Group of Students	Average Amount Game 1	Average Amount Game 2	Average Amount Game 3
Undergraduate students—Business track	2.15 €	3.38 €	5.78 €
Undergraduate students—Economics track	1.95 €	2.26 €	3.65 €
Master students—Business track	2.10 €	3.07 €	4.65 €
Master students—Economics track	1.50 €	1.54 €	2.34 €
Total	2.09 €	3.08 €	5.08 €

, which indicates for each group of students the average amount that the individuals of the group are willing to pay to participate in each of the three versions of the game (in particular, game 1 represents the version of the gamble that does not give an infinite expected payoff, game 2 refers to the truncated version of the game, and game 3 is the original version of the game).

These results are also presented in Figure 2.

The analysis of these elements allows for some interesting insights concerning the problem studied.

A first observation is the willingness of the business-oriented students to pay a higher amount to participate in the game than the economics-oriented students. This is an interesting result that does not emerge from the previous studies (for example, those of Neugebauer (2010), and of Bottom et al. (1989), which also involved students).

A second result is that younger students (i.e., those at the undergraduate level) are willing to pay a higher amount to participate in the game with respect to older students (i.e., those at the Master’s level), both in the business-oriented and economics-oriented groups. There is therefore an “age effect” (the lower the age, the higher the willingness to pay) that contrasts the one indicated by Neugebauer (2010), who noted, as shown above, that more senior individuals have a higher willingness to pay.

A further result is that for the first version of the game (i.e., with a finite expected value), business-oriented students are willing to pay an amount that is slightly higher than the expected value of the game (in this case, as shown in Section 3, the expected payoff is equal to EUR 2), while economics-oriented students are willing to pay an amount that is lower than this expected value.

Another conclusion is that all groups of students, at both undergraduate and Master’s levels, are willing to pay a sum lower than the expected value of the game for the truncated version of the game, which, in this case, is

$$E\left(X\right)=\sum _{i=1}^7{x}_i{p}_i=1·{\displaystyle \frac{1}{2}}+2·{\displaystyle \frac{1}{4}}+4·{\displaystyle \frac{1}{8}}+\dots +64·{\displaystyle \frac{1}{128}}=7·{\displaystyle \frac{1}{2}}=3.5$$

This sum ranges from EUR $1.54$ to EUR $3.38$, revealing that individuals expect that heads will appear for the first time after two or three tosses.

Finally, to play the complete version of the game, all students are willing to pay a sum that is higher than the cost of the truncated version but lower than the amount gained if heads appears for the first time after four tosses (EUR 8), which is dramatically lower than the potentially very large amount that the complete version of the game could enable them to win.

In general, therefore, the sums declared as amounts that individuals are willing to pay to enter the game reveal that they expect the game will end after three or four tosses of the coin (keeping in mind that in the complete version, the expected length of the game is two tosses, i.e., the gamble is expected to terminate after the second trial, as shown in Section 4).

Regarding the comparison with the other experimental studies considered in Section 5, the present research seems to give credit to the “expectancy heuristic” hypothesis, that is, the same conclusion obtained by Bottom et al. (1989), and it also aligns with the idea of Neugebauer (2010), that small probabilities are neglected. However, the average amount that individuals are willing to pay to enter the game is higher than in these other studies. A possible explanation of this result is the composition of the population of students involved in the present research, in which a large number of participants were business students that show more “aggressive” behavior with respect to other groups of students.

6.3. The Simulation of the Game

As reported in Hayden and Platt (2009), in a rare application of empirical methods, in 1777, Buffon hired a child to flip a coin until it came up heads and to do so 2048 times. In 1838, De Morgan added another 2048 data points. De Morgan and Buffon both argued that actual experience demonstrates that the true expected value of the gamble is quite low, justifying the low value placed on the game. More recently, computers have made it possible to simulate coin flips more rapidly, and although estimated values are higher, the fundamental result does not change. Importantly, the results of these empirical studies are generally greater than the bids people intuitively offer.

With these observations in mind, in the second part of this study, a computer simulation of the game was developed. Using Python (see the Appendix A for the code), a program was written to simulate throwing a coin. If heads appears for the first time at the i-th toss, the game ends, and a payoff of $2^{i-1}$ is registered; otherwise, a new toss of the coin is performed and the game continues until heads appears. Moreover, at the end of all the simulations, the average number of coin tosses and the average payoff obtained are computed. Different blocks of simulations were performed (first, a block of n = 10,000 simulations, followed by another block of 20,000 simulations, and so on), taking into account that a single simulation starts with the toss of a coin and ends when heads appears for the first time. At the end of each block of simulations (initially after 10,000 simulations, then after 20,000 simulations, and so on), the program computes the average payoff obtained, that is

$$\overline{p}\left(n\right)={\displaystyle \frac{totalsumofpayoffs}{n}}$$

It also computes the average number of coin tosses, that is

$$\overline{f}\left(n\right)={\displaystyle \frac{totalnumberofcointosses}{n}}$$

Finally, the program displays, for each block of simulations, the maximum number of coin tosses required in that block to obtain heads for the first time and therefore to end the game (this is a measure of the “extreme events” in the game).

6.4. Results and Discussion of the Simulations

The results of the simulations are reported in

Table 3. Results of the simulations.

Number of Simulations	Average Payoff	Average Number of Tosses	Maximum Number of Tosses
10,000	9.31 €	1.97	15
20,000	7.28 €	1.98	15
50,000	7.95 €	1.98	17
100,000	9.51 €	2	18
500,000	11.54 €	2	20
1,000,000	11.14 €	2	20

.

In this table, the first column indicates the number n of simulations performed, the second column reports the average payoff obtained $\overline{p}\left(n\right)$, the third column displays the average number of coin tosses $\overline{f}\left(n\right)$, and the fourth column reports the maximum number of coin tosses necessary to obtain heads for the first time.

The results are also represented in Figure 3.

The analysis of these data enables other interesting insights relevant to our problem.

First, it is possible to note that as n increases (starting from n = 10,000 to n = 1,000,000 simulations), the average number of coin tosses converges to two, which, in effect, is the expected value of the number of tosses for this problem (as shown in Section 4). For example, when n = 10,000 simulations are performed, the game ends after $1.97$ tosses on average; when n increases to 20,000 and then to 50,000 simulations, the game ends after $1.98$ tosses on average; and when the number of simulations is increased further to n = 100,000 and greater, the average number of coin tosses that are necessary to end the game is 2. The behavior of the participating individuals that emerges from our experiment is in line with these results. Indeed, as outlined above, the sums that participants in the study are willing to pay to enter the game reveal that they expect the game will end after three or four tosses of the coin.

A second result that emerges from the simulation is that the average payoff obtained from the game shows more erratic behavior. When n = 10,000 simulations are performed, the average payoff of the game is EUR $9.31$, while when the number of simulations increases to $n=20,000$ and then to $n=50,000$, the average payoff turns out to be EUR $7.28$ and EUR $7.95$, respectively, and when n increases further to $100,000$ and then to $1,000,000$, the average payoff rises to EUR $9.51$ and EUR $11.14$, respectively. All these results are larger than the average amount individuals taking part in the experiment are willing to pay to enter the game, and this can be considered evidence of the fact that individuals are risk-averse and not risk-neutral (the same conclusion obtained by Cox et al. (2009)).

The final result is that the maximum number of coin tosses performed in each simulation before obtaining heads for the first time demonstrates clear monotonic behavior with respect to the number of simulations. When n = 10,000 or $20,000$ simulations are performed, the maximum length of the game is 15 tosses; when $n=50,000$ and then $100,000$ simulations are considered, this length increases to 17 and 18 tosses; and when $n=500,000$ and 1,000,000 simulations are performed, the length reaches 20 tosses.

In general, the conclusion that emerges from these simulations is that the number of coin tosses after which the first head appears is quite small (in fact, the expected number of tosses to end the game is two, i.e., we expect that the game ends after two tosses of the coin) and, as a consequence, the payoff obtained is also small. In our study, for instance, the number of simulations ranges from n = 10,000 to n = 1,000,000, and the corresponding average number of coin flips ranges from $1.97$ to 2, while the average payoff ranges from EUR $7.28$ to EUR $11.54$. There are, of course, the “extreme events”, reported in the last column of the table, when heads appears for the first time only after or around 20 tosses, but these events are precisely “extreme” and rare. In this sense, the behavior of the participating individuals who, as shown in the previous experiment, are willing to pay only a small amount to enter the game, is consistent with the results of the simulations. In particular, this behavior is in line with the “expectancy heuristic” hypothesis (the same kind of conclusion obtained by Bottom et al. (1989)), and also with neglecting small probabilities (which was the main conclusion derived by Neugebauer (2010)).

7. Conclusions

For three centuries, the Saint Petersburg paradox has attracted the attention of many researchers, and different explanations have been proposed. However, few empirical studies have addressed the question. The goal of our research was to fill this empirical gap, obtaining some insights in order to better understand the behavior of individuals with respect to the prediction of the theory and to explain possible discrepancies between theoretical results and actual decisions.

In order to reach this objective, a survey experiment was first designed, in which a large number of students was involved. This is the main strength with respect to previous studies, which considered smaller numbers of individuals taking part in the game. During this step, the participants were asked to evaluate different versions of the Saint Petersburg game, declaring the amount they were willing to pay to enter the game. In the second part of the study, a number of simulations were performed, and the results were compared with the real behavior of the individuals.

The main conclusions are that there are differences between younger (undergraduate) and older (Master’s level) students and between business-oriented and economics-oriented students, and the amounts that participants in the survey are willing to pay to enter the gamble are quite small. In this sense, our results are compatible with the main empirical studies devoted to this topic: individuals seem to be risk-averse (as found by Cox et al. (2009)); they follow quite precisely the “expectancy heuristic“ hypothesis (as outlined by Bottom et al. (1989)), and they neglect small probabilities (as found by Neugebauer (2010)). With reference to the research question addressed, in particular, it turns out that in the version of the problem that does not give rise to infinite expected payoff, the behavior of the individuals is consistent with the theory, while in the version with infinite expected payoff, the behavior is similar in the complete and in the truncated game.

With respect to previous studies, our research has the advantage of considering a large number of participants and combining the survey with a simulation of the problem. On the contrary, a possible weakness of our work is its lack of statistical and econometrics results (which are present in other studies), which therefore represents the main possible direction for future research.

In conclusion, we want to stress the main point of this study: the experiment reported shows that the individuals involved in the game expect that heads will appear for the first time (and the game will end) after three or four tosses, and for this reason, they are not disposed to pay more than EUR 3 to EUR 5 to participate in the game. The simulations show that the average number of tosses is around two (that is, in fact, the expected value of the number of coin flips in this game), and therefore, the behavior of the individuals is consistent with this result and not paradoxical at all.

Put in other terms, the individuals (correctly) neglect the very small probabilities associated with very large payoffs that are responsible for the infinite expected value of the game (in the complete version) and consider such probabilities to be 0. The paradox of the Saint Petersburg game, therefore, stems not from the behavior of the individuals but from the mathematical structure of the problem, i.e., from the fact that for a particular payoff structure (i.e., when the payoffs double at each subsequent round of the game), the expected value turns out to be infinite. It is the divergence of the series

$$\sum _{i=1}^{\infty }{x}_i{p}_i=\sum _{i=1}^{\infty }{2}^{i-1}·{\displaystyle \frac{1}{2^i}}=\sum _{i=1}^{\infty }{\displaystyle \frac{1}{2}}$$

that makes the expected value of the game infinite, which creates the paradox, not the behavior of the individuals. In this sense, the paradox is intrinsic to the game (and the solutions proposed to solve it are not completely satisfactory, since in any case they determine a change in the original structure of the problem, as outlined by Yukalov (2021)), and the only thing we can recognize, as Martin (2008) concludes, is that “The St. Petersburg result is strange…The appropriate reaction might just be to try to accept the strange result”.