Petersburg Game Research Articles

Perplexing Expectations

This paper revisits the Pasadena game (Nover and Háyek 2004), a St Petersburg-like game whose expectation is undefined. We discuss serveral respects in which the Pasadena game is even more troublesome for decision theory than the St Petersburg game. Colyvan (2006) argues that the decision problem of whether or not to play the Pasadena game is ‘ill-posed’. He goes on to advocate a ‘pluralism’ regarding decision rules, which embraces dominance reasoning as well as maximizing expected utility. We rebut Colyvan’s argument, offering several considerations in favour of the Pasadena decision problem being well posed. To be sure, current decision theory, which is underpinned by various preference axioms, leaves indeterminate how one should value the Pasadena game. But we suggest that determinacy might be achieved by adding further preference axioms. We conclude by opening the door to a far greater plurality of decision rules. We suggest how the goal of unifying these rules might guide future research.

Laws of large numbers for cooperative St. Petersburg gamblers

General linear combinations of independent winnings in generalized \St~Petersburg games are interpreted as individual gains that result from pooling strategies of different cooperative players. A weak law of large numbers is proved for all such combinations, along with some almost sure results for the smallest and largest accumulation points, and a considerable body of earlier literature is fitted into this cooperative framework. Corresponding weak laws are also established, both conditionally and unconditionally, for random pooling strategies.

Simulating Daniel Bernoulli’s St. Petersburg game: Theoretical and empirical consistency

Russon and Chang simulated St. Petersburg games and found that their results were inconsistent with their theoretical predictions. In this article, the theoretical outcomes are derived this time using the methodology suggested by Daniel Bernoulli, and games are then simulated. When this is done, it is found that the theoretical and empirical results are consistent.

An Extension of the Kolmogorov–Feller Weak Law of Large Numbers with an Application to the St. Petersburg Game

The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables without finite mean is extended to a larger class of distributions, requiring regularly varying normalizing sequences. As an application we show that the weak law of large numbers for the St. Petersburg game is an immediate consequence of our result.

Solving Daniel Bernoulli’s St Petersburg paradox: The paradox which is not and never was

It has been accepted for over 270 years that the expected monetary value (EMV) of the St Petersburg game is infinite. Accepting this leads to a paradox; no reasonable person is prepared to pay the predicted large sum to play the game but will only pay, comparatively speaking, a very moderate amount. This paradox was ‘solved’ using cardinal utility. This article demonstrates that the EMV of the St Petersburg game is a function of the number of games played and is infinite only when an infinite number of games is played. Generally, the EMV is a very moderate amount, even when a large number of games is played. It is of the same order as people are prepared to offer to play the game. There is thus no paradox. Cardinal utility is not required to explain the behaviour of the reasonable person offering to play the game.

Merging to Semistable Laws

We show that, despite the lack of asymptotic distributions in the usual sense, the distribution functions of suitably centered and normalized partial sums of independent and identically distributed random variables from the domain of geometric partial attraction of a semistable law merge together uniformly with a family of semistable distribution functions. More generally, even the corresponding, possibly lightly trimmed sums merge. Analogous merge results hold also for sample extremes. The main result is illustrated on generalized St. Petersburg games.

Almost sure limit theorems for the St. Petersburg game

Abstract We show that the accumulated gain Sn and the maximal gain Mn in n St. Petersburg games satisfy almost sure limit theorems with nondegenerate limits, even though ordinary asymptotic distributions do not exist for Sn and Mn with any numerical centering and norming sequences.

The St. Petersburg paradox: A resolution for impatient risk seekers

If the St. Petersburg game is played in real time, gamblers will discount their possible payoffs subjectively due to impatience. Risk-neutral individuals and some risk seekers will not pay an infinite sum to play the game. For three different risk-seeking utility functions, this paper derives conditions under which a gambler, with each utility function, refuses to pay an infinite sum to play the game.

The St. Petersburg game and continued fractions

The St. Petersburg game is a well known example of a sequence of i.i.d. random variables with infinite expectation and was considered as a paradox since no single “fair” entry fee exists. This Note shows how the sequence of continued fraction digits of a random real number makes a reasonable choice of entry fees. Moreover, known results for continued fractions can be obtained for the St. Petersburg game using exactly the same proofs and these results explain exactly how the player is favoured even with a fair entry fee (thus resolving a point made by Aaronson).

A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games

Extending a result of Einmahl, Haeusler and Mason (1988), a characterization of the almost sure asymptotic stability of lightly trimmed sums of upper order statistics is given when the right tail of the underlying distribution with positive support is surrounded by tails that are regularly varying with the same index. The result is motivated by applications to cumulative gains in a sequence of generalized St. Petersburg games in which a fixed number of the largest gains of the player may be withheld.

Precision Calculation of Distributions for Trimmed Sums

Recursive methods are described for computing the frequency and distribution functions of trimmed sums of independent and identically distributed nonnegative integer-valued random variables. Surprisingly, for fixed arguments, these can be evaluated with just a finite number of arithmetic operations (and whatever else it takes to evaluate the common frequency function of the original summands). These methods give rise to very accurate computational algorithms that permit a delicate numerical investigation, herein described, of Feller's weak law of large numbers and its trimmed version for repeated St. Petersburg games. The performance of Stigler's theorem for the asymptotic distribution of trimmed sums is also investigated on the same example.

The limiting distribution of the St. Petersburg game

The St. Petersburg game is a well-known example of a random variable which has infinite expectation. Csörgő and Dodunekova have recently shown that the accumulated winnings do not have a limiting distribution, but that if measurements are taken at a subsequence b n {b_n} , then a limiting distribution exists exactly when the fractional parts of log 2 b n {\log _2}{b_n} approach a limit. In this paper the characteristic functions of these distributions are computed explicitly and found to be continuous, self-similar, nowhere differentiable functions.

Risk Aversion and Practical Expected Value: A Simulation of the St. Petersburg Game

Daniel Bernoulli 's classic St. Petersburg paradox is revisited in an attempt to resolve the intriguing issue; how should individual risk preference be differentiated? The authors argue that, unlike what is implied by the paradox, the fact that no person would pay a large sum of money in order to win a large payoff with a very small probability does not imply that all individuals are risk averse. The authors design a large-sample experimental test where the St. Petersburg game is simulated. The simulation results imply that the mathematical expectation of uncertain outcomes is not realized when associated with asymptotically diminishing probabilities. In such a case, it appears that there exists a "practical expected value " that is different from the theoretical expected value. The former, not the latter, must be used to determine individuals 'attitude toward risk.

Comment: The petersburg paradox. There we go again. A reply to Bierman and Rapoport

This note is a reply to the solution of a decision paradox by Harold Bierman, Jr. and Anatol Rapoport (Behavioral Science, October, 1989, 286–290), both expressing the need for psychological concepts to clarify the problem. It is argued that their games are special cases of the classical Petersburg game of which a solution, based exclusively on probability theory, is given by Feller (1970). Furthermore, it states that a serious practical problem remains to be solved.

Experts, Novices, and the St. Petersburg paradox: Is one solution enough?

AbstractAlthough the controversy over the correct solution to the St. Petersburg paradox continues in the decision making literature, few of the solutions have been empirically evaluated. Via the development of alternative versions of the St. Petersburg game, we were able to empirically test some of these solutions. Experts and novices behaved in accordance with Treisman's expectation heuristic when bidding for the right to play the various versions of the St. Petersburg game. When subjects were asked their preferences among the game versions. novices continued to behave in accordance with the expectation heuristic but a plurality of experts seemed to follow another strategy. This preference reversal and its implications and possible causes are thoroughly discussed. An alternative theory which mimicks the expectation heuristic is considered, and generalizations of the expectation heuristic and the St. Petersburg Paradox for z‐sided 'coins' (where z is any integer greater than or equal to 2) are presented. It appears that no one solution is yet rich enough for the St. Petersburg paradox.

Time, bounded utility, and the St. Petersburg paradox

The assumption of bounded utility function resolves the St. Petersburg paradox. The justification for such a bound is provided by Brito, who argues that limited time will bound the utility function. However, a reformulated St. Petersburg game, which is played for both money and time, effectively circumvents Brito's justification for a bound. Hence, no convincing justification for bounding the utility function yet exists.

A limit theorem which clarifies the ‘Petersburg Paradox'

The total gain, SN, in N successive Petersburg games is considered, and a limit theorem for SN/N – n when N = 2n and n → ∞is proved. The limit distribution can be determined numerically with good accuracy, and this fact provides a simple rule of thumb for determining a premium for the game, which is safe from the point of view of the casino.

A limit theorem which clarifies the ‘Petersburg Paradox'

The total gain, SN , in N successive Petersburg games is considered, and a limit theorem for SN/N – n when N = 2 n and n → ∞is proved. The limit distribution can be determined numerically with good accuracy, and this fact provides a simple rule of thumb for determining a premium for the game, which is safe from the point of view of the casino.

A Monte Carlo Simulation Related to the St. Petersburg Paradox

Most people would not be willing to pay very much to play. And yet, the expected payoff E[P] = 2*-i2*P/2*) is infinite. In [1], the author discusses how the (subjective) utility value of money may help to resolve this paradox. As indicated in ([3], Appendix to Chapter 3), Bernoulli suggested that one's utility value of monetary value x is B(x) = 5(lnx), whereas Cramer felt that the utility is C(x) =Jx . Therefore (as shown in [1]), the respective expected utilities of the St. Petersburg game are

On a Variant of the “Petersburg Game”

Previous article Next article On a Variant of the “Petersburg Game”T. L. RazinkovaT. L. Razinkovahttps://doi.org/10.1137/1115060PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout Previous article Next article FiguresRelatedReferencesCited byDetails Volume 15, Issue 3| 1970Theory of Probability & Its Applications History Submitted:01 July 1968Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1115060Article page range:pp. 538-540ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics