Petersburg Paradox Research Articles

A numerical evaluation of the Finite Monkeys Theorem

The Infinite Monkeys Theorem has long-established the eventual certainty of the complete works of William Shakespeare being reproduced by a monkey randomly pressing keys on a typewriter. This only considers the infinite limit, with either an infinite number of monkeys and/or an infinite time period of monkey labour. Here, we consider the Finite Monkeys Theorem and look at the probability of a given string being typed by one of a finite number of monkeys within a finite time allocation consistent with estimates for the lifespan of our universe. We also calculate the expected number of keystrokes until a target string would first be produced. Given the expected time until the heat death of the universe, we demonstrate that the widely-accepted conclusion from the Infinite Monkeys Theorem is, in fact, misleading in our finite universe. As such, this places the theorem in a class of probabilistic problems or paradoxes, including the St. Petersburg paradox, Zeno's dichotomy paradox and the Ross–Littlewood paradox wherein the infinite-resource conclusions directly contradict those obtained when considering limited resources, however sizeable.

Making valuations with the priority heuristic

The priority heuristic is a lexicographic semi-order for choosing between gambles. It has merits such as predicting, out-of-sample, people's majority choice more accurately than benchmarks such as prospect theory, having been axiomatized, and logically implying major violations of expected utility theory. The heuristic has shortcomings too, such as failing to account for individual differences and intricate choice patterns, and predicting less accurately than various model ensembles and neural networks in some environments. This note focuses on an important purported shortcoming of the heuristic, that it cannot produce valuations of gambles. I point out that the certainty equivalent of a gamble for the priority heuristic is known and suggest that this fact can be used to enhance the scope of the heuristic. Indeed, by making simple auxiliary assumptions and calculations, I demonstrate that the priority heuristic can explain the Saint Petersburg paradox and the equity premium puzzle, and to do so arguably more parsimoniously and plausibly than standard approaches.

A Guide to the Risk-Averse Gambler and Resolving the St. Petersburg Paradox Once and For All

Summary We use three kinds of computations: simulation, numeric, and symbolic, to guide risk-averse gamblers in general, and we offer particular advice on how to resolve the famous St. Petersburg paradox.

How to Save Pascal (and Ourselves) From the Mugger

Abstract In this article, we re-examine Pascal's Mugging, and argue that it is a deeper problem than the St. Petersburg paradox. We offer a way out that is consistent with classical decision theory. Specifically, we propose a “many muggers” response analogous to the “many gods” objection to Pascal's Wager. When a very tiny probability of a great reward becomes a salient outcome of a choice, such as in the offer of the mugger, it can be discounted on the condition that there are many other symmetric, non-salient rewards that one may receive if one chooses otherwise.

Understanding Ramsey-de Finetti Probabilities and the St. Petersburg Paradox

We provide simple interpretations of probability with a view towards its potential for real world (i.e., practical) applications. Our discussion is along the lines initiated by David Ramsey and Bruno de Finetti, though we do not provide a theory of probability but merely use their axioms and results to interpret mathematical probability. Subsequently, we provide a critique as well as a resolution of the well-known St. Petersburg paradox that is consistent with expected utility theory.

Rogue ocean waves and the St. Petersburg paradox.

Powerful rogue ocean waves have been objects of fascination for centuries. Elusive and awe-inspiring, with the potential to inflict catastrophic damage, rogue waves remain unpredictable and imperfectly understood. To gain further insight into their behavior, we analyzed 3 441 188 683 ocean surface waves to determine the statistical height distribution of the largest waves. We found that the distribution of rare events which resolves the St. Petersburg paradox also describes the relative height distribution of the largest waves. This result is expected to contribute to the modeling of ocean surface dynamics and improve the accuracy of marine weather forecasts.

St. Petersburg paradox: adoption of decisions on the basis of data mining and development of software in the sphere of business analytics

This article is devoted to analysis of models of St. Petersburg paradox, as well as development of software in the sphere of business analysis. This work is based on mathematical models using theories of probability and games as well as expert survey method. It is demonstrated that the St. Petersburg paradox is a mathematical problem of probability theory with artificial conditions. The influence of this problem on economical theory is exemplified by such provisions as the principle of diminishing marginal utility, the use of expected utility as criterion of decision adoption in uncertain environment, as well as foundations of microeconomics of insurance and risk management, theory of games and some approaches to financial simulation. Adoption of decisions on the basis of the St. Petersburg paradox is analyzed. Review of main decisions of the St. Petersburg paradox and their influence for economic theory has confirmed that the St. Petersburg paradox as a mathematical problem can be used as mathematical model upon implementation of financial simulation. Comparative analysis of available BI solutions has confirmed that most of them propose all major functions, and significant differences can be revealed in penetration of expanded functions.

A resolution of St. Petersburg paradox

The St. Petersburg paradox is the oldest paradox in decision theory and has played a pivotal role in the introduction of increasing concave utility functions embodying risk aversion and decreasing marginal utility of gains. All attempts to resolve it have considered some variants of the original set-up, but the original paradox has remained unresolved, while the proposed variants have introduced new complications and problems. Here a rigorous mathematical resolution of the St. Petersburg paradox is suggested based on a probabilistic approach to decision theory.

A St Petersburg Paradox for risky welfare aggregation

Abstract The principle of Anteriority says that prospects that are identical from the perspective of every possible person’s welfare are equally good overall. The principle enjoys prima facie plausibility, and has been employed for various theoretical purposes. Here it is shown using an analogue of the St Petersburg Paradox that Anteriority is inconsistent with central principles of axiology.

Extreme value theory and the St. Petersburg paradox in the failure statistics of wires

The fracture stress of materials typically depends on the sample size and is traditionally explained in terms of extreme value statistics. A recent work reported results on the carrying capacity of long polyamide and polyester wires and interpret the results in terms of a probabilistic argument known as the St. Petersburg paradox. Here, we show that the same results can be better explained in terms of extreme value statistics. We also discuss the relevance of rate dependent effects.

The Standard Model of Rational Risky Decision-Making

Expected utility theory (EUT) is currently the standard framework which formally defines rational decision-making under risky conditions. EUT uses a theoretical device called von Neumann–Morgenstern utility function, where concepts of function and random variable are employed in their pre-set-theoretic senses. Any von Neumann–Morgenstern utility function thus derived is claimed to transform a non-degenerate random variable into its certainty equivalent. However, there can be no certainty equivalent for a non-degenerate random variable by the set-theoretic definition of a random variable, whilst the continuity axiom of EUT implies the existence of such a certainty equivalent. This paper also demonstrates that rational behaviour under utility theory is incompatible with scarcity of resources, making behaviour consistent with EUT irrational and justifying persistent external inconsistencies of EUT. A brief description of a new paradigm which can resolve the problems of the standard paradigm is presented. These include resolutions of such anomalies as instant endowment effect, asymmetric valuation of gains and losses, intransitivity of preferences, profit puzzle as well as the St. Petersburg paradox.

Did COVID-19 Force a President to Face the St. Petersburg Paradox and Lose the White House?

It is said that a strategy to contain the COVID-19 pandemic is to “follow the Science”. This paper argues that following the Science, although very important, is secondary to following the probability of infection or death, however small that probability might appear to be at a given time. Following the Science is difficult, because it takes time to understand phenomena; and there are many hurdles to overcome in order to effectively disseminate it to the people. It is far easier for a Leader to invoke the St. Petersburg Paradox (SPP) and, by being a Teacher to the people, disseminate the dangers of staking too much by ignoring small probabilities. The SPP, which has remained, for more than 300 years now, an important and unsolved problem in Economics and Philosophy, vividly, and simply, teaches us to not assume that we will emerge victorious when facing small probabilities. It is argued that a Leader facing small probabilities, generated by this most formidable pathogen, can mitigate a very large loss by urging them to pay the relatively small cost of taking preventative measures to combat the pathogen. In the presence of such small probabilities, Science can never reach the heart of a nation as effectively, and speedily, as can a Leader invoking the SPP. Although it is human to be impatient, a small probability can grow explosively in a short time, leaving unimaginable losses in its wake, if there is no Teacher to alert us to that fact.

A Model of Anchoring and Adjustment for Decision-Making under Risk

We introduce a general model of anchoring and adjustment for decision-making under risk. To evaluate a lottery, agents first anchor on the simple mean of the lottery's outcomes, ignoring the outcomes' probabilities. Then, they adjust the anchor (insufficiently) in the direction of the lottery's expected utility. The resulting model implies behavior which is observationally similar to established prospect theory models. However, it is not rank-dependent and naturally applies to lotteries with any given number of outcomes, including continuous lotteries. We consider a number of applications to demonstrate the model's range. For the discrete case, we show how the model can explain four commonly-observed choice anomalies - both Allais paradoxes, the St. Petersburg paradox, event-splitting effects, and the effect of probability in the fourfold pattern of risk attitudes. For the continuous case, we show how the anchoring model can be applied to the equity premium puzzle. We also use historical data on asset returns to show that equity premiums can be explained by reasonable combinations of preference parameters. The anchoring model is an attractive and flexible tool that assumes people want to simplify complex decision processes.

Infinite Prospects*

AbstractPeople with the kind of preferences that give rise to the St. Petersburg paradox are problematic—but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the Countable Sure Thing Principle, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call Countable Independence. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have.

St. Petersburg Paradox and Failure Probability.

The St. Petersburg paradox provides a simple paradigm for systems that show sensitivity to rare events. Here, we demonstrate a physical realization of this paradox using tensile fracture, experimentally verifying for six decades of spatial and temporal data and two different materials that the fracture force depends logarithmically on the length of the fiber. The St. Petersburg model may be useful in a variety fields where failure and reliability are critical.

From the St. Petersburg paradox to the dismal theorem

AbstractThis paper aims to consider the meaning of the dismal theorem, as presented by Martin Weitzman [(2009) On modeling and interpreting the economics of catastrophic climate change. Review of Economics and Statistics91, 1–19]. The theorem states that a standard cost–benefit analysis breaks down if there is a possibility of catastrophes occurring. This result has a significant influence on debates regarding the economics of climate change. In this study, we present an intuitive similarity between the dismal theorem and the St. Petersburg paradox using a simple discrete probability distribution.

The St. Petersburg Paradox: A Fresh Algorithmic Perspective

The St. Petersburg paradox is a centuries-old puzzle concerning a lottery with infinite expected payoff on which people are only willing to pay a small amount to play. Despite many attempts and several proposals, no generally-accepted resolution is yet at hand. In a recent paper, we show that this paradox can be understood in terms of the mind optimally using its limited computational resources (Nobandegani et al. 2019). Specifically, we show that the St. Petersburg paradox can be accounted for by a variant of normative expected-utility valuation which acknowledges cognitive limitations: sample-based expected utility (Nobandegani et al. 2018). SbEU provides a unified, algorithmic explanation of major experimental findings on this paradox. We conclude by discussing the implications of our work for algorithmically understanding human cognition and for developing human-like artificial intelligence.

Resolution of St. Petersburg Paradox

The St. Petersburg paradox is the oldest paradox in decision theory and has played a pivotal role in the introduction of increasing concave utility functions embodying risk aversion and decreasing marginal utility of gains. All attempts to resolve it have considered some variants of the original set-up, but the original paradox has remained unresolved, while the proposed variants have introduced new complications and problems. Here a rigorous mathematical resolution of the St. Petersburg paradox is suggested based on a probabilistic approach to decision theory.

A Resource-Rational, Process-Level Account of the St. Petersburg Paradox.

The St. Petersburg paradox is a centuries-old philosophical puzzle concerning a lottery with infinite expected payoff for which people are only willing to pay a small amount to play. Despite many attempts and several proposals, no generally accepted resolution is yet at hand. In this work, we present the first resource-rational, process-level explanation of this paradox, demonstrating that it can be accounted for by a variant of normative expected utility valuation which acknowledges cognitive limitations. Specifically, we show that Nobandegani et al.'s (2018) metacognitively rational model, sample-based expected utility (SbEU), can account for major experimental findings on this paradox. Crucially, our resolution is consistent with two empirically well-supported assumptions: (a) People use only a few samples in probabilistic judgments and decision-making, and (b) people tend to overestimate the probability of extreme events in their judgment. Our work seeks to understand the St. Petersburg gamble as a particularly risky gamble whose process-level explanation is consistent with a broader process-level model of human decision-making under risk.

Методологические основания анализа, оценки и управления риском

The paper aims to represent the author’s methodological position that risk is a consequence of the decisions made by the person and is always associated with the person who not only makes a choice, but also evaluates the probabilities of possible events and associated losses. Taking a risk, the person chooses an alternative that is the result of her decision, although the possible results are not exactly known to her. The key to managing risk is the issue of measuring it. The law of large numbers by J. Bernoulli and related methodological and philosophical issues are considered. The significance of the St. Petersburg paradox for the modern understanding of risk is shown. Our thesis is that decision-making under the conditions of risk is not a person’s collision with circumstances independent of her, but a conscious and rational choice.