Petersburg Paradox Research Articles

THE RELATIONSHIP BETWEEN UTILITY AND RISK: EXAMPLES FROM ECONOMICS

When the issues in economics are analyzed deeply, the notion of utility is seen at the heart. As an important concept utility research has stayed popular so far. There are several research examining utility from different perspectives. One of the important topics that utility is associated with is risk preferences. Utility perception deviates from what economic theory predicts when it comes to make a choice among alternatives involving risk. Besides utility perception and risk preferences variate if the alternative is a monetary gain or a monetary loss violating principal rules on utility maximization. This study presents a detailed outlook of the concept of utility within the context of attitudes towards risk. It aims to explain that utility consideration differentiates according to the parameters of the situation rather than to the mathematical expected value calculation referring to the St. Petersburg Paradox.

From the St. Petersburg Paradox to the Dismal Theorem

This paper aims to consider the meaning of the dismal theorem, as is presented by Professor Martin Weitzman. 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 on 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 Need for Discontinuous Probability Weighting Functions: How Cumulative Prospect Theory is torn between the Allais Paradox and the St. Petersburg Paradox

Cumulative Prospect Theory (CPT) must embrace probability weighting functions with a discontinuity at probability zero to pass the two most prominent litmus tests for descriptive decision theories under risk: the Allais paradox and the St. Petersburg paradox. We prove in a nonparametric framework that, with continuous preference functions, CPT cannot explain both paradoxes simultaneously. Thus, Kahneman and Tversky’s (1979) originally proposed discontinuous probability weighting function has - when applied in a rank-dependent framework, of course - much more predictive power compared to all other popular, but continuous weighting functions, including e.g. Tversky and Kahneman's (1992) proposal. Neo-additive weighting functions constitute another parsimonious, yet promising class of discontinuous weighting functions. In other words, if we rashly restricted CPT to continuous preference functions we might erroneously jump to the conclusion that risk preferences are not stable over similar tasks or even reject CPT.

The median-based resolution of the St. Petersburg paradox

The St. Petersburg Paradox contributed to establishing expected utility theory by demonstrating that decision making based on the expectation (expected value, or mean, average) leads to an unreasonable behavior. Although the expected value is commonly used as an optimization criterion in various fields of mathematical sciences, such paradoxical problems as arbitrariness and intransitivity have led many researchers to forsake expected utility theory in search of more useful alternatives. Here we show an analytical solution of the St. Petersburg Paradox based on the median of the probability distribution. The present method provides a reasonable solution to any related problem. The median payout of repeated games suggests a new scaling relation in the limit of a large number of repetitions, while the game has no characteristic scale in terms of expected value.

Resolution of the St. Petersburg Paradox Using Von Mises’ Axiom of Randomness

In this article we will propose a completely new point of view for solving one of the most important paradoxes concerning game theory. The method used derives from the study of non-ergodic systems. This circumstance may create a dependency between results that are often extremely difficult to detect and quantify, such as in the field of finance. Consequently, the expected gain obtained from data that may be correlated has a statistical value that is difficult to determine, thus it cannot be used for decision-making purposes. Therefore, in this scenario, an alternative parameter to be use during the decision-making process must be found. The solution develop shifts the focus from the result to the strategy’s ability to operate in a cognitive way by exploiting useful information about the system. In order to determine from a mathematical point of view if a strategy is cognitive, we use Von Mises' axiom of randomness. Based on this axiom, the knowledge of useful information consequently generates results that cannot be reproduced randomly. Useful information in this case may be seen as a significant datum for the recipient, for their present or future decision-making process. In conclusion, the infinite behaviour in this paradox may be seen as an element capable of rendering the expected gain unusable for decision-making purposes. As a result, we are forced to face the problem by employing a different point of view. In order to do this we shift the focus from the result to the strategy’s ability to operate in a cognitive way by exploiting useful information about the system. Finally, by resolving the paradox from this new point of view, we will demonstrate that an expected gain that tends toward infinity is not always a consequence of a cognitive and non-random strategy. Therefore, this result leads us to define a hierarchy of values in decision-making, where the cognitive aspect, whose statistical consequence is a divergence from random behaviour, turns out to be more important than the expected gain.

Strong approximation and a central limit theorem for St. Petersburg sums

The St. Petersburg paradox (Bernoulli, 1738) concerns the fair entry fee in a game where the winnings are distributed as P(X=2k)=2−k,k=1,2,…. The tails of X are not regularly varying and the sequence Sn of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-Löf, 1985; Csörgő and Dodunekova, 1991). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that Sn can be approximated by a semistable Lévy process {L(n),n≥1} with a.s. error O(n(logn)1+ε) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.

Role of the St. Petersburg Paradox in Decision-Making

The article considers the appearance and modern interpretations of the St. Petersburg paradox. A review of the main solutions of the St. Petersburg paradox and an attempt to define their significance for the economic theory have been made. The impact of this problem on the economic theory can be seen by the example of such provisions as the principle of diminishing marginal utility, the use of expected utility as a criterion for decision-making in the context of uncertainty, as well as the basics of microeconomics of insurance and risk management, game theory, and some approaches to financial modeling.

Expected utility analysis of infinite compound lotteries

ABSTRACTLotteries can be used to model alternatives with uncertain outcomes. Decision theory uses compound ordinary lotteries to represent a structure of lotteries within lotteries, but can only rank the finite compound lottery structure. We expand upon this approach to introduce solutions for infinite compound ordinary lotteries (ICOL). We describe a novel procedure to simplify any ICOL as much as possible to a maximum reduced ICOL, which is not a unique representation. We limit our discussion to ICOLs of first order, which are defined as maximum reduced ICOLs with a single maximum reduced ICOL in their direct outcome. Two special cases of ICOLs of first order are discussed. These are recursive and semi-recursive ICOLs. We provide an analytical approach to find the expected utility of recursive ICOLs, and a numerical algorithm for semi-recursive ICOLs. We demonstrate our solution methods by evaluating example decision problems involving: a randomizing device with unsuccessful trials, the St. Petersburg paradox, and training with virtual reality.

The St. Petersburg paradox despite risk-seeking preferences: an experimental study

The St. Petersburg paradox is one of the oldest challenges of expected value theory. Thus far, explanations of the paradox aim at small probabilities being perceived as zero and the boundedness of utility of the outcome. This paper provides experimental results showing that neither diminishing marginal utility of the outcome nor perception of small probabilities can explain the paradox. We find that even in situations where subjects are risk-seeking, and zeroing-out small probabilities supports risk-taking, the St. Petersburg paradox exists. This indicates that the paradox cannot be resolved by the arguments advanced to date.

Замечание о Санкт-Петербургском парадоксе

St. Petersburg paradox, formulated by N. Bernoulli in the early 18th century, led to defining the utility function (D. Bernoulli, G. Cramer) as a way to resolve the paradox and played an important role in the development of decision making theory. In the 20th century, the paradox attracted the attention of many researchers, including Nobel Prize winners P. Samuelson, R. Aumann, L. Shapley. N. Bernoulli assumed that payments grow exponentially with the coin toss number. The growth rate of payments is higher than the exponential one in the generalized St. Petersburg paradox. The utility functions of Bernoulli and Cramer don't lead to the resolution of the paradox in this case. In 1934, K. Menger showed the necessity and sufficiency of the boundedness of the utility function for resolving of the generalized St. Petersburg paradox. A brief overview of the subject matter is given, as well as the autors' approach to resolving the classical paradox, based on discounting cash flows, in which the time intervals between consecutive coin tossings play a special role. The adaptation of the proposed approach to the generalized St. Petersburg paradox is also described. The proposed approach is an alternative to the traditional based utility function. It allows to solve, in particular, the inverse problem: to find (ambiguous solution) the moments of possible payments according to the set sizes of payments, the force of interest and the price of the game.

Demonetization: Did India Face the St. Petersburg Paradox and Lose?

In 2016, India demonetized 24% of its currency notes (viz. 86% in cash value) in circulation presuming this will remove “black” (illicit) and counterfeit cash holdings, and combat money laundering. This was the largest demonetization experiment in recent history. Although demonetization has occurred several times before, no unambiguous economic argument for, or against, it exists. This was a key enabler for demonetization, yet again. This paper argues that the St. Petersburg Paradox (SPP) provides a compelling argument against demonetization. Assuming the distribution of cash is lognormal, it is shown that the probability of black cash holdings will be small. If not, the holders would: a) be irrational because they are willing to accept, contrary to the SPP, the small probability of a large loss, by effectively perceiving it as zero, without using all means to immunize themselves against it; or b) be sure their cash can be legitimized via collusion with the State; or c) be sure they can incentivize law-abiding citizens to act as agents to legitimize the cash for a reasonable fee. Assuming rationality and no bureaucratic support, large probabilities of black cash holdings imply that many more law-abiding patriotic citizens have to be corruptible than seems rational.

R for Risk

The concepts of risk and uncertainty are rather subtle, and so are their applicability to the social sciences. By reviewing the historically significant example of the St. Petersburg Paradox, this note suggests that when mathematical models of rationality and intuitions about rationality conflict, sometimes the former must be bent to fit the latter. This line of reasoning is all the more relevant when confronting the kind of uncertainty which many refer to as “ambiguity”.

Intrinsic Uncertainty - An Explanation of the St. Petersburg Paradox

Consider any situation involving uncertainty, where the random variable of interest (e.g., payoff) is X. Let there exist a random variable, say Y, which represents the uncertainty intrinsic to the situation, and let there exist a function g such that X=g(Y). Our contention is that, once the intrinsic uncertainty has been identified, the relevant summary value is g(E(Y)), even though this may not be equal to the expectation E(g(Y)) of the variable of interest. We discuss two examples in which we can identify the random variable representing the intrinsic uncertainty. One of these examples is the St. Petersburg Paradox, where we argue that the intrinsic uncertainty is represented by the length of the game, and therefore, the variable of interest is the (finite) payoff received at the expected length of the game, as opposed to the infinite expected payoff of the game. As a result, we are able to resolve the St. Petersburg Paradox even under risk neutrality, i.e., without recourse to the strong risk aversion represented by a bounded utility function in the expected utility framework.

Given enough eyeballs, all bugs are shallow? Revisiting Eric Raymond with bug bounty programs

Bug bounty programs offer a modern platform for organizations to crowdsource their software security and for security researchers to be fairly rewarded for the vulnerabilities they find. Little is known however on the incentives set by bug bounty programs: How they drive new bug discoveries, and how they supposedly improve security through the progressive exhaustion of discoverable vulnerabilities. Here, we recognize that bug bounty programs create tensions, for organizations running them on the one hand, and for security researchers on the other hand. At the level of one bug bounty program, security researchers face a sort of St-Petersburg paradox: The probability of finding additional bugs decays fast, and thus can hardly be matched with a sufficient increase of monetary rewards. Furthermore, bug bounty program managers have an incentive to gather the largest possible crowd to ensure a larger pool of expertise, which in turn increases competition among security researchers. As a result, we find that researchers have high incentives to switch to newly launched programs, for which a reserve of low-hanging fruit vulnerabilities is still available. Our results inform on the technical and economic mechanisms underlying the dynamics of bug bounty program contributions, and may in turn help improve the mechanism design of bug bounty programs that get increasingly adopted by cybersecurity savvy organizations.

Entrance Fees and a Bayesian Approach to the St. Petersburg Paradox

In An Introduction to Probability Theory and its Applications, W. Feller established a way of ending the St. Petersburg paradox by the introduction of an entrance fee, and provided it for the case in which the game is played with a fair coin. A natural generalization of his method is to establish the entrance fee for the case in which the probability of heads is θ ( 0 < θ < 1 / 2 ) . The deduction of those fees is the main result of Section 2. We then propose a Bayesian approach to the problem. When the probability of heads is θ ( 1 / 2 < θ < 1 ) the expected gain of the St. Petersburg game is finite, therefore there is no paradox. However, if one takes θ as a random variable assuming values in ( 1 / 2 , 1 ) the paradox may hold, which is counter-intuitive. In Section 3 we determine necessary conditions for the absence of paradox in the Bayesian approach and in Section 4 we establish the entrance fee for the case in which θ is uniformly distributed in ( 1 / 2 , 1 ) , for in this case there is a paradox.

Strong approximation of the St. Petersburg game

ABSTRACTLet be i.i.d. random variables with and let . The properties of the sequence have received considerable attention in the literature in connection with the St. Petersburg paradox (Bernoulli 1738). Let be a semistable Lévy process with underlying Lévy measure . For a suitable version of and , we prove the strong approximation a.s. This provides the first example for a strong approximation theorem for partial sums of i.i.d. sequences not belonging to the domain of attraction of the normal or stable laws.

Demonetization: did India face the St. Petersburg paradox and lose

In 2016, India demonetized 24% of its currency notes (viz. 86% in cash value) in circulation presuming this will remove “black” (illicit) and counterfeit cash holdings, and combat money laundering. This was the largest demonetization experiment in recent history. Although demonetization has occurred several times before, no unambiguous economic argument for, or against, it exists. This was a key enabler for demonetization, yet again. This paper argues that the St. Petersburg Paradox (SPP) provides a compelling argument against demonetization. Assuming the distribution of cash is lognormal, it is shown that the probability of black cash holdings will be small. If not, the holders would: a) be irrational because they are willing to accept, contrary to the SPP, the small probability of a large loss, by effectively perceiving it as zero, without using all means to immunize themselves against it; or b) be sure their cash can be legitimized via collusion with the State; or c) be sure they can incentivize law-abiding citizens to act as agents to legitimize the cash for a reasonable fee. Assuming rationality and no bureaucratic support, large probabilities of black cash holdings imply that many more law-abiding patriotic citizens have to be corruptible than seems rational.

From St. Petersburg to Monte Carlo: A Resolution Rediscovered and Affirmed

The St. Petersburg paradox has served as an intriguing pedagogical device for almost 300 years, since it was first described by Daniel Bernoulli in 1738. Many explanations of the paradox have been offered, most involving either theories of individual behavior or the practical limitations of playing the game. We do not revisit any of the conventional explanations. Rather, we endeavor to demonstrate that the premise underlying the paradox – that the St. Petersburg game’s expected value is infinite – is mistaken. The analysis focuses on how the probability distribution of the game’s duration determines and limits the expected value. We do not claim precedence on so fundamental a resolution of the paradox. Our analysis is guided by the approach first described by the mathematician William Feller in 1945. We conduct a massive array of computer simulations within the framework suggested by Feller’s approach. The simulation results allow for an empirical evaluation of his coherent resolution.

A Central Limit Theorem for Costs in Bulinskaya’s Inventory Management Problem When Deliveries Face Delays

It is common in inventory theory to consider policies that minimize the expected cost of ordering and holding goods or materials. Nevertheless, the realized cost is a random variable, and, as the Saint Petersburg Paradox reminds us, the expected value does not always capture the full economic reality of a decision problem. Here we take the classic inventory model of Bulinskaya (1964), and, by proving an appropriate central limit theorem, we show in a reasonably rich (and practical) sense that the mean-optimal policies are economically appropriate. The motivation and the tools are applicable to a large class of Markov decision problems.

Bounded, Sigmoid Utility for Insurance Applications

Abstract Applying a well-known argument of Karl Menger to an insurance version of the St. Petersburg Paradox (in which the decision maker is confronted with losses, rather than gains), one can assert that von Neumann-Morgenstern utility functions are necessarily concave upward and bounded below as decision-maker wealth tends to negative infinity. However, this argument is subject to two potential criticisms: (1) infinite-mean losses do not exist in the real world; and (2) the St. Petersburg Paradox derives its force from empirical observation (i. e., that actual decision makers would not agree to an arbitrarily large insurance bid price to transfer an infinite-mean loss), and thus does not impart logical necessity. In the present article, these two criticisms are addressed in turn. We first show that, although infinite-mean insurance losses technically do not exist, they do provide a reasonable model for certain large (i. e., excess and reinsurance) property-liability indemnities. We then employ the Two-Envelope Paradox to demonstrate the logical necessity of concave-upward, lower-bounded utility for arbitrarily small (i. e., negative) values of wealth. Finally, we note that recognizing the bounded, sigmoid nature of utility functions challenges certain fundamental understandings in the economics of insurance demand, and can lead to vastly different conclusions regarding the bid price for insurance.