Petersburg Game Research Articles

Limit behaviors for a $\beta$-mixing sequence in the St. Petersburg game

Limit behaviors for a $\beta$-mixing sequence in the St. Petersburg game

Winning “Big” in the St. Petersburg Game

Winning “Big” in the St. Petersburg Game

The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences

This paper establishes complete convergence for weighted sums and the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables $$\{X,X_n,n\ge 1\}$$ with general normalizing constants under a moment condition that $$ER(X)<\infty $$ , where $$R(\cdot )$$ is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Stat Probab Lett 92:45–52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijn conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944–1946, 1995) on the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrative examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.

An asymmetric St. Petersburg game with trimming

Abstract Let Sn,n≥1, be the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that Sn∕(nlog2n)→ℙ1 as n→∞. In this paper we review some earlier results of ours and extend some of them as we consider an asymmetric St. Petersburg game, in which the distribution of the payoff X is given by ℙ(X=srk-1)=pqk-1,k=1,2,…, where p+q=1 and s,r&gt;0. Two main results are extensions of the Feller weak law and the convergence in distribution theorem of Martin-Löf (1985). Moreover, it is well known that almost-sure convergence fails, though Csörgő and Simons (1996) showed that almost-sure convergence holds for trimmed sums and also for sums trimmed by an arbitrary fixed number of maxima. In view of the discreteness of the distribution we focus on `max-trimmed sums', that is, on the sums trimmed by the random number of observations that are equal to the largest one, and prove limit theorems for simply trimmed sums, for max-trimmed sums, as well as for the `total maximum'. Analogues with respect to the random number of summands equal to the minimum are also obtained and, finally, for joint trimming.

A note on the asymptotics of the maxima for the St. Petersburg game

In this note, we consider the maxima of payoffs for the generalized St. Petersburg game. The maxima for the original St. Petersburg game cannot be normalized to converge to a nondegenerate limit distribution. However, tuning the parameters appearing in the generalization, we show the normalized maxima converge to the Fréchet distribution.

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 &lt; θ &lt; 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 &lt; θ &lt; 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.

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.

Fair pricing, and pricing paradoxes

The St Petersburg Paradox revolves round the determination of a fair price for playing the St Petersburg Game. According to the original formulation, the price for the game is infinite, and, therefore, paradoxical. Although the St Petersburg Paradox can be seen as concerning merely a game, Paul Samuelson (1977) calls it a “fascinating chapter in the history of ideas”, a chapter that gave rise to a considerable number of papers over more than 200 years involving fields such as probability theory and economics. In a paper in this journal, Vivian (2013) undertook a numerical investigation of the St Petersburg Game. In this paper, the central issue of the paradox is identified as that of fair (risk-neutral) pricing, which is fundamental in economics and finance and involves important concepts such as no arbitrage, discounting, and risk-neutral measures. The model for the St Petersburg Game as set out in this paper is new and analytical and resolves the so-called pricing paradox by applying a discounting procedure. In this framework, it is shown that there is in fact no infinite price paradox, and simple formulas for obtaining a finite price for the game are also provided.

Fair pricing, and pricing paradoxes

The St Petersburg Paradox revolves round the determination of a fair price for playing the St Petersburg Game. According to the original formulation, the price for the game is infinite, and, therefore, paradoxical. Although the St Petersburg Paradox can be seen as concerning merely a game, Paul Samuelson (1977) calls it a “fascinating chapter in the history of ideas”, a chapter that gave rise to a considerable number of papers over more than 200 years involving fields such as probability theory and economics. In a paper in this journal, Vivian (2013) undertook a numerical investigation of the St Petersburg Game. In this paper, the central issue of the paradox is identified as that of fair (risk-neutral) pricing, which is fundamental in economics and finance and involves important concepts such as no arbitrage, discounting, and risk-neutral measures. The model for the St Petersburg Game as set out in this paper is new and analytical and resolves the so-called pricing paradox by applying a discounting procedure. In this framework, it is shown that there is in fact no infinite price paradox, and simple formulas for obtaining a finite price for the game are also provided.

The St. Petersburg paradox and capital asset pricing

Durand (J Finance 12:348–363, 1957) shows that the classical St. Petersburg paradox can apply to the valuation of a firm whose dividends grow at a constant rate forever. To capture a more realistic pattern of dividends, we model the dividend growth rate as a mean reverting process, and then use the capital asset pricing model to derive the risk-adjusted present value. The model generates an equivalent St. Petersburg game. The long-run growth rate of the payoffs (dividends) is dominant in driving the value of the game (firm), and the condition under which the value is finite is less restrictive than that of the standard game.

Limit theorems for nonnegative independent random variables with truncation

We investigate asymptotic behavior of sums of independent and truncated random variables specified by P (0 \({\leqq X x) \asymp x^{-\alpha }}\) for \({\alpha > 0}\). By varying truncation levels we study strong laws of large numbers and central limit theorems. These are extensions of the results of Győrfi and Kevei [12] concerning the St. Petersburg game.

Limit distributions of generalized St. Petersburg games

The St. Petersburg game is known as a probability model with infinite expectation, and has some extensions. Gut and Martin-Löf (2013) studied convergence in distribution along a suitable specific subsequence under their generalization. In this article, we extend their results using residue analysis investigated by Vardi (1995). We give the explicit limit distributions via characteristic functions with gamma functions which become semi-stable or strictly semi-stable.

Extreme-trimmed St. Petersburg games

Let Sn, n≥1, describe the successive sums of the payoffs in the classical St. Petersburg game. Feller’s famous weak law, Feller (1945), states that Snnlog2n→p1 as n→∞. However, almost sure convergence fails, more precisely, lim supn→∞Snnlog2n=+∞ a.s. and lim infn→∞Snnlog2n=1 a.s. as n→∞. Csörgő and Simons (1996) have shown that almost sure convergence holds for trimmed sums, that is, for Sn−max1≤k≤nXk and, moreover, that this remains true if the sums are trimmed by an arbitrary fixed number of maximal sums. A predecessor of the present paper was devoted to sums trimmed by the random number of maximal summands. The present paper concerns analogs for the random number of summands equal to the minimum, as well as analogs for joint trimmings.

Considering the Pasadena “Paradox”

Nover and Hájek (2004) suggested a variant of the St Petersburg game which they dubbed the Pasadena game. They hold that their game ‘is more paradoxical than the St Petersburg game in several aspects’. The purpose of this article is to demonstrate theoretically and to validate by simulation, that their game does not lead to a paradox at all, let alone in the St Petersburg game sense. Their game does not produce inconsistencies in decision theory.

A Maxtrimmed St. Petersburg Game

Let S-n, n >= 1, describe the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that s(n)/n log(2)n ->(p) 1 as n -> infinity It is ...

On convergence of randomly indexed sequences; a counterexample based on the St. Petersburg game

If Y-1, Y-2,... is a sequence of random variables such that Y-n -> Y as n -> infinity, and {tau(t), t >= 0} is a family of indices such that tau(t) -> infinity t -> infinity, then ...

Ending the myth of the St Petersburg Paradox

Nicolas Bernoulli suggested the St Petersburg game, nearly 300 years ago, which is widely believed to produce a paradox in decision theory. This belief stems from a long standing mathematical error in the original calculation of the expected value of the game. This article argues that, in addition to the mathematical error, there are also methodological considerations which gave rise to the paradox. This article explains these considerations and why because of the modern computer, the same considerations, when correctly applied, also demonstrate that no paradox exists. Because of the longstanding belief that a paradox exists it is unlikely the mere mathematical correction will end the myth. The article explains why it is the methodological correction which will dispel the myth.

Capitalization in the St. Petersburg game

In spite of its infinite expectation value, the St. Petersburg game is not only a gamble without supply in the real world, but also one without demand at apparently very reasonable asking prices. We offer a rationalizing explanation of why the St. Petersburg bargain is unattractive on both sides (to both house and player) in the mid-range of prices (finite but upwards of about $4). Our analysis – featuring (1) the already-established fact that the average of finite ensembles of the St. Petersburg game grows with ensemble size but is unbounded, and (2) our own simulation data showing that the debt-to-entry fee ratio rises exponentially – explains why both house and player are quite rational in abstaining from the St. Petersburg game. The house will be unavoidably (and intentionally) exposed to very large ensembles (with very high averages, and so very costly to them), while contrariwise even the well-heeled player is not sufficiently capitalized (as our simulation data reveals) to be able to capture the potential gains from large-ensemble play. (Smaller ensembles, meanwhile, enjoy low means, as others have shown, and so are not worth paying more than $4 to play, even if a merchant were to offer them at such low prices per trial.) Both sides are consequently rational in abstaining from entry into the St. Petersburg market in the mid-range of asking prices. We utilize the concept of capitalization vis-à-vis a gamble to make this case. Classical analyses of this question have paid insufficient attention to the question of the propriety of using expected values to assess the St. Petersburg gamble. And extant analyses have not noted the average-maximum-debt-before-breaking-even figures, and so are incomplete.

Time Resolution of the St. Petersburg Paradox: A Rebuttal

Peters (2011) claims to provide a resolution of the three century old St. Petersburg paradox by using time averages and thereby avoiding the use of utility theory completely. Peters also claims to have found an error in Menger (1934, 1967) who established the vulnerability of any unbounded utility function to the St. Petersburg paradox. This paper argues that both these claims in Peters (2011) are incorrect. The time average argument can be circumvented by using a single random number (between zero and one) to represent the entire infinite sequence of coin tosses, or alternatively by applying a time reversal to the coin tossing. Menger’s proof can be reinstated by comparing the utility of playing the Super St. Petersburg game to the utility of an arbitrarily large sure payoff.