Laws of large numbers for cooperative St. Petersburg gamblers

Summary 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.     

Merging asymptotic expansions for semistable random variables

Merging asymptotic expansions are established for distribution functions from the domain of geometric partial attraction of a semistable law. The length of the expansion depends on the exponent of the semistable law and on the characteristic function of the underlying distribution. We obtain sufficient conditions for the quantile function in order to get real infinite asymptotic expansion. The results are generalizations of the existing theory in the stable case.     

Computational group theory in St. Petersburg

The paper is devoted to the history of computational group theory in St. Petersburg. Some features of this scientific area are outlined. We mention here some results of A. I. Skopin concerned with computations in Burnside groups of small prime-power exponents, as well as A. V. Rukolaine's work on the computation of ordinary characters of some finite simple groups. The note is dedicated to the 70th anniversary of A. I. Skopin's birth. Bibliography: 24titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 42–49.     

Merging and splitting in cooperative games: some (im)possibility results

Allocation rules for cooperative games can be manipulated by coalitions merging into single players, or, conversely, players splitting into a number of smaller units. This paper collects some (im)possibility results on merging- and splitting-proofness of (core) allocation rules for cooperative games with side-payments.     

Tail asymptotic expansions for L-statistics

We derive higher-order expansions of L-statistics of independent risks X ₁, …,X _n under conditions on the underlying distribution function F. The new results are applied to derive the asymptotic expansions of ratios of two kinds of risk measures, stop-loss premium and excess return on capital, respectively. Several examples and a Monte Carlo simulation study show the efficiency of our novel asymptotic expansions.     

A Maxtrimmed St. Petersburg Game

Let \(S_n\), \(n\ge 1\), describe the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that \(\frac{S_n}{n\log _2 n}\mathop {\rightarrow }\limits ^{p}1\) as \(n\rightarrow \infty \). It is also known that almost sure convergence fails. However, Csörg? and Simons (Stat Probab Lett 26:65–73, 1996) have shown that almost sure convergence holds for trimmed sums, that is, for \(S_n-\max _{1\le k\le n}X_k\). Since our actual distribution is discrete there may be ties. Our main focus in this paper is on the “maxtrimmed sum”, that is, the sum trimmed by the random number of observations that are equal to the largest one. We prove an analog of Martin-Löf’s (J Appl Probab 22:634–643, 1985) distributional limit theorem for maxtrimmed sums, but also for the simply trimmed ones, as well as for the “total maximum”. In a final section, we interpret these findings in terms of sums of (truncated) Poisson random variables.     

Large gains in the St. Petersburg game

We consider the accumulated gains of geometric size in the St. Petersburg game and study the logarithmic tail asymptotics of their distribution. To cite this article: G. Stoica, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Embedding and asymptotic expansions for martingales

The paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants (given the original martingale) that are explicitly given in terms of optional and predictable variations of the original process. Bartlett identities for the conditional cumulants are also found. A main corollary to these results is the establishment of second (and in some cases higher) order asymptotic expansions for martingales.Research supported in part by National Science Foundation grant DMS 93-05601 and Army Research Office grant DAAH04-1-0105     

Accessibility measures to nodes of directed graphs using solutions for generalized cooperative games

The aim of this paper consists of constructing accessibility measures to the nodes of directed graphs using methods of Game Theory. Since digraphs without a predefined game are considered, the main part of the paper is devoted to establish conditions on cooperative games so that they can be used to measure accessibility. Games that satisfy desirable properties are called test games. Each ranking on the nodes is then obtained according to a pair formed by a test game and a solution defined on cooperative games whose utilities are given on ordered coalitions. The solutions proposed here are extensions of the wide family of semivalues to games in generalized characteristic function form.     

Weak laws of large numbers for cooperative gamblers

Based on a stochastic extension of Karamata’s theory of slowly varying functions, necessary and sufficient conditions are established for weak laws of large numbers for arbitrary linear combinations of independent and identically distributed nonnegative random variables. The class of applicable distributions, herein described, extends beyond that for sample means, but even for sample means our theory offers new results concerning the characterization of explicit norming sequences. The general form of the latter characterization for linear combinations also yields a surprising new result in the theory of slow variation. Work was supported in part by the Hungarian Scientific Research Fund, Grant T048360, and carried out within the Analysis and Stochastics Research Group of the Hungarian Academy of Sciences.     

A note on the St. Petersburg paradox

((Without abstract.))     

The selectope for cooperative games

The selectope of a cooperative transferable utility game is the convex hull of the payoff vectors obtained by assigning the Harsanyi dividends of the coalitions to members determined by so-called selectors. The selectope is studied from a set-theoretic point of view, as superset of the core and of the Weber set; and from a value-theoretic point of view, as containing weighted Shapley values, random order values, and sharing values. Received May 1997/Revised version September 1999     

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).     

Complete asymptotic expansions for the two-sample problem

Let x₁, x₂,...,x_n and y₁, y₂,...,y_n be the results of two series of independent observations. Let us denote by F_{R 1} (x) and G_{R 2} (y) the empirical distribution functions constructed on the basis of the first and the second sample, respectively. Let us write This paper deals with a complete asymptotic expansion, for the case n₁=n, n₂=np, n of the probability in a power series 1/n, where p1 is a fixed integer, and ₁>0 and ₂> 0 are fixed positive numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 53, pp. 4–53, 1975.     

Refined asymptotic expansions for nonlocal diffusion equations

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u _t = J * u – u in the whole with an initial condition u(x, 0) = u ₀(x). Under suitable hypotheses on J (involving its Fourier transform) and u ₀, it is proved an expansion of the form

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.     

Simulation-based collaborative decision support for surge floods prevention in St. Petersburg

The paper aims to implement a simulation-based collaborative decision support approach for flood control management in application to St. Petersburg surge floods, which are prevented by a complex of dams with several large openings. Despite the evolution of the numerical hydrodynamic models, hardware performance and computer technologies the accurate forecasting of storm surges and decision support for gates maneuvering is still an important issue. The prospective architecture and principal solutions of Flood Warning System with the emphasis on the simulation-based approach and collaborative decision support system on the basis of e-Science platform CLAVIRE are considered.     

Bargaining in cooperative games

This paper introduces a new solution concept for cooperative games without sidepayments, called the aspiration bargaining set, and proves that this set is nonempty for a wide, and economically interesting, class of games.