A general solution concept for two-person,zero-sum games

In this paper, finitely additive mixed extensions of two-person zero-sum games are studied, where the players use probability contents as mixed strategies. It is well-known that symmetric finitely additive mixed extensions always have a saddle point, a finitely additive solution. We generalize this solution concept and assign a finitely additive solution to every bounded two-person, zero-sum game.     

The extent of non-conglomerability of finitely additive probabilities

Summary An arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive (PFA) one. The coefficient of the PFA probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition. If the probability is defined on a -field, the bound is sharp. Hence, non-conglomerability (or equivalently non-disintegrability) characterizes finitely as opposed to countably additive probability. Nonetheless, there exists a PFA probability which is simultaneously conglomerable over an arbitrary finite set of partitions.Neither conglomerability nor non-conglomerability in a given partition is closed under convex combinations. But the convex combination of PFA ultrafilter probabilities, each of which cannot be made conglomerable in a common margin, is singular with respect to any finitely additive probability that is conglomerable in that margin.     

Finitely additive and epsilon Nash equilibria

We prove the existence of a mixed strategy Nash equilibrium in normal form games when the space of mixed strategies consists of finitely additive probability measures. It is then proved that from this result an existence result for epsilon equilibria with countably additive mixed strategies can be obtained. These results are applied to the classic Cournot game.     

A general principle for limit theorems in finitely additive probability: The dependent case

In this paper we formulate and prove a general principle which enables us to deduce limit theorems for a sequence of random variables on a finitely additive probability space.     

Quasi-martingales with a linearly ordered index set

We consider quasi-martingales indexed by a linearly order set. We show that such processes are isomorphic to a given class of (finitely additive) measures. From this result we easily derive the classical theorem of Stricker as well as the decompositions of Riesz, Rao and the supermartingale decomposition of Doob and Meyer.     

On almost sure convergence in a finitely additive setting

In their book How To Gable If You Must, Dubins and Savage introduced finitely additive stochastic processes in discrete time and they obtained some results about finitely additive probability measures on infinite product spaces. In the paper, Some Finitely Additive Probability, Purves and Sudderth showed how to extend these finitely additive probability measures and it thus became possible to consider many of the classical strong convergence theorems. In this paper, we extend many of the classical strong convergence theorems to a finitely additive setting. Since all proofs in this paper are valid for a countably additive setting if we consider the problem on a coordinate representation process, the results in this paper are generalizations of the classical results on such a process. Some examples are also provided for contrasting a finitely additive setting with a countably additive setting.This research was based in part on the author's doctoral dissertation submitted to the University of Minnesota and was written with the partial support of the U.S. Army Grant DA-ARO-D-31-124-70-G-102     

Expected utility for nonstochastic risk

Stochastic random phenomena considered in von Neumann–Morgenstern utility theory constitute only a part of all possible random phenomena (Kolmogorov, 1986). We show that any sequence of observed consequences generates a corresponding sequence of frequency distributions, which in general does not have a single limit point but a non-empty closed limit set in the space of finitely additive probabilities. This approach to randomness allows to generalize the expected utility theory in order to cover decision problems under nonstochastic random events. We derive the maxmin expected utility representation for preferences over closed sets of probability measures. The derivation is based on the axiom of preference for stochastic risk, i.e. the decision maker wishes to reduce a set of probability distributions to a single one. This complements Gilboa and Schmeidler’s (1989) consideration of the maxmin expected utility rule with objective treatment of multiple priors.     

A Metric on Probabilities, and Products of Loeb Spaces

Two functions on finitely additive probability spaces that behavewell under products are introduced: discrepancy, which measureshow close one space comes to extending another, and bi-discrepancy,which is a pseudo-metric on the collection of all spaces ona given set, and a metric on the collection of complete spaces.These are then applied to show that the Loeb space of the internalproduct of two internal finitely additive probability spacesdepends only on the Loeb spaces of the two original internalspaces. Thus the notion of a Loeb product of two Loeb spacesis well defined. The Loeb operation induces an isometry fromthe nonstandard hull of the space of internal probability spaceson a given set to the space of Loeb spaces on that set, withthe metric of bi-discrepancy.     

Finitely additive,modular, and probability functions on pre-Semirings

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical results in probability theory such as the law of total probability, Bayes’ theorem, the equality of parallel systems, and Poincaré’s inclusion-exclusion theorem. While we prove that modular functions over a couple of known semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring (Id(D),+,?), where D is a Dedekind domain.     

A simple characterization of almost uniform convergence by stochastic convergence

Motivated by Egorov's theorem and the characterization of the equivalence of P-stochastic convergence and P-almost convergence by the property of the probability distribution P to be purely atomic and concentrated on a countable number of pairwise disjoint P-atoms (cf. [1], p. 68), it is proved that P-stochastic resp. P-almost convergence is equivalent to P-almost uniform convergence (cf. [2], p. 89/90) if and only if P is purely atomic and concentrated on a finite number of pairwise disjoint P-atoms. Furthermore, this property of P is equivalent to the condition that any P-stochastic convergent sequence admits a P-almost uniform convergent subsequence. Finally a proof is given that P is purely atomic and concentrated on a finite number of pairwise disjoint P-atoms if and only if there does not exist a purely finitely additive {0,1}-valued probability charge, which vanishes for all P-zero sets.     

Finitely additive measures on groups and rings

On topological groups a natural finitely additive measure can be defined via compactifications. It is closely related to Hartman's concept of uniform distribution on non-compact groups (cf. [3]). Applications to several situations are possible. Some results of M. Paštéka and other authors on uniform distribution with respect to translation invariant finitely additive probability measures on Dedekind domains are transferred to more general situations. Furthermore it is shown that the range of a polynomial of degree ≥2 on a ring of algebraic integers has measure 0.     

Uniquely Determined Uniform Probability on the Natural Numbers

In this paper, we address the problem of constructing a uniform probability measure on \({\mathbb {N}}\). Of course, this is not possible within the bounds of the Kolmogorov axioms, and we have to violate at least one axiom. We define a probability measure as a finitely additive measure assigning probability 1 to the whole space, on a domain which is closed under complements and finite disjoint unions. We introduce and motivate a notion of uniformity which we call weak thinnability, which is strictly stronger than extension of natural density. We construct a weakly thinnable probability measure, and we show that on its domain, which contains sets without natural density, probability is uniquely determined by weak thinnability. In this sense, we can assign uniform probabilities in a canonical way. We generalize this result to uniform probability measures on other metric spaces, including \({\mathbb {R}}^n\).     

The Aumann-Gould Integral

We introduce and study a multi-valued integral of Aumann-type where the selectors are Gould integrals of real functions with respect to finitely additive vector-valued measures. We also discuss the relationship between this kind of integral and the M-Gould set-valued integral with respect to a finitely additive multimeasure M. Received: November 17, 2006, Revised: September 17, 2007 and January, 8, 2008, Accepted: March 13, 2008.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Completeness and interpolation of almost‐everywhere quantification over finitely additive measures

We give an axiomatization of first‐order logic enriched with the almost‐everywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible extensions of these results to the almost‐everywhere quantifier over countably additive measures.     

A Yosida–Hewitt decomposition for totally monotone set functions on locally compact σ-compact topological spaces

We prove for totally monotone set functions defined on the set of Borel sets of a locally compact σ-compact topological space a similar decomposition theorem to the famous Yosida–Hewitt’s one for finitely additive measures. This way any totally monotone decomposes into a continuous part and a pathological part which vanishes on the compact subsets. We obtain as corollaries some decompositions for finitely additive set functions and for minitive set functions.     

Gamma-Compactification of Measurable Spaces

We define the gamma-compactification of an arbitrary measurable space and study its structure and properties in the general and topological cases. We introduce and study the notion of gamma-extension of a singleton in a topological space. We consider the procedure of extension of finitely additive measures from the original space to regular countably additive measures on the gamma-compactification of the space.     

关于A-收敛

设A={ai}（i=1）∞S_（e_1）~＋,其中S（e1）＋={x=（x（n））∈e1：‖x‖=1且x（n）≥0对任意的n∈N}.Banach空间X中的序列{x_n}称为A-收敛于x∈X是指对任意的ε〉0,→0当i→∞,其中A（ε）={n∈N：‖x_n-x‖≥ε}.这篇文章中,我们证明了该收敛可以用一个有限可加的概率测度加以刻画.我们对A-收敛与统计收敛的关系进行了讨论,证明了A-收敛为统计收敛完全取决于A的w~＊-拓扑性质.     

Lévy group and density measures

We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the Lévy group G of permutations of N. Using a new characterization of the Lévy group G we will prove that a finitely additive measure extends density if and only if it is G-invariant.