Countably Additive Research Articles

Gliding Hump Properties in Abstract Duality Pairs with Projections

Let E, G be Hausdorff topological vector spaces and let F be a vector space. Assume there is a bilinear operator : E X F →G such that : E →G is continuous for every y £ F. The triple E, F, G is called an abstract duality pair with respect to G or an abstract triple and is denoted by (E,F : G). If {Pj} is a sequence of continuous projections on E, then (E,F : G) is called an abstract triple with projections. Under appropriate gliding hump assumptions, a uniform bounded principle is established for bounded subsets ofE and pointwise bounded subsets of F. Under additional gliding hump assumptions, uniform convergent results are established for series ∑ ∞ j=1 when x varies over certain subsets of E and y varies over certain subsets of F. These results are used to establish uniform countable additivity results for bounded sets of indefinite vector valued integrals and bounded subsets of vector valued measures.

The problem of invoking infinite polytheisms: a response to Raphael Lataster and Herman Philipse

Raphael Lataster and Herman Philipse present an argument which they think decisively demonstrates polytheism over monotheism, if theism is assumed. Far from being decisive, the argument depends on very controversial and likely false assumptions about how to treat infinities in probability. Moreover, these problems are well known. Here, we focus on three objections. First, the authors rely on both countable additivity and the Principle of Indifference, which contradict each other. Second, the authors rely on a particular way of dividing up the possibility space, when there are equally as reasonable alternative divisions which give different answers to the one the authors arrived at. Third, the authors’ argument proves too much, giving us an argument against many positions we should not be able to argue against so easily.

Bayesian Humility

Say that an agent is epistemically humble if she is less than fully confident that her opinions will converge to the truth, given appropriate evidence. Is such humility rationally permissible? According to Gordon Belot’s orgulity argument: the answer is yes, but long-run convergence-to-the-truth theorems force Bayesians to answer no. That argument has no force against Bayesians who reject countable additivity as a requirement of rationality. Such Bayesians are free to count even extreme humility as rationally permissible. Furthermore, dropping countable additivity does not render Bayesianism more vulnerable to the charge that it is excessively subjective.

Decomposition of dyadic measures and unions of closed -sets for series in a Haar system

New properties of finitely additive set functions (quasi-measures) and Borel measures on dyadic product groups are established. The results obtained are applied to the theory of series in Haar systems — for example, for a broad family of classes of multiple Haar series on , a countable union of closed uniqueness sets is shown to be a uniqueness set too. Bibliography: 18 titles.

Compromising in bifocal distribution games: the average value

It is well known that, in distributions problems, fairness rarely leads to a single viewpoint (see, for instance, Young, Equity in theory and practice. Princeton University Press, Princeton, 1994). In this context, this paper provides interesting bases that support the simple and commonly observed behavior of reaching intermediate agreements when two prominent distribution proposals highlight a discrepancy in sharing resources. Specifically, we formalize such a conflicting situation by associating it with a ‘natural’ cooperative game, called bifocal distribution game, to show that both the Nucleolus (Schmeidler, SIAM J Appl Math 17:1163–1170, 1969) and the Shapley value (Shapley, Additive and non-additive set functions. Princeton University, Princeton, 1953a) agree on recommending the average of the two focal proposals. Furthermore, we analyze the interpretation of the previous result by means of axiomatic arguments.

When Can the Maximin Share Guarantee Be Guaranteed?

The fairness notion of maximin share (MMS) guarantee underlies a deployed algorithm for allocating indivisible goods under additive valuations. Our goal is to understand when we can expect to be able to give each player his MMS guarantee. Previous work has shown that such an MMS allocation may not exist, but the counterexample requires a number of goods that is exponential in the number of players; we give a new construction that uses only a linear number of goods. On the positive side, we formalize the intuition that these counterexamples are very delicate by designing an algorithm that provably finds an MMS allocation with high probability when valuations are drawn at random.

The theorem of Halmos and Savage under finite additivity

We prove an extension to the finitely additive setting of the theorem of Halmos and Savage. From this we deduce extensions of classical results of Drewnowski and of Yan as well as a new characterization of weak compactness in the space of finitely additive set functions.

We Turing Machines Can’t Even Be Locally Ideal Bayesians

Vann McGee has argued that, given certain background assumptions and an ought-implies-can thesis about norms of rationality, Bayesianism conflicts globally with computationalism due to the fact that Robinson arithmetic is essentially undecidable. I show how to sharpen McGee’s result using an additional fact from recursion theory—the existence of a computable sequence of computable realswith an uncomputable limit (a Specker sequence). In conjunction with the countable additivity requirement on probabilities, such a sequence can be used to construct a specific proposition to which Bayesianism requires an agent to assign uncomputable credence—yielding a local conflict with computationalism.

A Version of Komlós Theorem for Additive Set Functions

We provide a version of the celebrated theorem of Koml\'os in which, rather then random quantities, a sequence of finitely additive measures is considered. We obtain a form of the subsequence principle and some applications.

On J M Keynes’s Rejection of the Moscow School of Probability’s Limiting Frequency Approach to Probability and Kolmogorov’s Axiom of Additivity (Countable Additivity ): Non –Additivity was the fundamental, basic axiom upon which all of the Economics of Ke

<p>J M Keynes was an acknowledged, world renown, and internationally recognized expert in probability and statistics in the 1930’s based on his A Treatise on Probability (1921). . Keynes had been selected by statistics journals to serve as a referee during the 1930’s. It is, therefore, no surprise that he was selected as the referee by the League of Nations to review Jan Tinbergen’s work on business cycles that used an econometrics approach based on The Law of Large Numbers, the Central Limit Theorem, and the Gaussian (Normal) Distribution .The fundamental axiom used by Tinbergen was additivity . Kolmogorov and the Moscow School of Probability’s main innovation was to go from the axiom of additivity to the axiom of countable additivity. However, Keynes rejected additivity except in the special case that the weight of the evidence, w, which measured the relative completeness of the evidence ,defined on the closed unit interval [0,1],equaled 1 , approached 1,or approximated 1. Keynes also accepted goodness of fit tests, such as the Lexis –Q test, and exploratory data analysis as evidence that could be used to support using a particular probability distribution.</p>

On Multiple Keyword Sponsored Search Auctions with Budgets

We study multiple keyword sponsored search auctions with budgets. Each keyword has multiple ad slots with a click-through rate. The bidders have additive valuations, which are linear in the click-through rates, and budgets, which are restricting their overall payments. Additionally, the number of slots per keyword assigned to a bidder is bounded. We show the following results: (1) We give the first mechanism for multiple keywords, where click-through rates differ among slots. Our mechanism is incentive compatible in expectation, individually rational in expectation, and Pareto optimal. (2) We study the combinatorial setting, where each bidder is only interested in a subset of the keywords. We give an incentive compatible, individually rational, Pareto-optimal, and deterministic mechanism for identical click-through rates. (3) We give an impossibility result for incentive compatible, individually rational, Pareto-optimal, and deterministic mechanisms for bidders with diminishing marginal valuations.

Auctions for Heterogeneous Items and Budget Limits

We study individual rational, Pareto-optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. We consider settings with multiunit demand and additive valuations. For single-dimensional valuations we prove a positive result for randomized mechanisms, and a negative result for deterministic mechanisms. While the positive result allows for private budgets, the negative result is for public budgets. For multidimensional valuations and public budgets we prove an impossibility result that applies to deterministic and randomized mechanisms. Taken together this shows the power of randomization in certain settings with heterogeneous items, but it also shows its limitations.

Approximating the nash social welfare with indivisible items

In this letter we briefly discuss our main result from [Cole and Gkatzelis 2015]. Given a set of indivisible items and a set of agents having additive valuations, our goal is to allocate the items to the agents in order to maximize the geometric mean of the agents' valuations, i.e., the Nash social welfare. This problem is known to be NP-hard, and our main result is the first efficient constant-factor approximation algorithm for this objective.

On J M Keynes's Rejection of the Moscow School of Probability's Limiting Frequency Approach to Probability and Kolmogorov's Axiom of Additivity (Countable Additivity): Non-Additivity was the Fundamental, Basic Axiom upon Which All of the Economics of Keynes was Built

J M Keynes was an acknowledged, world renown, and internationally recognized expert in probability and statistics in the 1930’s based on his A Treatise on Probability (1921). Keynes had been selected by statistics journals to serve as a referee during the 1930’s. It is, therefore, no surprise that he was selected as the referee by the League of Nations to review Jan Tinbergen’s work on business cycles that used an econometrics approach based on The Law of Large Numbers, the Central Limit Theorem, and the Gaussian(Normal) Distribution. The fundamental axiom used by Tinbergen was additivity. Kolmogorov and the Moscow School of Probability’s main innovation was to go from the axiom of additivity to the axiom of countable additivity. However, Keynes rejected additivity except in the special case that the weight of the evidence, w, which measured the relative completeness of the evidence, defined on the closed unit interval [0,1], equaled 1, approached 1, or approximated 1. Keynes also accepted goodness of fit tests, such as the Lexis-Q test, and exploratory data analysis as evidence that could be used to support using a particular probability distribution. Keynes also rejected countable additivity for the same reasons he rejected additivity, in general. Keynes’s theory of probability and decision making is based on the interval valued approach of G. Boole, who also rejected additivity except as a limiting case. Thus, non-additivity and non-linearity, or imperfect and incomplete information, used as a shorthand description for economists, are the fundamental axioms on which the economics of Keynes, and Adam Smith before him, are based. Ergodicity is based on a measure function, which is based on the Kolmogorov and the Moscow School of Probability’s Axiom of Countable Additivity. Therefore, it is obvious that Keynes would reject ergodicity (and non ergodicity) as general results unless the researcher provided extensive exploratory data analysis and goodness of fit tests supporting their claims. Paul Davidson and the Post Keynesian School have never supplied a shred of evidence to support their claims. The only group of researchers to do this are the econophysics followers of Benoit Mandelbrot and Nicholas Nassim Taleb, who were the first to supply empirical evidence to support their analysis. Therefore, it is easy to completely reject the claims of Paul Davidson, made numerous times since 1983, that Keynes was unaware of the Kolmogorov-Moscow School of probability’s main result, which was to move from the axiom of additivity to the axiom of countable additivity, and then on to the concept of ergodicity, based on the definition of a measurable function. Keynes’s main axiom was non additivity, which leads directly to nonlinearity. Keynes’s conclusions regarding liquidity preference, his rejection of the conclusion of gross substitutability, and his rejection of the meaningfulness of the ergodic-non ergodic distinction in stochastic process theory, are all based on non additivity and non linearity. All classical and neoclassical results are special cases based on additivity and/or countable additivity that lead to the linearity of the normal distribution. There are no missing axioms from Keynes’s work. Paul Davidson has thus erred continuously over the last 33 years in confusing axioms with conclusions and claiming, without a shred of evidence, historical or otherwise, that Keynes was completely ignorant of the Kolmogorov-Moscow School of probability in the 1930’s. Keynes would never have been selected as the reviewer of Tinbergen’s work by the League of Nations if that was thought to be the case. The fundamental axioms of neoclassical economics are additivity and linearity. From these axioms, and no other axioms, one can derive the conclusions of the neutrality of money, gross substitutability, and ergodicity. One could use the assumption of perfect and complete information for all decision makers as a shorthand description of the Neoclassical position. All neoclassical schools base their analysis on additivity and linearity. Therefore, Neoclassical economics is a special, limiting case of Keynes’s more general theory based on the axioms of non-additivity and non-linearity.

Automated Design of Revenue-Maximizing Combinatorial Auctions

Designing optimal—that is, revenue-maximizing—combinatorial auctions (CAs) is an important elusive problem. It is unsolved even for two bidders and two items for sale. Rather than pursuing the manual approach of attempting to characterize the optimal CA, we introduce a family of CAs and then seek a high-revenue auction within that family. The family is based on bidder weighting and allocation boosting; we coin such CAs virtual valuations combinatorial auctions (VVCAs). VVCAs are the Vickrey-Clarke-Groves (VCG) mechanism executed on virtual valuations that are affine transformations of the bidders’ valuations. The auction family is parameterized by the coefficients in the transformations. The problem of designing a CA is thereby reduced to search in the parameter space of VVCA—or the more general space of affine maximizer auctions. We first construct VVCAs with logarithmic approximation guarantees in canonical special settings: (1) limited supply with additive valuations and (2) unlimited supply. In the main part of the paper, we develop algorithms that design high-revenue CAs for general valuations using samples from the prior distribution over bidders’ valuations. (Priors turn out to be necessary for achieving high revenue.) We prove properties of the problem that guide our design of algorithms. We then introduce a series of algorithms that use economic insights to guide the search and thus reduce the computational complexity. Experiments show that our algorithms create mechanisms that yield significantly higher revenue than the VCG and scale dramatically better than prior automated mechanism design algorithms. The algorithms yielded deterministic mechanisms with the highest known revenues for the settings tested, including the canonical setting with two bidders, two items, and uniform additive valuations. 1

Stresses in equilibrium configurations of inextensible nets with slack

The paper deals with nets formed by two families of fibers (cords) which can grow shorter but not longer, in a deformation. The nets are treated as two-dimensional continua in the three-dimensional space. The inextensibility condition places unilateral constraint on the partial derivatives y,1 and y,2 of the deformation of the form | y , 1 ( x ) | ≤ 1 , | y , 2 ( x ) | ≤ 1 , There is no deformation energy, the total energy reduces to the potential energy of the net under external forces. Equilibrium configurations are those of minimum energy. The stresses in equilibrium configurations thus reduce to the reactions to the constraints. Nonzero stresses occur only in tense regions where one or two constraints are satisfied with the equality sign. The paper follows the work of Paroni in treating the stress problem via the dual variational problem in the sense of convex analysis. Unlike in the work of Paroni, where stresses are modeled as finitely additive set functions, here a (perhaps more economic) choice of spaces is made that leads to more accessible stresses represented by (countably additive) measures. The present development is made possible by an observation, of independent value, that the space of measures with divergence measure is the dual of another Banach space, in the present context naturally interpreted as the space of strains. Our measures generalize stress fields represented by ordinary functions to account for stress concentrations along folded lines in tension, frequently occurring in equilibrium configurations of the net.

Dual representation of convex sets of probability measures on totally bounded spaces

Convex sets of probability measures, frequently encountered in probability theory and statistics, can be transparently analyzed by means of dual representations in a function space. This paper introduces totally bounded spaces, whose structure is defined by a set of bounded real-valued functions, as a general framework for studying such representations. The reinterpretation of classical theorems in this framework clarifies the role of compactness and leads to simple existence criteria. Applications include results on the existence of probability measures satisfying given sets of conditions and an equivalence of consistent preferences and families of probability measures. Moreover, countable additivity of probabilities is seen to be a consequence of elementary consistency assumptions.

Multivalent probability spaces

We give an abstract version of systems of sets similar to the λ-systems considered by Dynkin and other authors as a useful auxiliary tool. Our abstract version is of “Boolean” nature. This means, in particular, that the elements of the corresponding abstract algebra have no intrinsic set-theoretic structure. A natural system of axioms is specified. This system describes properties of two binary relations (inclusion and disjointness) and properties of two partial binary operations (addition and subtraction) closely connected with these binary relations. In particular, addition and subtraction are mutually inverse in a certain precisely specified sense. We state simple properties of these abstract Dynkin algebras and investigate extensions of such algebras by means of passages to certain limits (we refer to them as free extensions). The free extension of an abstract Dynkin algebra is closed under taking limits of monotone sequences of its elements. We prove that every (additive) probability on an abstract Dynkin algebra has a unique continuous (= countably additive) extension to the corresponding free extension of the algebra. This result, which disagrees with the usual difference between additivity and countable additivity, can be explained by the freeness of the extension under consideration.

On aggregation functions based on linguistically quantified propositions and finitely additive set functions

We study aggregation functions defined as convex combinations of the linguistically quantified propositions “at least k criteria are satisfied”. Our approach is similar to the TOWA function in spirit but instead of using a maxitive measure we propose to rely on a real-valued finitely additive set function. This assumption leads to a new framework. However, it is combinatorial by nature and, in general, it produces functions with high computational costs. Therefore, we analyze some particular settings and introduce new aggregation functions which can alleviate the combinatorial burden thanks to several combinatorial identities. These methods have interesting features and in particular, some of them make it possible to set different types of relationships between criteria by allowing the use of different t-norms. The interest of our proposals is illustrated with a famous example which cannot be modeled by classical aggregation functions such as the Choquet integral.

A simple and approximately optimal mechanism for an additive buyer

In this letter we briefly survey our main result from [Babaioff el al. 2014]: a simple and approximately revenue-optimal mechanism for a monopolist who wants to sell a variety of items to a single buyer with an additive valuation.