Monotone Independence Research Articles

Matricially free random variables

We show that the framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a concept which not only leads to a natural generalization of freeness, but also underlies other fundamental types of noncommutative independence, such as monotone independence and boolean independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial semicircle laws.

A combinatorial approach to monotonic independence over a C∗-algebra

The notion of monotonic independence, introduced by N. Muraki, is considered in a more general frame, similar to the construction of operator-valued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arsine distribution as well as a connection to operator-valued conditional freeness.

Multilinear function series in conditionally free probability with amalgamation

As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.

Revisiting Independence and Stochastic Dominance for Compound Lotteries

We establish mathematical equivalence between independence of irrelevant alternatives and monotonicity with respect to first order stochastic dominance. This formal equivalence result between the two principles is obtained under two key conditions. Firstly, for all natural numbers m, each principle is defined on the domain of compound lotteries with compoundness level m. Secondly, the standard concept of reduction of compound lotteries applies.

MONOTONIC INDEPENDENCE ASSOCIATED WITH PARTIALLY ORDERED SETS

A generalization of Muraki's notion of monotonic independence onto the case of partially ordered index set is given: algebras indexed by chains are monotonically independent, and algebras indexed by non-comparable elements are boolean independent. Examples of central limit theorem are shown in two cases. For the integral-points lattices ℕd the moments of the limit measure are related to the combinatorics of the finite heap-ordered labelled rooted trees (if d = 2). For the integral-points lattice ℕ × ℤd in Minkowski spacetime the limit measure is given by the recurrence of it's moments, which, for the case d = 1 is related to the inverse error function. Various formulas for computing mixed moments are shown to be related to the boolean-monotonic non-crossing pair partitions.

Only a Dictatorship is Efficient or Neutral

In many, if not most, elections, several different seats must be filled, so that a group of candidates, or an assembly, is selected. Typically in these elections, voters cast their ballots on a seat-by-seat basis. We show that seat-by-seat procedures are efficient or neutral only under extreme conditions. We then provide a domain restriction, called full separability, that ensures the efficiency of seat-by-seat election procedures. How should a voting system be judged? A time-honoured approach judges a system on the basis of the properties it satisfies. The literature on voting has considered myriad properties, including anonymity, neutrality, efficiency, independence of irrelevant alternatives, monotonicity, and Condorcet consistency. Although Arrow’s impossibility theorem (1963) famously warned that a given property may be more subtle and difficult to satisfy than is initially apparent, some of these properties are generally taken to be obviously desirable and easily satisfied, both in theory and in practice. Three such properties are efficiency, anonymity, and neutrality. After all, efficiency merely requires that, when all voters prefer outcome A to outcome B ,o utcome B not be chosen, while anonymity and neutrality only ask, respectively, that all voters and all outcomes be treated equally. Nevertheless, we argue in this paper that, while anonymity is a pervasive feature of political voting

Reference functions and solutions to bargaining problems with and without claims

Following an idea due to Thomson (Journal of Economic Theory, 1981, 25: 431–441) we examine the role of reference functions in the axiomatic approach to the solution of bargaining problems with and without claims. A reference function is a means of summarizing essential features of a bargaining problem. Axioms like Independence of Irrelevant Alternatives and Monotonicity are then reformulated with respect to this reference function. Under some weak conditions on the reference function we obtain characterizations of different parametrized classes of solutions. We present several examples of reference functions and thereby recover many well-known solutions to bargaining problems with and without claims.

MONOTONE INDEPENDENCE IS ASSOCIATIVE

It is shown that the notion of monotone independence introduced by Muraki and Lu leads to an associative product of quantum probability spaces. This product is then used to define monotone Lévy processes on dual semigroups.

MONOTONIC INDEPENDENCE, MONOTONIC CENTRAL LIMIT THEOREM AND MONOTONIC LAW OF SMALL NUMBERS

A notion of "monotonic independence" is formulated in the setting of C*-probability space. Based on this independence, a noncommutative central limit theorem and a noncommutative law of small numbers are given.

Distractor selection ratios

A latent variable representation for multiple-choice item and option characteristic curves is presented. Under standard assumptions of conditional independence of item responses and monotonicity of item characteristic curves, a criterion for distractors is proposed based on distractor selection ratios. A connection is made between the proposed criterion and the theory of individual choice behavior, providing new insight. The main results allow for the testing of the criterion from observable data without first specifying a parametric form for the characteristic curves. A series of examples apply the method.

Two characterizations of the uniform rule for division problems with single-peaked preferences

The uniform rule is considered to be the most important rule for the problem of allocating an amount of a perfectly divisible good between agents who have single-peaked preferences. The uniform rule was studied extensively in the literature and several characterizations were provided. The aim of this paper is to provide two different formulations and corresponding axiomatizations of the uniform rule. These formulations resemble the Nash and the lexicographic egalitarian bargaining solutions; the corresponding axiomatizations are based on axioms of independence of irrelevant alternatives and restricted monotonicity.

Two characterizations of the uniform rule for division problems with single-peaked preferences

The uniform rule is considered to be the most important rule for the problem of allocating an amount of a perfectly divisible good between agents who have single-peaked preferences. The uniform rule was studied extensively in the literature and several characterizations were provided. The aim of this paper is to provide two different formulations and corresponding axiomatizations of the uniform rule. These formulations resemble the Nash and the lexicographic egalitarian bargaining solutions; the corresponding axiomatizations are based on axioms of independence of irrelevant alternatives and restricted monotonicity.

Bargaining over the division of a shrinking pie: An axiomatic approach

The Bargaining Problem paradigm is extended to time-consuming conflict situations. Such a situation can be represented by a chain of bargaining domains, each representing the conflict at a different point in time. The solution function selects a point in the union of all these domains. We characterize a solution function which satisfies several requirements and explore its properties. One of the results is that an extension of the Adding requirement (Thomson-Myerson 1980) is enough, under some conditions, to yield a solution point, so there is no need to extend the stronger requirements of Independence of Irrelevant Alternatives (Nash 1950) or Monotonicity (Kalai-Smorodinsky 1975).

Risk aversion in n-person bargaining

Relations between risk properties of Pareto optimal n-person bargaining solutions are studied. The (weak) worse alternative property, risk sensitivity property, and risk profit opportunity property are considered, in particular relations between these properties, and between these properties and other properties such as independence of irrelevant alternatives and individual monotonicity. A distinction is made between bargaining games where all Pareto optimal outcomes are riskless, and bargaining games where Pareto optimal outcomes may represent the utilities of lotteries between riskless alternatives. In the first mentioned case, more general results can be obtained.

Truncated Egalitarian solutions

A family of solutions to the bargaining problem with a variable population, generalizing the Egalitarian solution, is introduced under the name of Truncated Egalitarian solutions. A Truncated Egalitarian solution solves small problems at thier Egalitarian outcome and larger problems at some fixed and therefore non-optimal point, violations of optimality being more severe for large than for small groups. Any solution satisfying Symmetry, Independence of Irrelevant Alternatives, Continuity and Monotonicity (when the claims of more agents have to be accomodated, all agents should sacrifice) coincides with a Truncated Egalitarian solution except perhaps when only two agents are involved, in which case utility substitutions are possible in some limited range.

The independence of certain axioms of structures in sets

The independence of the axioms for spans and the independence of the axioms for closure structures are usually taken for granted. In this paper, the author establishes the independence of monotonicity, extensiveness, idempotence, the exchange property, the property of having ∅ \emptyset as a fixed set and two covering properties ( α \alpha -character, with α \alpha being some cardinal number, and a covering property with respect to generators). The independence of the axioms for closure structures and spans follow immediately. It is shown that any proof of the independence of a given axiom must involve an example with certain restrictions on the cardinal number α \alpha .