Products of non-additive measures: a Fubini-like theorem

For non-additive set functions, the independent product, in general, is not unique and the Fubini theorem is restricted to slice-comonotonic functions. In this paper, we use the representation theorem of Gilboa and Schmeidler (Math Oper Res 20:197?C212, 1995) to extend the M?bius product for non-additive set functions to non-finite spaces. We extend the uniqueness result of Ghirardato (J Econ Theory 73:261?C291, 1997) for products of two belief functions and weaken the requirements on the marginals necessary to obtain the Fubini property in the product. More importantly, we show that for the M?bius product one side of the Fubini theorem holds for all integrable functions if one of the marginals either is a probability or a convex combination of a chain of unanimity games, i.e., we relax the requirement of slice-comonotonicity and enrich the set of possible applications.     

Re-examining the law of iterated expectations for Choquet decision makers

Yoo (Economic Letters 37:145–149, 1991) argues that the law of iterated expectations must be violated if the probability measure of a Choquet decision maker is non-additive. In this article, we prove the positive result that the law of iterated expectations is satisfied for Choquet decision makers whenever they update their non-additive beliefs in accordance with the Sarin and Wakker (Journal of Risk and Uncertainty 16:223–250, 1998) update rule. The formal key to this result is the act-dependence of the Sarin–Wakker update rule, which does not hold for the update rules considered by Yoo (1991).     

Parametric multi-attribute utility functions for optimal profit under risk constraints

We provide an economic interpretation of the practice consisting in incorporating risk measures as constraints in an expected prospect maximization problem. For what we call the infimum of expectations class of risk measures, we show that if the decision maker (DM) maximizes the expectation of a random prospect under constraint that the risk measure is bounded above, he then behaves as a “generalized expected utility maximizer” in the following sense. The DM exhibits ambiguity with respect to a family of utility functions defined on a larger set of decisions than the original one; he adopts pessimism and performs first a minimization of expected utility over this family, then performs a maximization over a new decisions set. This economic behaviour is called “maxmin under risk” and studied by Maccheroni (Econ Theory 19:823–831, 2002). As an application, we make the link between an expected prospect maximization problem, subject to conditional value-at-risk being less than a threshold value, and a non-expected utility economic formulation involving “loss aversion”-type utility functions.     

Great Expectations. Part I: On the Customizability of Generalized Expected Utility

We propose a generalization of expected utility that we call generalized EU (GEU), where a decision maker’s beliefs are represented by plausibility measures and the decision maker’s tastes are represented by general (i.e., not necessarily real-valued) utility functions. We show that every agent, “rational” or not, can be modeled as a GEU maximizer. We then show that we can customize GEU by selectively imposing just the constraints we want. In particular, we show how each of Savage’s postulates corresponds to constraints on GEU.     

Finding socially best spanning trees

This article combines Social Choice Theory with Discrete Optimization. We assume that individuals have preferences over edges of a graph that need to be aggregated. The goal is to find a socially “best” spanning tree in the graph. As ranking all spanning trees is becoming infeasible even for small numbers of vertices and/or edges of a graph, our interest lies in finding algorithms that determine a socially “best” spanning tree in a simple manner. This problem is closely related to the minimum (or maximum) spanning tree problem in Discrete Optimization. Our main result shows that for the various underlying ranking rules on the set of spanning trees discussed in this article, the sets of “best” spanning trees coincide. Moreover, a greedy algorithm based on a transitive group ranking on the set of edges will always provide such a “best” spanning tree.     

Dominance criteria for welfare comparisons: using equivalent income to describe differences in needs

The article demonstrates that the dominance approach—often used for the measurement of welfare in a population in which there are different household types (see e.g., Atkinson and Bourguignon, Arrow and the foundations of the theory of economic policy, 350–370, 1987)—can be based on explicit value judgments on the households’ living standard. We define living standard by equivalent income (functions) and consider classes of inequality averse social welfare functions: Welfare increases if the inequality of living standard is decreased. In this framework, we suggest three new dominance criteria and obtain characterizations of second degree stochastic dominance and of two criteria proposed by Bourguignon (Journal of Econometrics 42:67–80, 1989).     

Finite Alternating-Move Arbitration Schemes and the Equal Area Solution

We start by considering the Alternate Strike (AS) scheme, a real-life arbitration scheme where two parties select an arbitrator by alternately crossing off at each round one name from a given panel of arbitrators. We find out that the AS scheme is not invariant to “bad” alternatives. We then consider another alternating-move scheme, the Voting by Alternating Offers and Vetoes (VAOV) scheme, which is invariant to bad alternatives. We fully characterize the subgame perfect equilibrium outcome sets of these above two schemes in terms of the rankings of the parties over the alternatives only. We also identify some of the typical equilibria of these above two schemes. We then analyze two additional alternating-move schemes in which players’ current proposals have to either honor or enhance their previous proposals. We show that the first scheme’s equilibrium outcome set coincides with that of the AS scheme, and the equilibrium outcome set of the second scheme coincides with that of the VAOV scheme. Finally, it turns out that all schemes’ equilibrium outcome sets converge to the Equal Area solution’s outcome of cooperative bargaining problem, if the alternatives are distributed uniformly over the comprehensive utility possibility set and as the number of alternatives tends to infinity. Journal of Economic Literature Classification Number: C72.     

Delay in a bargaining game with contracts

In a multilateral bargaining game where a proposer and responders can set up a “principal–agent” relationship by means of binding cash-offer contracts, we show that there is a Markov SPE with a delay in reaching an agreement. We also show that all the individually rational and efficient payoffs can be supported by SPE.      

A Theory of Rational Choice under Ignorance

This paper contributes to a theory of rational choice for decision-makers with incomplete preferences due to partial ignorance, whose beliefs are representable as sets of acceptable priors. We focus on the limiting case of `Complete Ignorance' which can be viewed as reduced form of the general case of partial ignorance. Rationality is conceptualized in terms of a `Principle of Preference-Basedness', according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called `Simultaneous Expected Utility Maximization'. It can be interpreted as agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the (extremal) `acceptable priors' among which the decision maker has suspended judgment. An essential but non-standard feature of Simultaneous Expected Utility choices is their dependence on the entire choice set. This is justified by the conception of optimality as compromise rather than as superiority in pairwise comparisons.     

Expected Utility Consistent Extensions of Preferences

We consider the problem of extending a (complete) order over a set to its power set. The extension axioms we consider generate orderings over sets according to their expected utilities induced by some assignment of utilities over alternatives and probability distributions over sets. The model we propose gives a general and unified exposition of expected utility consistent extensions whilst it allows to emphasize various subtleties, the effects of which seem to be underestimated – particularly in the literature on strategy-proof social choice correspondences.      

Continuous Utility Functions Through Scales

We present here a direct elementary construction of continuous utility functions on perfectly separable totally preordered sets that does not make use of the well-known Debreu’s open gap lemma. This new construction leans on the concept of a separating countable decreasing scale. Starting from a perfectly separable totally ordered structure, we give an explicit construction of a separating countable decreasing scale, from which we show how to get a continuous utility map.      

Quasi-Bayesian Analysis Using Imprecise Probability Assessments And The Generalized Bayes’ Rule

The generalized Bayes’ rule (GBR) can be used to conduct ‘quasi-Bayesian’ analyses when prior beliefs are represented by imprecise probability models. We describe a procedure for deriving coherent imprecise probability models when the event space consists of a finite set of mutually exclusive and exhaustive events. The procedure is based on Walley’s theory of upper and lower prevision and employs simple linear programming models. We then describe how these models can be updated using Cozman’s linear programming formulation of the GBR. Examples are provided to demonstrate how the GBR can be applied in practice. These examples also illustrate the effects of prior imprecision and prior-data conflict on the precision of the posterior probability distribution.     

Notes on Theory and Practice in Social Work: a Comparative View

Please address reprint requests to Robert van Krieken, Department of Social Work, University of Sydney, Sydney, New South Wales 2006, Australia. Summary There are still a number of problems surrounding the relationshipbetween theories of social work practice and that practice itself.This paper examines the factors underlying those problems andemphasises their roots in the failure to examine the diversityof social work theories and practices. We refer to one attemptto resolve the problems—the distinction between ‘practicetheory’ and ‘theory of practice’—pointout some difficulties with it and suggest an alternative, three-folddistinction within theorising: between (a) materialist socialtheory, (b) strategic practice theory and (c) working concepts. As an example of how those distinctions can be used, we thenbriefly discuss the work of Oskar Negt and its introductioninto the Dutch welfare context, as it was the issues raisedby that which stimulated the ideas in this paper. We concludeby arguing that only this kind of perspective on theorisingcan produce ideas which are of real use to progressive socialwork practitioners.     

Elementary proof that mean–variance implies quadratic utility

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as “Buridan’s axiom” after the French philosopher’s fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only “pure” strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re-expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan’s axiom, nor any more fundamental axiom of utility theory.     

Risk-adjusted martingales and the design of “indifference” gambles

In the probability literature, a martingale is often referred to as a “fair game.” A martingale investment is a stochastic sequence of wealth levels, whose expected value at any future stage is equal to the investor’s current wealth. In decision theory, a risk neutral investor would therefore be indifferent between holding on to a martingale investment, and receiving its payoff at any future stage, or giving it up and maintaining his current wealth. But a risk-averse decision maker would not be indifferent between a martingale investment and his current wealth level, since he values uncertain deals less than their mean. A risk seeking decision maker, on the other hand, would readily accept a martingale investment in exchange for his current wealth, and would repeat this investment any number of times. These ideas lead us to introduce the notion of a “risk-adjusted martingale”; a stochastic sequence of wealth levels that a rational decision maker with any attitude toward risk would value constantly with time, and would be indifferent between receiving its pay-off at any future stage, or giving it up and maintaining his current wealth level. We show how to construct such risk-adjusted investments for any decision maker with a continuous monotonic utility function. The fundamental result we derive is that a pay-off structure of an investment (i) is a risk-adjusted martingale and (ii) can be represented by a lattice if and only if the pay-off functions are invariant transformations of the given utility function.     

Provability as a deontic notion

The purpose of this paper is to mark a significant difference between classical and several non-classical prepositional calculi. The argument presupposes familiarity with Kripke/Hintikka semantics for modal logic. The non-classical systems are Hintikka's logic of belief and alethic modal systems which have Kripke/Hintikka semantics.The difference is marked by showing that the semantic validity operator in classical logic behaves as a normal alethic necessity-operator while the non-classical semantic validity operators behave as normal deontic ought-operators. The crucial step is showing that a formula, valid by non-classical semantics, can be falsified. I show that the negation of a non-classical thesis can be added to a consistent set of formulae without making the set inconsistent or any other set inconsistent. This is shown by observing that consistent sets of formulae do not need to be related to other consistent sets by any of the alternativeness relations of Kripke/Hintikka model structures for non-classical systems.The deontic behavior of non-classical semantical validity operators is interpreted as showing that being a thesis of a non-classical system means, not that the thesis is a logical truth, but that the thesis is the content of a norm on how we ought to use, crucial terms such as believe and necessity.The basic idea of this essay was sketched at the 1970 Spring Meetings of the Association of Symbolic Logic on April 15 at Madison Wisconsin.     

On the Onassis problem

A decision problem is by convention characterized by its outcome matrix and by a subsequent utility evaluation. In trying to set up an outcome matrix based on wealth values it may occur that due to ambiguities inherent in the decision problem it is not clear which standard of value (or numéraire) should be used in order to measure wealth. A typical example of this kind is Stützel's so called Onassis Paradox. We show that problems of this kind can be solved within the conventional framework of decision theory. The analysis proceeds in two steps. First, state-dependent utility functions are derived; second, a model for evaluating these utility functions is presented.     

On the problem of imprecision

Among the problems raised by admitting statements which are neither true nor false is the problem of how we are to cope with vague concepts. One method of dealing with such concepts has been suggested by Rosser/Turquette (1952), i.e. the employment of a many valued set theory. It is our intention in this paper to discuss the use of many valued logics, especially the set theoretical proposals of Zadeh and Brown, in dealing with this problem. Towards this end, we shall pay close attention to the development of the concepts of fuzzy sets and fuzzy systems. In this regard, it will be argued that Zadeh had in mind a many valued logic where the connectives are interpreted according to Lukasiewicz and Tarski, whereas Brown's version may be based upon a Boolean-valued logic. Furthermore, we shall have occasion to distinguish between two sources of imprecision in specifying the membership of classes, i.e. conceptual vagueness and imprecision due to inexact measurement. Due to Suppes' work in this field, the latter sort of imprecision may be exemplified by providing a theory of the inexact measurement of subjective probability. Finally, we shall discuss some ways in which these two types of imprecision are intertwined.     

Evaluating Children's Services: Recent Conceptual and Methodological Developments

This article considers recent innovations and challenges withrespect to the evaluation of children’s services interventions.It sets out a method for designing and evaluating services thatis based on research evidence gathered on individual cases andthat permits revision of those services in the light of emergingresults. It starts by describing briefly different ways of identifyinga discrete group of children with similar needs. It then outlinesapproaches to ensuring that an intervention for those childrenis underpinned by theory and research evidence–includingthe idea of ‘logic modelling’–before discussingthe importance of capturing accurately how well the interventionis implemented. Experimental and non-experimental methods ofassessing an intervention’s effectiveness are considered,with particular emphasis on techniques for enhancing the causalinference that can be drawn from studies and the importanceof matching method and purpose. The article also demonstrateshow the principles and methods used in relation to evaluatingservices for groups can be applied in clinical work with individualcases, and identifies the benefits of this general approachto evaluation over and above the information that it provideson what works.     

'Anti-Oppressiveness': Critical Comments on a Discourse and its Context

This paper identifies and aims to explain an apparent dissonancebetween the dominance in Britain of ‘anti-oppressive’social work discourse and the socio-political context surroundingits use; a context often claimed to feature excessive regulationand control. Pursuing this, some politically radical aims associatedwith ‘anti-oppressiveness’ are spelt out, and thedifficulty of achieving these in an unconducive climate is discussed.Then, a distinction made by Robert Merton between ‘latent’and ‘manifest’ functions is used to suggest thatthe manifest radicalism of ‘anti-oppressive’ discoursecan helpfully be distinguished from some latent largely unrecognisedconsequences of its use – not consequences with politicallyradical impact, but with a social meaning congruent with a climateof control. It is concluded that the ‘success’ ofanti-oppressive discourse might well be viewed as requiringmore of the kind of critical analysis that the discourse itselfwas supposed to espouse.