Additive Representations of Preferences: A New Foundation of by P.P. Wakker

By P.P. Wakker

`This publication represents a really actual and critical contribution to the maths of choice concept.
The distinguishing characteristic of Wakker's method is that he has little need of a reference experiment.
There are extra effects aplenty and lots more and plenty for the choice theorist to contemplate. those effects also will support psychologists, economists and others formulate and try types of decision-making behaviour.
by means of any criteria its booklet is a landmark ... very good survey and precis of the literature on additive choice versions. every one web page represents the distillation of a lot idea. for this reason, it's a publication that rewards interpreting and rereading. Wakker's effects and techniques of evidence will enlighten learn into determination concept for plenty of years.'
British magazine of Mathematical and Statistical Psychology, forty three, 1990

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Assume that IIA holds, and let xORxl ... Rxm. We prove that not xmpxO. As in the proof of the above lemma, we may assume xj ". xj +l for all 0 S j S m-1. Hence Dl, ... ,Dm exist such that xj E C(Dj+l), xj +1 E ~+l for all 0 S j S m-1. If xm = xO, or not xmRx O, then also not xmpxO. So let us assume that xm ". , DO exists with xm E C(DO), xO E DO. For [not xmpx O] we must show that xO E C(DO). First we prove this for the case that t::. contains all two- and three-point subsets of X. Since {xj,xj+l } C Dj+l, by IIA: xj E C(xj,xj +l } for all 0 S j S m-1.

The major part of a statistical decision is then shifted to the choice of 'prior probabilities' and 'loss functions', matters outside the (classical) statistical model, and for that reason often called 'subjective'. See Berger&Wolpert(1984) and Edwards(1972) for more information on the likelihood principle. The sure-thing principle can be considered the adaptation to decision making under uncertainty of the 'independence condition' as formulated by von Neumann&Morgenstern(1944) for the context of decision making under risk (where alternatives are lotteries).

G. rj is IR~, or IR. Then all topological assumptions in the sequel are satisfied, and can be ignored. rj will always be endowed with the product topology, hence is connected (see Kelley, 1955, Chapter 3. problem 0) and separable too. The condition of topological separability has been added only for simplicity of presentation. 1). The crucial assumption is connectedness. It suffices for connectedness that any alternative can be transformed into any other alternative by a process of gradual changes (a 'continuous' process).

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