Independence, Additivity, Uncertainty by Karl Vind

By Karl Vind

The paintings in this e-book begun a long time in the past as an try to simplify and unify a few effects frequently taught in classes in mathematical economics. the industrial interpretation of the re­ sults have been representations of personal tastes as sums or integrals and the decomposition of personal tastes into utilities and percentages. It later became out that t.he process taken within the previous types have been additionally the right kind process in generalizing from personal tastes which have been overall preorders to personal tastes that have been no longer overall or tran­ sitive. a similar arithmetic might even in that state of affairs supply representations that have been additive. it will additionally supply decomposi­ tions the place innovations of uncertainty seemed. Early models of a few of the implications seemed as operating Pa­ pers No. one hundred thirty five, a hundred and forty, a hundred and fifty, and 176 from the guts for learn in administration technology, Berkeley. a primary model of chapters 2, four, 6, 7, and eight seemed 1969 with the identify" suggest Groupoids" [177]. they're basically unchanged -except for a few notes particularly in bankruptcy 6. one other model seemed 1990 as [178]. bankruptcy 10 comprises effects from an analogous types and from [181]. bankruptcy eleven by means of Birgit Grodal is predicated on [91] via Grodal and Jp,an-Francois Mertens. Chapters eleven and 12 - additionally through Birgit Gro­ dal -contains the consequences from the sooner types, yet were prolonged (by Karl Vind) take into consideration the recent corollaries of the consequences within the different chapters.

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Proof. Assume S connected, if (a, b) is a gap ]t-, a] = ]t-, b[ are both open and closed, so A connected implies that S has no gaps. Define the closed set B =I- 0,5 by B = {s E Sis t a, for all a E A} where A is a non-empty bounded set without supremum. B = UbEB {s E Sis :>- b} and therefore also open, again contradicting connectedness. To prove S connected, let A be a set such that A =I- 0, S. Let a E A and b tJ. A and for example b :>- a and let B = U [a, c] where the union is over the c's for which b :>- c and [a, c] C A.

Q (s', t') E Q (NOT (1,2» This means according to lemma 10 that there are total preorders ts and t:l' on 8 and T such that (s, t) E Q implies (s', t') E Q for s' ts 8and t' tT t. If assmnption A holds (8,ts) = (T,;:5T) and Q is the graph of the >-8 relation (denoted >-) on 8, so (8, t) E Q is equivalent to s >- t. (1,1') (s,t) E Q (S1,8~, t l , t~) f/. Q (NOT (1,1'» 44 5. 1. just R. (1,2') (s, t) E Q (Sl,S~, t1,t2) fj. Q (NOT (1,2')) Definition 84 (generalized Thomsen) Conditions NOT (1,2) and NOT (1,2') together is called the Generalized Thomt,en condition or just QT.

Let t. be a total preorder on Y, and assume factor independence and continuity. If for all i E I, Yi ~ y~, then y ~ y'. If for all i E I, Yi t. y~ and for some i E I, Yi i- y~, then y i- y'. Proof. Denote by n the set {I, 2"" ,n} C I. i- Y'1:8'l n Y rv i- Y'1:8'l n+1 Y - +Y' y rv so y ~ y' and analogously _ Lemma 8 Let Y = TIiEI Yi be a product set where I is countable. Let ~ be a total preorder on Y, and assume factor independence. Then continuity of t. implies that the convergence y' I:8'ln y -+ y' is 11niform in y E Y Proof.

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