There are many different ways to obtain a general characterization of Expected Utility Theory for arbitrary probability measures. In the following we sketch a rather new approach by Chatterjee and Krishna [CK]. The details of this derivation can be found in their article.
LetZbe a compact metric space. LetP.Z/be the set of all probability measures onZ.
The Independence Axiom can essentially be stated as in the finite case:
Axiom A.13 (Independence) Let p;q;r 2 P.Z/. Let p q and 2 .0; 1, then pC.1r/qC.1/r.
To state the Continuity Axiom we need to generalize the notion of continuity via the concept of open sets (compare Sect.A.3): we say that a functionfWX ! Y is continuousif, for all open setsUY, the setf1.U/is open.
Open sets inP.Z/can be defined via weak-?convergence (compare Sect.2.4.5):
first, define closed sets as sets which contain the limit of any converging sequence (compare Sect.A.3). Second, define open sets as complement to these closed sets.
We can do more and construct even a metricdthat measures the distance between
two probability measures and reflects the same convergence. Such a metric is given by the so-called “Wasserstein metric”, compare, e.g., [AGS05].
The Continuity Axiom then becomes:
Axiom A.14 (Continuity) The setsfq2P.Z/jqpgandfq2P.Z/jpqgare open.
Theorem A.15 Let Z be a compact metric space (e.g. a bounded and closed interval in R). Let be a preference relation, i.e., a complete and transitive relation onP.Z/, satisfying the Continuity and Independence Axioms, thencan be represented by a von Neumann-Morgenstern Expected Utility function uWZ!R.
To prove TheoremA.15, Chatterjee and Krishna use intermediate steps: they prove that the Independence Axiom together with the Continuity Axiom implies a new axiom, theTranslation Invariance Axiom. Together with the Continuity Axiom, this new axiom implies the existence of a EUT function, representing. Translation Invariance can be stated as follows:
Axiom A.16 (Translation Invariance) Let r be a signed measure on Z with aver- age r.Z/D0, in other words, let r be the difference of two probability measures on Z. Let p;q2P.Z/. Assume moreover that pCr;qCr2P.Z/. Then pq implies pCrqCr.
The intuition behind the translation invariance is that adding a signed measure r to a lottery does not change the preference relation. This means that making certain outcomes more likely, others less likelyin the same wayforpandq, does not change the original preference betweenp andq. This is morally the same as the Independence Axiom and mathematically at least close enough to show the equivalence of both axioms (under the condition of continuity) relatively easy.
How can we now use the Independence Axiom to construct an EUT function?
First, one can prove that the indifference sets under the preference relation are
“thin”. This means: for anyq 2 P.Z/and any" > 0there arep;r 2P.Z/which are “close” toq, i.e.,d.p;q/ < "andd.q;r/ < ", and thatpqr.
Second, one can show that, for anyp 2P.Z/, the contour setsfq 2P.Z/jq pg,fq2P.Z/jqpgandfq2P.Z/jq pgare all convex.
P.Z/is a convex subset of a vector space. We can pick a measureo2P.Z/and someız 2 P.Z/withız o. Let us choose moreover someq 2 P.Z/,q ¤ ız, q ¤ o. The structure of the indifference sets derived above allows us to find a continuous affine functionalfWP.Z/!Rsuch thatf.o/D 0,f.ız/D 1and such that the indifference set ofqis a contour set off, i.e., for allp 2P.Z/withq p we havef.q/ >f.p/.6
6More precisely, we first restrict ourselves to a finite dimensional subset, such that the existence of the affine functionalfcan be deduced from the Separating Hyperplane Theorem (see Sect.A.1).
Later one can show thatfis independent of the choice of this finite dimensional subset.
346 A Mathematics Using the translation invariance, one can show thatf reflects the preferences on all ofP.Z/. With a translation, we can also assume thatf is not only affine, but linear.
In the final step, we defineuWZ!Rbyu.z/WDf.ız/. We have to show that this definition is correct, i.e., that
U.p/WD Z
Z
u.z/dp.z/Df.p/:
This is easy to see for measures with finite support: letp D Pn
iD1piızi, then by linearity off,
Z
Z
u.z/dp.z/D Xn
iD1
piu.zi/D Xn
iD1
pif.zi/Df Xn
iD1
piızi
!
Df.p/:
We can approximate any measurep2P.Z/by measures with finite support. Since f is continuous, this proves thatU.p/ D f.p/for allp 2 P.Z/and thusuis an expected utility function representing the preference relation.
B
Teachers open the door. You enter it by yourself.
CHINESE PROVERB
The tests are meant to provide an immediate feedback when studying by yourself, hence we give solutions to all questions. Although some of the questions are tricky and require some thinking about the context of the chapter, the student should be able of answering most questions correctly after working through a chapter. If this is not the case, we would recommend to the reader, to study the chapter a little bit more in detail. A good result, however, can only ensure that the basic concepts have been understood and memorized.
Chapter2
Exercise: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Answers:
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348 B Solutions to Tests
Chapter3
Exercise: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Answers:
Chapter4
Exercise: 1 2 3 4 5 6 7 8 9 10
Answers:
Chapter5
Exercise: 1 2 3 4 5 6 Answers:
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