A challenge to the compound lottery axiom: A two-stage normative structure and comparison to other theories
Decisions among lotteries whose outcomes are themselves lotteries can be made by considering the collapsed lottery of the two stage lottery on the outcome space (view 1). Another approach is chosen by the authors: ``Especially when the underlying lottery uncertainties are resolved at different times or when they represent essentially different uncertainty concepts, a more complex model than that implied in view 1 is needed to fully capture the decision maker's primitive preference relationships.'' The authors prove a theorem which weakens the ``reduction of compound lotteries axiom''. Let \(Z\) be the outcome space and let \(D_2\) be a mixture set of Borel probability measures on the Borel \(\sigma\)-field of \(Z\) (second stage lotteries). Let \(D_1\) be a mixture space of Borel probability measures on the set \(X_2\) of all closed subsets of \(D_2\) (first stage lotteries). Let \(\succsim_i\) be a weak order on \(D_i\) which is continuous and fulfills strong independence \((i= 1,2)\). Furthermore, consistency is assumed, i.e.: For degenerate distributions \(d, d'\in D_1\) the preference \(d\succsim_1 d'\) implies consistent preference between the outcome probability distribution, i.e. if \(x\in X_2\), \(x'\in X_2\) are resultant states of \(d\) and \(d'\) respectively, then for every \(e'\in x'\) there exists an \(e\in x\) with \(e\succsim_2 e'\). Then there exist bounded functions \(u_2: Z\to \mathbb{R}\), \(u_1: X_2\to \mathbb{R}\) such that for \(d, d'\in D_i: d\succsim_i d'\Leftrightarrow E_d(u_i)\geq E_{d'}(u_i)\) \((i= 1,2)\). Moreover, there exists a unique continuous strictly increasing function \(\phi: \mathbb{R}\to \mathbb{R}\) s.t. \(u_1(d)= \phi(E_d(u_2))\) for all \(d\in D_2\); when the reduction of compound lotteries axiom is assumed \(\phi\) is linear but in general it is not. The paper contains a review of existing theories of two stage lottery. The results are compared to these theories.
scientific article; zbMATH DE number 710394
scientific article