Stochastic transitivity and quadratic representation functions
Let \(L\) be the set of finite lotteries with outcomes in \([0,M]\). Let \(\gtrsim\) be a complete, transitive, continuous, monotone (first order stochastic dominance) and strict quasi-concave relation on \(L\). It is assumed that comparing two lotteries \(X\) and \(Y\) the real choice set is the set of compound lotteries \(\{(X,\alpha; Y,1-\alpha)| \alpha\in [0,t]\}\) which can be identified -- assuming the reduction of compound lotteries axiom -- with the set \(\{\alpha X+ (1-\alpha)Y| \alpha\in [0,1]\}\subset L\). If \(V\) is the functional representing \(\gtrsim\) the decision maker will choose a compound lottery \((X,\alpha^*; Y,1- \alpha^*)\), where \(\alpha^*\) is determined by \(\max_ \alpha V(\alpha X+(1- \alpha)Y)\). Let \(C: L\times L\to \mathbb{R}\) be a mapping with \(C(X,Y)= \alpha^*\) if \(X\neq Y\), \(C(X,X)= {1\over 2}\). Then the relation \(\gtrsim\) implies a new relation \(\gtrsim^*\) on \(L\) by defining: \(X\gtrsim^* Y\Leftrightarrow C(X,Y)\gtrsim{1\over 2}\), \(X\sim^* Y\Leftrightarrow C(X,Y)= {1\over 2}\). It is shown: Either the relation \(\gtrsim^*\) is intransitive or the relation \(\gtrsim\) is quadratic (\(\gtrsim\) is quadratic if \(V(x_ 1,p_ 1;\dots; x_ n, p_ n)=\sum^ n_{i=1} \sum^ n_{j=i} p_ i p_ i\varphi(x_ i, x_ j)\) with a symmetric function \(\varphi\)). Since it is optimal for decision makers with a strict quasi-concave preference relation to randomize, the result shows that intransitivity must be observed if the preference relation \(\gtrsim\) is not quadratic. The proofs make strong use of geometrical considerations.
strict quasi- concave preference relation
scientific article; zbMATH DE number 554106