有理数概率下的效用表示定理

UTILITY REPRESENTATION WITH RATIONAL PROBABILITIES

Von Neum ann- Morgenstern的期望效用理论假设对所有的抽奖 (c1 ,p;c2 ,1- p) (以概率 p抽得结果 c1 ,以概率 1- p抽得结果 c2 )的偏好序在所有实数 p(0≤ p≤ 1)均有意义 ;而且期望效用理论基于一组公理 ,从而保证效用函数的存在性和正线性变换意义下的唯一性。然而 ,当概率为无理数时 ,对于抽奖就难以给出直观的解释 ,J.C.Shepherdson首先研究了基于有理数概率度量的效用理论。作者提出一组有理数概率下效用函数存在的公理 ,并证明该公理体系下的效用表示定理。

The expected utility theory of Von Neumann Morgenstern assumes that a preference order is defined for all lotteries (c 1, p; c 2, 1-p) (c 1 with probability p, c 2 with probability 1-p ) for all real p, 0≤p≤1 . And the expected utility theory is based on a set of axioms that assure the existence and uniqueness (up to a positive affine transformation) of utility for all real probability. But when the probability p is irrational, it is hard to interpret the lottery intuitively. J.C.Shepherdson first studies the utility theory based on rational probabilities. This paper puts forward a set of axioms and proves the existence and uniqueness of utility function with rational probabilities on the set of axioms.