### Abstract

This paper presents a new framework which generalizes the concept of conditional expectation to mean values which are implicitly defined as unique solutions to some functional equation. We call such a mean value an implicit mean. The implicit mean and its very special example, the quasi-linear mean, have been extensively applied to economics and decision theory. This paper provides a procedure of defining the conditional implicit mean and then analyzes its properties. In particular, we show that the conditional implicit mean is in general "biased" in the sense that an analogue of the law of iterated expectations does not hold and we characterize the quasi-linear mean as the only implicit mean which is "unbiased".

Original language | English |
---|---|

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Journal of Mathematical Economics |

Volume | 45 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2009 Jan 20 |

### Keywords

- Conditioning
- Implicit mean
- Law of iterated integrals

### ASJC Scopus subject areas

- Economics and Econometrics
- Applied Mathematics