Some Notes on Maximum Entropy Utility.

Eun Young Kim,Byeong Seok Ahn

doi:10.3390/e21070637

Eun Young Kim, Byeong Seok Ahn

Open Access

https://doi.org/10.3390/e21070637

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Abstract

The maximum entropy principle is effective in solving decision problems, especially when it is not possible to obtain sufficient information to induce a decision. Among others, the concept of maximum entropy is successfully used to obtain the maximum entropy utility which assigns cardinal utilities to ordered prospects (consequences). In some cases, however, the maximum entropy principle fails to produce a satisfactory result representing a set of partial preferences properly. Such a case occurs when incorporating ordered utility increments or uncertain probability to the well-known maximum entropy formulation. To overcome such a shortcoming, we propose a distance-based solution, so-called the centralized utility increments which are obtained by minimizing the expected quadratic distance to the set of vertices that varies upon partial preferences. Therefore, the proposed method seeks to determine utility increments that are adjusted to the center of the vertices. Other partial preferences about the prospects and their corresponding centralized utility increments are derived and compared to the maximum entropy utility.

Highlights

The maximum entropy principle is effective in solving decision problems, especially when it is not possible to obtain sufficient information to induce a decision [1,2,3,4,5]
Let us assume an ordered increasing utility (OIU) increment (in the latter part of the paper, we provide the ordered decreasing utility (ODU) increment defined by ∆u1 ≥ ∆u2 ≥ · · · ≥ ∆uK ):
We have shown two examples in which the maximum entropy principle works improperly when the utility simplex is restricted by additional partial preferences

Summary

Introduction

The maximum entropy principle is effective in solving decision problems, especially when it is not possible to obtain sufficient information to induce a decision [1,2,3,4,5]. An elegant approach to circumvent this problem is needed to solve real-world decision-making problems. To this end, Abbas [9] developed the maximum entropy approach to assigning cardinal utility to each prospect when only the ordered prospects are known. We doubt if the maximum entropy approach results in cardinal utilities representing a set of partial preferences properly where some other partial preferences about the prospects are incorporated. In another context of true maximum ignorance where the state of prior knowledge is not strong, the maximum a posteriori probability can be better estimated by classical Bayesian theory; it is not necessary to introduce a new and exotic approach such as maximum entropy [12]. We discuss the maximum entropy utility approach further using the notations and definitions from Abbas [9]

Does the Maximum Entropy Principle Always Guarantee a Good Solution?

Centralized Utility Increments

Conclusions