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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Relative utilitarianism ： ウィキペディア英語版 Relative utilitarianism Let $X$ be a set of possible `states of the world' or `alternatives'; society wishes to choose some state from $X$. Let $I$ be a finite set, representing a collection of people. For each $i\; \backslash in\; I$, let $u\_i:X\backslash longrightarrow\backslash mathbb$ be a ''utility function''. A ''social choice rule'' (or ''voting system'') is a mechanism which uses the data $(u\_i)\_$ to select some element(s) from $X$ which are `best' for society. (The basic problem of social choice theory is to disambiguate the word `best'.) The classic utilitarian social choice rule selects the element $x\; \backslash in\; X$ which maximizes the ''utilitarian sum'' $U(x):=\; \backslash sum\_\; u\_i(x).$ However, for this formula to make sense, we must assume that the utility functions $(u\_i)\_$ are both cardinal, and interpersonally comparable at a cardinal level. The notion that individuals have cardinal utility functions is not that problematic. Cardinal utility has been implicitly assumed in decision theory ever since Daniel Bernoulli's analysis of the Saint Petersburg Paradox. Rigorous mathematical theories of cardinal utility (with application to risky decision making) were developed by Frank P. Ramsey, Bruno de Finetti, von Neumann and Morgenstern, and Leonard Savage. However, in these theories, a person's utility function is only well-defined up to an `affine rescaling'. Thus, if the utility function $u\_i:X\backslash longrightarrow\; \backslash mathbb$ is valid description of her preferences, and if $r\_i,s\_i\backslash in\; \backslash mathbb$ are two constants with $s\_i>0$, then the `rescaled' utility function $v\_i(x)\; :=\; s\_i\backslash ,\; u\_i(x)\; +\; r\_i$ is an equally valid description of her preferences. If we define a new package of utility functions $(v\_i)\_$ using possibly different $r\_i\backslash in\; \backslash mathbb$ and $s\_i>0$ for all $i\; \backslash in\; I$, and we then consider the utilitarian sum $V(x):=\; \backslash sum\_\; v\_i(x),$ then in general, the maximizer of $V$ will ''not'' be the same as the maximizer of $U$. Thus, in a sense, classic utilitarian social choice is not well-defined within the standard model of cardinal utility used in decision theory, unless we specify some mechanism to `calibrate' the utility functions of the different individuals. ''Relative utilitarianism'' proposes a natural calibration mechanism. For every $i\; \backslash in\; I$, suppose that the values $m\_i\; \backslash \; :=\; \backslash \; \backslash min\_\; \backslash ,\; u\_i(x)\; \backslash quad\; \backslash mbox\backslash quad\; M\_i\; \backslash \; :=\; \backslash \; \backslash max\_\; \backslash ,\; u\_i(x)$ are well-defined. (For example, this will always be true if $X$ is finite, or if $X$ is a compact space and $u\_i$ is a continuous function.) Then define $w\_i(x)\; \backslash \; :=\; \backslash \; \backslash frac$ for all $x\backslash in\; X$. Thus, $w\_i:X\; \backslash longrightarrow\; \backslash mathbb$ is a `rescaled' utility function which has a minimum value of 0 and a maximum value of 1. The Relative Utilitarian social choice rule selects the element in $X$ which maximizes the utilitarian sum $W(x):=\; \backslash sum\_\; w\_i(x).$ As an abstract social choice function, relative utilitarianism has been analyzed by Cao (1982), Dhillon (1998), Karni (1998), Dhillon and Mertens (1999), Segal (2000), Sobel (2001) and Pivato (2008). (Cao (1982) refers to it as the `modified Thomson solution'.) When interpreted as a `voting rule', it is equivalent to Range voting. == See also == *Range voting 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Relative utilitarianism」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.