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Let <math>X</math> be a set of possible `states of the world' or `alternatives';  society wishes to choose some state from <math>X</math>.  Let <math>I</math> be a finite set, representing a collection of people.  For each <math>i \in I</math>, let <math>u_i:X\longrightarrow\mathbb{R}</math> be a ''[[utility|utility function]]''.  A ''[[social choice theory|social choice rule]]'' (or ''[[voting system]]'') is a mechanism which uses the data <math>(u_i)_{i \in I}</math> to select some element(s) from <math>X</math> which are `best' for society.  (The basic problem of [[social choice theory]] is to disambiguate the word `best'.)


The classic [[utilitarianism|utilitarian]] social choice rule selects the element <math>x \in X</math> which maximizes the ''utilitarian sum''
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<center><math> U(x):= \sum_{i\in I} u_i(x).</math></center>
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However, for this formula to make sense, we must assume that the utility functions <math>(u_i)_{i \in I}</math> are both [[cardinal utility|cardinal]], and [[social choice theory#Interpersonal utility comparison|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]], [[Coherence (philosophical gambling strategy)|Bruno de Finetti]],
[[Von Neumann–Morgenstern utility theorem|von Neumann and Morgenstern]], and [[Subjective expected utility|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 <math>u_i:X\longrightarrow \mathbb{R}</math> is valid description of her preferences, and if <math>r_i,s_i\in \mathbb{R}</math> are two constants with <math>s_i>0</math>, then the `rescaled' utility function <math>v_i(x) := s_i\, u_i(x) +  r_i</math> is an equally valid description of her preferences. If we define a new package of utility functions <math>(v_i)_{i\in I}</math> using possibly different <math>r_i\in \mathbb{R}</math> and <math>s_i>0</math> for all <math>i \in I</math>, and we then consider the utilitarian sum
 
<center><math> V(x):= \sum_{i\in I} v_i(x),</math></center>
 
then in general, the maximizer of <math>V</math> will ''not'' be the same as the maximizer of <math>U</math>.  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 <math>i \in I </math>, suppose that the values
 
<center><math> m_i \ := \ \min_{x \in X} \, u_i(x) \quad \mbox{and}\quad M_i \ := \ \max_{x \in X} \, u_i(x)</math></center>
 
are well-defined.  (For example, this will always be true if <math>X</math> is finite, or if <math>X</math> is a compact space and <math>u_i</math> is a continuous function.)  Then define
 
<center><math>w_i(x) \ :=  \ \frac{u_i(x) - m_i}{M_i - m_i}</math></center>
 
for all <math>x\in X</math>. Thus, <math>w_i:X \longrightarrow \mathbb{R}</math> 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 <math>X</math> which maximizes the utilitarian sum
 
<center><math> W(x):= \sum_{i\in I} w_i(x).</math></center>
 
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]]
 
== References ==
<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist}}
 
*{{Citation | last1=Cao| first1=Xiren|  title= Preference functions and bargaining solutions| journal= Proceeedings of the 21st IEEE Conference on Decision and Control|volume=1|year=1982|pages=164–171}}
 
*{{Citation | last1=Dhillon| first1=Amrita|  title=Extended Pareto rules and relative utilitarianism| journal=Social Choice and Welfare|volume=15|year=1998|pages=521–542 |doi=10.1007/s003550050121}}
 
*{{Citation | last1=Dhillon| first1=Amrita| last2=Mertens| first2=Jean-Francois| title=Relative utilitarianism| journal=Econometrica|volume=67|number=3|year=1999|pages=471–498 |doi=10.1111/1468-0262.00033}}
 
*{{Citation | last1=Karni| first1=Edi|title=Impartiality: definition and representation| journal=Econometrica|volume=66|number=6|year=1998|pages=1405–1415 |jstor=2999622}}
 
*{{Citation | last1=Pivato| first1=Marcus| title=Twofold optimality of the relative utilitarian bargaining solution| journal=Social Choice and Welfare| volume= 32 |number=1| year= 2008 | pages=79–92 |doi=10.1007/s00355-008-0313-0}}
 
*{{Citation | last1=Segal| first1=Uzi|  title=Let's agree that all dictatorships are equally bad| journal=Journal of Political Economy|volume=108|number=3|year=2000|pages=569–589 |doi=10.1086/262129}}
 
*{{Citation | last1=Sobel| first1=Joel|  title=Manipulation of preferences and relative utilitarianism| journal=Games and Economics Behaviour|volume=37|year=2001|pages=196–215 |doi=10.1006/game.2000.0839}}
 
== External links ==
 
[[Category:Economics]]
[[Category:Ethics]]
[[Category:Political science]]
[[Category:Social choice theory|*]]
[[Category:Voting systems]]
[[Category:Analysis of collective decision-making]]
[[Category:Utilitarianism]]
 
[[de:Sozialwahltheorie]]
[[it:Teoria della scelta sociale]]
[[ja:社会選択理論]]
[[pt:Teoria da escolha social]]

Latest revision as of 17:20, 4 June 2014


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