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{{No footnotes|date=April 2011}}
 
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''
 
<center><math> U(x):= \sum_{i\in I} u_i(x).</math></center>
 
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 mechanismFor 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]]

Revision as of 14:50, 22 September 2013

Template:No footnotes

Let be a set of possible `states of the world' or `alternatives'; society wishes to choose some state from . Let be a finite set, representing a collection of people. For each , let be a utility function. A social choice rule (or voting system) is a mechanism which uses the data to select some element(s) from 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 which maximizes the utilitarian sum

However, for this formula to make sense, we must assume that the utility functions 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 is valid description of her preferences, and if are two constants with , then the `rescaled' utility function is an equally valid description of her preferences. If we define a new package of utility functions using possibly different and for all , and we then consider the utilitarian sum

then in general, the maximizer of will not be the same as the maximizer of . 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 , suppose that the values

are well-defined. (For example, this will always be true if is finite, or if is a compact space and is a continuous function.) Then define

for all . Thus, 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 which maximizes the utilitarian sum

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

References

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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