Arrow's  impossibility theorem:

In an attempt to construct a consistent  social ranking of a set of alternatives on the basis of individual preferences over this set, Arrow obtained:

1)  an impossibility theorem;

2)  a  generalisation  of  the framework  of  welfare  economics,  covering  all collective decisions from political democracy and committee decisions to market allocation;  and

3)  an  axiomatic method which  sets a  standard  of  rigour  for  any  future endeavour.

Prof. Arrow pointed out that the construction  of social welfare function, which reflects  the  preferences  of  all  individuals  comprising the society,  is  an impossible task. His  main  contention  is  that  it  is  vefy  difficult to set  up reasonable democratic procedure for the aggregation of individual preferences into  a social  preference  for  making  a social choice. Arrow  has  proved  & general  theorem according  to which  it  is impossible  to  consmct  a  social ordering which  will  in  some way  reflect  the  individual ordering of all  the members of society.

While constructing  his  argument,  Arrow  has maintained  that  individual's ordering of social states  does not  depend exclusively upon  the  commodities consumed but  also  on  the  amounts of various types  of  collectives  such  as municipal services, parks, sanitation, erection of statues of  famous men, etc. In other words,  an  individual solely on  the  basis of her consumption cannot evaluate welfare results of collective activity; instead, individual ordering of social states will  depend  on  her own  consumption as  well  as  on  the consumption of others  in  a society. Individual ordering of  alternative  social states reflects her value judgments, which  are also called simply  'values7 by Arrow. According to him, it is ordering of social states according to the values
of  individuals  as  distinct from  the  individual  tastes, which  should  be determined for  the construction  of valid social welfare function.

The theorem's content, somewhat simplified, is as follows: A society needs to agree on a preference order among several different options. Each  individual in  the society has a particular personal preference order. The problem  is to find a general mechanism, called a social  choice function,  which  transforms the set of  preference orders, one  for  each  individual,  into a  global  societal preference order. This social choice  function  should have several desirable ("fair")  properties:

• Unrestricted domain  or  universality: the  social choice function should create  a deterministic, complete societal preference order from  every possible set of individual preference orders. (The vote must have a result that  ranks all possible choices relative to one another, the voting mechanism must be  able to process all possible sets of voter preferences, and it should  always give  the  same result for the same votes, without random selection.)
• Non-imposition or citizen sovereignty: every possible societal preference order should be  achievable by  some set  of  individual. preference orders. (Every result must be achievable somehow.)
• Non-dictatorship:  the social choice function should not  simply follow the preference order of a single individual while ignoring all others.
• Positive association of social and  individual values or Monotonicity:  if an individual modifies her preference order  by  promoting a certain option, then  the  societal preference order should respond only by promoting that same option or not  changing, never  by  placing  it  lower than before.  (An individual should not be able to hurt an option by ranking it higher.)
• Independence of relevant  tenatives: if we  restrict attention to a subset of options, and apply the social choice function only to those, then the result should  be  compatible with the outcome for the whole set  of  options. (Changes  in  individuals'  rankings of "irrelevant"  alternatives  [i.e.,  ones outside  the  subset] should have  no  impact on  the societal ranking of the "relevant"  subset.)