Not-That-Bright said:
You must forgive me as I fear your just trying to side-step the fact that you were repeatedly wrong with your dismissal of utilitarianism. Its' not archaic, discarded or outdated - could you please explain to me how you come to your decisions of what is right/wrong?
If you want to debate it, then debate it. Don't just throw up some theory which you know your opponent probably has not had any contact with then say 'disprove it or your idea is wrong' without properly looking into your claim, arrows' theorem in NO WAY affects utilitarianism and IMO it was a cheap tactic to skew the debate.
You have already been given the relevants proofs and information, I dont want to debate it becuase it is a non-argument.
http://en.wikipedia.org/wiki/Arrow's_Impossibility_Theorem
The need to aggregate
preferences occurs in many different disciplines: in
welfare economics, where one attempts to find an economic outcome which would be acceptable and stable; in decision making, where a person has to make a rational choice based on several criteria; and most naturally in
voting systems, which are mechanisms for extracting a decision from a multitude of voters' preferences.
The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options (outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of preferences on the set of outcomes. We are searching for a
preferential voting system, called a
social welfare function, which transforms the set of preferences into a single global societal preference order. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method:
- unrestricted domain or universality: the social welfare function should create a deterministic, complete societal preference order from every possible set of individual preference orders. In other words: 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 consistently give the same result for the same profile of votes—no randomness is allowed in the process.
- non-imposition or citizen sovereignty: every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is onto: It has an unrestricted target space.
- non-dictatorship: the social welfare function should not simply follow the preference order of a special individual while ignoring all others. This means that the social welfare function is sensitive to more than the wishes of a single voter.
- positive association of social and individual values or monotonicity: if an individual modifies his or 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 irrelevant alternatives: if we restrict attention to a subset of options and apply the social welfare 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 (ones outside the subset) should have no impact on the societal ranking of the relevant subset. This is a restriction on the sensitivity of the social welfare function.
Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social welfare function that satisfies all these conditions at once.
Another version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria with that of:
- unanimity or Pareto efficiency: if every individual prefers a certain option to another, then so must the resulting societal preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the preference profile.
This version of the theorem is stronger—has weaker conditions—since monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency, non-imposition, and independence of irrelevant alternatives together do not imply monotonicity.
et A be a set of
outcomes, N a number of
voters or
decision criteria. We shall denote the set of all full linear orderings of A by L(A) (this set is equivalent to the set S | A | of permutations on the elements of A). A
social welfare function is a function
which aggregates voters' preferences into a single preference order on A. The
n-tuple
of voter's preferences is called a
preference profile.
In its strongest and most simple form, Arrow's impossibility theorem states that whenever the set A of possible alternatives has more than 2 elements, then the following three conditions become
incompatible:
unanimity, or
Pareto efficiencyIf alternative
a is ranked above
b for all orderings
, then
a is ranked higher than
b by
. (Note that
unanimity implies
non-imposition). non-dictatorshipThere is no individual
i whose preferences always prevail. That is, there is no
such that
,
.
independence of irrelevant alternativesFor two preference profiles
and
such that for all individuals
i, alternatives
a and
b have the same order in
Ri as in
Si, alternatives
a and
b have the same order in
as in
.
Arrow's theorem is a mathematical result, but it is often expressed in a non-mathematical way, with a statement such as
"No voting method is fair",
"Every ranked voting method is flawed", or
"The only voting method that isn't flawed is a dictatorship". These statements are simplifications of Arrow's result which are not universally considered to be true. What Arrow's theorem does state is that a voting mechanism cannot comply with all of the conditions given above simultaneously.
Arrow did use the term "fair" to refer to his criteria. Indeed, the
Pareto principle, as well as the demand for non-imposition, seems trivial. As for the
independence of irrelevant alternatives (IIA) - suppose Dave, Chris, Bill and Agnes are running for office. And suppose Agnes has a clear advantage. Now according to Arrow's theorem, there could be a situation where if Dave steps out of the race, it will suddenly be Bill, and not Agnes, who would win the race. This would seem
"unfair" by many. And yet it can happen, and Arrow's theorem states that these "unfair" situations cannot be avoided in general, without relaxing some other criterion. Something has to give. So the important question to be asked, in light of Arrow's theorem is: which condition should be relaxed?
Various theorists and hobbyists have suggested weakening the IIA criterion as a way out of the paradox. Proponents of ranked voting methods contend that the IIA is an unreasonably strong criterion, which actually does not hold in most real-life situations. Indeed, the IIA criterion is the one breached in most useful
voting systems.
Advocates of this position point out that failure of the standard IIAC is trivially implied by the possibility of cyclic preferences. If voters cast ballots as follows...
7 votes for A > B > C
6 votes for B > C > A
5 votes for C > A > B
...then the net preference of the group is A > B > C > A. In this circumstance,
any system that picks a unique winner, and satisfies the very basic majoritarian rule that a candidate who receives a majority of all first-choice votes must win the election, will fail IIAC. Without loss of generality, consider that if a system currently picks A, and B drops out of the race, the remaining votes will be:
7 votes for A > C
11 votes for C > A
Thus, C will win, even though the change (B dropping out) concerned an "irrelevant" alternative candidate who did not win in the original circumstance.
Relaxing the IIA criterion, though popular, has a distinct disadvantage: it can result in
strategic voting, making the voting mechanism 'manipulable'. That is, any voting mechanism which is not IIA can yield a setup where some of the voters get a better result by mis-reporting their preferences (e.g. I prefer
a to
b to
c, but I claim I prefer
b to
c to
a). Clearly, any non-monotonic social welfare function is manipulable as well. If one uses a manipulable voting scheme in real life, one should expect some "dishonest" voting. What this means is that the real-life implementation of most voting mechanisms results in a complicated game of skill. The
Gibbard-Satterthwaite theorem, an attempt at weakening the conditions of Arrow's paradox, replaces the IIA criterion with a criterion of non-manipulability, only to reveal the same impossibility.
So, what Arrow's theorem really shows is that voting is a non-trivial game, and that
game theory should be used to predict the outcome of most voting mechanisms. This could be seen as a discouraging result, because a
game need not have efficient equilibria,
e.g., a ballot could result in an alternative nobody really wanted in the first place, yet everybody voted for.
Also, if you're after something a little more complex:
http://mason.gmu.edu/~atabarro/arrowstheorem.pdf
note page 11 especially