Since B eventually moves to the top of the societal preference, there must be some profile, number k, for which B moves above A in the societal rank.We call the voter whose ballot change causes this to happen the pivotal voter for B over A.
We then prove that this voter is a partial dictator (in a specific technical sense, described below).
Finally we conclude by showing that all of the partial dictators are the same person, hence this voter is a dictator.
In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.
The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem.
The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework.
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In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one.Say there are three choices for society, call them A, B, and C.Suppose first that everyone prefers option B the least: everyone prefers A to B, and everyone prefers C to B.More generally, changes in individuals' rankings of irrelevant alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset.For example, if candidate x ranks socially before candidate y, then x should rank socially before y even if a third candidate z is removed from participation.(See Remarks below.)If any 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.We are searching for a ranked voting electoral system, called a social welfare function (preference aggregation rule), which transforms the set of preferences (profile of preferences) into a single global societal preference order.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 (assumed to be a reasonable requirement of a fair electoral system) at once: The social preference between x and y should depend only on the individual preferences between x and y (pairwise independence).By unanimity, society must also prefer both A and C to B. On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity.Now arrange all the voters in some arbitrary but fixed order, and for each i let profile i be the same as profile 0, but move B to the top of the ballots for voters 1 through i.