We start with a set of at least two voters and a set of at least three candidates. Each voter ranks the set of candidates. Ties are allowed. The set of such rankings is called a "profile." We seek a function from the set of profiles to group rankings that is supposed to represent the collective rankings of each profile. Here are the properties that the function is required to satisfy:
Axiom 1. The group ranking should be a weak order.
Axiom 2. The function should satisfy monotonicity.
Axiom 3. The function should satisfy Pareto optimality.
Axiom 4. The function should satisfy independence of irrelevant alternatives (IIA).
Before explaining these axioms, just a few comments. Arrow’s original paper was in 1949. Since that time he revised it some, and lots of people have reformulated the axioms to suit themselves. Three formulations are in some of the reserve book readings. You are encouraged to look at Olinick, Roberts, and Luce and Raiffa.
Axiom 3, Pareto optimality: If all rankings in the profile are alike, then the group ranking is the same as these rankings.
Axiom 4, Independence of Irrelevant Alternatives: To determine whether x is preferred to y in the group ranking you need only consider how x is ranked relative to y in the profile—i.e., any modification of the profile that keeps the ranking of x and y unchanged will yield the same group ranking of x and y.
Axiom 2. Monotonicity: If two profiles are the same except that in the second one, candidate x is ranked higher than in the first, and if the group ranking for the first profile has x ranked above y, then the second ranking must also have x ranked above y. Thus, if x is a winner, moving x up in a voter’s ranking shouldn’t cause x to lose.
Axiom 1. Weak order: A ranking is a weak order if it is transitive, antisymmetric, and complete. (Complete to a combinatorialist is the same as "connected" to a logician, i.e., given any candidates x and y, each voter must prefer x to y, y to x, or be like them equally well.)
In the proof we define the term "decisive set" of voters. Given
a set of voters and a set of candidates, a set Dxy of voters is a
decisive set for candidate x over candidate y means that for any profile
in which all voters in Dxy prefer x to y, then in the group ranking x is
preferred to y.