Axiomatizing Majority Rule
Looking through my old files at 5AM this morning, I stumbled upon a paper that I wrote on Majority Rule. It occurred to me that this would have been a nice post on America’s Independence Day. Anyway, Majority Rule is one of the main tenets of a democratic republic and is an important piece in many governments, not just for the United States. I wrote this paper for a formal logic class I took while an sophomore undergraduate at BYU in 1997. At the time, I was really into formal logic and math — I still am, but I’m very, very rusty. Reading this paper now, for example, is super confusing to me. I’m just getting old and my interests have changed.
Formalizing a System such as Majority Rule also has many, many implications and applications: Algorithmic applications for Game Theory as well as in Computer Science are the most obvious ones. But, there is also many applications in broad mathematics, political science, law, government, linguistics and even sociology. Formalizing a System is important and helps us to better understand our detailed interactions as a community within or following a system.
Note: This paper does not address the complexities of socioeconomics or psychology. It is a simple axiomatization of a system. The non-quantitative pieces of a Majority Rule System, in my opinion, constitute the really complex pieces: sociology, economics, psychology, and politics. Math is easy; human stuff is super tough.
This post also assumes you have had formal training up to truth-functional logic, predicate logic, and metalogic. Also, if you are using Internet Explorer, the subset symbol (sideways U) will look like a square box. It’s really subset. There are no problems on the other browsers I tested this post on, but I strongly recommend making Flock your browser of choice.
Flock is great; it provides you the same browsing capabilities as Firefox, but allows for seamless integration with Flickr, del.icio.us, Photobucket, Your Blog, Shadows, and others. It’s fast, clean, and switching from Firefox is easy as you’ll see: the set-up is very user friendly and moves everything you were doing in Firefox or IE into Flock. Piece of cake. Check out Flock.
Think of this as a Geek’s Tribute to America — Without further ado, here is my Axiomatic Developement of The Majority Rule System:
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Why are we attracted to the majority rule system? Some argue that it is because the majority rule system is fair and effective. Proponents argue that it is fair because it satisfies two of its three main properties: option equality, voter equality, and sensitivity.
Option equality means that the options are equal, i.e., if the same amount of people vote for choice a, the results will switch. Voter equality means that no person’s vote is more influential than that of another’s vote. Furthermore, supporters of the majority rule system also argue that the system is effective because it satisfies the third property, sensitivity. Sensitivity means that if the voting is tied and one more person votes, then the option he votes for wins. This paper will show that the majority rule system does satisfy the three main properties mentioned above and, by doing so, proves to be both fair and effective. Note: this paper does not try to elaborate or address socio-political or socio-economical issues — such issues are appropriate and should be addressed in a more wholesale social scientific paper. I simply will show through the use of the proof that the Majority Rule System satisfies the properties it claims to uphold.
Before showing that majority rule satisfies option equality, voter equality, and sensitivity, I will need to set out my definitions and presuppositions. One major presupposition in the notion of majority rule is that of choice. Social scientists call the method of choosing a social choice function. A social choice function is a rule for choosing between options where some of the citizens prefer one option and others prefer the other. The majority rule system chooses one option over another when more citizens prefer the one option than the other. From this we get the definition of a social choice function (SCF):
SCF: C is a social choice function for options a and b and citizens V iff a≠b, and V is finite and non-empty, and for all disjoint subsets A and B of V, C(A, B) equals {a} or {b} or {a, b}.
We also get the definition of the majority rule social choice function:
mrSCF: C is a majority rule social choice function for options a and b and citizens V iff C is a social choice function for options a and b and citizens V, and for all disjoint subsets A and B of V, C(A, B)={a, b} if A≈B, C(A, B)={a} if A>B, and C(A, B)={b} if B>A.
In addition to the definition of majority rule social choice function, there are three properties of majority rule. The first is called option equality (OE). This may be defined thus:
OE: C satisfies option equality for options a and b and citizens V iff for all disjoint subsets A and B of V, C(A, B)={a} iff C(B, A)={b}.
The second property of majority rule is called voter equality (VE). Voter equality, defined, is:
VE: C satisfies voter equality for options a and b and citizens V iff for all disjoint subsets A and B of V, if A≈B, then C(A, B)=C(B, A).
The last property of majority rule is called sensitivity (S). This is defined as the following:
S: C satisfies sensitivity for options a and b and citizens V iff for all disjoint subsets A and B of V, if C(A, B)={a, b}, then for every non-empty subset X of V-(A∪B), C(A∪X, B)={a}.
As I proceed, please keep in mind the aim of this paper: to show that the majority rule social choice function satisfies the three properties of majority rule, namely, option equality, voter equality, and sensitivity. In other words,
Theorem: C is the Majority Rule social choice function for options a and b and citizens V iff C is a social choice function that satisfies Option Equality, Voter Equality, and Sensitivity for options a and b and citizens V.
To accomplish this aim, I must proceed in two phases. The first phase will contain the three properties of majority rule. The second phase will contain the three conditions of majority rule social choice function.
Phase I
1.1 (OE) C(A, B)={a} iff C(B, A)={b}
1.2 (VE) If A≈B, then C(A, B)=C(B, A)
1.3 (S) If C(A, B)={a, b}, then for every non-empty subset X of V-(A∪B), C(A∪X, B)={a}
Phase II
2.1 C(A, B)={a, b} if A≈B
2.2 C(A, B)={a} if A>B
2.3 C(A, B)={b} if B>A
What I have done by setting up both Phase I and II is to put the aim of this paper in a slightly different way: I now need to show how the elements in Phase I satisfy the elements in Phase II. To do this, I will assume the elements in Phase II and prove the elements in Phase I. Also, I need to show how the elements in Phase II satisfy those in Phase I. To do this, I will assume the elements in Phase I and prove the elements in Phase II. I can prove the theorem this way because it is a bi-conditional statement. That is, if I assume the left-hand side, the right-hand side should follow. Similarly, if I assume the right-hand side, the left-hand side should also follow. By showing how Phase I and II prove each other, I will ultimately show that the majority rule social choice function does satisfy option equality, voter equality, and sensitivity and hence demonstrate fairness and effectiveness.
Phase I
1.1: I need to show that C(A, B)={a} iff C(B, A)={b}.
Proof: Assume that C(A, B)={a}; show C(B, A)={b}. If A≈B, then C(A, B)={a, b}. If B>A, then C(A, B)={b}. By process of elimination and cases, since C(A, B)={a}, A>B. So, C(B, A)={b}. Now, assume the other side of the bi-conditional, i.e., C(B, A)={b}; show C(A, B)={a}. If A≈B, then C(A, B)={a, b}. If A>B, then C(A, B)={a}. By cases and process of elimination, B>A. So, C(A, B)={a}. Q.E.D.
1.2: If A≈B, then C(A, B)=C(B, A).
Proof: Assume A≈B; show C(A, B)=C(B, A). So, C(A, B)={a, b}. Also, C(B, A)={a, b}. Hence, C(A, B)=C(B, A). Q.E.D.
1.3: If C(A, B)={a, b}, then for every non-empty subset x of V-(A∪B), C(A∪X, B)={a}.
Proof: Assume C(A, B)={a, b}; show that for every non-empty subset X of V-(A∪B), C(A∪X, B)={a}. Since X≠∅ and x⊆(V-(A∪B)), A∪X>A. If A>B, then C(A, B)={a}.
If B>A, then C(A, B)={b}. So, by the process of elimination and cases, since C(A, B)={a, b}, A≈B. Hence, A∪X>B. Consequently, C(A∪X, B)={a}. Q.E.D.
Thus, by proving Option Equality, Voter Equality, and Sensitivity, I have shown that they are derived from the notion of majority social choice function. But I am only halfway finished.
I now need to prove the elements in Phase II. Remember, that in doing so, I will be assuming the elements in Phase I.
Phase II
2.1 If A≈B, then C(A, B)={a, b}.
Proof: Assume A≈B; show C(A, B)={a, b}. Since A≈B, by voter equality I get C(A, B)=C(B, A). Also, bear in mind the definition of a social choice function, namely, that C(A, B) must equal {a} or {b} or {a, b}. By option equality, C(A, B)={a} iff C(B, A)={b}. Yet, {a}≠{b}. So, C(A, b)={a, b}. Q.E.D.
2.2 If A>B, then C(A, B)={a}.
Proof: Assume A>B; show C(A, B)={a}. Sensitivity claims that if C(A, B)={a, b}, then for every non-empty subset X of V-(A∪B), C(A∪X, B)={a}. Let A’ be a subset of A such that A’≈B. Accordingly, by Phase 2.1, I get C(A’, B)={a, b}. So, C(A, B)={a}. Q.E.D.
2.3 If B>A, then C(A, B)={b}.
Proof: Assume B>A; show C(A, B)={b}. Sensitivity dictates that if C(A, B)={a, b}, then for every non-empty subset X of V-(A∪B), C(A∪X, B)={a}. Let B’ be a subset of B such that A≈B’. By Phase 2.1, I get C(B’, A)={a, b}. But this means that C(B, A)={a}. So, by option equality, C(A, B)={b}. Q.E.D.
Now, both Phases have been proven. Since this is true, I have shown that the majority rule social choice function does satisfy option equality, voter equality, and sensitivity. By satisfying these three properties, the majority rule proves to be both fair and effective.
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pete,
i can understand what you just said here in axiomizing majority rule. what’s the deal with choosing the u.s. president- the majority rule isnt really applicable, but a subset of the majority right– what’s the theory behind that idea of an electoral college choosing the president versus popular vote?
harry