Barack Obama, Super Delegates, and Set Theory

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About 10 years ago, I wrote a paper axiomatizing Majority Rule, using the Truth-Functional Logic and Set Theory; I explain the concepts of Social Choice Function, Majority Rule, Voter Equality, Sensitivity, and Option Equality. Today, I republish it here, and explain its application to the Democratic ticket between Barack Obama and HIlary Clinton and explain the Super Delegate concept and how the notion of Super Delegates are, in fact, Anti-Majority Rule because it violates the principle of Voter Equality.

There is a lot of concern lately regarding the 2008 Democratic ticket: The delegate contest between Barack Obama and Hilary Clinton is very close; if it continues to be as close as it has been, the contest could be decided by the Super-Delegates. This is hotly debated because some camps believe that the Super Delegates are not democratic — that is, the voice of the people is heard through the primary and caucuses, but "power" is voiced via the Super-Delegate votes.

In what follows, I republish a paper I wrote about 10 years ago while I was an undergraduate student. I submitted this paper to the Notre Dame Journal of Formal Logic; this paper was accepted, but I received a "revise-and-resubmit." I didn’t revise or resubmit — I was pretty busy trying to support a wife and two kids and get through my undergraduate studies as quickly as I could. Looking back, I wish I had; that would’ve been cool to say that I was published in some fancy/geeky logic journal. Oh well.

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Majority Rule consists of three main properties:

Option Equality
Voter Equality
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. 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 show through the use of proof theory 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 and, in Truth-Functional Logic, I am allowed to assume deduction by cases. 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.

Super-Delegates Violate Voter Equality

Super-Delegates, defined (taken from CNN):

Rationale for Super-Delegates
The Democratic Party established this system in part in response to the nomination of George McGovern in 1972. McGovern took only one state and had only 37.5 percent of the popular vote. Then in 1976, Jimmy Carter was a dark-horse candidate with little national experience.
The purpose of the super-delegate system is to act as a check on ideologically extreme or inexperienced candidates. It also gives power to people who have a vested interested in party policies: elected leaders. Because the primary and caucus voters do not have to be active members of the party (in New Hampshire they can sign up and sign out going-and-coming at the polls), the super-delegate system has been called a safety-valve.
Importance of Super-Delegates
The Democratic Party allocates delegates based on a state’s Presidential vote in the prior three elections and the number of electors. In addition, states that hold their primaries or caucuses later in the cycle receive bonus delegates.
It has been 30 years since the Democratic Party had a cliffhanger going into the Convention. If there is no clear winner after state primaries and caucuses, then the super-delegates — who are bound only by their consciences — will decide the nominee.

I recognize that, in a stale-mate, we need a mechanism to break the tie-breaker. In response to the stale-mate, the Super-Delegate concept makes sense. But, what is not in alignment with the Majority Rule principle of Voter Equality is that the Super Delegates can vote based on their conscience. If the Super-Delegates were constrained to vote for the candidate that received the most popular votes, then the Super-Delegate concept would satisfy both Voter Equality and Sensitivity. Otherwise, the Super-Delegate concept remains to be Anti-Majority Rule because it precisely violates the principle of Voter Equality.