# Social decision rules and Condorcet’s Jury Theorem

Thoughts from Microeconomics Lecture 1, Cambridge Economics Part IIA (Michaelmas)

#### Social decision rules

Dr. Elliott introduced four social decision rules in this lecture. Social decision rules are rules by which a society collectively makes decisions, e.g. whether or not they should exit a union. The four decision rules introduced were

1. Majority Rule (get >50% of the vote to win),
2. Super Majority Rule (get at least k votes more than the 50% mark to win, where k>0),
3. Weighted Majority Rule (each person’t vote is weighted, weights are real so they can be positive, negative or zero), and
4. Dictatorship (one person’s vote determines the outcome).

Each person can choose to vote for (1) or against (-1) or abstain (0). Society collectively can choose to act for or against or to abstain from acting.

Thoughts: All four rules are versions of a weighted super majority rule (where k >=0). In (1) and (2), everyone’s votes are weighted equally. In (1) and (3), k=0. In (4), everyone except the dictator’s vote weights are equal to 0 the dictator’s vote has a weight of 1, and k=0 (in the standard case).

#### Axiomatic approach

There were four axioms that together uniquely characterised Majority Rule (Theorem, May 1952). They are:

1. Anonymity,
2. Positive Responsiveness
3. Universal Domain and
4. Neutrality.

Thoughts What’s interesting these axioms’ correspondence to symmetries. Neutrality basically means the function is odd. Anonymity corresponds to some symmetry in all dimensions, with each vote being a dimension. Positive Responsiveness can be thought of as an outward direction applied the function (if going toward the origin is going ‘inwards’). Universal Domain is a domain-wide symmetry in that the entire domain is included.

Note also that even if I assign a person’s vote a weight of zero, my decision rule still satisfies the Universal Domain axiom because I’m still considering them.

The four axioms are also easy to remember: their first letters spell out APUN (A Pun).

#### Condorcet’s Jury Theorem

The premise of this theorem is there is some external state X that is either true or false. A jury of people will then vote on whether they think X is true or false, e.g. if they think  a person is guilty of a crime or not based on evidence offered.

The theorem states that, assuming (1) competence (people make decisions better than random) and (2) conditional independence in voting, as the number of people in the jury n increases, the probability of obtaining the correct decision p increases. Moreover, as n tends to infinity, p tends to 1.

Thoughts One of my early thoughts was ‘How will I remember these assumptions?’ Fortunately, both assumptions and the theorem all begin with the letter C, so it’s easy. Actually, all three begin with ‘co’. And co means together. So all three – together – begin with ‘co’. That’s cool, no? 🙂

There is also a stronger result in this theorem about p being fixed which I might comment on later.

PS: As you may have noticed, I am not defining these concepts in detail – that would take too much space and be unnecessary – there are plenty of places online where you can find definitions. I am only pointing out my observations.