Social Choice Theory: Setup and Aggregation Rules

Social Choice Theory: Setup and Aggregation Rules

The Problem of Collective Choice

In Lecture 1 we studied individual decision-making: a single agent with rational preferences chooses the alternative that maximizes her utility. But most political decisions are made collectively. A legislature passes a bill, voters choose a president, a committee selects a policy. In each case, many individuals with potentially different preferences must arrive at a single collective choice. How should that aggregation work?

Social choice theory, developed formally by Kenneth Arrow in the early 1950s, provides the framework for studying this question rigorously. The goal is to find a rule that maps a profile of individual preferences to a social preference — ideally one that is itself rational and satisfies other desirable properties.

The Formal Setup

We have $N$ voters, indexed $1, 2, \ldots, N$, and a set of alternatives $X$. Each voter $i$ has a preference relation $\succsim_i$ over $X$, which we assume is rational (complete, reflexive, transitive). As in Lecture 1, $x \succsim_i y$ means "voter $i$ considers $x$ at least as good as $y$," with strict preference $\succ_i$ and indifference $\sim_i$ derived in the usual way.

A preference profile $(\succsim_1, \succsim_2, \ldots, \succsim_N)$ is a list recording each voter's preference relation. It encodes all the information about individual preferences in the electorate.

A social choice function $F$ maps each preference profile to a social preference relation $\succsim_S$. We write:

$F(\succsim_1, \ldots, \succsim_N) = \succsim_S$

The social preference $\succsim_S$ is meant to represent the "collective will" of the group. We write $x \succsim_S y$ to mean "society considers $x$ at least as good as $y$," with strict social preference $\succ_S$ and social indifference $\sim_S$ defined analogously.

A Running Example

To make these ideas concrete, consider three voters and three candidates: Gore (G), Bush (B), and Nader (N). Suppose the three voters have the following strict rankings from best to worst:

• Voter 1: Gore $\succ_1$ Nader $\succ_1$ Bush
• Voter 2: Gore $\succ_2$ Bush $\succ_2$ Nader
• Voter 3: Bush $\succ_3$ Nader $\succ_3$ Gore

We will apply four different aggregation rules to this profile and see what social preference each produces.

Four Aggregation Rules

Majority rule: Under majority rule, $x \succsim_S y$ if and only if at least half of voters prefer $x$ to $y$ (i.e., $x \succsim_i y$). Applying this to our example: Gore vs. Bush — voters 1 and 2 prefer Gore, voter 3 prefers Bush, so Gore $\succ_S$ Bush. Gore vs. Nader — voters 1 and 2 prefer Gore, so Gore $\succ_S$ Nader. Bush vs. Nader — voters 2 and 3 prefer Bush, so Bush $\succ_S$ Nader. The resulting social ranking is Gore $\succ_S$ Bush $\succ_S$ Nader.

Unanimity rule: Under unanimity rule, $x \succsim_S y$ only if every single voter prefers $x$ to $y$. Gore vs. Bush: voters 1 and 2 prefer Gore, voter 3 prefers Bush — no unanimous agreement, so Gore and Bush are incomparable under the social preference. Gore vs. Nader: voters 1 and 2 prefer Gore, voter 3 prefers Nader — again incomparable. Bush vs. Nader: voters 2 and 3 prefer Bush, voter 1 prefers Nader — incomparable. Notice that unanimity allows incomparability: the social preference relation simply does not rank these pairs. Incomparability is a logically valid output of a social choice function — the rule is simply silent on those comparisons.

Borda count: The Borda count assigns numerical scores by rank position. With three candidates, a voter gives 3 points to her top choice, 2 to her second choice, and 1 to her last choice. We sum the points each candidate receives across all voters:

$$\begin{aligned} \text{Gore:} & \quad 3 + 3 + 1 = 7 \\ \text{Bush:} & \quad 1 + 2 + 3 = 6 \\ \text{Nader:} & \quad 2 + 1 + 2 = 5 \end{aligned}$$

Wait — let us redo this carefully. Voter 1 ranks: Gore (3), Nader (2), Bush (1). Voter 2 ranks: Gore (3), Bush (2), Nader (1). Voter 3 ranks: Bush (3), Nader (2), Gore (1). Summing: Gore gets $3+3+1=7$, Bush gets $1+2+3=6$, Nader gets $2+1+2=5$. Social ranking: Gore $\succ_S$ Bush $\succ_S$ Nader.

Dictatorship: Under a dictatorship, the social preference is simply one designated voter's preference relation. If voter 2 is the dictator, the social ranking is Gore $\succ_S$ Bush $\succ_S$ Nader (matching voter 2's ranking exactly). If voter 3 is the dictator, the social ranking is Bush $\succ_S$ Nader $\succ_S$ Gore.

Three Axioms for Evaluating Aggregation Rules

Now we want to evaluate these rules. Are they satisfactory as methods for aggregating preferences? We introduce three axioms that capture basic desiderata.

Completeness (C): The social preference $\succsim_S$ is complete — for every pair $x, y \in X$, either $x \succsim_S y$ or $y \succsim_S x$ (or both). The social preference can always rank any pair. This axiom ensures the social preference is a well-defined ordering that can guide collective choice.

Pareto (P): If every voter strictly prefers $x$ to $y$ — that is, $x \succ_i y$ for all $i$ — then society strictly prefers $x$ to $y$: $x \succ_S y$. This is a minimal responsiveness condition. It says that if everyone agrees, society should agree too. A rule that ignores unanimous preferences seems hard to defend.

No Dictator (ND): There is no voter $d$ such that, for every preference profile and every pair $x, y$, $x \succ_d y$ implies $x \succ_S y$. In other words, there is no single individual whose strict preferences always determine the social strict preference, regardless of what everyone else thinks. Democracy should not reduce to one person's rule.

Checking the Rules

Let us now check each rule against each axiom.

Majority rule satisfies both Pareto and No Dictator. Pareto is clear: if everyone prefers $x$ to $y$, a fortiori a majority does, so $x \succ_S y$. No Dictator: majority rule is symmetric in voters, so no single voter can determine outcomes against the will of all others. Completeness is also satisfied pairwise — for any pair $x, y$, at least half prefer $x$ or at least half prefer $y$. However, as we shall see in Lecture 3, majority rule can violate transitivity: even if each individual preference relation is transitive, the social preference produced by majority rule can cycle. This is Condorcet's paradox.

Unanimity rule violates Completeness. As we saw in the example, unanimity can leave pairs of alternatives incomparable when voters disagree. Since incomparability means the social preference fails to rank those alternatives, completeness is violated. Unanimity does satisfy Pareto trivially (if everyone prefers $x$, unanimity gives $x \succ_S y$) and No Dictator (one voter's preference alone is not enough to establish social preference — all must agree).

Borda count satisfies Completeness (there are always well-defined Borda scores), Pareto (if everyone prefers $x$ to $y$, $x$ gets a strictly higher rank from every voter, so its total score exceeds $y$'s), and No Dictator (the scores aggregate all voters equally). Its problematic axiom is one we have not yet introduced — Independence of Irrelevant Alternatives — which is the subject of Lecture 3.

Dictatorship satisfies Completeness (the dictator's preferences are complete) and Pareto (if everyone prefers $x$ to $y$, so does the dictator, so $x \succ_S y$). It obviously violates No Dictator by construction.

The picture that emerges is that each rule has a flaw. No rule in our list satisfies all three axioms plus every other reasonable property we might want. This is not a coincidence — it is the content of Arrow's theorem, to which we turn in Lecture 3.