Course Overview and Decision Theory

Course Overview and Decision Theory

What This Course Is About

This course sits at the intersection of economics and political science. We use the formal tools of economic theory — preferences, utility functions, equilibrium — to analyze political phenomena: voting, elections, collective decision-making, and information aggregation. The central question running through the entire semester is: how do groups of individuals with different interests and different information make collective decisions, and how well do those collective decisions serve the group?

We proceed through three broad modules. First, social choice theory asks what it means to aggregate individual preferences into a collective preference and what Arrow's impossibility theorem tells us about the fundamental limits of that project. Second, electoral competition asks how candidates choose platforms and what equilibrium policies look like when candidates care about winning, about policy, or are uncertain about what voters want. Third, information aggregation asks whether voting can pool dispersed private information into a correct collective decision — and which voting rules are best suited for that purpose.

Throughout, the policy space is one-dimensional. We write $X = [-1, 1]$, where $-1$ represents the most left-wing policy and $+1$ the most right-wing. This is a modeling simplification that keeps the mathematics tractable while capturing the most important political intuitions.

Choice Sets and Preference Relations

The most basic objects in this course are a choice set and a preference relation. The choice set $X$ is simply the collection of alternatives under consideration — in our setting, the interval of possible policies $[-1, 1]$.

A preference relation $\succsim$ over $X$ describes how an individual compares any two alternatives. We write $x \succsim y$ to mean "$x$ is at least as good as $y$" in the eyes of that individual. This is a subjective statement — it reflects the individual's values and evaluation, not any objective fact about the policies $x$ and $y$.

From the weak preference relation $\succsim$, we derive two further relations:

Strict preference: We write $x \succ y$ — read "$x$ is strictly preferred to $y$" — when $x \succsim y$ but not $y \succsim x$. The individual prefers $x$ and does not consider $y$ to be at least as good.

Indifference: We write $x \sim y$ when both $x \succsim y$ and $y \succsim x$. The individual sees no difference between the two alternatives.

It is worth emphasizing the subjectivity point. Whether a particular immigration policy is good or bad is not a mathematical fact — it depends on each individual's circumstances, values, and beliefs. Our framework takes preferences as given and studies their collective implications. We do not adjudicate between preferences.

The Rationality Axioms

Not every preference relation is well-behaved. We impose three axioms that together define what we mean by rational preferences.

Completeness: For all $x, y \in X$, either $x \succsim y$ or $y \succsim x$ (or both). The individual can always compare any two alternatives. There is no pair about which she simply has no opinion. Without completeness, utility maximization would be ill-defined — the individual could not always choose.

Reflexivity: For all $x \in X$, $x \succsim x$. Every alternative is at least as good as itself. This is almost tautological but is needed for the formal development.

Transitivity: For all $x, y, z \in X$: if $x \succsim y$ and $y \succsim z$, then $x \succsim z$. Transitivity also holds in two other forms: if $x \succ y$ and $y \succ z$ then $x \succ z$, and if $x \sim y$ and $y \sim z$ then $x \sim z$. Transitivity is the most substantive axiom — it says that preferences are internally consistent. An agent whose preferences cycle (preferring $x$ to $y$, $y$ to $z$, and $z$ to $x$) can be exploited through a sequence of trades that leaves her strictly worse off.

Together, these three axioms define rational preferences. The central representation theorem — which we state without proof — says that rational preferences can always be encoded by a utility function.

Utility Functions and Utility Maximization

A utility function $u: X \to \mathbb{R}$ represents the preference relation $\succsim$ if, for all $x, y \in X$:

$x \succsim y \iff u(x) \geq u(y)$

The utility function encodes the preference ordering: a higher utility value means the alternative is weakly preferred. The existence of such a function is guaranteed by the rationality axioms (under mild continuity conditions on $X$).

An important point: utility functions are ordinal, not cardinal. Only the ranking matters, not the specific numerical values. If $u$ represents $\succsim$, so does any strictly increasing transformation $f(u(\cdot))$. The statement $u(x) = 7$ does not mean the individual experiences "7 units of happiness" — it just means the individual prefers $x$ to any $y$ with $u(y) < 7$.

Given a utility function, the individual's choice problem is to select the alternative with the highest utility:

$x^* = \arg\max_{x \in X}\, u(x)$

This is the fundamental behavioral assumption throughout the course: rational agents maximize utility.

Two Applications: Politician and Voter Utility

Two specific utility functions appear repeatedly in this course, and it is worth introducing them now.

Politician (candidate) utility: Candidate $i$ has a bliss point $z_i \in [-1, 1]$ — her most preferred policy. Her utility from policy $x$ being implemented is:

$u_i(x) = -|x - z_i|$

This is a tent-shaped function peaked at $z_i$. The further the implemented policy from $z_i$, the lower the candidate's payoff. In the electoral competition models, we assume the Democrat's bliss point satisfies $z_D < v^M$ and the Republican's satisfies $z_R > v^M$, reflecting genuine ideological differences.

Voter utility: Voter $i$ has an ideal point $v_i \in [-1, 1]$ and utility:

$u_i(x) = -(x - v_i)^2$

This quadratic loss function gives symmetric, single-peaked preferences. The voter is equally unhappy about deviations to the left and to the right of $v_i$, with unhappiness growing as the square of the distance from the ideal point. A common alternative is the absolute value $u_i(x) = -|x - v_i|$, which has the same single-peaked structure with linear rather than quadratic loss.

Both specifications share the key property that preferences are single-peaked: there is a unique most-preferred policy, and utility decreases monotonically as the policy moves away from that point in either direction. This structure is central to the Median Voter Theorem developed in Lecture 4.

Looking Ahead

The decision-theoretic framework introduced here — choice sets, preference relations, rationality axioms, utility maximization — provides the foundation for everything that follows. In Lecture 2, we ask what happens when we try to aggregate the preferences of many individuals into a single collective preference. The answer, which Arrow established in 1951, turns out to be deeply surprising.