Grad Notes

A growing collection of self-contained handouts on the mathematics needed to read formal political-economy theory and to write it. In the spirit of Kim Border’s mathematical economics notes, but oriented to political scientists, with examples drawn from across political science and political economy. Each handout is independent and reference-able on its own; the numbering below is a suggested reading order rather than a strict prerequisite chain.

Foundations

Language and logic

The first formal vocabulary the project assumes — propositions, connectives, valuations, and what follows from what.

Preface

Project framing, conventions, scope, and what is assumed. Background: algebra and a year of calculus.
Propositional logic

The simplest formal system in which the notion of “follows from” can be made precise. Covers syntax (atomic propositions, connectives, well-formed formulas), semantics (valuations, truth-in-a-valuation), tautology / contradiction / satisfiability, logical equivalence, and logical consequence.

Sets and structures

Sets and the basic structural vocabulary built on them — set operations, relations, orders, and cardinality.

Naive set theory

Sets and set operations with the Venn-diagram picture; indexed unions and intersections; Cartesian products and tuples; disjoint unions; relations and equivalence classes; functions. Closes with Russell’s paradox and the axiom of separation.
Order theory

Preorders, partial orders, and total orders, with the strict-and-symmetric decomposition; Hasse diagrams; bounds, maxima vs. suprema, and the maximum-versus-maximal distinction; lattices; order-preserving functions. The headline application is the utility representation of preferences, with lex preferences as the canonical no-utility-representation counterexample.
Cardinality and infinity

Comparing the sizes of infinite sets via bijection; Schröder–Bernstein; the countability of \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{N} \times \mathbb{N}\); Cantor’s diagonal argument for the uncountability of \(\mathbb{R}\); Cantor’s theorem and the unbounded hierarchy of cardinals. Closes with the discrete-versus-continuous distinction underlying applied modeling.

Analysis

Real analysis on \(\mathbb{R}^n\) — the structural backbone of every existence-and-comparative-statics argument in formal political-economy theory.

Sequences and limits

\(\epsilon\)-\(N\) convergence and the least upper bound property as the structural fact distinguishing \(\mathbb{R}\) from \(\mathbb{Q}\); the construction of \(\mathbb{R}\) from \(\mathbb{Q}\). The completeness-equivalence triangle (LUB / monotone convergence / Cauchy / Bolzano–Weierstrass), and \(\limsup\) / \(\liminf\) for bounded sequences that don’t converge.
Open and closed sets

\(\mathbb{R}^n\) with the Euclidean metric; open and closed sets and their algebra; closure, interior, boundary, accumulation points, with the sequential characterization of closedness. Compactness via open covers, Heine–Borel, and the sequential equivalent; connectedness in \(\mathbb{R}\) (= intervals) and convex-implies-connected in \(\mathbb{R}^n\).
Continuity

\(\epsilon\)-\(\delta\) continuity at a point with the sequential and topological characterizations; the algebra of continuous functions. The intermediate value theorem (the topological reason market-clearing arguments work) and the extreme value theorem (the topological reason optimization problems have solutions); uniform continuity and the Heine–Cantor theorem.

Probability and measure

Measure-theoretic probability, from \(\sigma\)-algebras through the central limit theorem — the foundations underlying polling, learning, and any stochastic model.

Probability spaces and measures

\(\sigma\)-algebras as the right collection of subsets to assign probabilities to, given that the full power set \(\mathcal{P}(\mathbb{R})\) is too rich to admit a translation-invariant countably additive measure; measures and the Kolmogorov axioms. Basic measure properties (monotonicity, subadditivity, continuity from above and below); Lebesgue measure on \(\mathbb{R}\); independence, conditional probability, and Bayes’ rule.
Integration

Riemann integration with rigor (the partition definition, the FTC, the Dirichlet-function failure of Riemann integrability); Lebesgue integration built in three stages on a measure space. Monotone convergence, Fatou’s lemma, dominated convergence; the Riemann–Lebesgue agreement theorem and Lebesgue’s characterization of Riemann-integrability (“bounded and continuous almost everywhere”).
Random variables and expectations

Random variables as measurable functions; distributions and CDFs (discrete and absolutely continuous); expectation as the Lebesgue integral against the probability measure. Variance and covariance; the standard inequalities (Markov, Chebyshev, Jensen, Cauchy–Schwarz); conditional expectation via Radon–Nikodym, with the tower property as the workhorse identity.
Convergence and limit theorems

The four modes of convergence (almost sure, in probability, in \(L^p\), in distribution) and their logical relations; the Borel–Cantelli lemmas. The weak and strong laws of large numbers; the classical i.i.d. central limit theorem.

Formal logic, deeper

Proof systems, first-order logic, and the model-theoretic vocabulary in which formal political-economy modeling already implicitly operates.

Proof systems

Natural deduction with introduction and elimination rules and assumption discharge, in Gentzen-style horizontal-line presentation; sequent calculus and Hilbert systems noted in passing. Soundness and completeness for propositional logic via Lindenbaum’s lemma and the canonical-valuation construction; compactness as a corollary of completeness.
First-order logic

Signatures, terms, formulas (free vs. bound, sentences); structures and Tarski’s recursive definition of satisfaction. Many-sorted FOL with sorted variables and structures-with-multiple-carriers as the natural setting for political-economy modeling, where the universe of discourse is typed (voters, candidates, strategies, alternatives) rather than a single undifferentiated set; Gödel’s completeness theorem stated and contrasted with the incompleteness theorems.
Model theory and modeling

Theories as sets of sentences, the \(\mathrm{Mod}\) / \(\mathrm{Th}\) Galois correspondence, elementary equivalence and isomorphism, complete theories, and the semantic view of theories (Suppes, van Fraassen). The aim is to align the model-theoretic vocabulary with the way political scientists already use the word “model” — every Cournot duopoly, agenda-setter model, signaling game is a structure or class of structures in this sense.

Linear algebra

Linear algebra over \(\mathbb{R}\) — vector spaces, linear maps, eigenvalues, quadratic forms — the foundation under spatial-voting models, network analysis, Markov chains, and the second-order conditions of optimization.

Vector spaces and linear maps

Vector spaces with \(\mathbb{R}^n\), function spaces \(\mathbb{R}^X\), and polynomials as canonical examples; linear independence, span, basis, dimension (with the well-definedness theorem); linear maps with kernel, image, and the rank-nullity theorem. Matrices as encodings of linear maps under chosen bases (with matrix multiplication as composition, change of basis as conjugation); inner products, Cauchy–Schwarz, Gram–Schmidt, orthogonal projection.
Eigenvalue theory and quadratic forms

Eigenvalues and eigenvectors with the characteristic-polynomial criterion; diagonalization (with the defective-matrix and Jordan caveat); symmetric matrices and the spectral theorem (real eigenvalues, orthonormal eigenbasis); quadratic forms and definiteness, with eigenvalue-based characterization and Sylvester’s criterion via leading principal minors. Stochastic matrices and the Perron–Frobenius theorem with the irreducibility and aperiodicity refinement that delivers Markov-chain convergence to a unique stationary distribution.

Optimization

Optimization on \(\mathbb{R}^n\) — the differentiation, convexity, static, and dynamic apparatus that runs under every comparative-statics, equilibrium-existence, and dynamic-policy argument in formal political-economy theory.

Differentiation in \(\mathbb{R}^n\)

Partial derivatives and the gradient with the directional-derivative formula and the gradient-as-direction-of-steepest-ascent reading; total derivatives, Jacobians, the chain rule; higher-order derivatives and Hessians with Clairaut’s symmetry-of-mixed-partials theorem. Taylor’s theorem to second order as the local-quadratic-approximation underlying SOCs; the implicit function theorem with the explicit comparative-statics formula \(D \mathbf{g} = -[D_{\mathbf{y}} \mathbf{F}]^{-1} D_{\mathbf{x}} \mathbf{F}\), set up as the formal engine of comparative statics in equilibrium analysis.
Convex sets and concave functions

Convex sets and the canonical political-economy convex sets (the probability simplex, mixed-strategy spaces, budget sets, intersections); convex hull; the separating hyperplane theorem with the supporting-hyperplane corollary. Concave and convex functions, with the local-implies-global maximum theorem; first-order (gradient-inequality) and second-order (Hessian-negative-semidefinite) characterizations of concavity; quasi-concavity with the upper-contour-set definition, concave-implies-quasi-concave, and the bordered-Hessian sufficient condition.
Static optimization

Unconstrained optimization with FOC and SOC; equality-constrained problems via Lagrange multipliers, with the shadow-price interpretation; inequality-constrained problems via the Karush–Kuhn–Tucker conditions and complementary slackness. Existence theorems (Weierstrass via compactness) and Berge’s theorem of the maximum, framed as the structural backbone of equilibrium-existence arguments in game theory; the envelope theorem (unconstrained and constrained) as the engine of comparative statics.
Dynamic optimization

Finite-horizon dynamic programming and the Bellman recursion (with backward induction); infinite-horizon discounted problems with stationary value function and stationary optimal policy. The Bellman equation recast as the fixed-point equation \(V^* = T V^*\) for the Bellman operator \(T\) on bounded functions under the sup norm, with Banach’s contraction mapping theorem delivering convergence of value-function iteration from any starting guess; the envelope condition (Benveniste–Scheinkman) as the dynamic analogue of the static envelope theorem.

Markov chains

Markov chains as the natural framework for stochastic political-economy dynamics whose future depends on the past only through the present, plus Markov decision processes as the natural extension to stochastic dynamic-decision settings.

Markov chains and Markov decision processes

Markov chains and the Markov property on a finite or countable state space, transition matrices, \(n\)-step probabilities, Chapman–Kolmogorov; reachability, communicating classes, irreducibility, periodicity, recurrence; stationary distributions and the existence/uniqueness/convergence theorem (proved via Perron–Frobenius from the eigenvalue handout), with reversibility, detailed balance, and the spectral-gap-based mixing-time bound. Markov decision processes as Markov chains with decisions, with the stochastic Bellman operator a contraction of modulus \(\beta\) by the same Banach argument as the dynamic-optimization handout — the deterministic continuation \(V^*(f(s, a))\) replaced by the conditional-expected continuation \(\sum_{s'} P(s' \mid s, a) V^*(s')\).

Decision Theory

Individual choice over deterministic alternatives, over lotteries with given probabilities, and over acts when the probabilities themselves are not pinned down — the three escalating positions on the epistemic state of the chooser.

Choice under certainty

Choice functions and choice correspondences on finite menus, with budget families as the cross-menu observation structure; rationalizability; revealed weak and strict preference; WARP and the Arrow–Sen rationalization theorem; SARP and Houthakker’s theorem. The preference–utility–choice triangle as the synthesis section, with each arrow’s failure mode named (lex preferences for \(\succsim \to u\), empty argmax for \(u \to C\), Condorcet cycle for \(C \to \succsim\)).
Choice under risk

Lotteries on a finite outcome space (\(\Delta(X)\) as a simplex); the von Neumann–Morgenstern axioms with independence as the load-bearing axiom; the expected-utility theorem with full proof in three steps (best/worst extraction, indifference probability via continuity, linear extension via independence). The Allais paradox as a structural failure of independence; prospect theory (S-shaped value function, probability weighting, loss aversion); rank-dependent expected utility as the EU-respecting weakening that preserves stochastic dominance.
Choice under ambiguity

Savage’s subjective expected utility framework with the P1–P7 axioms and the representation theorem; Anscombe–Aumann acts (functions from states to lotteries) as the lighter axiomatic path that builds on the previous handout’s vNM machinery. The Ellsberg paradox in the three-color version, with Knightian uncertainty as the distinction the Bayesian programme had foreclosed and Ellsberg reopened; Gilboa–Schmeidler maxmin EU as the centerpiece, with Schmeidler’s Choquet expected utility and the Choquet–maxmin equivalence theorem on convex capacities tying the two multi-prior frameworks together.

Welfare and Bargaining

Two handouts on the welfare-and-bargaining branch of applied political-economy modeling: efficiency and welfare-function characterizations of Pareto optima, and axiomatic bargaining solutions defined on the Pareto set with disagreement points (Nash, Kalai–Smorodinsky).

Pareto optimality and welfare functions

Pareto dominance as a strict partial order on the alternative space (ordinal, not requiring interpersonal utility comparisons); existence of Pareto optima under compactness and continuity, via the Weierstrass-on-the-unweighted-utilitarian-sum proof; sufficient conditions for Pareto optimality via positive-weight welfare-function maximization. The welfare-function characterization theorem — every Pareto optimum is a non-negative-weighted-sum welfare-function maximizer under convexity and concavity, proved via separating hyperplane on the utility-imputation set \(\mathcal{U}\) and the strictly-dominating cone \(\mathcal{V}\) — with utilitarianism, Rawlsian maximin, and Bergson–Samuelson welfare functions as the substantive landscape of welfare aggregation.
Axiomatic bargaining

Bargaining problems formalized as a pair \((S, d)\) with \(S\) a non-empty compact convex set of feasible utility profiles and \(d\) a disagreement point strictly Pareto-dominated by some feasible alternative, with the Fearon (1995) war-bargaining setup running as the canonical example throughout. The Nash bargaining solution as the maximizer of \((u_A - d_A)(u_B - d_B)\), with Nash’s axiomatic characterization (Pareto efficiency, symmetry, scale invariance, IIA) and the asymmetric / generalized Nash solution with bargaining weights for asymmetric institutional power; the Kalai–Smorodinsky alternative weakening of IIA in favor of monotonicity; the welfare-function reading of bargaining solutions as a logarithmic Bergson–Samuelson welfare function anchored at the disagreement point, bridging back to the previous handout.