# Student Resources

## Lecture notes

“Basic Concepts in Mathematical Analysis: A Tourist Brochure” A math survey that starts from the basics.

## Surveys in political economy

"Introduction to the Formal Political Theorist's Basic Toolkit of Rational Choice Models" (2019) The title says it all: notes for graduate students giving a run down of a number of basic models in rational choice modeling.

“The Political Economy of Dynamic Elections: A Survey and Some New Results” (2017) with Cesar Martinelli.

“Probabilistic Voting in the Spatial Model of Elections: The Theory of Office-Motivated Candidates” (2005) in *Social Choice and Strategic Decisions: Essays in Honor of Jeffrey S. Banks,*David Austen-Smith and John Duggan, eds., New York: Springer. Here is the last version prior to publication.

“A Note on Backward Induction, Iterative Elimination of Weakly Dominated Strategies, and Voting in Binary Agendas" (2003) This note was written in 2003 but did not include figures, which were at that time done by hand. Since that writing, the open questions I raised were closed by Patrick Hummel, ``Iterative Elimination of Weakly Dominated Strategies in Binary Voting Agendas with Sequential Voting,'' Social Choice and Welfare, 31: 257--269. This note is in the original form except that I have included figures (omitting two that were redundant) and added margin comments reflecting Hummel's findings. No other modifications (typo corrections or otherwise) were made.

"Equilibrium Equivalence Under Expected Plurality and Probability of Winning Maximization" (2000) Incomplete: one section missing.

## Technical stuff

"Theorems of the Alternative: An Essay in Memory of Kim Border" (2021) This note collects a number of theorems of the alternative, which inform us about solutions to systems of linear equalities and inequalities. To each primary system is associated a secondary system, and the results establish two stark alternatives: either the primary system has a solution, or the secondary system does, but not both. I first learned of these results from Kim Border, who was one of my advisors at Caltech. Kim passed away on November 19, 2020. He dedicated his career to helping other scholars do better work, and this modest note is dedicated to his memory.

"Elementary Proofs of Tests for Definiteness of a Matrix in Terms of Principal Minors" (2020) This note gives alternative (and I think nice) proofs that a matrix is positive definite if and only if its leading principal minors are positive, and that it is positive semi-definite if and only if all its principal minors are non-negative. The proofs use elementary matrix algebra and results from optimization theory. A byproduct is a bonus theorem that gives another test for positive semi-definiteness that requires calculation of fewer determinants.

"A One-page Statement and Proof of Arrow's Theorem" (2020) As if the world needed another short proof of Arrow's theorem... I couldn't help myself. This version significantly shortens and simplifies the previous one: by exploiting symmetry arguments (across groups and alternatives), the proof uses only two preference profiles. There are other short proofs, but I've also tried to make this as simple as possible.

"Continuity Properties of the Pareto Correspondence in the Spatial Model of Politics" (2019) This note contains reflections on the upper and lower hemicontinuity of the Pareto correspondence, with an emphasis on lower hemicontinuity, which is generally harder to come by. These results are doubtless known in some form (at least in the setting of an exchange economy), but they were fun to work out.

"A Quadratic Triangle Inequality" (2019) A very short, simple version of the triangle inequality that holds for weighted sums of squared norms of vectors. This is something that came up in a different project, and I thought it might be of small, but positive, interest.

"A Conditional Maximum Theorem" (2019) with Paulo Barelli. This note contains a version of the maximum theorem that permits non-compact feasible sets by strengthening the continuity assumption on the objective function being maximized.

"A Note on Continuity Properties of Parameterized Solutions to a Class of Equations" (2019) This is a short note looking at solutions to a integral equation and examining continuity properties of those solutions with the topology of pointwise convergence. The key is to impose uniform continuity over compact sets over the integrand. This will be of limited interest, but I wanted to write this down.

“A Simple Example of a Finite Stochastic Game with Sequential Moves and No Pure-strategy Stationary Markov Perfect Equilibrium” (2016) A small example that might be interesting to someone.

“A Note on the Pareto Manifold in the Spatial Model of Politics” (2016) I provide sufficient conditions for the Pareto set to exhibit a manifold structure near a Pareto optimal alternative -- the analysis takes utility functions as given, rather than making genericity claims. I then investigate a natural parameterization of the Pareto manifold and the corresponding parameterized utilities. In contrast to Smale (1976), I do not assume an economic environment; rather, I consider a general model that includes the spatial model of politics as a special case.

"Making Determinants Less Weird" (2010) Just some notes I wrote giving some intuition behind determinants, including a geometric proof of Cramer's rule. Sorry the notes are so terse.

"A Graphical Perspective on Kolmogorov's Inequality" (2010) Nothing deep here, but some nice 3-d pictures (check out p.2). I actually first drew them in grad school to enhance my intuition for the result.

"A Note on Sensitivity Analysis for Local Solutions of Non-Linear Programs" (2007) with Tasos Kalandrakis.

## Presentations and other information for grad students

"Bilateral Lobbying: Political Influence as a First Price Auction" (2018) A short talk on a current project for prospective grad students.

“A Comparison of PhD Programs in Formal Political Theory” (2015) This report summarizes my findings on training in formal political theory in five PhD programs, in addition to the University of Rochester, in political science. It represents my best effort to form a picture from the websites of these departments.

“The Formal Political Theory Field at Rochester” (2015) Slides from a graduate recruiting seminar. Here are some slides from a seminar I gave in 2013.

"Formal Political Theory" (2006) Slides from a graduate recruiting seminar.