# John Duggan's Research Site

## Bio

I am a professor of political science and economics at the University of Rochester, and a research associate of the W. Allen Wallis Institute of Political Economy. I am currently the Chair of the Department of Political Science, after serving as: co-Chair of the Department in 2022--23, Chair for the period 2018--21, Director of the Wallis Institute during 2002--2012, and co-managing editor of Social Choice and Welfare during 2008--2015. I received my PhD in social science from the California Institute of Technology in 1995. My specializations are game theory, political economy, and social choice theory. My current work is on equilibrium existence in non-cooperative games, dynamic models of bargaining and elections, multi-dimensional spatial models of political competition, and informational aspects of voting and elections.

Stuck on a problem...

## About my research

A mathematical model is a lens that can be helpful in aiding our understanding of a phenomenon, and as such, all models incorporate structure and assumptions that can color or distort that understanding. I tend to favor transparency, in the form of less restrictive assumptions, over resolution, even if that means sacrificing empirical implications for general theoretical propositions. The main focus of my research has been the effect of dynamic and informational incentives on electoral and legislative politics, often using concepts from social choice theory as a benchmark for comparison and, sometimes, as a modeling tool. I've found political economy to be a rich source for theoretical problems: it turns out that discontinuities and non-convexities, which are often ruled out by assumption in economic theory, are a fundamental feature of politics. In some cases, this has led me to questions in pure game theory or social choice, e.g., the existence of equilibria in dynamic games, or the possibility of acyclic preference aggregation in the context of Arrow's impossibility theorem. Jeff Banks once told me that a theorist should follow his or her instincts: a problem should be chosen for its innate interest, not because it's easy or popular; in the long run, the interesting problems will reveal a more durable importance. The contents of this website reflect my attempt to pursue interesting problems, and to make a modest contribution to the field of formal political economy along the way.

## Recent work

In this section, I provide links to papers and other content added in the last year or so. Dates indicate the date of the post and sometimes differ from the date of the linked paper. NB: additions have been slow (due to chair responsibilities and book projects), but more are on the way!

October 15 "Relaxing Continuous Differentiability: Some Interesting Examples" Continuously differentiable functions have nice properties. They are continuous; the contour set at point with non-zero gradient has a nice manifold structure; Lagrange's theorem establishes a first order condition for constrained maximization problems at solutions satisfying the constraint qualification; and around any solution at which a system of equations is non-singular, the implicit function theorem yields a unique selection of solutions, and this selection is differentiable. This note shows that these properties do not hold when continuous differentiability is relaxed to directional differentiability.

October 1, 2024 "Accountability in Markovian Elections" (with Jean Guillaume Forand) We study the ability of elections to solve the dynamic programming problem of a representative voter in a dynamic citizen-candidate framework. We assume a finite number of states and general utilities and state transition probability, thereby allowing for non-trivial dynamics. We show that politicians may manipulate the state to affect their electoral prospects, resulting in suboptimal payoffs for the voter; and we show that for equilibria satisfying a reciprocity condition, equilibria become approximately optimal as the voter becomes patient. The latest version exploits results on undiscounted dynamic programming to examine in detail the structure of equilibria as the players become patient (see our note for details).

October 1, 2024 "A Note on Farsighted Decision Making in Stationary Dynamic Programming Problems" (with Jean Guillaume Forand) We write up some notes on undiscounted dynamic programming. Each optimal policy rule determines a unique ergodic distribution, and we show that as the discount factor goes to one, the limit of these ergodic distributions is optimal: there is no other policy rule that generates an ergodic distribution with a higher expected payoff for the decision maker.

February 16, 2021 “Lobbying and Policy Extremism in Repeated Elections" (with Peter Bils and Gleason Judd) We investigate the effect of lobbying on policy choices in an infinite-horizon model of elections. We find that when the effectiveness of money is fixed, if office incentives become large, then policy choices converge to the median. However, if office incentives are fixed and the effectiveness of money becomes large, then polarized equilibria that exhibit arbitrarily extreme policy choices by all politician types can be supported. The new version contains numerical examples to illustrate the non-obvious implications of dynamic incentives in the model. Here are some slides from a talk. This paper is forthcoming at the Journal of Economic Theory.

January 20, 2021 "Theorems of the Alternative: An Essay in Memory of Kim Border" 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.

December 27, 2020 "Elementary Proofs of Tests for Definiteness of a Matrix in Terms of Principal Minors" 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.

November 11, 2020 “Subgame-Perfect Equilibrium in Games with Almost Perfect Information: Dispensing with Public Randomization” (with Paulo Barelli) Harris, Reny, and Robson (1995) show that correlated subgame perfect equilibrium exist in a general class of dynamic games. We show that when nature’s moves are atomless, every such equilibrium can be de-correlated: there is a payoff-equivalent subgame-perfect equilibrium of the original game. As a corollary, we obtain an existence result of He and Sun (2020) for subgame perfect equilibria in games with atomless moves by nature. This paper is forthcoming at Theoretical Economics.

August 30, 2020 "Representative Voting Games" (with Jean Guillaume Forand) We examine policy outcomes in dynamic elections in which the identity of a representative voter depends on an evolving state variable. We show that a subset of "reelection balanced" equilibria of the electoral model match the stationary Markov equilibria of a stochastic game played among representative voters, if politicians are sufficiently office motivated. We also provide general conditions for existence of a representative voter in each state. This paper is forthcoming at Social Choice and Welfare.

July 18, 2020 "A One-page Statement and Proof of Arrow's Theorem" 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.

July 18, 2020 "A Model of Interest Group Influence and Campaign Advertising" (with Zuheir Desai) We propose a model in which interest groups contribute money for campaign advertising, which informs a proportion of voters about the policy position of the challenger. The model admits the possibility of positive and negative advertising (to help or hurt the challenger, respectively), but we show that in a given equilibrium, only one type of advertising is possible; the sort realized depends on voters' priors over the challenger. In a positive advertising equilibrium, challengers in a centrally located "funding interval" receive contributions, and in a negative advertising equilibrium, extreme challengers are targeted. This paper is published in the Quarterly Journal of Political Science, 16: 105-137.

July 18, 2020 “Electoral Accountability and Responsive Democracy” (with Cesar Martinelli) We analyze a canonical two-period model of elections in which politicians’ preferences and actions are imperfectly observed by voters, i.e., elections are subject to adverse selection and moral hazard. We establish existence of electoral equilibrium, and we give a characterization of equilibria. We show that as politicians become more office motivated, policy is responsive to voter preferences in the sense that the expected level of effort exerted by politicians in the first period becomes arbitrarily large. We allow for the possibility that excessive effort is harmful to voters, and thus the welfare implications of the analysis depend on the properties of voter preferences. This paper is published in The Economic Journal, 130: 675-715.

July 18, 2020 "Lobbying as a Multidimensional Tug of War" (with Jacque Gao) We analyze lobbying as a contest in which each lobbyist exerts effort, and effort levels continuously determine a policy outcome in a multidimensional space. We solve for the unique pure strategy equilibrium, and we examine comparative statics with respect to a preference parameter, e.g., if lobbyists become more sensitive to large policy losses, then the equilibrium outcome converges to the Rawlsian policy, which maximizes the payoff of the worst-off lobbyist. The model has the additional interpretation of committee deliberation, in which members each attempt to pull the outcome in their preferred directions. This paper is published in Social Choice and Welfare, 54: 141-166.

January 28, 2020 "A Formal Theory of Democratic Deliberation" (with Hun Chung) We provide a framework to model three forms of democratic deliberation: myopic discussion, constructive discussion, and debate. The debate game has a unique subgame perfect equilibrium outcome, which is a compromise of the preferences of the participants. In contrast to the first two forms, debate always concludes with a single outcome (rather than a cycle) and this outcome is path independent (does not depend on the status quo). Here is a link to the technical appendix. This paper is published in the American Political Science Review, 114: 14-35.

October 1, 2019 "Continuity Properties of the Pareto Correspondence in the Spatial Model of Politics" 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.

September 9, 2019 "Introduction to the Formal Political Theorist's Basic Toolkit of Rational Choice Models" The title says it all: notes for graduate students giving a run down of a number of basic models in rational choice modeling.

August 24, 2019 "A `Conditional' Maximum Theorem" (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.

August 24, 2019 "A Quadratic Triangle Inequality" 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.

August 24, 2019 "A Note on Continuity Properties of Parameterized Solutions to a Class of Equations" 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.