# John Duggan's Research Site

## Bio

I am a professor of political science and economics at the University of Rochester, chair of the Department of Political Science, and a research associate of the W. Allen Wallis Institute of Political Economy. I was director of the Wallis Institute for the period 2002--2012, and I was co-managing editor for the journal *Social Choice and Welfare* for the period 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.

## 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 trading off statistical analysis of testable hypotheses 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's choice of problems 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 theoretical 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.

**October 20, 2019 **“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.

**October 8, 2019** "A One-page Proof of Arrow's Theorem (in 10 point font with appropriate margins)" As if the world needed another short proof of Arrow's theorem... I couldn't help myself. This was recently updated to correct an error in the last step.

**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 27, 2019** "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 forthcoming in the *American Political Science Review.*

**September 10, 2019** "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 forthcoming in *Social Choice and Welfare*.

**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 26, 2019** "A Model of Interest Group Influence in Electoral Campaigns" (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.

**August 24, 2019** “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 (2016) for subgame perfect equilibria in games with atomless moves by nature.

**August 24, 2019** "Accountability via Delegation in Dynamic Elections" (with Jean Guillaume Forand) We study the ability of elections to solve the dynamic programming problem of a representative voter. We examine conditions under which the corresponding politician type is a "faithful delegate," and the possibility that delegation allows the voter to achieve optimal policies. We demonstrate the possibility of political failures due to existence of suboptimal equilibria, and we provide relatively narrow conditions precluding such equilibria.

**August 24, 2019** "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.

**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.

**August 24, 2019** “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. Here are some slides from a talk.

**September 6, 2018 ** "Thoughts on Modeling in Formal Political Science" Some notes giving a very introductory overview of formal political theory.

**August 27, 2018 ** "Necessary Gradient Restrictions at the Core of a Voting Rule" This paper strengthens known gradient restrictions for the core of a voting rule. For the special case of majority rule with an even number of voters, it implies that if a core point is the ideal point of one voter, then for every pair of voters whose gradients do not point in opposite directions, there is a third voter whose gradient is semi-positively dependent with the gradients of the first two. A difficulty in the proof of a result by Schofield (1983) is identified, and a counterexample is presented. This paper is published in *Journal of Mathematical Economics, *79: 1--9. Here is the last version before publication.

**August 27, 2018** “Directional Equilibria” (with Hun Chung) We propose a new solution concept (directional equilibrium) for the multidimensional spatial model. We establish connections the core, Pareto optimality, existence, and generic local uniqueness and stability, and we provide non-cooperative foundations. The main update is a generalization of Theorem 1, which shows that if an alternative belongs to the majority core, then it is a directional equilibrium; the proof uses the gradient restriction provided in "Necessary Gradient Restrictions at the Core of a Voting Rule." Here is the last version before publication; here is a working paper with results on generic properties; here are some slides from a longer talk; and here are slides from a shorter talk. This paper is published in *Journal of Theoretical Politics, *30: 272--305.

**August 27, 2018** “Existence and Indeterminacy of Markovian Equilibria in Dynamic Bargaining Games” (with Vincent Anesi) We show that if certain gradient restrictions hold at an alternative, then when players are sufficiently patient, there is a continuum of equilibria with absorbing sets arbitrarily close to that alternative. This paper is published in *Theoretical Economics*, 13: 505--525 Here is a longer working paper with a result that delivers a continuum of Pareto efficient equilibria. Here are some slides from a talk.

**June 13, 2018** "Existence of Markov Electoral Equilibria" (with Jean Guillaume Forand) We prove existence and upper hemicontinuity of a form of stationary equilibrium in a dynamic model of elections with a discrete (countable) state space and general policies and preferences..

**June 12, 2018** "The Power of Commitment in Dynamic Elections" (with Jean Guillaume Forand) This paper studies the role of commitment in achieving responsive policy outcomes in a dynamic model of elections. We show that with positive commitment power, a plan can be implemented in symmetric equilibrium if and only if it is optimal for the representative voter. For non-symmetric equilibria, we give conditions under which there exists a faithful delegate, and we provide further conditions under which such a delegate disciplines the choices of other politician types, so that all politicians approximately solve the representative dynamic programming problem. This paper is superceded by our 2019 paper, "Accountability via Delegation in Dynamic Elections."

**May 7, 2018** "Bilateral Lobbying: Political Influence as a First Price Auction" This paper models lobbying as a type of auction, in which lobbyists simultaneously make offers consisting of a policy and transfer, and the politician chooses either one of the offers or implements the status quo. Pure strategy equilibria exist generally. With two lobbyists, the equilibrium outcome is "constrained efficient," and with multiple lobbyists, the possible winners are maximal with respect to an acyclic competitiveness relation. In the spatial model with quadratic utility, there is always an equilibrium in which the lobbyist furthest from the politician wins. Here are some slides from a talk.

**April 10, 2018** "Extreme Agenda Setting Power in Dynamic Bargaining Games" with Zizhen Ma. We investigate a canonical model of bargaining with a fixed agenda setter, and we show that when players are impatient or the set of alternatives is one-dimesional, the equilibrium outcome from the static model obtains; but when players are patient and the alternatives are multidimensional, the equilibrium outcome typically converges to the ideal point of the agenda setter. Here is a working paper with additional results on boundary equilibria and genericity in the Whitney topology. Here are some slides from a talk.

**January 9, 2018** "Weak Rationalizability and Arrovian Impossibility Results for Responsive Social Choice" I give representation theorems for choice functions satisfying weak rationality conditions, and I use these to deduce impossibility theorems for preference aggregation satisfying responsiveness conditions along with weak (or no!) rationality conditions. This paper is published in the special issue of *Public Choice* in honor of Kenneth Arrow, 179: 7--40. Here is the last version before publication.

**May 31, 2017** “The Political Economy of Dynamic Elections: Accountability, Commitment, and Responsiveness” (with Cesar Martinelli) This article is published in the *Journal of Economic Literature*, 55: 916--984. It contains a survey and synthesis of the literature on electoral accountability, i.e., repeated elections in which politicians cannot commit their policy choices. We focus on conditions under which policy choices of office holders nevertheless reflect the preferences of the median voter.

**May 31, 2017** “The Political Economy of Dynamic Elections: A Survey and Some New Results” (with Cesar Martinelli) This is a longer version of the above *JEL* paper with expanded discussion and technical detail, including many proofs omitted from the shorter paper.