PS 507 Social Choice and Democracy

Spring 2019

Officially titled "Seminar in Positive Political Theory," this course is an introduction to "social choice theory," that is, the axiomatic theory of preference aggregation, and to the problems of implementing desired modes of preference aggregation. It gives specific attention to the implications of preference aggregation theory for real-life democracy.

This website will be updated to reflect current plans with respect to coverage and schedule, assignments, and any relevant links to materials associated with the course. All readings, apart from the required texts, can be obtained either via the links shown or in the class shared folder on Box.

Course Outline and Approximate Schedule

0. Introductory Session

Jan. 16

Please read the following and try the associated exercises for the first two prior to our initial class meeting on Wednesday Jan. 16.

  • Calvert, "Relations on Sets" supplementary notes (in shared folder).
    • Even if you already know these basic mechanics (drawn from set theory and from classic consumer theory), please try your hand at the several exercises contained in the reading. (You don't have to turn them in, but I'll be taking questions.)
    • You may skip the final section, on "Partially Ordered Sets," since we probably won't use that material in this course<\li>
  • Calvert, "The Condorcet Paradox and the Impossibility of Nice Social Choice," supplementary notes (in shared folder).
    • An introduction to the idea of impossibility theorems in preference aggregation theory. It makes reference to the third reading below, which we will then discuss in more detail.
    • Exercise: Can you formulate a version of Theorem 1 after weakening the definition of "respecting majority preferences'' to require only that, if more individuals strictly prefer x to y than prefer y to x, society does not strictly prefer y to x? The initial problem is, can we get away with declaring social indifference between x and y whenever a contradiction arises between transitivity and respecting majority preference?
  • Kenneth May, "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision." (Available both in shared folder and online via JSTOR.)


1.  The General Possibility Theorem

Jan. 23

Social choice theory began with what Kenneth J. Arrow (in Social Choice and Individual Values, 1951) called his "general possibility theorem," which established that well behaved (according to Arrow) social choice is impossible. His set-up is fundamental to the field: (1) preference aggregation functions; (2) axioms that qualify such a function as well-behaved; and (3) methods of prooving the incompatibility of those axioms. Since Arrow, the methods of proof have been tightened up and alternatives devised, and trade-offs between the axioms explored.

Assigned readings:

  • Julian H. Blau, "A Direct Proof of Arrow's Theorem." Econometrica, Vol. 40, No. 1 (Jan., 1972), pp. 61-67.   Click here to obtain via JSTOR.
    • This is a classic proof whose approach is often used. Sen's general characterization of such proofs goes as follows:
      • First, a field-expansion lemma: a group decisive between some pair of alternatives is decisive over every pair.
      • Second, a group-contraction lemma: a decisive group has a decisive subgroup.
      • Since, by the unanimity axiom, the whole group is decisive; and since the whole group is assumed finite; then some single individual is decisive between every pair of alternatives---a dictator. This description is from p. 4 of Amartya Sen, "Rationality and Social Choice." American Economic Review, Vol. 85, No. 1 (Mar., 1995), pp. 1-24, available via JSTOR and shared folder
  • John Geanakoplos, "Three Brief Proofs of Arrrow's Impossibility Theorem." Economic Theory, Vol. 26 (2005), pp. 211-215. shared folder

Assigned exercises on Arrow's theorem and Blau's proof: to be turned in in class.


2.  Other applications of the Arrow set-up: Strategic Voting, Liberalism, and Judgment Aggregation

Jan. 30 & Feb. 6

For Jan. 30:

  • Amartya Sen, "The Impossibility of a Paretian Liberal." Journal of Political Economy Vol. 78, No. 1 (Jan.-Feb., 1970), pp. 152-157.   Click here to obtain via JSTOR.
    • Exercise: Reformulate Sen's conditions U, P, and L explicitly in terms of a social decision function, rather than the social welfare function.
  • Alan Gibbard, "Manipulation of voting schemes", Econometrica, Vol. 41 (1973), pp. 587-601.  Click here to obtain via JSTOR.
  • Revised edition 01/29/2019 Supplementary notes: Calvert, notes on "Comparing Impossibility Theorems" (6 pages). Click here to obtain online.

For Feb. 6:

  • Problem to turn in: From Sen's paper last week, state and prove a version of Sen's Theorem II that focuses on a "collective choice rule" (a mapping from preference profiles to a complete, reflexive social preference relation) rather than on a "social decision function."
    • Hints: the point is to make explicit the role of an acyclicity condition: there is no "cycle" of P-relations of any finite length. So xPy and yPz and zPw implies NOT wPx, and so forth.
    • The proof should look a LOT like Sen's original proof.
  • Christian List and Philip Pettit, "Aggregating Sets of Judgments: An Impossibility Result." Economics and Philosophy, Vol. 18 (2002), pp. 89-110.   Click here to obtain online. Also in shared folder.
    • Our main event will be proving Theorem 1; the proof is given in the Appendix.
    • Errata: On page 98, line 2, "the following table" refers to Table 1 on the previous page. On page 109, line 6 from bottom of main text, "with the following properties:" refers to Table 3 at the top of the page.
    • Exercise: What are the column totals in Table 3?
  • Christian List, "The theory of judgment aggregation: an introductory review." Synthese Vol. 187, No. 1 (July 2012), pp. 179-207.   Click here to obtain via JSTOR.
    • Main technical event: the "sketch proof" of Theorem 3a in section 4.4. But more generally, read to understand how List claims to extend the earlier List & Pettit result.
  • Skim to understand application of Arrow's theorem: Matthew L. Spitzer, "Radio Formats by Administrative Choice." University of Chicago Law Review Vol. 47, No. 4 (Summer, 1980), pp. 647-687.   Click here to obtain via JSTOR.


3.  Instability in Multidimensional Voting Models

Feb. 13

Richard D. McKelvey, "Intransitivities in multidimensional voting models and some implications for agenda control." Journal of Economic Theory, Vol. 12, Issue 3 (June 1976), pp. 472-482.   Click here to obtain online.

  • Calvert, "Mathematical notes on the McKelvey paper," supplementary notes (in shared folder)

Feb. 20

Norman Schofield, "Instability of Simple Dynamic Games." Review of Economic Studies, Vol. 45, No. 3 (Oct. 1978), pp. 575-594.   Click here to obtain via JSTOR.

  • Calvert, "Mathematical notes on the Schofield paper," supplementary notes (in shared folder)


4.  Social Choice and Democracy

Feb. 27; Mar. 6, 20, 27; Apr. 3

Feb. 27

  • Riker, Liberalism Against Populism.

Mar. 6

  • Mackie, "All Men are Liars: Is Democracy Meaningless?" In Jon Elster, ed., Deliberative Democracy (Cambridge Univ. Press, 1998, pp. 69-96)
  • Keith Dowding, "Can Populism Be Defended? William Riker, Gerry Mackie and the Interpretation of Democracy." Government and Opposition, Vol. 41, No. 3 (2006), pp. 327-346. Click here to obtain online from Cambridge Press.

Mar. 20

  • Partha Dasgupta and Eric Maskin, "On the Robustness of Majority Rule." Journal of the European Economic Association Vol. 6, Issue 5 (Sept. 2008), pp. 949-973. Discussed in Knight & Johnson.
  • Knight & Johnson, The Priority of Democracy selections from chapters 1 & 4.

Mar. 27

  • Anthony McGann, The Logic of Democracy: Reconciling Equality, Deliberation, and Minority Representation. Univ. of Michigan Press (2006).

Apr. 3

  • Sean Ingham, Rule by Multiple Majorities: A New Theory of Popular Control, manuscript; forthcoming, Cambridge Univ. Press (2019). In shared folder on Box.


Units we might cover all or part of in April:

5. Covering and Agendas

  • Nicholas R. Miller, "Graph-theoretical approaches to the theory of voting." American Journal of Political Science, Vol. 21, No. 4 (Nov. 1977), pp. 769-803. Click here to obtain via JSTOR.
  • Nicholas R. Miller, "A new solution set for tournaments and majority voting: Further graph-theoretic approaches to the theory of voting."  American Journal of Political Science Vol. 24, No. 1 (Feb. 1980), pp. 68-96. Click here to obtain via JSTOR.
  • Nicholas R. Miller, "The Covering Relation in Tournaments: Two Corrections."  American Journal of Political Science Vol. 27, No. 2 (May, 1980), pp. 382-385. Click here to obtain via JSTOR.
  • Kenneth A. Shepsle and Barry R. Weingast, "Uncovered sets and sophisticated voting outcomes with implications for agenda institutions."  American Journal of Political Science Vol. 28, No. 1 (Feb. 1984), pp 49-74. Click here to obtain via JSTOR.
  • Jeffrey S. Banks, "Sophisticated voting outcomes and agenda control." Social Choice and Welfare Vol. 1 (1985), pp. 295-306. Click here to obtain via JSTOR.
  • Richard D. McKelvey, "Covering, Dominance, and the Institution-Free Properties of Social Choice." American Journal of Political Science Vol. 30, No. 2 (May 1986), pp. 283-314. Click here to obtain via JSTOR.


6. Models of Political Argument

  • John W. Patty, "Arguments-Based Collective Choice." Journal of Theoretical Politics Vol. 20, No. 4 (2008), pp. 379-414.   Click here to obtain online.
  • Christian List, "Group Communication and the Transformation of Judgments: An Impossibility Result." Journal of Political Philosophy Volume 19, Number 1, 2011, pp. 1–27.
  • John W. Patty and Elizabeth Maggie Penn, "A social choice theory of legitimacy." Social Choice and Welfare, Vol. 36, No. 3/4 (Apr. 2011), pp. 365-382. Online here.
  • David van Mill, "The Possibility of Rational Outcomes from Democratic Discourse and Procedures." Journal of Politics Vol. 58, No. 3 (Aug., 1996), pp. 734-752. Online here.


7. Implementation and Mechanism Design

  • Introduction only:  Eric Maskin, "Nash Equilibrium and Welfare Optimality," Review of Economic Studies, Vol. 66 (1999), pp. 23-38. Click here to obtain via JSTOR.
  • R. Repullo, "A Simple Proof of Maskin’s Theorem on Nash Implementation," Social Choice and Welfare, Vol. 4 (1987), pp. 39-41.  To be supplied.
  • Roger Myerson, "Bayesian Equilibrium and Incentive-Compatility:  An Introduction."  In L. Hurwicz, D. Schmeidler, and H. Sonnenschein, eds., Social Goals and Social Organization. Cambridge Univ. Press (1985).
  • Jeffrey S. Banks and Randall L. Calvert, "A Battleof-the-Sexes Game with Incomplete Information."  Games and Economic Behavior Vol. 4 (1992), pp. 347-372.  Click here to obtain via ScienceDirect.


8.  Ongoing Political Coalitions

  • Dennis Epple and Michael H. Riordan, "Cooperation and Punishment under Repeated Majority Voting." Public Choice Vol. 55, No. 1/2 (Sep., 1987), pp. 41-73.   Click here to obtain via JSTOR.
  • Kathleen Bawn,"Constructing 'Us': Ideology, Coalition Politics, and False Consciousness." American Journal of Political Science, Vol. 43, No. 2 (Apr. 1999): pp. 303-34.   Click here to obtain via JSTOR.
  • David P. Baron, "A Noncooperative Theory of Legislative Coalitions." American Journal of Political Science Vol. 33, No. 4 (Nov., 1989), pp. 1048-1084.   Click here to obtain via JSTOR.
    • Especially relevant is the material from page 1064 to the end, where coalition discipline is not assumed but has to be maintained through suitable repeated-game equilibrium strategies.
  • David P. Baron and John A. Ferejohn, "Bargaining in Legislatures." American Political Science Review Vol. 83, No. 4 (Dec., 1989), pp. 1181-1206.   Click here to obtain via JSTOR.
    • Especially relevant is their "folk theorem" result, Proposition 2 on p. 1189. (The results for which this paper is usually cited, however, are those that follow Proposition 2, which add the assumption of "stationary" equilibria and hence rule out repeated-game retaliation schemes.


This page written by Randall Calvert  2019
Email comments and questions to calvert at

Wednesdays 3:00-5:00pm, Seigle 210

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Textbooks (not in bookstore -- obtain on your own)

William H. Riker, Liberalism against Populism:  A Confrontation between the Theory of Democracy and the Theory of Social Choice (W.H. Freeman & Co., 1982). (Republished in 1988 by Waveland Press, identical except for cover.)
Anthony McGann, The Logic of Democracy: Reconciling Equality, Deliberation, and Minority Representation. Univ. of Michigan Press (2006).

Course Requirements

With the exception of a couple of less technical works, we'll work through the most important technical details of each assigned reading in class. For each, we'll take turns presenting individual theorems and proofs. I will call on you in class; everybody is responsible for everything. I may also assign problems that contribute to the reading.

By the end of the semester, each student should turn in a paper addressing a social choice theory problem you have identified (in consultation with me) formulated, and made initial attempts at solving. Your paper should include a literature review relating your problem to the existing literature. The topics for these papers should be set by the middle of the semester.