Lecturer: Eric Pacuit ( website)
NEWSWelcome to Social Choice Theory for Logicians! Please check back to this website as I will post information about the course as it becomes available.
Venue: North American Summer School for Logic, Language and Information
Meeting Times: 3:00 - 4:30 (Day 1, Day 2, Day 3, Day 4, Day 5)
Location: University of Texas at Austin
Social Choice Theory is the formal analysis of collective decision making. A growing number of logical systems incorporate insights and ideas from this important field. This course will introduce the key results (including proofs) and the main research themes of Social Choice Theory. The primary objective is to introduce the main mathematical methods and conceptual ideas found in the Social Choice literature. I will also pay special attention to recent logical systems that have been developed to reason about group decision making and how social choice-style analyses are being used by logicians. Specific topics include (a schedule and list of topics is available below)
- Proofs of key Social Choice results (eg., Arrow's Impossibility Theorem, Sen's Impossibility of the Paretian Liberal, and the Gibbard-Satterthwaite Theorem),
- Axiomatic characterizations of voting methods (May's Characterization of the majority rule, Maskin's Characterization of majority rule, Fishburn's Characterization of Approval Voting, Young's Characterization of scoring rules, Saari's characterization of Borda Count),
- Voting paradoxes (eg., Condorcet's paradox, Anscombe's Paradox, the No-Show Paradox),
- Generalizations of the classic framework (eg., assuming there are infinitely many voters, Saari's geometric approach to social choice, judgement aggregation), and
- Modal preference logics for reasoning about multiagent preference aggregation.
- Applications of Arrow's Theorem outside of the theory of group decision making
The course will not only provide a broad overview of the field of Social Choice from a logicians perspective, but will also discuss key technical results of particular interest to logicians. The main goal is to provide a solid foundation for students that want to incorporate results and ideas from Social Choice Theory into their field of study.
Reading MaterialThe course will be based on the following articles (during the course I will fill in many of the missing details and touch on some additional topics)
- EP (2011). Voting Methods, Stanford Encyclopedia of Philosophy.
- U. Endriss (2011). Logic and Social Choice Theory, in Logic and Philosophy Today, J. van Benthem and A. Gupta, eds., College Publications.
- C. List (2011). The Theory of Judgement Aggregation: An Introductory Review, forthcoming in Synthese (see also the course notes by D. Grossi and G. Pigozzi)
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ScheduleBelow is a schedule for the course (which is subject to change) that will contain links to any handouts, slides and relevant papers for each lecture.
| Day 1
June 18, 2011
|Introduction and Motivation (slides): Much of our daily lives are spent taking part in various types of social procedures. Examples range from voting in a national election to deliberating with others in small committees. Many interesting issues arise when we carefully examine our group decision-making processes. For example, suppose that a group of friends are deciding where to go for dinner. If everyone agrees on which restaurant is best, then it is obvious where to go. But, how should the friends decide where to go if they have different opinions about which restaurant is best? Can we always find a choice that is `fair' taking into account everyone's opinions or must we choose one person to act as a `dictator'? In this introductory lecture, I motivate and discuss the central problems of social choice theory and highlight a few key paradoxes. Background reading includes:
Three impossibility results (handout): In the second part of the lecture, I state and prove three key impossibility results: Arrow's Theorem, the Muller-Satterthwaite Theorem and Sen's Impossibility Theorem. A nice overview of these fundamental theorems can be found in:
| Day 2
June 19, 2011
|During the first half of the class, we went carefully through the proof of Arrow's Theorem. We then introduced the main conceptual and technical issues for characterization of voting methods. For more background reading, see Sections 2 & 3 of my Voting Methodsarticle. (slides)|
| Day 3
June 20, 2011
|The lecture today will cover two general topics (slides):
Characterizing Voting Methods: I state and prove two characterizations of majority rule:
We then turn to an alternative way of characterizing voting methods in terms of This approach to characterizing voting methods is nicely summarized in:
Strategic manipulation: In the discussions, we assumed that the voters are reporting their true preferences. However, there are instances where it is in a voter's interest to misrepresent her true preferences or otherwise "game the system". We discuss these general situations and state and prove the fundamental result: The Gibbard-Satterthwaite Theorem (see section 2.3 of Endriss' paper for details).
| Day 4
June 21, 2011
The lecture today will briefly introduce three different topics (slides):
Condorcet Jury Theorem: The approach discussed up to now is to analyze voting methods in terms of “fairness criteria” that ensure that a given method is sensitive to all of the voters' opinions in the right way. However, one may not be interested only in whether a collective decision was arrived at “in the right way,” but in whether or not the collective decision is correct. The most well-known analysis comes from the writings of Condorcet who shoed that if there are only two options, then majority rule is, in fact, the best procedure from an epistemic point of view.
Judgement Aggregation: Judgement aggregation considers situations where we want to aggregate voters' judgements about various interconnected propositions. There is now an extensive literature on judgement aggregation. We will summarize the key results of this area.
| Day 5
June 22, 2011
| The final lecture (slides) will discuss preference modal logics and other logical frameworks for reasoning about social choice theory (including generalizations to infinite populations). We will also discuss various applications of impossibility results (especially Arrow's Theorem) outside of social choice theory.
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Below is a list of some additional reading material related to some of the topics we will discuss in this course. This is not a complete list of all relevant material, but a reasonably large sampling.
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Recent courses and seminars (contains links to relevant papers)
- T. Ågotnes, W. van der Hoek, and M. Wooldridge. On the Logic of Preference and Judgment Aggregation. Autonomous Agents and Multiagent Systems, 22(1):4-30, 2011.
- H. Andreka, M. Ryan and P.-Y. Schobbens, Operators and Laws for Combining Preference Relations, Journal of Logic and Computation, volume 12, 2002.
- Michel Balinski and Rida Laraki, A theory of measuring, electing and ranking, Proceeding of the National Academy of Sciences USA, vol. 104, no. 21, pp. 8720-8725 May 22, 2007.
- Steven Brams, Marc Kilgour and William Zwicker, The Paradox of Multiple Elections, Social Choice and Welfare, Volume 15, Number 2, pgs. 211 - 236, 1998
- Keith Dowding and Maartin Van Hees, In Praise of Manipulation, British Journal of Political Science, Volume 38, pgs. 1 - 15, 2007.
- Peter C. Fishburn and Steven J. Brams, Paradoxes of Preferential Voting, Mathematics Magazine, Vol. 56, No. 4, 1983, pp. 207-214.
- J. Geanakoplos, Three Brief Proofs of Arrow’s Impossibility Theorem, Economic Theory, 26(1): 211-215, 2005.
- U. Grandi and U. Endriss. First-Order Logic Formalisation of Arrow's Theorem. Proc. 2nd International Workshop on Logic, Rationality and Interaction (LORI-2009), 2009
- Stephan Hartmann, Gabriella Pigozzi and Jan Sprenger, Reliable Methods of Judgement Aggregation, 2008
- M. Van Hees, The limits of epistemic democracy, Social Choice and Welfare, Volume 28, Number 4, 2007.
- M. Pauly, On the Role of Language in Social Choice Theory, Synthese, 163(2):227-243, 2008.
- Lingfang Li and Donald Saari, Sen's Theorem: Geometric Proof and New Interpretations, Social Choice and Welfare, Volume 31, Number 3, 2008
- David Makinson, Combinatorial versus decision-theoretic components of impossibility theorems, Theory and Decision, Volume 40, pgs. 181 - 190, 1996
- Amartya Sen, The Possibility of Social Choice, Nobel Prize Lecture, 1998
- Patrick Suppes, The pre-history of Kenneth Arrow's social choice and individual values, Social Choice and Welfare, Volume 25, pgs. 319 - 329, 2005
- Nicolas Troquard, Wiebe van der Hoek and Michael Wooldridge, Reasoning about social choice functions, Journal of Philosophical Logic, 40: pgs. 473 - 498, 2011.
- K. Arrow, Social Choice and Individual Values, Yale University Press, 2nd edition, 1970.
- Steven Brams, Mathematics and Democracy, Princeton University Press, 2007.
- Wulf Gaertner, A Primer in Social Choice Theory, Oxford University Press, 2006.
- I. McClean and A. Urken, Classics of Social Choice, University of Michigan Press, 1995.
- S. Nitzan, Collective Preference and Choice, Cambridge: Cambridge University Press, 2010.
- W. Poundstone, Gaming the Vote: Why Elections aren't Fair (and What We Can Do About It), Hill and Wang, 2008
- M. Regenwetter, B. Grofman, A.A.J. Marley, I. Tsetlin, Behavioral Social Choice, Cambridge University Press, 2006.
- Donald Saari, Disposing Dictators, Demystifying Voting Paradoxes: Social Choice Analysis, Cambridge University Press, 2008.
- Donald Saari, Basic Geometry of Voting, Springer, 2003.
- Y. Shoham and K. Leyton-Brown, Chapter 9: Aggregation Preferences: Social Choice, in Multiagent Systems: Algorithmic, Game Theoretic and Logical Foundations, Cambridge University Press, 2009.
- Alan D. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press, 2005.
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Additional Informationloriweb.org: a web portal with a number of important resources (call for papers, conference announcements, available positions, general discussions, etc.).
Recent courses and seminars (contains links to relevant papers)
- Rationality (Tilburg University, Spring 2011)
- Ulle Endriss' Computational Social Choice Seminar focusing on logic and social choice theory (ILLC, 2011).
- Seminar on Voting Theory (Stanford, Fall 2008)
- COMSOC: Computational Social Choice, a bi-annual conference on computational issues in social choice (the next one is in 2012 in Krakow, Poland).
- LGS: Logic, Games and Social Choice, a bi-annual conference focusing primarily on logic and social choice theory
- International Meeting of the Social Choice and Welfare Society, bi-annual conference on social choice theory.
- TARK: Theoretical Aspects of Rationality and Knowledge is a bi-annual conference on the interdisciplinary issues involving reasoning about rationality, knowledge and game theory.
- LOFT: Logic and the Foundations of Game and Decision Theory is a bi-annual conference which focuses, in part, on applications of formal epistemology in game and decision theory.