SOCIAL CHOICE (6.154.11) Norman Scho…eld Center in Political Economy Washington University in Saint Louis, MO 63130 USA Phone: 314 935 4774 Fax: 314 935 4156 E-mail: scho…
[email protected] Key words: Impossibility Theorem, Cycles, Nakamura Theorem, Voting.. Contents 1 Introduction 1.1 Rational Choice 1.2 The Theory of Social Choice 1.3 Restrictions on the Set of Alternatives 1.4 Structural Stability of the Core 2 Social Choice 2.1 Preference Relations 2.2 Social Preference Functions 2.3 Arrowian Impossibility Theorems 2.4 Power and Rationality 2.5 Choice Functions 3 Voting Rules 3.1 Simple Binary Preferences Functions 3.2 Acyclic Voting Rules on Restricted Sets of Alternatives 4 Conclusion Acknowledgement Glossary Bibliography 1 Summary Arrows Impossibility implies that any social choice procedure that is rational and satis…es the Pareto condition will exhibit a dictator, an individual able to control social decisions. If instead all that we require is the procedure gives rise to an equilibrium, core outcome, then this can be guaranteed by requiring a collegium, a group of individuals who together exercise a veto. On the other hand, any voting rule without a collegium is classi…ed by a number,v; called the Nakumura number. If the number of alternatives does not exceed v; then an equilibrium can always be guaranteed. In the case that the alternatives comprise a subset of Euclden spce, of dimension w; then an equilibrium can be guaranteed as long as w v 2: In general, however, majority rule has Nakumura number of 3, so an equilibrium can only be guaranteed in one dimension.