Effectivity Functions in Social Choice

The present book treats a highly specialized topic, namely effec tivity functions, which are a tool for describing the power structure implicit in social choice situations of various kind. One of the ad vantages of effectivity functions is that they seem to contain exactly the information which is needed in several problems of implementa tion, that is in designing the rules for individual behaviour given that this behaviour at equilibrium should result in a prescribed functional connection between preferences and outcome. We shall be interested in both formal properties of effectiv ity functions and applications of them in social choice theory, and among such applications in particular the implementation problem. This choice of emphasis necessarily means that some other topics are treated only superficially or not at all. We do not attempt to cover all contributions to the field, rather we try to put some of the results together in order to get a reasonably coherent theory about the role of the power structure in cooperative implementation. The authors are indebted to many persons for assistance and advice during the work on this book. In particular, we would like to thank Peter Fristrup and Bodil Hansen for critical reading of the manuscript, and Lene Petersen for typesetting in '.lEX.

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1. Introduction.- 1. Collective decisions and power structure.- 2. Social choice.- 3. Games and game forms.- 4. The effectivity functions of a game form.- 5. Effectivity functions and cooperative games.- 6. An overview of the book.- 7. Notes.- 2. The structure of effectivity functions.- 1. Introduction.- 2. Fundamental concepts.- 3. Polarity.- 4. Selected families of effectivity functions.- 5. Topological effectivity functions.- 6. Notes.- 3. Stability.- 1. Introduction.- 2. The core of an effectivity function.- 3. Stable effectivity functions.- 4. Stability of convex effectivity functions.- 5. Acyclicity and stability.- 6. Stable veto functions.- 7. Notes.- 4. Implementation.- 1. Introduction.- 2. Implementation of social choice correspondences by game forms.- 3. Monotonic social choice functions.- 4. Notes.- 5. Implementation in strong Nash equilibrium.- 1. Strong Nash equilibrium.- 2. The effectivity functions of a strongly Nash implementable social choice correspondence.- 3. The fundamental theorem.- 4. The nucleus.- 5. The supernucleus.- 6. The subcore.- 7. A characterization of strong Nash implementable social choice correspondences.- 8. Notes.- 6. Exact and strong consistency.- 1. Self-implementation.- 2. Successive elimination and exact and.- consistency.- 3. The Holzman conditions.- 4. Notes.- 7. Implementation in coalition proof Nash equilibrium.- 1. Coalition proof Nash equilibrium.- 2. Implementation in coalition proof Nash equilibrium.- 3. Uniform domination.- 4. U-effectiveness and indirect u-domination.- 5. Notes.- 8. Other effectivity functions.- 1. Introduction.- 2. The ?-effectivity function.- 3. Cooperative equilibria of type 1.- 4. Equilibria of type 2 and 3.- 5. Notes.- 9. Value of effectivity functions.- 1. Introduction.- 2. Effectivity functionswith convex feasible sets.- 3. Correlated strategies and effectivity functions.- 4. Effectivity functions of repeated games.- 5. Notes.- References.