Coalitional Games
GPEM,
Universitat Pompeu Fabra, Academic year 1998/99
Homework
General Description
Game theory is a branch of mathematics which is used in modelling
situations in which players with conflicting interests interact.
Coalitional Games are games in which the possibilities of the
players are described by the available resources of different groups
(coalitions) of players.
The course Coalitional Games will discuss the major principles of this
branch of game theory, and some of its applications to economics as well
as to other fields of social science. In the first part of the course
we shall study the core of transferable utility games. Among other
things, the relations between the core and price-equilibrium will
receive a considerable attention.
The second part of the course will concentrate on the axiomatic
approach to solution concepts. We shall discuss the major developments
of the last fifteen years or so, highlighting the centrality of the
consistency axiom, and of the core as a solution concept. Within the
axiomatic approach will shall also discuss the prekernel and the Shapley
value.
In the rest of the course depending on time constraints and popular
demand, we may discus in greater depth a topic related to the main parts
of the course. Possible candidates include non-transferable utility
games, and the Nash bargaining problem; cores of large games and
economies, and concepts of perfect competition; matching problems;
applications to local public goods; the nucleolus and the bargaining
set.
Part 1: The Core of Transferable Utility Games
- Transferable utility games: Basic definitions and examples.
- The core of a game: Definition, non-emptiness (Bondareva-Shapley).
- Markets and market games: Competitive equilibrium, existence of equilibrium, the core of a market game and competitive equilibria of related markets. Equivalence of market games and totally balanced games.
- Coalition Structure: Core with coalition structure; superadditive covers, equal treatment properties.
- Credibility and the core: Ray's and Greenberg's characterizations of the core.
Part 2: The Axiomatic Approach to Solution Concepts
- Consistency and the Core: Axiomatic approach, the reduced game property, Peleg's characterization of the core.
- The prekernel and the converse reduced game property. Application: the bankruptcy problem.
- Coalition structure and the reduced game property.
- The Shapley value. Shapley's axioms; random order formula; potential. Applications: voting, oligopoly.
Bibliography
We shall not follow a particular textbook. Most of the first part of the course is covered by the textbooks mentioned below, but most of the second part is not covered by textbooks. Thus, in the second part
journal articles will have a larger role.
Textbooks
- Robert J. Aumann,
Lectures on Game Theory, Westview Press, Boulder, Colorado, 1989.
- Martin J. Osborne and Ariel Rubinstein,
A Course in Game Theory, MIT Press, Cambridge MA, 1994.
- Martin Shubik, Game Theory in the Social Sciences : Concepts and Solutions,
The MIT Press, Cambridge, MA, reprint edition, 1985.
- Martin Shubik, A Game-Theoretic Approach to the Political Economy, (Game Theory in
the Social Sciences, Vol 2), The MIT Press, Cambridge, MA, 1984.
Articles
- Aumann R.J. and J.H. Dreze (1974), "Cooperative Games with Coalition Structures," International journal of Game Theory 3:217-237.
- Aumann R.J. and M. Maschler (1985), "Game Theoretic Analysis of a bankruptcy Problem from the Talmud," Journal of Economic Theory 36:195-213.
- Hart S. and A. Mas-Colell (1989) "Potential, Value, and Consistency," Econometrica 57:589-614.
- Owen G. (1992) "The Assignment Game: The Reduced Game," Annales d'Economie et de Statistique 25/26:71-79.
- Peleg B. (1986) "On the Reduced Game Property and its Converse," International Journal of Game Theory 15:187-200.
- Ray D. (1989) "Credible Coalitions and the Core," International Journal of Game Theory 18:185-187.
- Scarf H. E. (1986) "Notes on the Core of a Productive Economy," in W. Hildenbrand and A. Mas-Colell Contributions to Mathematical Economics In Honor of Gérard Debreu, North-Holland, Amsterdam, pp. 401-429.
- Shapley L.S. (1967) "On Balanced Sets and Cores," Naval Research Logistics Quarterly 14:453-460.
- Shapley L.S. and M. Shubik (1969) "On Market Games," Journal of Economic Theory 1:9-25.
- Shapley L.S. and M. Shubik (1972) "The Assignment Game I: The Core," International Journal of Game Theory 1:111-130.