coalitional games

Graduate program in economics, Brown University, Academic year 1999/2000

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 topics in economic theory (EC219) 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 recent fifteen years or so, highlighting the centrality of the consistency axiom, and of the core as a solution concept. Within the axiomatic approach we will also discuss the prekernel and the Shapley value, and some of their applications in economics, political science, and Jewish law.

In the third part of the course, we will discus general coalitional games, which include the Nash bargaining problem and non-transferable utility games, among other models. In this part of the course we will check to what extent results of transferable utility games can be generalized.

**Transferable utility games:**basic definitions and examples.**The core**of a game: definition, non-emptiness (Bondareva-Shapley).**Markets and market games, part I:**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.**Market and market games, part II:**increasing returns to scale, distributive production functions, Scarf's quasi-equilibrium, and the core.**Coalition structure:**Core with coalition structure; superadditive covers, equal treatment properties.**Credibility**and the core: Ray's and Greenberg's characterizations of the core.

**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 Talmudic bankruptcy problem.**Coalition structure and the reduced game property.****The Shapley value.**Shapley's axioms; random order formula; potential. Applications: voting, oligopoly, and the Talmudic bankruptcy problem.

**General coalitional games:**basic definitions, and the distinction between non-transferable utility games. Economic models as coalitional games.**The Nash bargaining problem**and the Nash solution. Comparison with other solutions.**The core of general coalitional games**and its relation to exchange and production economies.**The axiomatic approach applied to general coalitional games.**

We shall not follow a particular textbook. Most of the first part and some of the third part are 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.

- 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.

- 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.
- Binmore K., A. Rubinstein and A. Wolinsky (1986) "The Nash Bargaining Solution in Economic Modelling," Rand Journal of Economics 17:176-188.
- Davis and M. Maschler (1965) "The Kernel of a Cooperative Game," Naval Research Logistics Quarterly 12:223-259.
- Hart S. and A. Mas-Colell (1989) "Potential, Value, and Consistency," Econometrica 57:589-614.
- Lensberg T. (1988) "Stability and the Nash Solution," Journal of Economic Theory 45:330-341.
- Maschler M. (1992) "The Bargaining Set, Kernel and Nucleolus," in R.J. Aumann and S. Hart, Handbook of Game Theory with Economic Applications (vol. I), Elsevier.
- Nash J. (1950) "The Bargaining Problem," Econometrica 18:155-162.
- 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. (1953) "A Value for
`n`-Person Games," in A.W. Tucker and R.D. Luce Contributions to the Theory of Games IV, Princeton, Princeton University Press, pp. 307-317. - 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.
- Schmeidler D. (1969) "The Nucleolus of a Characteristic Function Game," SIAM Journal of Applied Mathematics 17:1163-1170.