This course will focus on cooperative game theory and its applications, and build up some proof techniques.

 Prerequisites: A solid knowledge of microeconomics analysis and familiarity with logic analysis are essential. A basic command in mathematical analysis will be helpful.

 Requirements: This course is built around the reading and analysis of journal articles. Requirements include solving problem sets and submitting two (nicely printed) referee reports on papers not assigned as required readings but related to this course. You need to discuss with me before you decide on which papers you are interested and would like to provide your reports.

This course will consider cooperative game theory and its related resource allocation problems, and study the allocation rules or decision rules from the normative and strategic viewpoints. A number of properties motivated by fairness or justice principles will be discussed and introduced. We then compare allocation rules on the basis of these properties. More importantly, we are interested in the implications of the properties, when imposed singly or in various combinations. The ultimate object is to trace out the boundary between those combinations that are compatible and those that are not, and when compatible, to give as explicit as possible a description of the family of rules satisfying them.

2.1 Transferable utility games
We depart from the classical theory of transferable utility (TU) games. Several TU game solution concepts will be introduced here such as Shapley value, the nucleolus, and etc. These solution concepts will be discussed throughout this class, and more importantly, they will be applied to resource allocation problems that we will encounter during this class. Some references and textbooks might be helpful.

References
Moulin H (1988) Axioms of cooperative decision making, Cambridge University Press,
Cambridge
Moulin H (1995) Cooperative microeconomics: a game-theoretic introduction, Princeton
University Press, Princeton, New Jersey
Shapley LS (1953) \A value for N-person games," in Contributions to the Theory of Game
II,HW Kuhn and AW Tucker, eds. Annals of Mathematics Studies 28: 251-263
Schmeidler D (1969) \The nucleolus of a characteristic function game," SIAM Journal on
Applied Mathematics 17: 1163-1170
Megiddo N (1974) \On the non-monotonicity of the bargaining set, the kernel, and the
nucleolus of a game," SIAM Journal of Applied Mathematics 27: 335-338 The three
musketeers: four classical solutions to bankruptcy problems. Math Soc Sci 42:
307-328
Dutta B and Ray D (1989) \A concept of Egalitarianism under Participation Constrains,"
Econometrica 57(3): 615-636


2.2 Bankruptcy problems
Bankruptcy problem is the problem of distributing an innitely divisible and homogeneous resource among agents having conflicting claims on it. The question is: how should the resource be attributed? A \rule" is a function that associates with each bankruptcy problem a division of the amount available, called an \awards vector." A number of desirable properties of rules have been formulated for this problem, and motivated by various fairness principles. The literature devoted to the search for rules satisfying these properties, singly and in various combinations, is initiated by O'Neill (1982). For a comprehensive survey of this literature, see Thomson (2003). We will examine several division rules such as the Talmud, the proportional rule, and others in light of these properties. Some references and books might be useful to understand the subject.

References
Aumann RJ, Maschler M (1985) \Game theoretic analysis of a bankruptcy problem from
the Talmud," Journal of Economic Theory 36: 195-213
Moulin H (2000) \Priority rules and other asymmetric rationing methods," Econometrica
68: 643-684
O'Neill B (1982) \A problem of rights arbitration from the Talmud," Mathematical Social
Sciences 2: 345-371
Thomson W (2000) Consistent allocation rules. Mimeo, University of Rochester, U.S.A.
ThomsonW(2001) How to Divide when There Isn't Enough. Book manuscript, University
of Rochester, U.S.A.
Thomson W (2003) \Axiomatic analysis of bankruptcy and taxation problems: a survey,"
Mathematical Social Sciences 45: 249-297
Young P (1987) \On dividing an amount according to individual claims or liabilities,"
Mathematics of Operation Research 12: 398-414
Young P (1988) \Distributive justice in taxation," Journal Economic Theory 43: 321-335


2.3 Cost-sharing problems
This is the problem of allocating the cost of a public facility among agents using it jointly. Agents are ordered in terms of their needs for the facility. The facility should be built so as to satisfy the agent with the greatest need. A division rule is a function that associates with each problem of this kind a contributions vector. We ask "how" to perform the division of the cost among these agents using it, and "what" the most desirable way to allocate the cost is. This class of cost allocation problems is formalized by Littlechild and Owen (1973), and has two main real-life applications. The first one is the so-called airport problem. Namely, a group of airlines share a runway. To accommodate the planes of airline i, the runway must be of a certain length, and ci denote the cost of such runway. A runway accommodating airlines in a group S would cost max cj such that j in S. How should the cost of the runway be distributed among all airlines using it? The second one is the so-called irrigation problem, which is to distribute the cost of maintaining an irrigation ditch. Consider ranchers distributed along a ditch. The rancher closest to the headgate only needs that the first segment be maintained, the second closest rancher needs that the first two segments be maintained, and so on. How should the ranchers share the cost of maintaining the ditch? The following are some references that might help understand the material in this cost sharing problems.

References
Aadland D, Kolpin V (1998) \Shared irrigation costs: An empirical and axiomatic analysis,"
Mathematical Social Sciences 35: 203-218.
Littlechild SC, (1974) \A simple expression for the nucleolus in a special case," International
Journal of Game Theory 3: 21-29.
Littlechild SC, Owen G (1973) \A simple expression for the Shapley value in a special
case," Management Science 3: 370-372.
Littlechild SC, Owen G (1976) \A further note on the nucleolus of the airport game,"
International Journal of Game Theory 5: 91-95.
Littlechild SC, Thompson GF (1977) \Aircraft landing fees: a game theory approach,"
Bell Journal of Economics 8: 186-204.
O'Neill B (1982) \A problem of rights arbitration from the Talmud," Mathematical Social
Sciences 2: 345-371.
Potters J, Sudholter P (1999) \Airport problems and consistent allocation rules," Mathematical
Social Sciences 38: 83-102.
Schmeidler D (1969) \The nucleolus of a characteristic function games," SIAM Journal
on Applied Mathematics 17: 1163-1170.
Sonmez T (1994) \Population monotonicity of the nucleolus on a class of public good
problems," mimeo, University of Rochester, Rochester, NY, USA
ThomsonW(2000) \Consistent allocation rules," mimeo, University of Rochester, Rochester,
NY, USA
||||||(2004) \Cost allocation and airport problems," mimeo, University of Rochester,
Rochester, NY, USA