Abstract
The common features of a cooperative game model—such as the model of a game with transferable utility in Chap. 9—include: the abstraction from a detailed description of the strategic possibilities of a player; instead, a detailed description of what players and coalitions can attain in terms of outcomes or utilities; solution concepts based on strategic considerations and/or considerations of fairness, equity, efficiency, etc.; if possible, an axiomatic characterization of such solution concepts. For instance, one can argue that the core for TU-games is based on strategic considerations whereas the Shapley value is based on a combination of efficiency and symmetry or fairness with respect to contributions. The latter is made precise by an axiomatic characterization as in Problem 9.17.
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- 1.
We restrict attention here to two-person bargaining problems. For n-person bargaining problems and, more generally, NTU-games, see the Notes section at the end of the chapter and Chap. 21
- 2.
A subset of \(\mathbb{R}^{k}\) is convex if with each pair of points in the set also the line segment connecting these points is in the set. A set is closed if it contains its boundary or, equivalently, if for every sequence of points in the set that converges to a point that limit point is also in the set. It is bounded if there is a number M > 0 such that | x i | ≤ M for all points \(\mathbf{x}\) in the set and all coordinates i.
- 3.
The usual assumption is that the utility functions are expected utility functions, which uniquely represent preferences up to choice of origin and scale.
- 4.
\(\mathbb{R}_{+}^{2}:=\{\mathbf{ x} = (x_{1},x_{2}) \in \mathbb{R}^{2}\mid x_{1},\ x_{2} \geq 0\}\).
- 5.
Check that there are 34 possible different matchings for this problem.
- 6.
Hence, by definition players in coalitions can only possibly improve by exchanging their initially owned houses, not the houses they acquired after the exchange has taken place.
References
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Nash, J. F. (1950). The bargaining problem. Econometrica, 18, 155–162.
Osborne, M. J. (2004). An introduction to game theory. New York: Oxford University Press.
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Peters, H. (2015). Cooperative Game Models. In: Game Theory. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46950-7_10
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