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Lecture 11 Coalitional Games and the Core2010 2

Lecture 11 Coalitional Games and the Core2010 2 - ECON...

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ECON 4109H LECTURE 11 Coalitional Games and the Core December 8, 2010 ECON 4109H LECTURE 11

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Coalitional Games A Coalitional game is a model of interacting decision-makers that focuses on behavior of groups of players. For every group of players, there is a set of actions ... not only for individuals. A group of players is a coalition . The group of all players is the grand coalition . An outcome of a coalitional game consists of a partition of the set of players into groups, together with an action for each group in the partition. In general a player may care about the actions of all coalitions that form. We consider only the case in which players care only about what happens within their own coalition. ECON 4109H LECTURE 11
Coalitional Games Definition A coalitional game consists of: 1 a set of players 2 for each coalition, a set of actions 3 for each player, preferences over the set of all actions of all coalitions of which she is a member. The grand coalition N is the set of all players. ECON 4109H LECTURE 11

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Two-Player Unanimity Game Two people together can produce a unit of output, which they may share in any way they wish. Each player cares only about the amount of output she receives, and prefers more to less. Players The two people: 1 and 2. Actions Each player alone has a single action, which yields no output. The set of actions of the coalition { 1, 2 } of both players is the set of all pairs ( x 1 , x 2 ) of nonnegative numbers such that x 1 + x 2 = 1 (the set of divisions of one unit of output between the two players.) Preferences Each player’s preferences are represented by the amount of output she obtains. ECON 4109H LECTURE 11
S -allocations In many examples, each coalition controls the allocation of some good. Each action of a coalition S in such a game is a distribution among the members of S of the good that S controls. Such an allocation is an S - allocation . An N -allocation is simply an allocation . ECON 4109H LECTURE 11

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Landowner and Workers A landowner’s estate, when used by k people, produces output f ( k + 1 ) of food, where f is an increasing function. Total number of workers is m . The landowner cares only about the amount of output she receives, preferring more to less. Same is true for workers. ECON 4109H LECTURE 11
Landowner and Workers Players The landowner and m workers. Actions A coalition consisting solely of workers has a single action in which no member receives any output. The set of actions of a coalition S consisting of landowners and k workers is the set of all S -allocations of the output f ( k + 1 ) among members of S . Preferences Each player’s preferences are represented by the amount of output she obtains. ECON 4109H LECTURE 11

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Three-player Majority Game Three people have access to one unit of output. A majority–two or three people–may control the allocation of this output.
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Lecture 11 Coalitional Games and the Core2010 2 - ECON...

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