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Solutions Manual to Accompany Game Theory: An Introduction, 2nd Edition by E. N. Barron

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Cooperative Games

6.1   Coalitions and Characteristic Functions

Problems

6.1 A stag-hunt game has characteristic function v(S) = α|S|, SN, v(N) = 1, where images.

(a) Find the normalized stag-hunt characteristic function.

6.1.a Answer: The normalized function is images, and v′(N) = 1.

(b) Find C(0) using the normalization.

6.1.b Answer: The core using v′ is

images

The unnormalized allocations satisfy

images

Since 1 – nα > 0, xi – α ≥ 0 and images. In terms of the unnormalized allocations, images.

6.2 A customer wants to buy a bolt and a nut for the bolt. There are three players but player 1 owns the bolt and players 2 and 3 each own a nut. A bolt together with a nut is worth 5 but is worthless otherwise. Also, a nut without a bolt is worthless. Define a characteristic function for this game and verify that it is superadditive. ...

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