Solutions Manual to Accompany Game Theory: An Introduction, 2nd Edition

Cooperative Games

6.1   Coalitions and Characteristic Functions

Problems

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

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

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

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

6.1.b Answer: The core using v′ is

The unnormalized allocations satisfy

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

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. ...