6.1 Coalitions and Characteristic Functions
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, xi – 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. ...
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