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Note

Please see the appendix to this chapter for an example that highlights how the Shapley value of this simple game is calculated.

This table gives the list of winning coalitions for this game and 1 means that the corresponding world region is decisive for the corresponding coalition while 0 means that the corresponding world region is not decisive. Considering the world region a, its marginal contribution is positive for one and only one coalition: (a,c,d). Then, applying the formula given in (3) for world region a, we have (3-1)1(5-3)1 _ 1 . a 5! 30

that the Shapley value is the same for world region c and d, even though the population of region d is larger. This result describes a fundamental concept behind the Shapley value: since player d has no greater opportunity than c to form a minimal winning coalition, he must have the same share as player c in a bargaining game. In contrast, the "vote vector" result—payoffs strictly proportional to each player's share of the total population, i.e., no world

( 5 15 20 25 35 ^ region has majority power on his own—is -,-,—,-—,- .

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