The Deegan-Packel power index

Another index of voting power that has received some attention in the literature is that proposed by Deegan and Packel (1978). Their measure is based on the notion of minimal winning coalitions (MWCs), which are coalitions that become losing if any single voter is removed.3 Following Riker's (1962) size principle, they argue that coalitions exceeding the minimal winning ones will not form (why bother recruiting more supporters if your coalition is already winning?) They also assume that each MWC is equally likely, and that members of MWCs split any gains equally.

These assumptions uniquely determine the following power index. Suppose that player i is a member of the minimal winning coalitions

{S1(S2,...,SK }, which have s1(s2,...,sK members respectively. Then the total Deegan-Packel (DP) power of i is:

and the Deegan-Packel power index of i is:

Returning to our example, note that the minimal winning coalitions are {A, B},{ B, C}, and {C, A}. Therefore,

and so:

which, once again, is the same as ф and фBz (again, this need not always be the case).

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