Power Indices by Modified Multilinear Approximation

Program ipmmle

(Go straight to data input screen.)

This page enables you calculate the Banzhaf, Penrose and Coleman power indices for a large voting body, for example the hundreds or thousands of ordinary shareholders of a large public corporation. The program ipmmle employs a combination of direct enumeration and approximation based on a probabilistic voting model. It can be applied to large voting bodies of any size (either in terms of number of players or in terms of votes). The voting weights and quota are not restricted to being integers.

The program computes the following for each member:
Penrose index or Absolute Banzhaf index:  the number of swings divided by the number of possible voting outcomes among the other members.

Normalised Banzhaf index:  the number of swings as a proportion of the total number of swings for all members. The indices sum to 1 over all members.

Power to Prevent Action (Coleman):  the number of swings divided by the number of voting outcomes that lead to a decision. A measure of a member's  blocking power.

Power to Initiate Action (Coleman):  the number of swings divided by the number of outcomes that do not produce a decision. A measure of the member's power to get its way within the voting body.
The algorithm uses the modification of the approximation method of multilinear extensions of Guillermo Owen (see 1995) described in Leech (2003). It is necessary to divide the players into two groups:  "major" and "minor". The "major" players are a small number including the largest and the "minor" ones are the remainder. The algorithm trades off speed against accuracy: the larger is the number of "major" players assumed the slower but more accurate the algorithm. In practice, in most applications, the number of major players need not be greater than 10 in order to achieve high accuracy. The weights must be entered in decreasing size order.

References:
Penrose (1946), Banzhaf (1965), Coleman (1971), Owen (1972, 1975a, 1975b, 1995), Leech (2003; see also 2001, 2002b, 2002c for applications).



Data Input for ipmmle

Enter your data in the boxes below.

(The numbers are examples which can be overwritten.)

      
Number of "Major" Players
:
           Number of "Minor" Players:

Quota:                                

Weights
: (Paste the values of weights with no punctuation and with a space between each weight. The weights must be in decreasing size order beginning with the largest.)



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