Wim Ruitenburg's Spring 2011 MATH 1300-101

Last updated: January 2011
Comments and suggestions: Email wimr@mscs.mu.edu

From the chapter we can learn:

Voting power distribution systems where all parties P_i have a vote weight w_i, and where it takes quota amount q to pass a motion.
Examples of possible anarchy.
Examples of guaranteed gridlock.
When does a sytem force unanimity?
What is dictatorial power?
What is veto power?
What is a dummy party?

Here are examples of attempts to quantify the power of the parties.

The Banzhaf power index.
List all winning coalitions. If there are 4 parties, then there are 2^4 = 2*2*2*2 = 16 possible coalitions. Inside each winning coalition underline the critical parties. For each party P_i count the number of times it occurs underlined, to get sum B_i. Add the total number of all underlined parties in all winning coalitions to get sum T. Note that T also equals the sum of the B_i. Then party P_i has as power index quotient beta_i = B_i / T.
The Shapley-Shubik power index.
List all winning coalition formations. If there are 4 parties, then there are 4! = 4*3*2*1 = 24 possible coalition formations. Compute the SS_i as we did for the B_i before, except that there is another definition of being a critical party. Now T equals the number of possible coalition formations as well as equals the sum of the SS_i. Party P_i has as power index quotient sigma_i = SS_i / T.