Weighted Voting Systems Chapter 2 Objective Calculate the

Weighted Voting Systems Chapter 2 Objective: Calculate the Shapley-Shubik Power Index for a weighted voting system. Learn additional notation and terminology for sequential coalitions.

�Write two or three sentences comparing and contrasting the Banzhaf vs. Shapley-Shubik Power Distributions. Closing Product Exit Ticket

� Sequential Coalitions

�Consider the following weighted voting system [4: 3, 2, 1]. ◦ How many sequential coalitions are there? ◦ List all of the sequential coalitions. Example # 1

�Find 11!/10! �Find 101!/99! �Find (9!+11!)/10! Example # 2

�A player in a sequential coalition that turns it from a losing to a winning coalition. �There is only one per coalition. �Find it by adding up weights from Left to Right. Pivotal Player

�Consider the following weighted voting system [4: 3, 2, 1]. ◦ Find the pivotal player in each sequential coalition. Example # 3

�A mathematical measurement of a player’s power in a weighted voting system based on how many times that player is the pivotal player in a sequential coalition. � σ1 = # of time Player 1 is the pivotal player N! Pronounced ‘sigma-one’ Shapely-Shubik Power Index

�A complete list of the power indexes � σ1 , σ2 , σ3, …… σn �The sum of all the σ’s is equal to 1. (or 100% if using percentages. Shapley-Shubik Power Distribution

i. iii. iv. v. Make a list of all sequential coalitions. Determine the pivotal player of each coalition. Count the number of times each player is pivotal. (SS 1 for P 1, SS 2 for P 2, etc. ) Find the total number of sequential coaltions (N!) Find the ratio for each player (σ1 = SS 1/N!) Calculation

Do the next one on your own.

Consider the weighted voting system [95: 65, 30, 25]. Find the following: a) List all of the sequential coalitions. b) Circle the pivotal player in each coalition. c) Calculate the Shapley-Shubik Power Distribution. Example #4

Discuss with a neighbor some of the similarities and differences between Banzhaf and S-S Power Index. Then write a couple of sentences that summarizes what you discussed. Consider the following ….