×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

by: Amy Brogan

11

0

2

# Week 5: Weighted Voting Systems Part 2 MATH 1014

Marketplace > University of Cincinnati > Mathematics (M) > MATH 1014 > Week 5 Weighted Voting Systems Part 2
Amy Brogan
UC
GPA 3.7

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

The Banzhaf Power Index and how to quantify the number of sets depending on the number of voters.
COURSE
Mathematics of Social Choice
PROF.
Mary Koshar
TYPE
Class Notes
PAGES
2
WORDS
CONCEPTS
Banzhaf power index combinations permutations sets voting weighted nCk
KARMA
25 ?

## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by Amy Brogan on Saturday February 13, 2016. The Class Notes belongs to MATH 1014 at University of Cincinnati taught by Mary Koshar in Spring 2016. Since its upload, it has received 11 views. For similar materials see Mathematics of Social Choice in Mathematics (M) at University of Cincinnati.

×

## Reviews for Week 5: Weighted Voting Systems Part 2

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/13/16
Koshar Amy Brogan February 12, 2016 Week 5: Weighted Voting Systems Part 2 We have learned that a voting group with 5 voters has 120 permutation sets. With the Shapely-Shubik index, power is the number of times a voter is pivotal in a permutation. But what if the voters are only voting yes/no, and the order doesn’t matter. How do we quantify a combination voting set? This is called a coalition, and it’s used in the Banzhaf Power Index.  Coalition: o (general) a group of people with a common cause o (math) a combination of votes (vs. a permutation) Ex 1: [16: 10, 8, 6, 5, 3] Some Outcomes: {10, 6, 3} {3} {8, 6, 5, 3} Are these winning combinations (do they add up or over the quota)? Yes, no, yes. Which voters are critical to pass the motion? In the first set, if w10 changed their mind and voted no, then it would not pass, so w10 is critical. If w6 changed their mind, the same thing, so w6 is critical. But if w3 changed their mind, the motion would still carry, so they are not critical. Ex 2: [13: 11, 8, 5] What are all winning sets for this group? (They’re combination, so order doesn’t matter) {11, 8} {11, 5} {8, 5} and {11, 8, 5} What about listing all possible coalitions? This is easiest to do it by number of voters: 1 Voter Combination: {11} {8} {5} 2 Voters Combination: {11, 8} {11, 5} {8, 5} 3 Voters Combination: {11, 8, 5} That’s 7 combinations, but we also have to remember there is an empty set { } where nobody votes yes, so 8 combinations. With Banzhaf the number of times each voter is critical is 11:2, 8:2, and 5:2.  Critical: when the voter’s weight in a winning coalition is required to make it winning  Power: a voter has power when s/he is critical in some winning coalition  Goal: find the number of times a voter is critical Since there are only two possible answers (yes and no) we can calculate the possible number of coalitions with 2n 2^n n is the number of voters Ex 3: [6: 5, 4, 2] All coalitions: 1: {5} {4} {2} 2: {5, 4} {5, 2} {4, 2} 3: {5, 4, 2} +{ } for 8; the winning coalitions are highlighted, and the critical voters are underlined 5 is critical 2 times, as are 4 and 2. The total number critical voters is 6 (2+2+2) so the power index for each is 2/6, or simplified 1/3. The end “fractions” have to add up to one. If they don’t, check your work. The power index is written as before: (1/3, 1/3, 1/3) Ex 4: [10: 10, 6, 5] 1: {10} {6} {5} 2: {10, 6} {10, 5} {6, 5} 3: {10, 6, 5} 10: 3, 6:1, 5:1  5 critical positions  (3/5, 1/5, 1/5) Ex 5: [18: 12, 8, 8, 6] 2 4= 16 coalitions 1 voter has n sets, and n voters has 1 set when we are writing out all possible coalitions, but how do we calculate the middle sets? nC k = # of permutations of k from n voters = n! (multiplied by) 1 = n! __ ways to order k (n-k)! k! k!(n-k!) Ex 6: C 3 = There are 20 coalitions for sets of 3 voters. Ex 7: [12: 8, 6, 3, 3, 1, 1] – a lot of coalitions, what about just finding the number of sets of 4? 7C 4 = There are 35 coalitions for sets of 4 voters.

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Steve Martinelli UC Los Angeles

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Kyle Maynard Purdue

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made \$280 on my first study guide!"

Bentley McCaw University of Florida

Forbes

#### "Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com