2. 1 Message to the user... The most effective way to use a PowerPoint slide show is to go to “SLIDE SHOW” on the top of the toolbar, and choose “VIEW SHOW” from the pull down menu. OR, using the shortcut toolbar on the bottom left, choose the rightmost icon (“SLIDE SHOW”) Use the spacebar, enter key or mouse to move through the slide show. Use the backspace key to undo the last animation on a slide TEACHERS : If using this show as part of a lecture, it is helpful to go to “PRINT” in the “FILE” menu and use the drop down menu at the bottom left: “PRINT WHAT.” Printing the “OUTLINE VIEW” will be helpful if you intend to view many slides with your class; or you can print a particular slide to use as a handout. (Many shows will include sound… you may want to turn on your speakers!) Weighted voting systems: Show #1 of 4

3. 2 WEIGHTED VOTING SYSTEMS sets of voters (slide #3--17) the power of coalitions (#18--23) power of individual voters (#24--26)

4. 3 WEIGHTED VOTING SYSTEMS Example : A certain small business has 4 shareholders. ALICE owns 45 shares of stock BRADLEY owns 30 shares CHARLES owns 15 shares DOROTHY owns 10 shares

5. 4 WEIGHTED VOTING SYSTEMS ALICE owns 45 shares of stock BRADLEY owns 30 shares CHARLES owns 15 shares DOROTHY owns 10 shares Each stockholder attends monthly meetings, at which they are able to vote on matters that affect the business. We will look at 3 ways that the voting might be accomplished, and discuss the “fairness” of each of the voting systems considered.

6. 5 WEIGHTED VOTING SYSTEMS WVS NOTATION: A wvs with “n” voters... [q: w 1, w 2, w 3, … w n ] where q = quota (number of votes needed to pass a motion) and w n = the weight of voter n (number of votes that particular voter has) NOTE: We will say that the wvs is valid as long as the QUOTA is at least a *simple majority and the QUOTA is not more than the total number of votes possible.

7. 6 WEIGHTED VOTING SYSTEMS WVS NOTATION: A wvs with “n” voters... [q: w 1, w 2, w 3, … w n ] where q = quota & w n = the weight of voter n NOTE: We will say that the wvs is valid as long as the QUOTA is at least a simple majority and the QUOTA is not more than the total number of votes possible. NOTE also… The voting system is made up of the actual voters… (ex) {Alice, Brad, Chuck, Dorothy} We will not allow the voters to simply be “named” by their # of votes. i.e. If Alice has 45 votes, we won’t call her “voter 45” … we’ll still call her Alice!!!

8. 7 WEIGHTED VOTING SYSTEMS ALICE owns 45 shares of stock BRADLEY owns 30 shares CHARLES owns 15 shares DOROTHY owns 10 shares We will look at 3 ways that the voting might be accomplished, and discuss the “fairness” of each of the voting systems considered. Wvs#1 The number of votes each person has (weight of each voter) is equal to the number of shares of stock (s)he owns. And the number of votes needed to pass a measure (quota) is the simple majority.

9. 8 WEIGHTED VOTING SYSTEMS ALICE owns 45 shares of stock BRADLEY owns 30 shares CHARLES owns 15 shares DOROTHY owns 10 shares Wvs#1 FIND the simple majority: 45+30+15+10 = 100 total votes. The simple majority is the first integer that is more than half of 100 (the total # of votes possible). That’s 51 votes! (note: a vote of 50 to 50 would be a tie… a vote of 51 to 49 would produce a winner & loser) Wvs#1 The number of votes each person has (weight of each voter) is equal to the number of shares of stock (s)he owns. And the number of votes needed to pass a measure (quota) is the simple majority.

10. 9 WEIGHTED VOTING SYSTEMS WVS #1 shorthand notation: [51: 45, 30, 15, 10] Let’s use some shorthand notation for the voters’ too… so that we can take a look at the whole system, and how the voters might vote on certain issues. We’ll look at the set of voters as follows: {A, B, C, D} ALICE owns 45 shares of stock BRADLEY owns 30 shares CHARLES owns 15 shares DOROTHY owns 10 shares

11. 10 WEIGHTED VOTING SYSTEMS How the voting could work... The next set of slides will deal with looking at all of the possibilities for a vote on a motion. We will assume that no voter is allowed to abstain from voting. When voters join together on one side of an issue (FOR or AGAINST), they will form a subset of voters called a COALITION. After you have done this look at the workings of a weighted voting system, you will NOT have to do this for any other problem. This is the BIG PICTURE… so you get a feel for the possibilities.

12. 11 Let’s do an exhaustive look at all of the possible ways voting could work here. (**handout**) Coalition voting FOR the motion Coalition voting AGAINST the motion weightstatusweightstatus Motion PASS or FAIL BD__ ABCD45lose55blockFAIL BACD30lose70blockFAIL CABD15lose85blockFAIL DABC10lose90blockFAIL ABCD75win25losePASS AC__ AD__ BC__ ABC__ CD__ BCD__ ABD__ ACD__ noneABCD0lose100blockFAIL ABCD__

13. 12 Coalition voting FOR the motion Coalition voting AGAINST the motion weightstatusweightstatus Motion PASS or FAIL BD__ ABCD45lose55blockFAIL BACD30lose70blockFAIL CABD15lose85blockFAIL DABC10lose90blockFAIL ABCD75win25losePASS AC__ AD__ BC__ ABC__ CD__ BCD__ ABD__ ACD__ noneABCD0lose100blockFAIL ABCD__

14. 13 WEIGHTED VOTING SYSTEMS [51: 45,30,15,10] BDAC40lose60blockFAILACBD60win40losePASSADBC55win45losePASSBCAD45lose55blockFAILABCD90win10losePASSCDAB25lose75blockFAILBCDA55win45losePASSABDC85win15losePASSACDB70win30losePASSCoalition voting FOR the motion Coalition voting AGAINST the motion weightstatusweightstatus Motion PASS or FAIL ABCD45lose55blockFAIL BACD30lose70blockFAIL CABD15lose85blockFAIL DABC10lose90blockFAIL ABCD75win25losePASS noneABCD0lose100blockFAIL ABCDnone100win0losePASS

15. 14 WEIGHTED VOTING SYSTEMS Notice that there are 8 winning coalitions; And that ALICE is in 7 out of the 8 winning coalitions, While the other three voters are each in 5 out of 8 winning coalitions. Brad might not be too happy about that! He’s got a lot more votes than Chuck and Dot… Maybe if the QUOTA was higher, he would gain some “power?” So, for WVS#2, let’s make the QUOTA higher than just a simple majority of the votes… And for WVS#3, we’ll try something different...

16. 15 WVS#2: The number of votes will be the same as in WVS#1, but the QUOTA will be 70 votes. (70 votes needed to pass a motion). WVS#2: [70: 45, 30, 15, 10] ALICE owns 45 shares BRADLEY owns 30 shares CHARLES owns 15 shares DOROTHY owns 10 shares

17. 16 WVS#3: The number votes will be the tens digit of the # of shares of stock, and the QUOTA will be the simple majority. WVS#3: [?: 4, 3, 1, 1] the simple majority will be the first integer that is more than half of: 4+3+1+1 = 9 9/2 = 4.5 5 votes will be the simple majority. (notice that a vote of 5 to 4 would have a winner and loser.) [5: 4, 3, 1, 1] ALICE owns 45 shares BRADLEY owns 30 shares CHARLES owns 15 shares DOROTHY owns 10 shares

18. 17 Complete wvs charts If available, use the handout(s) and complete a chart that looks at the way each of these wvs work. If a handout is not available, create the charts on your own. Notice the similarities and differences in the STATUS and PASS/FAIL columns You should see that in Method #1 and Method #3, the STATUS columns are the same for corresponding coalitions. These are called EQUIVALENT wvs

19. 18 WVS power Notice that a coalition of voters can either: WIN LOSE or *BLOCK when voting on a motion Winning Coalitions have a weight that is at least the quota. They pass a motion. Losing Coalitions have a weight that is less than the quota. *Blocking Coalitions have enough votes to keep their opponents from passing a motion.

20. 19 WVS power Winning Coalitions have a weight that is at least the quota. Losing Coalitions have a weight that is less than the quota. *Blocking Coalitions have enough votes to keep their opponents from winning. How many votes are needed to BLOCK a measure?? If you know the TOTAL # of votes in the system and you know the QUOTA It is always true that a LOSING coalition will have a weight that is less than or equal to the TOTAL - QUOTA So, in order to BLOCK, a coalition must have MORE THAN the (TOTAL - QUOTA)

21. 20 WVS power How many votes are needed to BLOCK a measure?? In order to BLOCK, a coalition must have MORE THAN the (TOTAL - QUOTA) (ex) In WVS#1, the total # votes = 100 the quota = 51 A coalition with (100 - 51) = 49 votes is a losing coalition (in fact, any coalition with less than 49 votes loses, too!) Any coalition (voting AGAINST a motion) that has MORE THAN 49 votes can BLOCK a motion from being passed.

22. 21 WVS power (ex) In WVS#1, Any coalition with MORE THAN 49 votes can BLOCK a motion from being passed. REMEMBER… A BLOCKING COALITION is voting AGAINST the motion under consideration. This coalition might have a weight equal to the quota, but it might not! It just depends on how many votes the coalition voting FOR the motion has!

23. 22 WVS power In WVS#2, how many votes are needed to BLOCK a measure from being passed? The quota is 70 The total # of votes is 100 To BLOCK a motion, a coalition must have MORE THAN (100 - 70) that is MORE THAN 30 votes. (Look at the coalitions you’ve marked BLOCK to confirm this)

24. 23 WVS power Your text does not mention BLOCKING coalitions However, in order to thoroughly discuss POWER in a wvs, it is important to remember that whatever coalition has enough votes to attain their goal, has some type of power. A WINNING COALITION: has the power to PASS a measure. A BLOCKING COALITION: has the power to prohibit a measure from being passed (blocking power.)

25. 24 WVS power Winning or Blocking Power is related to the status of the entire coalition. We must also discuss the POWER OF INDIVIDUAL VOTERS in the system and within their coalitions. A voter whose weight is greater than or equal to the quota is called a DICTATOR. A DICTATOR is a one-person winning coalition.

26. 25 WVS power A voter whose weight is greater than or equal to the quota is called a DICTATOR. A DICTATOR is a one-person winning coalition. A voter has VETO POWER if (s)he can prevent a measure from being passed even if all others vote for the measure. A voter with VETO POWER is a one-person blocking coalition. A voter whose vote is NEVER necessary in any winning coalition is called a DUMMY VOTER (if that voter leaves, the remaining members will still make up a winning coalition).

27. 26 WVS: The power of a voter... In any wvs, there will always be the argument about “fairness.” Do some voters have too much power? Are there any voters who have no power at all? How can we describe the “power of a voter” in a system? We will look at TWO measures of power. The first is called the BANZHAF POWER INDEX. This index measures the power of a voter by describing how many winning coalitions the voter is critical for the success of.

28. 27 End of show #1 Going on?... Weighted voting system: Show #2: Banzhaf Power Distribution