This situation where each voter is not equal in the number of votes they control is called:

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "This situation where each voter is not equal in the number of votes they control is called:"

Transcription

1 Finite Math A Chapter 2, Weighted Voting Systems 1 Discrete Mathematics Notes Chapter 2: Weighted Voting Systems The Power Game Academic Standards: PS.ED.2: Use election theory techniques to analyze election data. Use weighted voting techniques to decide voting power within a group. ONE PERSON ONE VOTE is an democratic idea of equality But what if the voters are not PEOPLE but are governments? countries? states? If the institutions are not equal, then the number of votes they control should not be equal. The United Nations Security Council 15 voting nations: 5 permanent members (Britain, China, France, Russia, United States), 10 nonpermanent members appointed for a 2-year rotation. Permanent members have more votes than non permanent members. Stock Holders/Shareholders: The more stock you own, the more say you have in decision making for the company. The Electoral College Each state gets a number of electors (votes) equal to the number of Senators plus the number of Representatives in Congress. California has 55 votes but North Dakota only has 3 votes. Each state is a voter but states with heavy concentration of population receive a bigger vote. This situation where each voter is not equal in the number of votes they control is called: 2.1 An Introduction to weighted voting Important terms: : A voting situation where voters are not necessarily equal in the number of votes they control. : A vote with only two choices. (usually yes/no) : The voters (symbolized by P 1, P 2, P 3, etc.) : The number of votes a player controls. : The smallest number of votes required to pass a motion.

2 Finite Math A Chapter 2, Weighted Voting Systems 2 Notation Example 1. [14: 8, 6, 5, 1] [q: w 1, w 2, w 3,..., w n ] q = w s = quota = total votes = Player 1 (P 1) = controls votes / has a weight of Player 2 (P 2) = controls votes Player 3 (P 3) = controls votes Player 4 (P 4) = controls vote Example 2. Given the weighted voting system [16: 8, 6, 4, 4, 3, 1], state the following: The number of players: The weight of P 5: The total number of votes: The minimum % of the quota to nearest whole %: Common Types of Quotas: Simple majority/strict majority Two-thirds majority Unanimity U.S. Senate: Simple Majority to pass an ordinary law (51 votes) 60 votes to stop a filibuster 2/3 of the votes (67) to override a presidential veto. Weighted Voting Issues Example 3: Four partners decide to start a business. P 1 buys 8 shares, P 2 buys 7 shares, P 3 buys 3 shares and P 4 buys 2 shares. One share = one vote. a. The quota is set at two-thirds of the total number of votes. Describe as a weighted voting system. b. The partnership above decides the quota is too high and changes the quota to 10 votes. c. The partnership above decides to make the quota equal to 21 votes.

3 Finite Math A Chapter 2, Weighted Voting Systems 3 For a weighted voting system to be legal: the quota must be at least a and no more than V Symbolically: If V w1 w2 w3... wn, then q V 2 Example 3d. What if our partnership changed the quota to 19? 4. [q: 7, 2, 1, 1, 1] What is the smallest legal quota? What is the largest legal quota? What is the value of the quota if at least two-thirds of the votes are required to pass a motion? What is the value of the quota if more than three-fourths of the votes are required to pass? 5. A committee has 4 members (P 1, P 2, P 3, P 4). P 1 has twice as many votes as P 2. P 2 has twice as many votes as P 3. P 3 and P 4 have the same number of votes. The quota is 49. Describe the weighted voting system using the notation [q: w1, w2, w3, w4] given the definitions of quota below. (Hint: write the weighted voting system as [49: 4x, 2x, x, x] and then solve for x. a) The quota is a simple majority b) The quota is more than three-fourths

4 Finite Math A Chapter 2, Weighted Voting Systems 4 Dictators, Dummies, and Veto Power Example 6: [11: 12, 5, 4] What do you notice about P 1? P 1 has all the power P 2 and P 3 have no power Note: If any player is a dictator, then EVERY OTHER PLAYER is a dummy. Even if there is no dictator, there may still be dummies. Example 7: [30: 10, 10, 10, 9] Example 8: [12: 9, 5, 4, 2] Is there a dictator? If P 1 chooses to vote against the motion, can the other players combine weight to meet the quota? If a player is not a dictator, but the other players cannot meet the quota without his votes, we say he has veto power. Sometimes, more than one player will have veto power. Example 9. Determine which players, if any, are: dictators, veto power, dummies a) [15: 16, 8, 4, 1] b) [18: 16, 8, 4, 1] c) [24: 16, 8, 4, 1] Example 10. Consider [q: 8, 4, 2]. Find the smallest value of q for which a) all three players have veto power b) P 2 has veto power, but P 3 does not c) P 3 is the only dummy

5 Finite Math A Chapter 2, Weighted Voting Systems 5 2.2/2.3 The Banzhaf Power Index Who is the most POWERFUL player? : A group of players who choose to vote together : The set of all voters. This represents a unanimous vote. Weight of the coalition: Winning coalitions Losing coalitions : Any player who MUST BE PRESENT in a winning coalition in order for it to remain a winning coalition. Note: If you subtract the critical player s votes from the coalition, the number of votes drops below the quota. Example 1: Find the critical player or critical players in each of the following coalitions. [15: 13, 9, 5, 2] a) {P 1, P 4} b) {P 2, P 3, P 4} c) {P 3, P 4} d) {P 1, P 2, P 3} [51: 30, 25, 25, 20] a) {P 1, P 3} b) {P 1, P 2, P 3} c) {P 2, P 3, P 4} d) {P 2, P 3} The Banzhaf Power Index: A player s power is proportional to the number of coalitions for which that player is critical. The more often a player is critical, the more power he holds.

6 Finite Math A Chapter 2, Weighted Voting Systems 6 Calculate the Banzhaf Power Index: Idea: Step 1: Make a list of all WINNING coalitions. Step 2: Determine which players are critical in each coalition. (circle, underline, highlight) Step 3: Count the total number of times each player is critical Step 4: Add the total number of times each player is critical to find the grand total number of critical players. The Banzhaf Power INDEX number for each player = step 3 step 4 The Banzhaf Power DISTRIBUTION for the weighted voting system is the % of power each player holds. Example 2: Find the Banzhaf Power index for the weighted voting system: [101: 99, 98, 3] Example 3: Find the Banzhaf Power Distribution for [4: 3, 2, 1]

7 Finite Math A Chapter 2, Weighted Voting Systems 7 How do you know you have all the possible coalitions written down? If n = number of players in a weighted voting system, Be systematic or use the formula! Then the number of possible coalitions is: 2 n 1 How many coalitions if 4 players? How many coalitions if 5 players? Example 4: Find the Banzhaf Power Distribution for [6: 4, 3, 2, 1] Banzhaf Coalitions: 4 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P1, P2, P4} {P3} {P1,P4} {P1, P3, P4} {P4} {P2,P3} {P2, P3, P4} {P2,P4} {P1, P2, P3, P4} {P3,P4} Example 5: Consider the weighted voting system [q: 8, 4, 2, 1]. Find the Banzhaf Power Distribution of this weighted voting system when: a) q = 8 b) q = 10 c) q = 14

8 Finite Math A Chapter 2, Weighted Voting Systems 8 What I expect to see for work on your homework: 1. Write down all possible coalitions and cross off losers OR just the winning coalitions. 2. Critical Players should be circled or underlined. 3. Show fraction of BPI for each player AND calculate the % for BPD. Possible Coalitions: Use these to help you: Banzhaf Coalitions: 3 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P3} {P2,P3} Banzhaf Coalitions: 4 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P1, P2, P4} {P3} {P1,P4} {P1, P3, P4} {P4} {P2,P3} {P2, P3, P4} {P2,P4} {P1, P2, P3, P4} {P3,P4} Where weighted voting systems/banzhaf are used: Banzhaf is used to QUANTIFY the amount of power each player holds. 1. Nassau County Board of Supervisors (see p. 55): Votes were given to districts according to population and quota was simple majority. [58: 31, 31, 28, 21, 2, 2] Banzhaf showed that two of the six counties actually had no voting power that they were actually dummy voters. Final result: 1993 court decision abolishing weighted voting in New York States. Districts were created of roughly the same population and each given one voted. 2. United Nations Security Council: Banzhaf shows that a permanent member of the council holds more than 10 times the amount of power as one of the non-permanent members. There are 5 permanent members (Britain, China, France, Russia, US) and 10 non-permanent members. This voting arrangement may change as others are being considered for permanent membership. 3. European Union Banzhaf quantifies the amount of power each nation has and shows that smaller nations such as Luxembourg and Malta still hold some power.

9 Finite Math A Chapter 2, Weighted Voting Systems 9 2.4/2.5 The Shapley Shubik Power Index: The Shapley-Shubik Power Index: A player s power is proportional to the number of sequential-coalitions for which that player is pivotal. The more times a player is pivotal, the more power he holds. Sequential coalition: Banzhaf: { P1, P2, P3} Shapley-Shubik: P 1, P 3, P 2 These 3 players decide to vote together. These 3 players decide to vote together. They form a coalition. P1 votes 1 st, P3 votes 2 nd, P2 votes 3 rd. Order listed in the { } doesn t matter. They form a sequential coalition. Order listed in the is important. Pivotal player: Example: Find the Pivotal Player 1. Given the weighted voting system [5: 3,2,1,1} find the pivotal player for the given sequential coalition. a) [P 1,P 4,P 3,P 2] b) [P 3,P 1,P 2,P 4] c) [P 4,P 3,P 2,P 1]

10 Finite Math A Chapter 2, Weighted Voting Systems 10 Counting Sequential Coalitions: List the possible sequence for 3 players. How many are there? How many sequential coalitions are there for 4 players? For 5 players? Multiplication Rule: If there are m ways to do task 1 and n ways to do task 2, then there are mxn ways to do both tasks together. Shapley-Shubik Power Distribution Factorials: If N= the number of players, then the number of sequential coalitions is N! N! = N x (N-1) x... x 3 x 2 x 1 Step 1: Make a list of all sequential coalitions Step 2: For each sequential coalition, determine the pivotal player. Step 3: For each player, count the number of times they are pivotal and divide by the number of sequential coalitions. Calculate the %. Example 2: Find the Shapely Shubik Power Distribution for [4: 3, 2, 1] Sequential Coalitions: 3 Players [P1,P2,P3] [P1,P3,P2] [P2,P1,P3] [P2,P3,P1] [P3,P1,P2] [P3,P2,P1]

11 Finite Math A Chapter 2, Weighted Voting Systems 11 Example 3: Find the Shapley-Shubik Power Distribution for [6: 4, 3, 2, 1] Sequential Coalitions: 4 Players [P 1,P 2,P 3,P 4] [P 2,P 1,P 3,P 4] [P 3,P 1,P 2,P 4] [P 4,P 1,P 2,P 3] [P 1,P 2,P 4,P 3] [P 2,P 1,P 4,P 3] [P 3,P 1,P 4,P 2] [P 4,P 1,P 3,P 2] [P 1,P 3,P 2,P 4] [P 2,P 3,P 1,P 4] [P 3,P 2,P 1,P 4] [P 4,P 2,P 1,P 3] [P 1,P 3,P 4,P 2] [P 2,P 3,P 4,P 1] [P 3,P 2,P 4,P 1] [P 4,P 2,P 3,P 1] [P 1,P 4,P 2,P 3] [P 2,P 4,P 1,P 3] [P 3,P 4,P 1,P 2] [P 4,P 3,P 1,P 2] [P 1,P 4,P 3,P 2] [P 2,P 4,P 3,P 1] [P 3,P 4,P 2,P 1] [P 4,P 3,P 2,P 1] Example 4: Find the Shapley-Shubik Power Distribution for [10: 8, 4, 2, 1] Sequential Coalitions: 4 Players [P 1,P 2,P 3,P 4] [P 2,P 1,P 3,P 4] [P 3,P 1,P 2,P 4] [P 4,P 1,P 2,P 3] [P 1,P 2,P 4,P 3] [P 2,P 1,P 4,P 3] [P 3,P 1,P 4,P 2] [P 4,P 1,P 3,P 2] [P 1,P 3,P 2,P 4] [P 2,P 3,P 1,P 4] [P 3,P 2,P 1,P 4] [P 4,P 2,P 1,P 3] [P 1,P 3,P 4,P 2] [P 2,P 3,P 4,P 1] [P 3,P 2,P 4,P 1] [P 4,P 2,P 3,P 1] [P 1,P 4,P 2,P 3] [P 2,P 4,P 1,P 3] [P 3,P 4,P 1,P 2] [P 4,P 3,P 1,P 2] [P 1,P 4,P 3,P 2] [P 2,P 4,P 3,P 1] [P 3,P 4,P 2,P 1] [P 4,P 3,P 2,P 1]

This situation where each voter is not equal in the number of votes they control is called:

This situation where each voter is not equal in the number of votes they control is called: Finite Mathematics Notes Chapter 2: The Mathematics of Power (Weighted Voting) Academic Standards: PS.ED.2: Use election theory techniques to analyze election data. Use weighted voting techniques to decide

More information

The Mathematics of Power: Weighted Voting

The Mathematics of Power: Weighted Voting MATH 110 Week 2 Chapter 2 Worksheet The Mathematics of Power: Weighted Voting NAME The Electoral College offers a classic illustration of weighted voting. The Electoral College consists of 51 voters (the

More information

In this lecture, we will explore weighted voting systems further. Examples of shortcuts to determining winning coalitions and critical players.

In this lecture, we will explore weighted voting systems further. Examples of shortcuts to determining winning coalitions and critical players. In this lecture, we will explore weighted voting systems further. Examples of shortcuts to determining winning coalitions and critical players. Determining winning coalitions, critical players, and power

More information

Check off these skills when you feel that you have mastered them. Identify if a dictator exists in a given weighted voting system.

Check off these skills when you feel that you have mastered them. Identify if a dictator exists in a given weighted voting system. Chapter Objectives Check off these skills when you feel that you have mastered them. Interpret the symbolic notation for a weighted voting system by identifying the quota, number of voters, and the number

More information

Chapter 11. Weighted Voting Systems. For All Practical Purposes: Effective Teaching

Chapter 11. Weighted Voting Systems. For All Practical Purposes: Effective Teaching Chapter Weighted Voting Systems For All Practical Purposes: Effective Teaching In observing other faculty or TA s, if you discover a teaching technique that you feel was particularly effective, don t hesitate

More information

Lecture 7 A Special Class of TU games: Voting Games

Lecture 7 A Special Class of TU games: Voting Games Lecture 7 A Special Class of TU games: Voting Games The formation of coalitions is usual in parliaments or assemblies. It is therefore interesting to consider a particular class of coalitional games that

More information

2 The Mathematics of Power. 2.1 An Introduction to Weighted Voting 2.2 The Banzhaf Power Index. Topic 2 // Lesson 02

2 The Mathematics of Power. 2.1 An Introduction to Weighted Voting 2.2 The Banzhaf Power Index. Topic 2 // Lesson 02 2 The Mathematics of Power 2.1 An Introduction to Weighted Voting 2.2 The Banzhaf Power Index Topic 2 // Lesson 02 Excursions in Modern Mathematics, 7e: 2.2-2 Weighted Voting In weighted voting the player

More information

Homework 4 solutions

Homework 4 solutions Homework 4 solutions ASSIGNMENT: exercises 2, 3, 4, 8, and 17 in Chapter 2, (pp. 65 68). Solution to Exercise 2. A coalition that has exactly 12 votes is winning because it meets the quota. This coalition

More information

Math for Liberal Arts MAT 110: Chapter 12 Notes

Math for Liberal Arts MAT 110: Chapter 12 Notes Math for Liberal Arts MAT 110: Chapter 12 Notes Voting Methods David J. Gisch Voting: Does the Majority Always Rule? Choosing a Winner In elections with more then 2 candidates, there are several acceptable

More information

Warm-up Day 3 Given these preference schedules, identify the Plurality, Borda, Runoff, Sequential Runoff, and Condorcet winners.

Warm-up Day 3 Given these preference schedules, identify the Plurality, Borda, Runoff, Sequential Runoff, and Condorcet winners. Warm-up Day 3 Given these preference schedules, identify the Plurality, Borda, Runoff, Sequential Runoff, and Condorcet winners. Plurality: Borda: Runoff: Seq. Runoff: Condorcet: Warm-Up Continues -> Warm-up

More information

Fairness Criteria. Review: Election Methods

Fairness Criteria. Review: Election Methods Review: Election Methods Plurality method: the candidate with a plurality of votes wins. Plurality-with-elimination method (Instant runoff): Eliminate the candidate with the fewest first place votes. Keep

More information

12.3 Weighted Voting Systems

12.3 Weighted Voting Systems 12.3 Weighted Voting Systems There are different voting systems to the ones we've looked at. Instead of focusing on the candidates, let's focus on the voters. In a weighted voting system, the votes of

More information

A Mathematical View on Voting and Power

A Mathematical View on Voting and Power A Mathematical View on Voting and Power Werner Kirsch Abstract. In this article we describe some concepts, ideas and results from the mathematical theory of voting. We give a mathematical description of

More information

On Axiomatization of Power Index of Veto

On Axiomatization of Power Index of Veto On Axiomatization of Power Index of Veto Jacek Mercik Wroclaw University of Technology, Wroclaw, Poland jacek.mercik@pwr.wroc.pl Abstract. Relations between all constitutional and government organs must

More information

Kybernetika. František Turnovec Fair majorities in proportional voting. Terms of use: Persistent URL:

Kybernetika. František Turnovec Fair majorities in proportional voting. Terms of use: Persistent URL: Kybernetika František Turnovec Fair majorities in proportional voting Kybernetika, Vol. 49 (2013), No. 3, 498--505 Persistent URL: http://dml.cz/dmlcz/143361 Terms of use: Institute of Information Theory

More information

NOTES. Power Distribution in Four-Player Weighted Voting Systems

NOTES. Power Distribution in Four-Player Weighted Voting Systems NOTES Power Distribution in Four-Player Weighted Voting Systems JOHN TOLLE Carnegie Mellon University Pittsburgh, PA 15213-3890 tolle@qwes,math.cmu.edu The Hometown Muckraker is a small newspaper with

More information

Warm-up Day 3. Phones OFF and in pockets! 1) Given these preference schedules, identify the Condorcet, Runoff, and Sequential Runoff winners.

Warm-up Day 3. Phones OFF and in pockets! 1) Given these preference schedules, identify the Condorcet, Runoff, and Sequential Runoff winners. Warm-up Day 3 1) Given these preference schedules, identify the Condorcet, Runoff, and Sequential Runoff winners. Phones OFF and in pockets! Condorcet: Runoff: Seq. Runoff: 2) If each voter approves of

More information

MATH 1340 Mathematics & Politics

MATH 1340 Mathematics & Politics MATH 1340 Mathematics & Politics Lecture 1 June 22, 2015 Slides prepared by Iian Smythe for MATH 1340, Summer 2015, at Cornell University 1 Course Information Instructor: Iian Smythe ismythe@math.cornell.edu

More information

Weighted Voting. Lecture 12 Section 2.1. Robb T. Koether. Hampden-Sydney College. Fri, Sep 15, 2017

Weighted Voting. Lecture 12 Section 2.1. Robb T. Koether. Hampden-Sydney College. Fri, Sep 15, 2017 Weighted Voting Lecture 12 Section 2.1 Robb T. Koether Hampden-Sydney College Fri, Sep 15, 2017 Robb T. Koether (Hampden-Sydney College) Weighted Voting Fri, Sep 15, 2017 1 / 20 1 Introductory Example

More information

Thema Working Paper n Université de Cergy Pontoise, France

Thema Working Paper n Université de Cergy Pontoise, France Thema Working Paper n 2011-13 Université de Cergy Pontoise, France A comparison between the methods of apportionment using power indices: the case of the U.S. presidential elections Fabrice Barthelemy

More information

Louis M. Edwards Mathematics Super Bowl Valencia Community College -- April 30, 2004

Louis M. Edwards Mathematics Super Bowl Valencia Community College -- April 30, 2004 Practice Round 1. The overall average in an algebra class is described in the syllabus as a weighted average of homework, tests, and the final exam. The homework counts 10%, the three tests each count

More information

Weighted Voting. Lecture 13 Section 2.1. Robb T. Koether. Hampden-Sydney College. Mon, Feb 12, 2018

Weighted Voting. Lecture 13 Section 2.1. Robb T. Koether. Hampden-Sydney College. Mon, Feb 12, 2018 Weighted Voting Lecture 13 Section 2.1 Robb T. Koether Hampden-Sydney College Mon, Feb 12, 2018 Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, 2018 1 / 20 1 Introductory Example

More information

APPLICATION: PIVOTAL POLITICS

APPLICATION: PIVOTAL POLITICS APPLICATION: PIVOTAL POLITICS 1 A. Goals Pivotal Politics 1. Want to apply game theory to the legislative process to determine: 1. which outcomes are in SPE, and 2. which status quos would not change in

More information

BOOK REVIEW BY DAVID RAMSEY, UNIVERSITY OF LIMERICK, IRELAND

BOOK REVIEW BY DAVID RAMSEY, UNIVERSITY OF LIMERICK, IRELAND B A D A N I A O P E R A C Y J N E I D E C Y Z J E Nr 2 2008 BOOK REVIEW BY DAVID RAMSEY, UNIVERSITY OF LIMERICK, IRELAND Power, Freedom and Voting Essays in honour of Manfred J. Holler Edited by Matthew

More information

Power in Voting Games and Canadian Politics

Power in Voting Games and Canadian Politics Power in Voting Games and Canadian Politics Chris Nicola December 27, 2006 Abstract In this work we examine power measures used in the analysis of voting games to quantify power. We consider both weighted

More information

Math of Election APPORTIONMENT

Math of Election APPORTIONMENT Math of Election APPORTIONMENT Alfonso Gracia-Saz, Ari Nieh, Mira Bernstein Canada/USA Mathcamp 2017 Apportionment refers to any of the following, equivalent mathematical problems: We want to elect a Congress

More information

Seminar on Applications of Mathematics: Voting. EDB Hong Kong Science Museum,

Seminar on Applications of Mathematics: Voting. EDB Hong Kong Science Museum, Seminar on pplications of Mathematics: Voting ED Hong Kong Science Museum, 2-2-2009 Ng Tuen Wai, Department of Mathematics, HKU http://hkumath.hku.hk/~ntw/voting(ed2-2-2009).pdf Outline Examples of voting

More information

A comparison between the methods of apportionment using power indices: the case of the U.S. presidential election

A comparison between the methods of apportionment using power indices: the case of the U.S. presidential election A comparison between the methods of apportionment using power indices: the case of the U.S. presidential election Fabrice BARTHÉLÉMY and Mathieu MARTIN THEMA University of Cergy Pontoise 33 boulevard du

More information

A Geometric and Combinatorial Interpretation of Weighted Games

A Geometric and Combinatorial Interpretation of Weighted Games A Geometric and Combinatorial Interpretation of Weighted Games Sarah K. Mason and R. Jason Parsley Winston Salem, NC Clemson Mini-Conference on Discrete Mathematics and Algorithms 17 October 2014 Types

More information

Voting and Apportionment(Due with Final Exam)

Voting and Apportionment(Due with Final Exam) Voting and Apportionment(Due with Final Exam) The XYZ Takeaway W Affair. 1. Consider the following preference table for candidates x, y, z, and w. Number of votes 200 150 250 300 100 First choice z y x

More information

The Electoral College

The Electoral College The Electoral College What is the Electoral College Simple way of thinking about it: The States Elect the President.. Even though we can tally up a national popular vote, there are really 50 separate elections

More information

The Impact of Turkey s Membership on EU Voting. Richard Baldwin and Mika Widgrén. Abstract

The Impact of Turkey s Membership on EU Voting. Richard Baldwin and Mika Widgrén. Abstract Centre for European Policy Studies CEPS Policy Brief No. 62/February 2005 The Impact of Turkey s Membership on EU Voting Richard Baldwin and Mika Widgrén Abstract Thinking ahead for Europe This policy

More information

Mathematics of the Electoral College. Robbie Robinson Professor of Mathematics The George Washington University

Mathematics of the Electoral College. Robbie Robinson Professor of Mathematics The George Washington University Mathematics of the Electoral College Robbie Robinson Professor of Mathematics The George Washington University Overview Is the US President elected directly? No. The president is elected by electors who

More information

Coalitional Game Theory

Coalitional Game Theory Coalitional Game Theory Game Theory Algorithmic Game Theory 1 TOC Coalitional Games Fair Division and Shapley Value Stable Division and the Core Concept ε-core, Least core & Nucleolus Reading: Chapter

More information

A Theory of Spoils Systems. Roy Gardner. September 1985

A Theory of Spoils Systems. Roy Gardner. September 1985 A Theory of Spoils Systems Roy Gardner September 1985 Revised October 1986 A Theory of the Spoils System Roy Gardner ABSTRACT In a spoils system, it is axiomatic that "to the winners go the spoils." This

More information

The Election What is the function of the electoral college today? What are the flaws in the electoral college?

The Election What is the function of the electoral college today? What are the flaws in the electoral college? S E C T I O N 5 The Election What is the function of the electoral college today? What are the flaws in the electoral college? What are the advantages and disadvantages of proposed reforms in the electoral

More information

Three Branches, One Government

Three Branches, One Government Three Branches, One Government This game can be played by groups of two to three students or be used by individual students for practice and review. Purpose: to review the work of the executive, legislative,

More information

For the Encyclopedia of Power, ed. by Keith Dowding (SAGE Publications) Nicholas R. Miller 3/28/07. Voting Power in the U.S.

For the Encyclopedia of Power, ed. by Keith Dowding (SAGE Publications) Nicholas R. Miller 3/28/07. Voting Power in the U.S. For the Encyclopedia of Power, ed. by Keith Dowding (SAGE Publications) Nicholas R. Miller 3/28/07 Voting Power in the U.S. Electoral College The President of the United States is elected, not by a direct

More information

Who benefits from the US withdrawal of the Kyoto protocol?

Who benefits from the US withdrawal of the Kyoto protocol? Who benefits from the US withdrawal of the Kyoto protocol? Rahhal Lahrach CREM, University of Caen Jérôme Le Tensorer CREM, University of Caen Vincent Merlin CREM, University of Caen and CNRS 15th October

More information

An empirical comparison of the performance of classical power indices. Dennis Leech

An empirical comparison of the performance of classical power indices. Dennis Leech LSE Research Online Article (refereed) An empirical comparison of the performance of classical power indices Dennis Leech LSE has developed LSE Research Online so that users may access research output

More information

The Integer Arithmetic of Legislative Dynamics

The Integer Arithmetic of Legislative Dynamics The Integer Arithmetic of Legislative Dynamics Kenneth Benoit Trinity College Dublin Michael Laver New York University July 8, 2005 Abstract Every legislature may be defined by a finite integer partition

More information

due date: Monday, August 29 (first day of school) estimated time: 3-4 hours (for planning purposes only; work until you finish)

due date: Monday, August 29 (first day of school) estimated time: 3-4 hours (for planning purposes only; work until you finish) AP Government Summer Work 2016 due date: Monday, August 29 (first day of school) estimated time: 3-4 hours (for planning purposes only; work until you finish) Your assignment is to read the U. S. Constitution

More information

Turkey: Economic Reform and Accession to the European Union

Turkey: Economic Reform and Accession to the European Union Turkey: Economic Reform and Accession to the European Union Editors Bernard Hoekman and Sübidey Togan A copublication of the World Bank and the Centre for Economic Policy Research 2005 The International

More information

AP Comparative Government and Politics

AP Comparative Government and Politics 2018 AP Comparative Government and Politics Free-Response Questions College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central

More information

DE FACTO CONTROL: APPLYING GAME THEORY TO THE LAW ON CORPORATE NATIONALITY. By Russell Stanley Q. Geronimo *

DE FACTO CONTROL: APPLYING GAME THEORY TO THE LAW ON CORPORATE NATIONALITY. By Russell Stanley Q. Geronimo * INTRODUCTION DE FACTO CONTROL: APPLYING GAME THEORY TO THE LAW ON CORPORATE NATIONALITY By Russell Stanley Q. Geronimo * One unexamined assumption in foreign ownership regulation is the notion that majority

More information

A priori veto power of the president of Poland Jacek W. Mercik 12

A priori veto power of the president of Poland Jacek W. Mercik 12 A priori veto power of the president of Poland Jacek W. Mercik 12 Summary: the a priori power of the president of Poland, lower chamber of parliament (Sejm) and upper chamber of parliament (Senate) in

More information

Two-dimensional voting bodies: The case of European Parliament

Two-dimensional voting bodies: The case of European Parliament 1 Introduction Two-dimensional voting bodies: The case of European Parliament František Turnovec 1 Abstract. By a two-dimensional voting body we mean the following: the body is elected in several regional

More information

Notes for Session 7 Basic Voting Theory and Arrow s Theorem

Notes for Session 7 Basic Voting Theory and Arrow s Theorem Notes for Session 7 Basic Voting Theory and Arrow s Theorem We follow up the Impossibility (Session 6) of pooling expert probabilities, while preserving unanimities in both unconditional and conditional

More information

Lecture 8 A Special Class of TU games: Voting Games

Lecture 8 A Special Class of TU games: Voting Games Lecture 8 A Special Class of TU games: Voting Games The formation of coalitions is usual in parliaments or assemblies. It is therefore interesting to consider a particular class of coalitional games that

More information

Article I. Article III. Article IV. Article V. Article VI. Article VII

Article I. Article III. Article IV. Article V. Article VI. Article VII Directions: Read the U.S. Constitution and complete the following questions directly on this handout. Be sure to identify the location of each answer in the Constitution (example: Article I, Section 3,

More information

A New Method of the Single Transferable Vote and its Axiomatic Justification

A New Method of the Single Transferable Vote and its Axiomatic Justification A New Method of the Single Transferable Vote and its Axiomatic Justification Fuad Aleskerov ab Alexander Karpov a a National Research University Higher School of Economics 20 Myasnitskaya str., 101000

More information

POWER VOTING. Degree Thesis BY NIKÉ S. PANTA. BSc Mathematics Mathematical Analyst Specialisation. Supervisor:

POWER VOTING. Degree Thesis BY NIKÉ S. PANTA. BSc Mathematics Mathematical Analyst Specialisation. Supervisor: POWER VOTING Degree Thesis BY NIKÉ S. PANTA BSc Mathematics Mathematical Analyst Specialisation Supervisor: László Varga, assistant lecturer Department of Probability Theory and Statistics Eötvös Loránd

More information

Chapter 1 Practice Test Questions

Chapter 1 Practice Test Questions 0728 Finite Math Chapter 1 Practice Test Questions VOCABULARY. On the exam, be prepared to match the correct definition to the following terms: 1) Voting Elements: Single-choice ballot, preference ballot,

More information

Campaign Strategy Script

Campaign Strategy Script Campaign Strategy Script SHOT / TITLE DESCRIPTION 1. 00:00 Animated Open Animated Open 2. 00:07 Stacey on the street STACEY ON CAMERA: HI, I M STACEY DELIKAT. IN THE FINAL WEEKS LEADING UP TO THE ELECTIONS,

More information

AP Government and Politics THE US CONSTITUTION STUDY GUIDE Available at:

AP Government and Politics THE US CONSTITUTION STUDY GUIDE Available at: Name Class Period AP Government and Politics THE US CONSTITUTION STUDY GUIDE Available at: www.constitutioncenter.org PART I: THE OVERALL STRUCTURE OF THE CONSTITUTION A. Read each article of the Constitution.

More information

Class Period THE US CONSTITUTION. 2. Compare Article I with Article II. Which article is longer and more detailed? WHY do you suppose it s longer?

Class Period THE US CONSTITUTION. 2. Compare Article I with Article II. Which article is longer and more detailed? WHY do you suppose it s longer? Name Class Period AP GOVERNMENT there s a copy of the Constitution online at http://bit.ly/1j4mbqa or http://bit.ly/1dlarv1 THE US CONSTITUTION 1. Read each article of the Constitution. Summarize the general

More information

Volkswagen vs. Porsche. A Power-Index Analysis.

Volkswagen vs. Porsche. A Power-Index Analysis. Volkswagen vs. Porsche. A Power-Index Analysis. Roland Kirstein July 13, 2009 Abstract If Porsche had completed the takeover of Volkswagen, the superisory board of Porsche SE would have consisted of three

More information

One Man, Votes: A Mathematical Analysis of the Electoral College

One Man, Votes: A Mathematical Analysis of the Electoral College Volume 13 Issue 2 Article 3 1968 One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College John F. Banzhaf III Follow this and additional works at: http://digitalcommons.law.villanova.edu/vlr

More information

AP US Government & Politics Summer Assignment 2017

AP US Government & Politics Summer Assignment 2017 Name Date: AP US Government & Politics Summer Assignment 2017 This summer assignment will analyze the US Constitution which will prepare you for the first unit of study in the course. The first unit explores

More information

Chapter 9: Social Choice: The Impossible Dream

Chapter 9: Social Choice: The Impossible Dream Chapter 9: Social Choice: The Impossible Dream The application of mathematics to the study of human beings their behavior, values, interactions, conflicts, and methods of making decisions is generally

More information

Unit: The Legislative Branch

Unit: The Legislative Branch - two houses. Name: Date: Period: Unit: The Legislative Branch Part One: How Congress is Organized Gerrymandering- to a state into an odd-shaped district for reasons. - people in a representative s district.

More information

Allocation of Seats and Voting Power in the Norwegian Parliament

Allocation of Seats and Voting Power in the Norwegian Parliament Allocation of Seats and Voting Power in the Norwegian Parliament By Jon Kåsa Abstract: In recent years there seems to be a trend in Norwegian politics that larger parties are getting bigger while smaller

More information

Two-Tier Voting: Solving the Inverse Power Problem and Measuring Inequality

Two-Tier Voting: Solving the Inverse Power Problem and Measuring Inequality Two-Tier Voting: Solving the Inverse Power Problem and Measuring Inequality Matthias Weber Amsterdam School of Economics (CREED) and Tinbergen Institute February 19, 2015 Abstract There are many situations

More information

REPRESENTATIVE DEMOCRACY - HOW TO ACHIEVE IT

REPRESENTATIVE DEMOCRACY - HOW TO ACHIEVE IT - 30 - REPRESENTATIVE DEMOCRACY - HOW TO ACHIEVE IT Representative democracy implies, inter alia, that the representatives of the people represent or act as an embodiment of the democratic will. Under

More information

The actual midterm will probably not be multiple choice. You should also study your notes, the textbook, and the homework.

The actual midterm will probably not be multiple choice. You should also study your notes, the textbook, and the homework. Math 101 Practice First Midterm The actual midterm will probably not be multiple choice. You should also study your notes, the textbook, and the homework. Answers are on the last page. MULTIPLE CHOICE.

More information

This assignment must be completed in your own words. Copying or sharing answers is unacceptable and will face academic dishonesty consequences.

This assignment must be completed in your own words. Copying or sharing answers is unacceptable and will face academic dishonesty consequences. This assignment must be completed in your own words. Copying or sharing answers is unacceptable and will face academic dishonesty consequences. Directions: Read the U.S. Constitution and complete the following

More information

Annick Laruelle and Federico Valenciano: Voting and collective decision-making

Annick Laruelle and Federico Valenciano: Voting and collective decision-making Soc Choice Welf (2012) 38:161 179 DOI 10.1007/s00355-010-0484-3 REVIEW ESSAY Annick Laruelle and Federico Valenciano: Voting and collective decision-making Cambridge University Press, Cambridge, 2008 Ines

More information

Constitution Quest PART I - THE OVERALL STRUCTURE OF THE CONSTITUTION

Constitution Quest PART I - THE OVERALL STRUCTURE OF THE CONSTITUTION Constitution Quest Directions : Read the U.S. Constitution and complete the following questions directly on this handout legibly. This is due on the second week of class and you will be responsible for

More information

WARWICK ECONOMIC RESEARCH PAPERS

WARWICK ECONOMIC RESEARCH PAPERS Voting Power in the Governance of the International Monetary Fund Dennis Leech No 583 WARWICK ECONOMIC RESEARCH PAPERS DEPARTMENT OF ECONOMICS VOTING POWER IN THE GOVERNANCE OF THE INTERNATIONAL MONETARY

More information

Presidential Election Democrat Grover Cleveland versus Benjamin Harrison. ************************************ Difference of 100,456

Presidential Election Democrat Grover Cleveland versus Benjamin Harrison. ************************************ Difference of 100,456 Presidential Election 1886 Democrat Grover Cleveland versus Benjamin Harrison Cleveland 5,540,309 Harrison 5,439,853 ************************************ Difference of 100,456 Electoral College Cleveland

More information

THE PRO S AND CON S OF THE ELECTORAL COLLEGE SYSTEM

THE PRO S AND CON S OF THE ELECTORAL COLLEGE SYSTEM High School: U.S. Government Background Information THE PRO S AND CON S OF THE ELECTORAL COLLEGE SYSTEM There have, in its 200-year history, been a number of critics and proposed reforms to the Electoral

More information

A Simulative Approach for Evaluating Electoral Systems

A Simulative Approach for Evaluating Electoral Systems A Simulative Approach for Evaluating Electoral Systems 1 A Simulative Approach for Evaluating Electoral Systems Vito Fragnelli Università del Piemonte Orientale Dipartimento di Scienze e Tecnologie Avanzate

More information

Interdisciplinary Teaching Grant Proposal. Applicants:

Interdisciplinary Teaching Grant Proposal. Applicants: Interdisciplinary Teaching Grant Proposal Applicants: Core Faculty Professor Ron Cytron, Department of Computer Science, School of Engineering Professor Maggie Penn, Department of Political Science, College

More information

Standard Voting Power Indexes Do Not Work: An Empirical Analysis

Standard Voting Power Indexes Do Not Work: An Empirical Analysis B.J.Pol.S. 34, 657 674 Copyright 2004 Cambridge University Press DOI: 10.1017/S0007123404000237 Printed in the United Kingdom Standard Voting Power Indexes Do Not Work: An Empirical Analysis ANDREW GELMAN,

More information

Math 13 Liberal Arts Math HW7 Chapter Give an example of a weighted voting system that has a dummy voter but no dictator that is not [6:5,3,1].

Math 13 Liberal Arts Math HW7 Chapter Give an example of a weighted voting system that has a dummy voter but no dictator that is not [6:5,3,1]. Math 13 Liberal Arts Math HW7 Chapter 11 1. Give an example of a weighted voting system that has a dummy voter but no dictator that is not [6:5,3,1]. 2. Explain why the weighted voting system [13: 10,

More information

MATH 1340 Mathematics & Politics

MATH 1340 Mathematics & Politics MATH 1340 Mathematics & Politics Lecture 13 July 9, 2015 Slides prepared by Iian Smythe for MATH 1340, Summer 2015, at Cornell University 1 Apportionment A survey 2 All legislative Powers herein granted

More information

House Copy OLS Copy Public Copy For Official House Use BILL NO. Date of Intro. Ref.

House Copy OLS Copy Public Copy For Official House Use BILL NO. Date of Intro. Ref. 2/01/2019 RMK BPU# G:\CMUSGOV\N04\2019\LEGISLATION\N04_0011.DOCX SG 223 SR 281 TR 076 DR F CR 33 House Copy OLS Copy Public Copy For Official House Use BILL NO. Date of Intro. Ref. NOTE TO SPONSOR Notify

More information

POLITICAL LITERACY. Unit 1

POLITICAL LITERACY. Unit 1 POLITICAL LITERACY Unit 1 STATE, NATION, REGIME State = Country (must meet 4 criteria or conditions) Permanent population Defined territory Organized government Sovereignty ultimate political authority

More information

Compare the vote Level 3

Compare the vote Level 3 Compare the vote Level 3 Elections and voting Not all elections are the same. We use different voting systems to choose who will represent us in various parliaments and elected assemblies, in the UK and

More information

An Overview on Power Indices

An Overview on Power Indices An Overview on Power Indices Vito Fragnelli Università del Piemonte Orientale vito.fragnelli@uniupo.it Elche - 2 NOVEMBER 2015 An Overview on Power Indices 2 Summary The Setting The Basic Tools The Survey

More information

AP Gov - Plank Summer Assignment - The Constitution Name: Prd:

AP Gov - Plank Summer Assignment - The Constitution Name: Prd: AP Gov - Plank Summer Assignment - The Constitution Name: Prd: You do NOT need a textbook to complete this assignment. Use the attached PDF of the Constitution. In order to have the necessary background

More information

Compare the vote Level 1

Compare the vote Level 1 Compare the vote Level 1 Elections and voting Not all elections are the same. We use different voting systems to choose who will represent us in various parliaments and elected assemblies, in the UK and

More information

The United States Constitution & The Illinois Constitution. Study Guide

The United States Constitution & The Illinois Constitution. Study Guide The United States Constitution & The Illinois Constitution Study Guide Test Date: Thursday, October 7, 2010 www.studystack.com/menu-279563 Separation of Powers: Checks & Balances Executive Legislative

More information

MATH4999 Capstone Projects in Mathematics and Economics Topic 3 Voting methods and social choice theory

MATH4999 Capstone Projects in Mathematics and Economics Topic 3 Voting methods and social choice theory MATH4999 Capstone Projects in Mathematics and Economics Topic 3 Voting methods and social choice theory 3.1 Social choice procedures Plurality voting Borda count Elimination procedures Sequential pairwise

More information

Buying Supermajorities

Buying Supermajorities Presenter: Jordan Ou Tim Groseclose 1 James M. Snyder, Jr. 2 1 Ohio State University 2 Massachusetts Institute of Technology March 6, 2014 Introduction Introduction Motivation and Implication Critical

More information

EU Decision-making and the Allocation of Responsibility

EU Decision-making and the Allocation of Responsibility EU Decision-making and the Allocation of Responsibility Manfred J. Holler * First version: February 3, 2011 Revised: May 10, 2011 Second Revision: March 2012 Prepared for the Research Handbook on the Economics

More information

U.S. Constitution TEST. Notecards

U.S. Constitution TEST. Notecards U.S. Constitution TEST Notecards How many senators does each state have? Two What are the three branches of government? - Legislative Branch - Executive Branch - Judicial Branch Who is known as the Father

More information

Voting and Apportionment(Due by Nov. 25)

Voting and Apportionment(Due by Nov. 25) Voting and Apportionment(Due by Nov. 25) The XYZ Takeaway W Affair. 1. Consider the following preference table for candidates x, y, z, and w. Number of votes 200 150 250 300 100 First choice z y x w y

More information

YORKTOWN HIGH SCHOOL 5200 Yorktown Boulevard Arlington, Virginia June 7, Dear Future AP Government Student,

YORKTOWN HIGH SCHOOL 5200 Yorktown Boulevard Arlington, Virginia June 7, Dear Future AP Government Student, YORKTOWN HIGH SCHOOL 5200 Yorktown Boulevard Arlington, Virginia 22207 June 7, 2017 Dear Future AP Government Student, Greetings from Ms. Boudalis, Mr. Mandel, and Mr. Zito! In a few short months, one

More information

Why are there only two major parties in US? [party attachments below]

Why are there only two major parties in US? [party attachments below] Why are there only two major parties in US? [party attachments below] A. Institutional Constraints on 3 rd Parties 1. Election System Single-member districts (SMDs) Winner-take-all first-past-the-post

More information

THE SOUTH AUSTRALIAN LEGISLATIVE COUNCIL: POSSIBLE CHANGES TO ITS ELECTORAL SYSTEM

THE SOUTH AUSTRALIAN LEGISLATIVE COUNCIL: POSSIBLE CHANGES TO ITS ELECTORAL SYSTEM PARLIAMENTARY LIBRARY OF SOUTH AUSTRALIA THE SOUTH AUSTRALIAN LEGISLATIVE COUNCIL: POSSIBLE CHANGES TO ITS ELECTORAL SYSTEM BY JENNI NEWTON-FARRELLY INFORMATION PAPER 17 2000, Parliamentary Library of

More information

SHAPLEY VALUE 1. Sergiu Hart 2

SHAPLEY VALUE 1. Sergiu Hart 2 SHAPLEY VALUE 1 Sergiu Hart 2 Abstract: The Shapley value is an a priori evaluation of the prospects of a player in a multi-person game. Introduced by Lloyd S. Shapley in 1953, it has become a central

More information

2009BargagliottieMAA6up.pdf 1

2009BargagliottieMAA6up.pdf 1 A COURSE IN QUANTITATIVE AND POLITICAL LITERACY Kira Hamman Pennsylvania State University, Mont Alto Why? What possessed me to try this? What? What s the curriculum? Who? Should you do it, too? 2009BargagliottieMAA6up.pdf

More information

Fair Division in Theory and Practice

Fair Division in Theory and Practice Fair Division in Theory and Practice Ron Cytron (Computer Science) Maggie Penn (Political Science) Lecture 5b: Alternative Voting Systems 1 Increasing minority representation Public bodies (juries, legislatures,

More information

Nine of the 13 states had to approve the Constitution in. order for it to be the law of the land. This happened on June 21,

Nine of the 13 states had to approve the Constitution in. order for it to be the law of the land. This happened on June 21, Task 1: Read Nine of the 13 states had to approve the Constitution in order for it to be the law of the land. This happened on June 21, 1788 when New Hampshire ratified it. The government of the United

More information

IMF Governance and the Political Economy of a Consolidated European Seat

IMF Governance and the Political Economy of a Consolidated European Seat 10 IMF Governance and the Political Economy of a Consolidated European Seat LORENZO BINI SMAGHI During recent years, IMF governance has increasingly become a topic of public discussion. 1 Europe s position

More information

Voting power in the Electoral College: The noncompetitive states count, too

Voting power in the Electoral College: The noncompetitive states count, too MPRA Munich Personal RePEc Archive Voting power in the Electoral College: The noncompetitive states count, too Steven J Brams and D. Marc Kilgour New York University May 2014 Online at http://mpra.ub.uni-muenchen.de/56582/

More information

1 von :46

1 von :46 1 von 10 13.11.2012 09:46 1996-2005 Thomas Bräuninger and Thomas König Department of Politics and Management University of Konstanz, Germany Download IOP 2.0, click here Release 5/05 Download previous

More information

Part Seven: Public Policy

Part Seven: Public Policy Part Seven: Public Policy Justice is itself the great standing policy of civil society; and any eminent departure from it, under any circumstances, lies under the suspicion of being no policy at all. Edmund

More information

Article I: Sec 1: Sec 2: Sec 3: Sec 4: Sec 5: Sec 6: Sec 7: Sec 8: Sec 9: Sec. 10: Article II: Sec 1: Sec 2:

Article I: Sec 1: Sec 2: Sec 3: Sec 4: Sec 5: Sec 6: Sec 7: Sec 8: Sec 9: Sec. 10: Article II: Sec 1: Sec 2: THE US CONSTITUTION STUDY GUIDE Directions: Read the US Constitution and complete the following questions PART I: THE OVERALL STRUCTURE OF THE CONSTITUTION 1. Read each article of the Constitution. Summarize

More information