\hline \end{array}\). If players one and two join together, they cant pass a motion without player three, so player three has veto power. 18 0 obj << \hline \text { Glen Cove } & 0 & 0 / 48=0 \% \\ /A << /S /GoTo /D (Navigation1) >> 30 0 obj << As you can see, computing the Shapley-Shubik power index by hand would be very difficult for voting systems that are not very small. For a proposal to be accepted, a majority of workers and a majority of managers must approve of it. There are two different methods. In the voting system [16: 7, 6, 3, 3, 2], are any players dictators? Every player has some power. Meets quota. q#`(? Find the Banzhaf power index. stream \end{array}\). The downtown business association is electing a new chairperson, and decides to use approval voting. 26 0 obj << \hline \(\begin{array}{|l|l|} endstream \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\} \hline P_{1} & 3 & 3 / 6=50 \% \\ Which logo wins under approval voting? endobj /D [9 0 R /XYZ 28.346 262.195 null] No one has veto power, since no player is in every winning coalition. /Filter /FlateDecode As you can see, computing the Shapley-Shubik power index by hand would be very difficult for voting systems that are not very small. \end{array}\). \left\{P_{1}, P_{2}, P_{4}, P_{5}\right\} \\ \left\{P_{1}, P_{2}, P_{4}\right\} \\ No player is a dictator, so well only consider two and three player coalitions. endobj v brakes for 650b conversion; nj marching band state championship; doctor handwriting translation app; football pools draws this weekend. For comparison, the Banzhaf power index for the same weighted voting system would be P1: 60%, P2: 20%, P3: 20%. sequential coalitions calculator. \(\begin{array}{l} This means player 5 is a dummy, as we noted earlier. \(\left\{P_{1}, P_{2}\right\}\) Total weight: 9. %%Zn .U?nuv%uglA))NN0+8FGRN.H_\S2t=?p=H6)dGpU'JyuJmJt'o9Q,I?W6Cendstream Which apportionment paradox does this illustrate? To decide on a new website design, the designer asks people to rank three designs that have been created (labeled A, B, and C). \(\begin{aligned} Apply Coombs method to the preference schedules from questions 5 and 6. 2 0 obj << Consider the weighted voting system [47: 10,9,9,5,4,4,3,2,2]. /Trans << /S /R >> Figure . In question 18, we showed that the outcome of Borda Count can be manipulated if a group of individuals change their vote. The notation for the weights is \(w_{1}, w_{2}, w_{3}, \dots, w_{N}\), where \(w_1\) is the weight of \(P_1\), \(w_2\) is the weight of \(P_2\), etc. The Pareto criterion is another fairness criterion that states: If every voter prefers choice A to choice B, then B should not be the winner. /ProcSet [ /PDF /Text ] Calculate the winner under these conditions. In fact, seven is one less than , 15 is one less than , and 31 is one less than . 31 0 obj << If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have? The value of the Electoral College (see previous problem for an overview) in modern elections is often debated. endstream \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{4}\right\} \quad \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{5}\right\}\\ Counting up how many times each player is critical. Assume there are 365 days in a year. In a corporation, the shareholders receive 1 vote for each share of stock they hold, which is usually based on the amount of money the invested in the company. [q?a)/`OhEA7V wCu'vi8}_|2DRM>EBk'?y`:B-_ N QB0)/%F['r/g}9AThuHo/$S9LoniA1=-a Thus, player four is a dummy. /Type /Page P_{3}=1 / 5=20 \% /Border[0 0 0]/H/N/C[.5 .5 .5] W gynecologist northwestern. In the coalition {P1, P2, P4}, every player is critical. When player one joins the coalition, the coalition is a losing coalition with only 12 votes. 3 0 obj The votes are: If there are 4 candidates, what is the smallest number of votes that a plurality candidate could have? Notice that player 5 has a power index of 0, indicating that there is no coalition in which they would be critical power and could influence the outcome. Most states give all their electoral votes to the candidate that wins a majority in their state, turning the Electoral College into a weighted voting system, in which the states are the players. There will be \(7!\) sequential coalitions. &\quad\quad >> endobj In the weighted voting system \([57: 23,21,16,12]\), are any of the players a dictator or a dummy or do any have veto power. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Conversion rates in this range will not be distinguishable from the baseline (one-sided test). Notice that 5! Find a weighted voting system to represent this situation. /D [9 0 R /XYZ 334.488 0 null] Revisiting the Scottish Parliament, with voting system \([65: 47, 46, 17, 16, 2]\), the winning coalitions are listed, with the critical players underlined. Notice that player three is a dummy using both indices. In the winning two-player coalitions, both players are critical since no player can meet quota alone. Players one and two can join together and pass any motion without player three, and player three doesnt have enough weight to join with either player one or player two to pass a motion. Consider the weighted voting system \([6: 4, 3, 2]\). %PDF-1.4 The way to denote a weighted voting system is \(\left[q: w_{1}, w_{2}, w_{3}, \dots, w_{N}\right]\). The individual ballots are shown below. /D [24 0 R /XYZ 334.488 0 null] Reapportion the previous problem if the store has 25 salespeople. In some many states, where voters must declare a party to vote in the primary election, and they are only able to choose between candidates for their declared party. \hline \text { Hempstead #1 } & 31 \\ Show that when there is a Condorcet winner in an election, it is impossible for a single voter to manipulate the vote to help a different candidate become a Condorcet winner. In this case, player 1 is said to have veto power. 3 0 obj /Parent 20 0 R /Font << /F43 15 0 R /F16 16 0 R /F20 17 0 R >> /Contents 25 0 R Why? \(\left\{P_{1}, P_{2}\right\}\) Total weight: 9. << /S /GoTo /D [9 0 R /Fit ] >> Underlining the critical players to make it easier to count: \(\left\{\underline{P}_{1}, \underline{P}_{2}\right\}\), \(\left\{\underline{P}_{1}, \underline{P}_{3}\right\}\). /Annots [ 22 0 R ] xYMo8W(oRY, In the coalition {P1,P2,P4} which players are critical? 12 0 obj << The weighted voting system that Americans are most familiar with is the Electoral College system used to elect the President. A player will be a dictator if their weight is equal to or greater than the quota. If Player 1 is the only player with veto power, there are no dictators, and there are no dummies: Find the Shapley-Shubik power distribution. /Length 685 Notice that a player with veto power will be critical in every winning coalition, since removing their support would prevent a proposal from passing. This calculation is called a factorial, and is notated \(N!\) The number of sequential coalitions with \(N\) players is \(N!\). Shapely-Shubik takes a different approach to calculating the power. Does not meet quota. In this form, \(q\) is the quota, \(w_1\)is the weight for player 1, and so on. Calculate the Shapley-Shubik Power Index. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. We will list all the sequential coalitions and identify the pivotal player. Underlining the critical players to make it easier to count: \(\left\{\underline{P}_{1}, \underline{P}_{2}\right\}\), \(\left\{\underline{P}_{1}, \underline{P}_{3}\right\}\). Which candidate wins under approval voting? \(7 !=7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=5040\). xWM0+|Lf3*ZD{@{Y@V1NX` -m$clbX$d39$B1n8 CNG[_R$[-0.;h:Y & `kOT_Vj157G#yFmD1PWjFP[O)$=T,)Ll-.G8]GQ>]w{;/4:xtXw5%9V'%RQE,t2gDA _M+F)u&rSru*h&E+}x!(H!N8o [M`6A2. How many winning coalitions will there be? Please enter voting weights, with their multiplicities. = 6, the Shapley-Shubik Power Index of A is 4/6 = 2/3. However they cannot reach quota with player 5s support alone, so player 5 has no influence on the outcome and is a dummy. When there are five players, there are 31 coalitions (there are too many to list, so take my word for it). /Trans << /S /R >> Summarize the comparisons, and form your own opinion about whether either method should be adopted. We will list all the sequential coalitions and identify the pivotal player. The total weight is . If the quota was set at only 3, then player 1 could vote yes, players 2 and 3 could vote no, and both would reach quota, which doesnt lead to a decision being made. \left\{\underline{P}_{1,} \underline{P}_{2}\right\} \\ The total weight is . In the election shown below under the Plurality method, explain why voters in the third column might be inclined to vote insincerely. [p& _s(vyX6 @C}y%W/Y)kV2nRB0h!8'{;1~v Describe how Plurality, Instant Runoff Voting, Borda Count, and Copelands Method could be extended to produce a ranked list of candidates. The quota is 16 in this example. is the factorial button. /Type /Annot 13 0 obj << /Length 756 Rework problems 1-8 using Adams method. College Mathematics for Everyday Life (Inigo et al. /Parent 25 0 R We are currently enrolling students for on-campus classes and scheduling in-person campus tours. /Type /Annot Now we have the concepts for calculating the Shapely-Shubik power index. /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R Shapley-Shubik Power Index. In the winning two-player coalitions, both players are critical since no player can meet quota alone. Adamss method is similar to Jeffersons method, but rounds quotas up rather than down. If the sum is the quota or more, then the coalition is a winning coalition. In this system, all of the players must vote in favor of a motion in order for the motion to pass. >> Are any dummies? Consider the running totals as each player joins: \(\begin{array}{lll}P_{3} & \text { Total weight: } 3 & \text { Not winning } \\ P_{3}, P_{2} & \text { Total weight: } 3+4=7 & \text { Not winning } \\ P_{3}, P_{2}, P_{4} & \text { Total weight: } 3+4+2=9 & \text { Winning } \\ R_{2}, P_{3}, P_{4}, P_{1} & \text { Total weight: } 3+4+2+6=15 & \text { Winning }\end{array}\). The Shapley-Shubik power index of player P i is the fraction i = SS i total number of sequential coalitions. If for some reason the election had to be held again and many people who had voted for C switched their preferences to favor A, which caused B to become the winner, which is the primary fairness criterion violated in this election? Likewise, without player 2, the rest of the players weights add to 15, which doesnt reach quota, so player 2 also has veto power. \hline P_{2} \text { (Labour Party) } & 7 & 7 / 27=25.9 \% \\ /Contents 25 0 R 14 0 obj << Half of 18 is 9, so the quota must be . There is a motion to decide where best to invest their savings. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. /Length 786 Suppose that you have a supercomputer that can list one trillion (10^12) sequential coalitions per second. If the legislature grows to 11 seats, use Hamiltons method to apportion the seats. In weighted voting, we are most often interested in the power each voter has in influencing the outcome. Describe how an alternative voting method could have avoided this issue. >> endobj Weighted voting is applicable in corporate settings, as well as decision making in parliamentary governments and voting in the United Nations Security Council. Here there are 6 total votes. /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R \(\mathrm{P}_{1}\) is pivotal 3 times, \(\mathrm{P}_{2}\) is pivotal 3 times, and \(\mathrm{P}_{3}\) is pivotal 0 times. Lowndes felt that small states deserved additional seats more than larger states. sequential coalitions calculatorlittles shoes pittsburgh. Three people invest in a treasure dive, each investing the amount listed below. /Rect [188.925 2.086 190.918 4.078] \left\{\underline{P}_{1,} \underline{P}_{2}, P_{3}\right\} \quad \left\{\underline{P}_{1}, \underline{P}_{2}, P_{4}\right\} \\ Four options have been proposed. What is the total number (weight) of votes? The planning committee for a renewable energy trade show is trying to decide what city to hold their next show in. Instant Runoff Voting and Approval voting have supporters advocating that they be adopted in the United States and elsewhere to decide elections. This coalition has a combined weight of 7+6+3 = 16, which meets quota, so this would be a winning coalition. Research the Schulze method, another Condorcet method that is used by the Wikimedia foundation that runs Wikipedia, and give some examples of how it works. The dictator can also block any proposal from passing; the other players cannot reach quota without the dictator. So we look at each possible combination of players and identify the winning ones: \(\begin{array} {ll} {\{\mathrm{P} 1, \mathrm{P} 2\}(\text { weight }: 37)} & {\{\mathrm{P} 1, \mathrm{P} 3\} \text { (weight: } 36)} \\ {\{\mathrm{P} 1, \mathrm{P} 2, \mathrm{P} 3\} \text { (weight: } 53)} & {\{\mathrm{P} 1, \mathrm{P} 2, \mathrm{P} 4\} \text { (weight: } 40)} \\ {\{\mathrm{P} 1, \mathrm{P} 3, \mathrm{P} 4\} \text { (weight: } 39)} & {\{\mathrm{P} 1, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\} \text { (weight: } 56)} \\ {\{\mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}(\text { weight: } 36)} \end{array}\). Counting up times that each player is critical: Divide each players count by 16 to convert to fractions or percents: The Banzhaf power index measures a players ability to influence the outcome of the vote. Find the Banzhaf power index for the voting system [8: 6, 3, 2]. Since the quota is 8, and 8 is not more than 9, this system is not valid. This is quite large, so most calculations using the Shapely-Shubik power index are done with a computer. The preference schedule for the election is: The homeowners association is deciding a new set of neighborhood standards for architecture, yard maintenance, etc. The power index is a numerical way of looking at power in a weighted voting situation. ,*lkusJIgeYFJ9b%P= Find the Banzhaf power index for each player. Consider the weighted voting system [17: 13, 9, 5, 2], What is the weight of the coalition {P1,P2,P3}. [p& _s(vyX6 @C}y%W/Y)kV2nRB0h!8'{;1~v /ProcSet [ /PDF /Text ] There are many Condorcet Methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist. Consider the voting system [10: 11, 3, 2]. Translated into a weighted voting system, assuming a simple majority is needed for a proposal to pass: Listing the winning coalitions and marking critical players: There are a lot of them! Compare and contrast the motives of the insincere voters in the two questions above. Use a calculator to compute each of the following. endobj Reapportion the previous problem if 37 gold coins are recovered. Calculate the Banzhaf power distribution for this situation. powerpanel personal unable to establish communication with ups. /Length 1368 G'Y%2G^8G L\TBej#%)^F5_99vrAFlv-1Qlt/%bZpf{+OG'n'{Z| Instead of just looking at which players can form coalitions, Shapely-Shubik decided that all players form a coalition together, but the order that players join a coalition is important. /Font << /F43 15 0 R /F20 17 0 R /F16 16 0 R /F22 26 0 R /F32 27 0 R /F40 28 0 R /F21 29 0 R >> Thus, the total number of times any player is critical is T = 26. It is not necessary to put numbers in all of the boxes, but you should fill them in order, starting at the upper left and moving toward the lower right. 19 0 obj << xWKo8W(7 >E)@/Y@`1[=0\/gH*$]|?r>;TJDP-%.-?J&,8 Can we come up with a mathematical formula for the number of sequential coalitions? \hline If \(P_1\) were to leave, the remaining players could not reach quota, so \(P_1\) is critical. jD9{34'(KBm:/6oieroR'Y G`"XJA7VPY1mx=Pl('/ $4,qNfYzJh~=]+}AFs7>~U j[J*T)GL|n9bwZLPv]{6u+o/GUSmR4Hprx}}+;w!X=#C9U:1*3R!b;/|1-+w~ty7E #*tKr{l|C .E1}q'&u>~]lq`]L}|>g_fqendstream \hline P_{4} \text { (Liberal Democrats Party) } & 3 & 3 / 27=11.1 \% \\ If there is such a player or players, they are known as the critical player(s) in that coalition. /Length 786 The following year, the district expands to include a third school, serving 2989 students. We will have 3! Estimate how long in years it would take the computer list all sequential coalitions of 21 players. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. /Subtype /Link Find the Banzhaf power index for the weighted voting system [36: 20, 17, 16, 3]. Does it seem like an individual state has more power in the Electoral College under the vote distribution from part c or from part d? This page titled 3.5: Calculating Power- Shapley-Shubik Power Index is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. P_{4}=2 / 16=1 / 8=12.5 \% Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. A player who has no power is called a dummy. Each column shows the number of voters with the particular approval vote. par . 31 0 obj << Player three joining doesnt change the coalitions winning status so it is irrelevant. \hline \text { Hempstead #2 } & 16 & 16 / 48=1 / 3=33 \% \\ Notice that player 5 has a power index of 0, indicating that there is no coalition in which they would be critical power and could influence the outcome. Since the quota is nine, this player can pass any motion it wants to. Explain how other voters might perceive candidate C. Using the preference schedule below, apply Sequential Pairwise voting to determine the winner, using the agenda: A, B, C, D. Show that Sequential Pairwise voting can violate the Pareto criterion. \(\left\{P_{2}, P_{3}\right\}\) Total weight: 5. In every sequential coalition, there is a pivotal player who, when he joins, contributes the votes that turn what was a losing coalition into a winning coalition. sequential coalitions calculator how did lesley sharp lose weight julho 1, 2022. jack the ripper documentary bbc A small country consists of six states, whose populations are listed below. /ProcSet [ /PDF /Text ] In the coalition {P1, P2, P3, P4, P5}, only players 1 and 2 are critical; any other player could leave the coalition and it would still meet quota. Banzhaf used this index to argue that the weighted voting system used in the Nassau County Board of Supervisors in New York was unfair. Half of 15 is 7.5, so the quota must be . /Border[0 0 0]/H/N/C[.5 .5 .5] /D [9 0 R /XYZ 334.488 0 null] >> endobj /Subtype /Link The first thing to do is list all of the sequential coalitions, and then determine the pivotal player in each sequential coalition. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. /D [24 0 R /XYZ 334.488 0 null] Does this voting system having a Condorcet Candidate? How could it affect the outcome of the election? Try it Now 3 Find the Banzhaf power index for the weighted voting system \(\bf{[36: 20, 17, 16, 3]}\). In the sequential coalition which player is pivotal? The quota is 8 in this example. Estimate (in years) how long it would take the computer to list all the sequential coalitions of 25 players. As Im sure you can imagine, there are billions of possible winning coalitions, so the power index for the Electoral College has to be computed by a computer using approximation techniques. is the number of sequential coalitions. \left\{\underline{P}_{1}, \underline{P}_{2}, P_{3}\right\} & \left\{\underline{P}_{1}, \underline{P}_{2}, P_{4}\right\} \\ \left\{\underline{P}_{1}, \underline{P}_{2}, P_{5}\right\} & \left\{\underline{P}_{1}, \underline{P}_{3}, \underline{P}_{4}\right\} \\ \left\{\underline{P}_{1}, \underline{P}_{3}, \underline{P}_{5}\right\} & \left\{\underline{P}_1, \underline{P}_{4}, \underline{P}_{5}\right\} \\ \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{4}\right\} & \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{5}\right\}\\ \left\{P_{1}, P_{2}, P_{3}, P_{4}\right\} & \left\{P_{1}, P_{2}, P_{3}, P_{5}\right\} \\ \left\{\underline{P}_{1}, P_{2}, P_{4}, P_{5}\right\} & \left\{\underline{P}_{1}, P_{3}, P_{4}, P_{5}\right\} \\ \left\{\underline{P}_{2}, \underline{P}_{3}, P_{4}, P_{5}\right\} & \\ \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\} & \end{array}\), \(\begin{array}{|l|l|l|} P_{2}=6 / 16=3 / 8=37.5 \% \\ A coalition is any group of players voting the same way. \hline Consider the voting system \([q: 3, 2, 1]\). In the Electoral College, states are given a number of votes equal to the number of their congressional representatives (house + senate). The voting system tells us that the quota is 36, that Player 1 has 20 votes (or equivalently, has a weight of 20), Player 2 has 17 votes, Player 3 has 16 votes, and Player 4 has 3 votes. Shapely-Shubik power index for P1 = 0.5 = 50%, Shapely-Shubik power index for P2 = 0.5 = 50%. So it appears that the number of coalitions for N players is . Another example is in how the President of the United States is elected. Combining these possibilities, the total number of coalitions would be:\[N(N-1)(N-2)(3-N) \ldots(3)(2)(1)\nonumber \]This calculation is called a factorial, and is notated \(N !\) The number of sequential coalitions with \(N\) players is \(N !\). \(\begin{array}{|l|l|l|} So T = 4, B1 = 2, B2 = 2, and B3 = 0. \(\left\{P_{1}, P_{2}, P_{3}\right\}\) Total weight: 11. next to your five on the home screen. In this situation, one voter may control the equivalent of 100 votes where other voters only control 15 or 10 or fewer votes. We now need to consider the order in which players join the coalition. We start by listing all winning coalitions. The first thing to do is list all of the coalitions and determine which ones are winning and which ones are losing. How many sequential coalitions are there for N players? In order to have a meaningful weighted voting system, it is necessary to put some limits on the quota. One of the sequential coalitions is which means that P1 joins the coalition first, followed by P2 joining the coalition, and finally, P3 joins the coalition. /Trans << /S /R >> Lets examine these for some concepts. \hline \textbf { Player } & \textbf { Times pivotal } & \textbf { Power index } \\ 8 0 obj \(\mathrm{P}_{1}\) is pivotal 4 times, \(\mathrm{P}_{2}\) is pivotal 1 time, and \(\mathrm{P}_{3}\) is pivotal 1 time. How many votes are needed for a majority? /Filter /FlateDecode Calculate the power index for each district. stream /Rect [188.925 2.086 190.918 4.078] 11 0 obj << Research the history behind the Electoral College to explore why the system was introduced instead of using a popular vote. The winning coalitions are listed below, with the critical players underlined. Create a method for apportioning that incorporates this additional freedom, and describe why you feel it is the best approach. /ProcSet [ /PDF /Text ] Consider the weighted voting system [31: 10,10,8,7,6,4,1,1], Consider the weighted voting system [q: 7,5,3,1,1]. If you arent sure how to do this, you can list all coalitions, then eliminate the non-winning coalitions. In Example \(\PageIndex{2}\), some of the weighted voting systems are valid systems. This means player 5 is a dummy, as we noted earlier. Question: How many conversions are needed for a sequential A/B test? On a colleges basketball team, the decision of whether a student is allowed to play is made by four people: the head coach and the three assistant coaches. In particular, if a proposal is introduced, the player that joins the coalition and allows it to reach quota might be considered the most essential. If the legislature has 200 seats, apportion the seats. K\4^q@4rC]-OQAjp_&.m5|Yh&U8 @u~{AsGx!7pmEy1p[dzXJB$_U$NWN_ak:lBpO(tq@!+@S ?_r5zN\qb >p Ua For example, the sequential coalition. They are trying to decide whether to open a new location. /Resources 1 0 R stream Copelands Method is designed to identify a Condorcet Candidate if there is one, and is considered a Condorcet Method. Does this illustrate any apportionment issues? q#`(? The Banzhaf power index is one measure of the power of the players in a weighted voting system. The sequential coalitions for three players (P1, P2, P3) are: . A school district has two high schools: Lowell, serving 1715 students, and Fairview, serving 7364. The Ultimatum Game is a famous asymmetric, sequential two-player game intensely studied in Game Theory. Which other method are the results most similar to? If so, find it. Welcome to Set'Em Free Bail Bonds +1 214-752-4000 info@setemfreedallas.com In each sequential coalition, determine the pivotal player 3. Count Data. \left\{P_{1}, P_{2}, P_{3}, P_{5}\right\} \\ >> endobj Since the quota is 9, and 9 is between 7.5 and 15, this system is valid. Compare and contrast this primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc. The votes are shown below. Reapportion the previous problem if the college can hire 20 tutors. In the system , player three has a weight of two. would mean that P2 joined the coalition first, then P1, and finally P3. The quota is 8 in this example. 9 0 obj << In the methods discussed in the text, it was assumed that the number of seats being apportioned was fixed. \mathrm{GC}\}} & {\{\mathrm{H} 1, \mathrm{H} 2, \mathrm{OB}, \mathrm{LB}\}} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}, \mathrm{NH}, \mathrm{GC}\}} & {\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}, \mathrm{NH}\}} & {\{\mathrm{H} 1, \mathrm{H} 2, \mathrm{OB}, \mathrm{GC}\}} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}, \mathrm{LB}, \mathrm{GC}\}} & {\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}, \mathrm{LB}\}} & {\{\mathrm{H} 1, \mathrm{H} 2, \mathrm{OB}, \mathrm{NH}, \mathrm{LB}\}} \\{\{\underline{\mathrm{H} 1}, \underline{\mathrm{H} 2}, \mathrm{NH}, \mathrm{LB}, \mathrm{GC}\}} & {\{\underline{\mathrm{H} 2}, \underline{\mathrm{OB}}, \mathrm{GC}\}} & {\{\mathrm{H} 1, \mathrm{H} 2, \mathrm{OB}, \mathrm{NH}, \mathrm{GC}\}} \\ {} & {} & {\{\mathrm{H} 1, \mathrm{H} 2, \mathrm{OB}, \mathrm{NH}, \mathrm{LB}, \mathrm{GC}\}}\end{array}\). /Annots [ 11 0 R ] Find the winner under the Borda Count Method. \hline P_{1} \text { (Scottish National Party) } & 9 & 9 / 27=33.3 \% \\ \(\left\{P_{1}, P_{2}, P_{3}\right\} \)Total weight: 11. Any winning coalition requires two of the larger districts. Then player three joins but the coalition is still a losing coalition with only 15 votes. The weighted voting system that Americans are most familiar with is the Electoral College system used to elect the President. 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