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Math Exam Math Exam Document Transcript

  • 2. In this situation Mr.X was smart to use the mean because it would show the average of the scores on the 8 test scores rather than just using the median which would be the middle test score number. The advantage of using the mean is because it’s a average score shows you how well the students are doing overall. However the disadvantage is a student did well on all 7 of the test but the last and final test he took he did poorly it would hinder his final score. The advantage of Mr. Y using the median would be that even if there was a bad test score it wouldn’t be factored into the average because using his method it just the middle number. The disadvantages is that it’s harder to tell how the student are doing in the class because its not the overall percentage of all the test combined. 3. I believe that Condorcet’s method would work the best because the winner is whoever would win against every other candidate in a one-on-one contest using majority rule. Majority rule just tells you who are favored the most so if you put each candidate head to head it’s a good way to eliminating all the ones that would lose and keeping only the winners to continue on in the next face off. The least effective method I believe is the plurality voting. In this system only first place votes are considered and not the second and third choices of the voters. This isn’t fair because a candidate with the most votes wins even though they may have several less then ½ the total votes that where cast. For example in the 2000 presidential election plurality voting failed to satisfy the condorect winner criterion. 5. The advantages of the Shapley-Shubik are that in a decision making body it is natural some voters will be on both extremes. And those who are less inclined to extreme positions will be the pivotal voters therefore they will have more power. Hence the advantage is that the normal voters (voters that on either extreme of the voting spectrum.) will have the pivotal vote and have the advantage of winning. The disadvantage is that the voters who are on the extreme of the voting spectrum will not have that much power in the voting system. (example on example page) 6. The advantages of the Banzhaf index is works on the model that all of the participants of the voting system operate unpredictably without consulting one other. The disadvantage is that if a voter changes from yes to no it would cause measures to fail. Also if there is a land slide no status is then assigned to anyone. (example on example page) 9. For two parties the adjusted winner procedure produces an allocation based on each players assignment of 100 points over the items to be divided that has the following properties. • The allocation is equitable: Both players receive the same number of points. • The allocation is envy-free: Neither player would be happier with what the other received. • The allocation is Pareto-optimal: No other allocation, arrived at by any means, can make one party better off without making the other party worse off.
  • Example Page 1. Shapley-Shubik and Banzhaf index examples. Varying Decision Rules in the Council of Ministers As the decision rules affect the distribution of a priori voting power, it is in the interests of existing and prospective members to discover which rules would maximize their influence in the Council of Ministers. We have above reported the 70 percent and simple majority rules. The differences between voting power distributions under these two rules seem marginal. To the extent that they differ, they would seem to suggest that the current 70 percent rule is slightly biased in favour of the ten-vote countries at the expense of the smaller ones. Figure 1 depicts the Shapley-Shubik index values - expressed in percentages of countries with ten votes (represented by the curve V10), five votes (V5), three votes (V3) and two votes (V2) respectively in the current Side 277 Fig. 2. The Banzhaf Index Values (in Percentages) of Countries with 2 Votes (V2), 3 Votes (V3), 5 Votes (V5) and 10 Votes (V10) in the Current 12-Member EC Council of Ministers as Functions of the Decision Rule.
  • Fig. 3. The Shapley-Shubik Index Values (in Percentages) of Countries with 3 Votes (V3), 4 Votes (V4), 5 Votes (V5) and 10 Votes (V10) in a Hypothetical 15-Member EC Council of Ministers as Functions of the Decision Rule. 12-member EC. These values have been calculated as functions of alternativedecisionrules are expressed as the number of votes that are needed to carry a motion. The corresponding Banzhaf index values are indicated in Figure 2. Figures 3 and 4 give the power index curves for the Side 278 Fig. 4. The Banzhaf Index Values (in Percentages) of Countries with 3 Votes (V3), 4 Votes (V4), 5 Votes (V5) and 10 Votes (V10) in a Hypothetical 15-Member EC Council of Ministers as Functions of the Decision Rule.
  • hypothetical EC in which Austria and Finland have 3 votes and Sweden 4 votes. Thus, the V3 and V4 curves represent the power index values of Austria, Finland and Sweden respectively. Figures 1 and 2 show that the variations in the power index values in the interval from simple majority to 70 percent rule are in general marginal. The ten-vote countries do well in terms of a priori voting power inasmuch as their share of voting power exceeds their relative share of votes in this interval. The voting power maximum, however, is well outside this interval. The Shapley-Shubik index reaches its maximum when the player in question a vetoer. This is not always the case with, the Banzhaf index although Figure 2 does not indicate this (Laakso 1978). In the 15-member EC the Banzhaf index values of the ten-vote countries diminish with an increase in the decision rule, whereas these values for smaller countries increase. In terms of the Shapley-Shubik index the variation in the 50-70 percent interval is very small. The ten vote countries maximize their Shapley-Shubik index value at the decision rule where no coalition that does not include all ten member countries is not winning. We see that the behaviour of the two power indices is markedly different in the 15-member EC. The difference, however, appears only in the range of very large qualified majorities.