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
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
• 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
• 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.
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
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
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.