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# Shapley-Shubik Distribution, Weighted Voting Systems and Combinations

1) Find the Shapley- Shubik distribution (in percent) for the system (7: 5,3,2,1,).

2) In a cafeteria every meal has a 5 choice main course, a 3 choice salad, and a 2 choice soup, and a 4 choice dessert . If some one ate a different complete meal every day, how many days could he eat before he would start to repeat a meal?

3) How many tickets for president / vice president/ secretary / and treasurer can be made from the numbers of 11 students? (None can hold more than one office.)

3b) How many different 5 member scout divisions can be made from a 20 man military division?

4) For each weighted voting system find all dictators, veto power players, and dummies ( 13: 8,3,3,2,1,)

5) Solve: 2( 3y+4)-5_y-2)=3y-6(y-3)

#### Solution Preview

1) FIND THE SHAPLEY-SHUBIK DISTRIBUTION (IN PERCENTS) FOR THE SYSTEM ( 7: 5,3,2,1)

What are the sequential coalitions? With four players, there are 4! coalitions = 24 coalitions. To show you how this is done in long form, I'll list all the sequential coalitions and then comment on who is the pivotal voter in each one. That means, if the players vote in the order listed, which player's vote is pivotal for causing a winning coalition?

5,3,2,1 (3 is pivotal)
5,3,1,2 (3 is pivotal)
5,2,3,1 (2 is pivotal)
5,2,1,3 (2 is pivotal)
5,1,3,2 (3 is pivotal)
5,1,2,3 (2 is pivotal)

3,2,5,1 (5 is pivotal)
3,2,1,5 (5 is pivotal)
3,5,2,1 (5 is pivotal)
3,5,1,2 (5 is pivotal)
3,1,2,5 (5 is pivotal)
3,1,5,2 (5 is pivotal)

2,3,1,5 (5 is pivotal)
2,3,5,1 (5 is pivotal)
2,5,3,1 (5 is pivotal)
2,5,1,3 (5 is pivotal)
2,1,5,3 (5 is pivotal)
2,1,3,5 (5 is pivotal)

1,2,3,5 (5 is pivotal)
1,2,5,3 (5 is pivotal)
1,3,2,5 (5 is pivotal)
1,3,5,2 (5 is pivotal)
1,5,3,2 (3 is pivotal)
1,5,2,3 (2 is pivotal)

Now, let's add them up. In how many sequential coalitions is the player with 5 votes pivotal? 16. Therefore, his is 16/24 = 66.67% approximately.

In how many coalitions is the player with 3 votes pivotal? 4. Therefore, his is 4/24 = 16.67%

In how many coalitions is the player with 2 votes pivotal? 4. Therefore, his is 4/24 = 16.67%

In how many coalitions is the player with 1 vote pivotal? None. Therefore, his is 0/24 = 0%

2) IN A CAFETERIA EVERY MEAL HAS A 5 CHOICE MAIN COURSE, A 3 CHOICE SALAD , AND A 2 CHOICE SOUP, AND A 4 CHOICE DESSERT . IF SOMEONE ATE A DIFFERENT COMPLETE MEAL EVERY DAY, HOW MANY ...

#### Solution Summary

Shapley-Shubik Distribution, Weighted Voting Systems and Combinations are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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