**Simple Practice**

Prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, NY 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk/__

1. i. Use Adams, Jefferson, and Webster to apportion a legislature of 20 seats. Each of four regions is entitled to a minimum of at least one seat. The populations of these regions is shown below. The apportionment should be done "twice:"

a. Using a divisor and rounding (state what divisor you used so that when you rounded as the method requires that 20 seats were distributed). (If a tie occurs when 20 seats are given out, say so.)

b. Using the table method check if you get the same answer as you did for above for Adams, Jefferson, and Webster. For each of these methods indicate the order in which the seats are given out via the table method; when there is a tie, indicate this by having the states share these seats. Thus, here for example, seats 1-4 will be shared.

A | B | C | D | |

Population | 12300 | 8500 | 7000 | 2200 |

ii. Use Hamilton's method to apportion 20 seats using the data above.

2. Given the weighted voting game: [14; 9, 7, 6, 5]

a. List the minimal winning coalitions

b. Determine the Shapley power of each of the players.

c. Determine the Banzhaf power of each of the players.

3. Given the election below:

i. Compute the pairwise vote matrix for the election above.

ii. Decide the winner (if there is one) of the election using:

a. Plurality

b. Run-off

c. Sequential Run-off

d. Borda Count

e. Condorcet

f. Sequential run-off based on the Borda Count (Baldwin)

g. Coombs

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