Homework 5: Game Theory (2017)

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

1. Determine the i. Coleman power, ii. Banzhaf power, iii. Shapley-Shubik power and iv. Deegan -Packel for the weighted voting game below, whose players are 1, 2, 3, 4 and where the quota is 15.

[15; 9, 7, 6, 4]

The weights are listed in the order of the players' names. State what the winning coalitions and minimal winning coalitions for this game are.

2. The tables below give strict preferences (without ties) for five men and five women. Men ranking women first, and women ranking men second. Thus, m1 ranks w5 as th best while w3 ranks m1 second best.

a. Determine the stable matching which is male optimal.

Determine for each man, the rank the woman he is paired with and for each woman the rank of the man she is paired with.

b. Determine the stable matching which is female optimal.

Determine for each man, the rank the woman he is paired with and for each woman the rank of the man she is paired with.

Note: In the tables below the columns are headed by RANKS rather than by the names of those being ranked.

Boys rank girls

1st 2nd 3rd 4th 5th

m1  w2 w3 w1 w5 w4
m2  w2 w1 w3 w5 w4
m3  w2 w1 w5 w3 w4
m4  w5 w3 w4 w2 w1
m5  w3 w4 w5 w2 w1

girls rank boys

1st 2nd 3rd 4th 5th

w1  m2 m1 m3 m5 m4
w2  m2 m3 m1 m5 m4
w3  m5 m1 m3 m2 m4
w4  m5 m2 m1 m4 m3
w5  m2 m4 m5 m1 m3