Mathematical Modeling: Table of Weighted Voting Games with Four or Fewer Players


prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

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


Here is a table of the weighted voting games with four or fewer players which do not have dummy players or multiple "dictators."

What is shown are minimal winning coalitions, weighted voting representation (remember, there are many other such representations) and Shapley-Shubik Power vectors. You may wish as an exercise to find the Banzhaf Power vectors for the same games.

One player:

MW: {1}; [1; 1]; Shapley-Shubik: (1)

Two players:

MW: {1, 2}; [2; 1, 1]; Shapley-Shubik: (1/2, 1/2)

Three players:

NW: {1, 2, 3 }; Shapley-Shubik: (1/3, 1/3, 1/3)

MW: {1,2}, {1, 3}, {2,3}; [2; 1, 1, 1]; Shapley -Shubik: (1/3, 1/3, 1/3)

MW: {1,2}, {1,3}; [3; 2, 1, 1,]; Shapley-Shubik; (4/6, 1/6,1/6)

Four Players

MW: {1, 2, 3, 4}; [4; 1, 1, 1, 1]; Shapley-Shubik: (1/4, 1/4, 1/4, 1/4)

MW: {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}; [3; 1, 1, 1, 1]; Shapley-Shubik: (1/4,1/4,1/4,1/4)

MW: {1,2}, {1,3}, {1,4}, {2, 3, 4}; [3; 2, 1, 1, 1]; Shapley-Shubik: (3/6, 1/6, 1/6, 1/6)

MW: {1, 2, 3}, {1, 2, 4}, {1, 3, 4}; [4; 2, 1, 1, 1]; Shapley-Shubik: (3/6, 1/6, 1/6, 1/6)

MW: {1, 2}, {1, 3}, {1, 4}; [4; 3, 1, 1, 1]; Shapley-Shubik: (9/12, 1/12, 1/12; 1/12)

MW: {1,2}, {1, 3, 4}, {2, 3, 4}; [ 4; 2, 2, 1, 1}; Shapley-Shubik: (2/6, 2/6, 1/6, 1/6)

MW: {1,2, 3}, {1, 2, 4}; [5; 2, 1, 1, 1]; Shapley-Shubik: (5/12, 5/12, 1/12, 1/12)

MW: {1, 2},. {1, 3, 4}; [5; 3, 2, 1, 1]; Shapley-Shubik: (7/12, 3/12, 1/12, 1/12)

MW: {1,2}, {1,3}, {2, 3, 4}; [5; 3, 2, 2, 1]; Shapley-Shubik: (5/12, 3/12, 3/12, 1/12)

It is interesting to note that there can two Shapley-Shubik vectors which are identical even though the games are not equivalent, In one case above this happens when the weights for the players are the same and only the quota has changed.