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Election Theory A Tale of Two Countries or Voting Power Comes To the Rescue!

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Presentation on theme: "Election Theory A Tale of Two Countries or Voting Power Comes To the Rescue!"— Presentation transcript:

1 Election Theory A Tale of Two Countries or Voting Power Comes To the Rescue!

2 A Tale of Two Countries The U. S. after the November 2008 elections: House of Representatives Democratic: 257 Republican: 178 Senate Democratic: 57 Republican: 41 Independent: 2 What would happen if everyone voted along party lines?

3 A Tale of Two Countries Germany after the September 2009 elections: Christian Democrats/Christian Social Union(CDU/CSU): 225 Social Democrats (SPD): 222 Free Democrats (FDP): 61 Left Party: 54 Greens: 51 What would happen if everyone voted along party lines?

4 A Tale of Two Countries We can calculate a Power Index to determine the relative voting power of each subgroup.

5 A Tale of Two Countries We can calculate a Power Index to determine the relative voting power of each subgroup. 1.Determine the number of votes needed to win.

6 A Tale of Two Countries We can calculate a Power Index to determine the relative voting power of each subgroup. 1.Determine the number of votes needed to win. 2.Consider all of the possible coalitions, or combinations of members.

7 A Tale of Two Countries We can calculate a Power Index to determine the relative voting power of each subgroup. 1.Determine the number of votes needed to win. 2.Consider all of the possible coalitions, or combinations of members. 3.Determine which of those coalitions have enough votes to win—winning coalitions.

8 A Tale of Two Countries We can calculate a Power Index to determine the relative voting power of each subgroup. 1.Determine the number of votes needed to win. 2.Consider all of the possible coalitions, or combinations of members. 3.Determine which of those coalitions have enough votes to win—winning coalitions. 4.The power index for a given member is the number of winning coalitions in which that member is required.

9 A Tale of Two Countries Germany after the September 2009 elections: Christian Democrats/Christian Social Union(CDU/CSU): 225 Social Democrats (SPD): 222 Free Democrats (FDP): 61 Left Party: 54 Greens: 51 What is the power index for each party?

10 A Tale of Two Countries Christian Democrats/Christian Social Union(CDU/CSU): 225 Social Democrats (SPD): 222 Free Democrats (FDP): 61 Left Party: 54 Greens: 51 1.Total number of votes = 613. Unless stated otherwise, assume that more than half of the total votes is required for passing. In this case, 613/2 = 306.5, so 307 votes are required to pass.

11 A Tale of Two Countries Christian Democrats/Christian Social Union(CDU/CSU): 225 Social Democrats (SPD): 222 Free Democrats (FDP): 61 Left Party: 54 Greens: 51 2. Possible coalitions include: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E

12 A Tale of Two Countries Christian Democrats/Christian Social Union(CDU/CSU): 225 Social Democrats (SPD): 222 Free Democrats (FDP): 61 Left Party: 54 Greens: 51 3. Determine the winning coalitions: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E Example: {A,B} is a winning coalition because A + B = 225 + 222 = 447 > 307

13 A Tale of Two Countries CDU/CSU: 225 SPD: 222 FDP: 61 Left Party: 54 Greens: 51 4. Determine the power index for each member: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E Start with A. For each coalition that requires A in order to keep it a winning coalition, add 1 to A’s power index. For example, if you remove A from {A,B}, then that is no longer a winning coalition—A gets 1 point. If you remove A from {A,B,C,D,E}, though, the coalition still wins even without A.

14 A Tale of Two Countries CDU/CSU: 225 SPD: 222 FDP: 61 Left Party: 54 Greens: 51 4. Determine the power index for each member: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E A is a critical member of 8 winning coalitions, so it has a power index of 8.

15 A Tale of Two Countries CDU/CSU: 225 SPD: 222 FDP: 61 Left Party: 54 Greens: 51 4. Determine the power index for each member: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E B is a critical member of 8 winning coalitions, so it has a power index of 8.

16 A Tale of Two Countries CDU/CSU: 225 SPD: 222 FDP: 61 Left Party: 54 Greens: 51 4. Determine the power index for each member: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E C is a critical member of 4 winning coalitions, so it has a power index of 4.

17 A Tale of Two Countries CDU/CSU: 225 SPD: 222 FDP: 61 Left Party: 54 Greens: 51 4. Determine the power index for each member: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E D is a critical member of 4 winning coalitions, so it has a power index of 4.

18 A Tale of Two Countries CDU/CSU: 225 SPD: 222 FDP: 61 Left Party: 54 Greens: 51 4. Determine the power index for each member: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E E is a critical member of 4 winning coalitions, so it has a power index of 4.

19 A Tale of Two Countries CDU/CSU: 225 SPD: 222 FDP: 61 Left Party: 54 Greens: 51 4. Determine the power index for each member: {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E} {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E} A B C D E Compare the power index of each party. This gives us a picture of the party’s relative strength. 8 8 4 4 4

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