Congressional voting systems
How is this different than what we just learned with elections?
1) Limited to 2­candidate systems (usually Yes/No, Pro/Con votes like for
a bill)
2) Some voters have more votes than others
3) And we stipulate how many votes makes something pass.
So its way, way different.
Definitions:
Players
The individuals that are allowed to only vote Yea or Nay on any vote (no
apportionment). We refer to each vote as a Motion. We assume
everyone votes and no one abstains.
Weights
The number of votes that each player will get.
Quota
minimum number of votes required to pass a motion.
(edit: if the number is a mixed number (3.5, 7.2, etc.), we clearly must
round up or we dont satisfy the definition.)
this is sometimes a majority but other times a greater number (60 votes to
move bill out of Caucus in US, ⅔ in the Carolina senate). This number
usually only makes sense if its between 50% of votes and 100% of votes.
Notation:
q ­ Quota
P1, P2, …, PN ­ Players
w1, w2, …, wN ­ Weights for Players
Voting System (and all data necessary) expressed as:
Example:
[14: 8, 7, 3, 2]
Example:(gridlock)
Same votes.
Set q = 21.
???
Example:(Impossible)
Same votes.
Set q = 10.
????
Example:(Few votes, much power)
[19: 8,7,3,2]
All votes matter.
We require unanimity
Equivalent to
[4:1,1,1,1]
Example:(Many votes, few power)
[30: 10, 10, 10, 9]
“Silent partner”
e.g. Shark Tank: Silent partner gets equity but has no say.
Definition:
Dictator
A Player’s weight (at most 1) is bigger than or equal to the quota.
Example:
[11: 12, 5, 4]
e.g. Shark Tank: You still get to control your company.
Definition:
Veto Power
A player with weight w has veto power if and only if w<q, V­w<q (where
V = w1+ w2 + … + wN)
Example:
[12:9,5,4,2]
Check P1 satisfies having veto power.
Definition:
Coalition
Any set of players who might join forces and vote the same way.
● Coalitions may be 1 player, or all players (Grand Coalition).
● Since there are only two sides, we have Winning Coalition and a
Losing Coalition.
Critical Player (of a Winning Coalition)
Any player of a Winning Coalition is critical if that player’s vote is
necessary to meet the quota.
(If W is the weight of the coalition, v the weight of the critical player, then
W­v < q)
Example:
Critical Count
Given all possible Winning Coalitions, with all known Critical Players for
each coalition, the Critical Count of a player P is the number of Winning
Coalitions that Player is a Critical Player of.
Example:
Definitions:
Banzhaf Power Index:
The ratio of a Player’s Critical Count (B) over (Edit: THE SUM OF ALL
CRITICAL COUNTS B1+B2+B3+...BN) (T).
Banzhaf Power Distribution
The collection of Banzhaf Power Indices for all players.
Algorithm (BANZHAF POWER DISTRIBUTION):
1. Make a list of all possible winning coalitions.
2. For all of these coalitions, determine which are the critical players.
3. Find the critical counts
4. Find
5. Compute the Banzhaf power indices:
Example: (WNBA Draft)
Example:
Definition:
Dummy
A player with votes but zero power
Review:
Players, weights, quotas
Coalitions, critical players, Banzhaf Power Index
HW: 29, 30, 33, 35
Won’t be quizzed, but I may adjust these problems for test.
2.3) SHAPLEY­SHUBIK POWER
Definition:
Sequential Coalition:
An ordered list of the players.
e.g. < P_1, P_2, …, P_N>
factorial (!):
For any positive integer N, “N factorial” = N! = 1*2*3*…*(N­1)*N
Pivotal Player
Given a sequential coalition, the pivotal player is the first player in the
sequence such that the player and all previous players match or exceed
the quota.
Example:
[4: 3, 2,1]
<P1,P2,P3> <P2,P1,P3> <P3,P1,P2>
<P1,P3,P2> <P2,P3,P1> <P3,P2,P1>
Pivotal Count
The number assigned to each player, consisting of the number of
unique sequential coalitions for which the player is the Pivotal Player.
Example:
Shapley­Shubik Power Index
of a Player is the ratio of the Player’s Pivotal Count over the total Pivotal
Count for all players.
Example:
Shapley­Shubik Power Distribution
The ordered list of all Shapley­Shubik Power Indices.
Example:
Algorithm (Shapley­Shubik Power Distribution):
1. Make a list of the N! sequential coalitions with N players
2. In each sequential coalition determine the pivotal player.
3. Find the pivotal counts SS1, SS2, …, SSN
4. compute the SSPIs:
Example(WNBA Draft Revisited)
[6:4,3,2,1]
How many?