5 Concluding remarks
Application 2. The partisan bias
We can apply the result of Lemma 2 to measure the “partisan bias,” which is the deviation between the proportion of seats held by a party in the final legislature at thet+ 1-th decision level and that of the votes its members received at the polls. For the measurement, we rearrange voters with each favorite policy to those who are supporting parties of xj ∈ {0,1}, as in the application in Gilligan and Matsusaka. Here, the population of voters is stillN and the number of hierarchical levels is t+ 1.
Let voters who prefer party 0 to party 1 be called partisan. The fraction of voters favoring party 1 is V = N1 PN
i=1xi and the fraction favoring party 0 is 1−V. The party affiliation of representative k at the t+ 1-th decision level is xkt+1,1 ∈ {0,1}, corresponding to the median representative in thek-th district at the t-th decision level. The fraction of seats held by party 1 and party 0 in the final legislature is Lt = K1
t
PKt
k=1xkt+1,1 and 1 −Lt, respectively. Then, the partisan bias can be defined as βt = |Lt− V|. If there is no bias, then the fraction of
seats held by party 1 is the same as the fraction of supporters in voters, and Lt =V. Since a representative at thet+ 1-th decision level needs at least
s≡ 1
supporters at the voters level, as shown in the proof of Lemma 2, we obtain the next proposition of the partisan bias.
Proposition 6. Let Assumption 1 hold. Let V be the fraction of voters who support party 1, and suppose there are enough party 1 voters to elect at least one representative of party 1 to the t+ 1-th decision level, but not to elect all representatives: 12(KKt−1
t + 1)12(KKt−2
Proof. Since one representative at thet+ 1-th decision level needs at leasts voters, when there arePKt
i=1xit+1,1representatives at thet+ 1-th decision level, the minimum number of supporters at the voters level is sPKt
i=1xit+1,1, which is equal to or less than the actual number of voters supporting party 1: sPKt
Using formula (4) and subtracting V from both sides, we get βt =Lt−V ≤
The maximum partisan bias βtis the ratio between the relative position of the median, who is the final representative at thet+ 1-th decision level, and that at the voters level. In fact, this proposition is a generalization of the partisan biasβ calculated in Proposition 4 of Gilligan and Matsusaka, and corresponds to the case of t = 1 in our model. Gilligan and Matsusaka point out that increasing the number of seatsK1in the legislature decreases the partisan bias, holding the number of voters constant at N. However, one fact needs to be added to their finding. For
instance, applying the case of t = 1 in our example 1, while both {K1 = 3, K2 = 1} and {K1 = 9, K2 = 1}reach the same result (i.e., the 10-th voter becomes the final representative), the maximum partisan biases are (10/272/3 −1)V = 0.8V and (10/275/9 −1)V = 0.5V, respectively.
Thus, the legislature with fewer seats has a larger bias than that with more seats, even when the same representative is elected as the final representative in both, with gerrymandering.
Example
Example 3. Let us again consider the example of N = 27, t = 2, and the voters set N = {1,2,3, . . . ,25,26,27}. In this case, the median of all voters is 14 and the most extremely liberal voter who is electable as the final representative is j3,1∗ = 12(3 + 1)· 12(93 + 1)· 12(279 + 1) = 8.
Lemma 4 and Proposition 4 state that voters between 8 and 14, who belong to either the fourth or the fifth district at the first decision level, from the definition ofmi, are electable as the final representative in this example.
Sliding voters: We have to slide voters by zero, one, two, three, and four positions to place voters 8,9,10,11,12,13,14 at the fourth and fifth district median positions, since m12−1(KN
1−1) =
4−1
2 (279 −1) = 3, following Table 8. Then, we have Table 10.
Table 10: Placing voters between 8 and 14 on district median at the first decision level
Sliding District number
by 1 2 3 4 5 6 7 8 9
0 {1,2,27} {3,4,26} {5,6,25} {7, 8,24} {9,10,23} {11,12,22} {13,14,21} {15,16,20} {17,18,19}
↓ ↓
1 {1,2,3} {4,5,27} {6,7,26} {8,9,25} {10,11,24} {12,13,23} {14,15,22} {16,17,21} {18,19,20}
↓ ↓
2 {1,2,3} {4,5,6} {7,8,27} {9,10,26} {11,12,25} {13,14,24} {15,16,23} {17,18,22} {19,20,21}
↓ ↓
3 {1,2,3} {4,5,6} {7,8,9} {10,11,27} {12,13,26} {14,15,25} {16,17,24} {18,19,23} {20,21,22}
↓ ↓
4 {1,2,3} {4,5,6} {7,8,9} {10,11,12} {13, 14,27} {15,16,26} {17,18,25} {19,20,24} {21,22,23}
the final decision levels, we need to slide two positions. Then, we have the representatives {2,5,8,10,12,14,16,18,20}. At the second decision level, we need to slide representatives by one position:
Sliding by 1 2 3
1 {2,5,8} {10,12,20} {14,16,18}
Now we have representatives {5,12,16} in the final decision level. Lastly, 12 is elected as the final representative.
Next, if we want voter 14 to be elected as a district representative in the second through the final decision levels, we need to slide four positions, which is full sliding at the first decision level. Then, we have the representatives {2,5,8,11,14,16,18,20,22}. At the second level, we need to slide representatives by one position, which is also full sliding, for representative 14 to belong to the median district:
Sliding by 1 2 3
1 {2,5,8} {11,14,22} {16,18,20}
At the final level, we have {5,14,18}, and voter 14 is elected as the final representative. In this case, voters slide fully at each decision level so that all voters before voter 14 are lined up consecutively in ascending order.
Lastly, if we want voter 11 to be elected as a district representative in the second through the final decision levels, we need to slide voters either one, three, or four positions. If we choose to slide by one position, we have the representatives {2,5,7,9,11,13,15,17,19}, and if we choose three positions, we have {2,5,8,11,13,15,17,19,21}. Sliding by four positions is similar to three positions because 11 is in the fourth district. When sliding by one position, at the second decision level, representatives need to slide by one position. When sliding by three positions, they need to slide by zero positions:
Sliding by
first level second level 1 2 3
1 1 {2,5,7} {9,11,19} {13,15,17}
3 0 {2,5,21} {8,11,19} {13,15,17}.
Since 11 appears as the district median of the fourth and fifth districts when sliding by one, three, and four positions, they achieve the same result at the first decision level. Thus, we can obtain the same result at the final decision level.
Acknowledgments
This project started when Kobayashi of Hosei University visited the Corvinus University of Budapest as part of an exchange program from February 20 to March 30, 2012. Kobayashi thanks the Corvinus University of Budapest for their financial support and their hospital-ity. We thank Paolo Balduzzi, P´eter Bir´o, Yoich Hizen, Hiroyuki Ozaki and the participants of seminars and conferences for their helpful comments and useful conversations. Kobayashi grate-fully acknowledges the financial support from JSPS KAKENHI (Grant Number 25870793) and Tasn´adi gratefully acknowledges the financial support from the Hungarian Scientific Research Fund (OTKA K-112975).
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