Leadership & Management

A Numerical Comparison Among Relaxed Divisor Methods with 3 Bias Measurements

Description
The aim of this research is to find the fairest apportionment methodIn this researchwe estimate the biases of apportionment methods based on the Stolarsky meanknow as relaxed divisor methodsWe use three bias measurementsthe Balinski and Young
Published
of 12
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  3    ∗  †∗   †   .   25  3  Balinski  Young  Ernst  A Numerical Comparison Among Relaxed Divisor Methodswith 3 Bias Measurements Sumachaya Harnsukworapanich ∗ Tetsuo Ichimori †∗ Graduate School of Information Science and TechnologyOsaka Institute of Technology † Faculty of Information Science and Technology,Osaka Institute of Technology Abstract.  The aim of this research is to find the fairest apportionment method  Inthis research  we estimate the biases of apportionment methods based on the Stolarskymean  know as relaxed divisor methods  We use three bias measurements  the Balinskiand Young measurement  the Ernst measurement and the B measurement which is a newmeasurement created by the researchers to compare with the other bias values  All threemeasurements are compared together  and the results show that our measurement producessimilar results to the other two measurements  In addition  the Webster method gives thelowest bias value  compared to the other methods  1.    1  3  1  Balinski  Young  [1]  2  Ernst  [2]  )  3  [3]  1   ─ 21 ─ 日本応用数理学会論文誌Vol. 26, No. 1, 2016, pp. 21~32   2  3  4  3  5  2.    Alabama  *1  *2  [1]  5  8  s  i   p i  >  0   p  h  i  q i  =  hp i /  p  d  ( a )  d  ( a )  a  a  a  ≤  d  ( a )  ≤  a + 1   y  >  0  a  d  ( a − 1)  <  y  <  d  ( a )   y  a  0  <  y  <  d  (0)   y  0  a   y  =  d  ( a )   y  a  a + 1  d  ( a )   x  >  0   p i /  x  d  ( a )  a i  a i  h  a i  i  a i  h   x  a i  h   x  [1]  d  ( a )  =  a  Adams  Table  3.    Stolarsky  [6]  *1  *2  2 22  日本応用数理学会論文誌 Vol. 26, No. 1, 2016   ─ 22 ─  Table 1. The Rounding functions of 5 traditional divisor methods  Method    Adams Dean Hill Webster Je ff  erson d  ( a ) :  a  a ( a + 1) a + 12 √  a ( a + 1)  a +  12  a + 12  0  <  x  <  y  Stolarsky   M  θ  (  x ,  y )  = (  y θ  −  x θ  θ  (  y −  x ) ) 1 θ  − 1  [9]   x  <  M  θ  (  x ,  y )  <  y  θ   lim θ  →−∞  M  θ  (  x ,  y )  =  x  lim θ  → + ∞  M  θ  (  x ,  y )  =  y  θ    M  − 1 (  x ,  y )  =  √   xy   M  2 (  x ,  y )  =  (  x +  y ) / 2  lim θ  → 0  M  θ  (  x ,  y )  =  (  y −  x ) / (ln  y − ln  x )  lim θ  → 1  M  θ  (  x ,  y )  =  (1 / e )(  y  y /  x  x ) 1 / (  y −  x )  identric  d  (0)  =  lim  x → + 0  M  θ  (  x ,  x + 1)  1  a  d  ( a )  =  M  θ  ( a , a + 1)  d  θ  ( a )  [4  5  7  8]  1  θ   θ   → −∞  Adams  θ   =  − 4  Dean  [8]  θ   =  − 1  Hill  θ   =  2  Webster   θ   =  0  TS  [10]  θ   =  1  Theil  [11]  θ   →  + ∞  Je ff  ersons  Table 2  4.    3  Balinski and Young  BY)  Ernst  ER)  [3]  (B)  3 3つの尺度を用いた緩和除数方式の偏りの計測 23   ─ 23 ─  Table 2. The values of parameter   θ   and their corresponding methods  θ   :  −∞ − 4  − 1 0 1 2  + ∞ Method : Adams Dean Hill TS Theil Webster Je ff  erson 4.1 BY   Balinski and Young  q i  a i  50  1  50  50  3  1  3  1  3  1   L  =  { 1 , . . . , 17 }  S   =  { 34 , . . . , 50 }  θ   − 12  + 12  25  a i  ( θ  )  θ   i  Balinski and Young[1]   BY   ( θ  )  =  k  S   ( θ  ) k   L  ( θ  )  −  1(4.1)   L  k   L  ( θ  )  = ∑ i ∈  L a i  ( θ  ) ∑ i ∈  L q i  S    k  S   ( θ  )  = ∑ i ∈ S  a i  ( θ  ) ∑ i ∈ S  q i  4 24  日本応用数理学会論文誌 Vol. 26, No. 1, 2016   ─ 24 ─  4.2 ER  ER   q i  a i  BY  Ernst[2]   ER ( θ  )  =  1  − k  ′ S   ( θ  ) k  ′  L  ( θ  )(4.2)   L  k  ′  L  ( θ  )  = ∑ i ∈  L q i a i  ( θ  )  S    k  ′ S   ( θ  )  = ∑ i ∈ S  q i a i  ( θ  )  4.3 B  BY  ER   [3]  B  d  θ   ( a )  q i  [ q i ] θ   [ q i ] θ   q i  a  q i  <  d  θ   ( a )  [ q i ] θ   =  a  q i  ≥  d  θ   ( a )  [ q i ] θ   =  a + 1  q i   B ( θ  )  = s ∑ i = 1 [ q i ] θ   −  h (4.3)  4.4    2010  5 3つの尺度を用いた緩和除数方式の偏りの計測 25   ─ 25 ─
Search
Similar documents
View more...
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks