Why one would accept Voting Theory for Democracy and reject the Penrose Square Root Weights

Munich Personal RePEc Archive

Why one would accept Voting Theory for Democracy and reject the Penrose

Square Root Weights

Colignatus, Thomas

Thomas Cool Consultancy Econometrics

6 July 2007

Online at https://mpra.ub.uni-muenchen.de/3885/

MPRA Paper No. 3885, posted 06 Jul 2007 UTC

! " " # $% # " " & " ' %(&

) $* " + " # &, " ' + #

- " %. " / 0 -%/0 # " &, # 1 " 0

" %(& " 23 $! " # " " '

! 4( # %(& # + " # " " 5 #

" %(& " "

" " " 6 3 7 8 9 #

" " % %(& " " ) " -%/0

" + # " + + # " +

" " "

! " ) " + " # &, " 3

$% # " " & " ' 3 # " %(& : ; " " #

%(&

$ " " + # 8 # 1 " %

" # " " 1 " % " +

+ # " # < " + " #

5 " + " " " # # #

# # " " " . " " #

; =- " > %. " / * > %

) " # '

" " " " "

" 23 $! " # " " '3 # "

! 4( " # ! 4( " ) # %(&3 "3

+ " 3 # %(& " ) ! 4( 6 +

" " 3 3 #

+

5 "# " "; ## " + "

" " ; - " 3 5 # #

8 # " # ";

" " " # " " " %(& +

" 3 3 " " " 3

" + + 5 3 # + " +

3 " + " 3 " " # + #

" + + " # " # # + # "

; ? 3 # " (/6 -&3

# & % " " 3 # # " "

!" !" #"$ " % &

5 # 3 " 8 ! 4(3 " + # %(&3 " 3

# " " " ! 4(

" # " # " " # - " %. " / 0

-%/0

'

# " # ! 4( " # " " ## "

#" " ; + " @ + " #

" - " " "

( " " # " " A "

2B 23 2B $ # ' " " " . " "

" " " # ) 3

## # " !

@ # " " " " "

" # # "

" # " " + "

" " " 3 3 # " C+ " "3 $ 3 + '

" " " " " "

@ " + #" " " "

1 ) " " " ;

" "+ + +

7 " # " 3

# $ " + ' 3 # # " 3

" C " " # " D 3

# # " # " 3 " # " # " ! 4(

" 8 " "

0 " 3 ! 4( " +

$ " ' " + " " # #

+ " " 8 " + ## " 5

3 " C " # A " 3 + "

" " ; " ; "

" " # " " " # " # "

!" !" #"$ " % & (

+ ! 4( " 3 ! 4( 8

@ # " "

! 4( " #

3 # " # " 3 " C " 3

" # " 3 " 3 " # # " 3 # "

" # " " @ " " # " "

" " 3 5 " " #

@ " + "

" "

" "

! 4( + " # 2BB 3 ?

# " # # " "+ 1 . > "

- : " " "

"3 & % " " "

: ) " ! 4( " :"" 2B? 3 2B?23

2B E " " # #

# " + # " " ! 4( " "

" " # " :"" > ## # 9 "

4 @ - # " " &

" # + " " A + "3

" " # :"" > # # 3 "

# " " " "

% + ; # + " " " # +

" "+ " + ; 9 "

4 @ - # " " 0

+ 3 " # "

4" " " 8 # #

$0 " " " + D 6 + ## "

# + " # 3 " +

" " " + " " " " " 5 " "

3 " + " " # " 3 " + " " 8 3

" " " # "" + # #

@ + " " @ + 3 " "

9 " 4 @ - 5 " # " + " " 3

# " + " > + ; # # " +

) !" !" #"$ " % &

> + # ## " " # "

/ " + " # " " # "

" '

3 ! 4( (/6 -& " + " " % +

) " " " " "

# " # # " + " 3 "

% " # "

3 @ ; ##

" : 3 " # " " % # "

# # " + " " "

: " "+ 3 " 3 # " @ " #"

+ " / " "

3 " " @ + + ## " #

/ " # " " @ 3 +

@ + "" " " # ## 8 3 + " + # "

; " + + " " "

# " 3 + " @ " " + "

" " " " " C " + " # "

" " # " " " C

" ; + "" "

" #" # " #" . 3

" " " # # " 4"

# + # " # "

" " 3 # " " " " ## "

" + " " " " " " "

9 " 4 @ - # @ + " F

## " " "" " " " @ +

+ + ## " 4"

" " # ## "

" C 4" $ ' G " C " "

% . 3 4" " " "

" " "# @ + +

! 4( " " " " " " ) + #

+ ) + " G " "

: " # " : " "

!" !" #"$ " % & *

"; " " : " ";

@ # # " " + .

+ ,

: " " + ; &, # 1 " #

1 H 3 6 " I 3? 3

3 22 " 5 ,%3 " # " A 3

,< " " 5 1

" " " * - "

# " "# 4 9 8 # 555 # " " + "

4 " " 3 " # - " %. " / 0 +

6 3 7 8 9 # H " " # % 8 ;

J 8; ; " # " 6 H " " "

" 7 " D "" " # 6 H # "

%. " / 0 5 ; 6 H #

7 " D "" " " # 5 + ;

" " # 7 " D 7 " " #" #

3 %(& " 3 " " ; "

%(& : " " * 3 "

" # " # 3 . "

6 H " - " "

" + " 9 8 #3 9 8 # "

# - " > - " 3 5 + # "

- " 9 8 # + " " @

" ## " - " "

$* ' " $ ' ## -%/0

- " " # #" $ " 3 + ' " 3

$ + + " 8 # &, # ' "

" # $ + ' $- " +

"' < ! 4( $+ "' # 88 # "

#" . # " " # "C " $- " + "'

# !" !" #"$ " % &

" " " + $- " - "' # 88 $+ "' -" " "

# # " . # ) "

" " " + . " "

3 " " " " + " ;

. # ) " " " C "

" " # " 3 # "

" : " "

" " " ; #" 7 :"" >

"

, " " + 3 + " . $- " "'

" " " " "

. " " # 8 : " + 3 "

" " " 3 " 3 " " "

" # 6 H "" " 7 " D

" 8 # " # - " %. " / 0 "

" " " > " %(&

" # " # " -%/0 #" " # " + + $

3 + ' #" " +

" " " G . " " $ " " '

" + $ " " '3 " ; # " # &,3

# " " " "

" " " " : " % 8 ; J 8; ;

" K . # ) " $. ' # "

" . " # " &, . " " 5

$- " + "' " 5 @

" " "

6 3 7 8 9 # H " " # -%/0 " # " "

# " $! " @ # 9 8 # " " + 3

@ " 3 #" + " . ; 3 "

+ " # + +

) " + " " " 2 5 " "

" " ) #" + " ,%

& " # " # + + " @

" " 2 " + 3 " " 3 . " C" "

!" !" #"$ " % & !

+ " " # " " ) " : " 3

+ " @ ; + " " ) " " " "#

" " " # " "

" " + 3 ) " " # +

" " " # " " + "

" " " @ " " '

7 " D $5 " " . " C"

# " + " A + " + "

@ " " 3 # " " " " +

+ " " " " + " " ' : 3 5

; " # 6 H 3 ; " # " # 9 -%3

7 " D "" "

0 " # %. " / 0 " "

6 H

% 8 ; J 8; ;

$< L " M " " " + 3 " + "

" " " + " " #" + " " #

+ " " # +

#" + + '

" . " + % N

J "+ 2 " " B HK

? 2K > ; F $9 " O F >

# " C O' % N J " $% 3 " " "

" + 3 > + ; " " " " " " + " '

- " @ @ " > #

# # " : #

" > " " 6 H " 5 "+

6 H " # %. " / 0 "

" + " 3 "" " " "

" . # " "

"

7 " D 2H

- !" !" #"$ " % &

$ + # " # "

" . # ## " " &+ ; "" " " #

" . @ # +

" # " " " 3 "" " " # "

" " + # # " G "

" + " "" "

" 5 " 3 0 3 # # " 3

" 3 5 "

" " " #

" " + . " C" "

'

" " " " # "

#" " " ; # "

" # " 6 H

" + " + + #" " + " ;

" " # " F 3 7 " D "

" " " . # $ " '

" 6 H +

: "+ " "

" # " %(& " " &, # 1 "

&, " -1 % " #

; + 3 # " + " 5

" . " # " " 3 "

6 3 7 8 9 # H A + "3 " # " &,

# 1 " " 1 " % +

: " "+ - " 3 " 6 H 3

> 5 " $ "

; ' " + G " "

" " " " * ;

" # " ) " 3 "+

" " " : 3 " " " "

"" " # %. " / 0 " # "

+ " " . " " " #"

# # " " # @ " + "

!" !" #"$ " % & .

$

0 - " " " # "" " + # +

" + " " " + " # - " *

" "

* # " " ; " " # %(&

$ " " + # 8 # 1 " %

" # " " 1 " % " +

+ # " # < " + " #

5 " + " " " # # #

# # " " " . " " #

; =- " > %. " / * > %

) " # '

0 " ; " " # %(&

5 " + " " . - " " "+ %. "

/ 0 6 3 7 8 9 # H 3 ; 3 ; ;

" # " " # 9 -%3 "" " 7 " D

# " . - " " "+

" " 3 " " + " " 5

"+ ; " # %(& " 3 % 8 ;

J 8; ; 7 " D ; " "

@ " 3 # " # + # " # "

" " " " "

" " $ " # " " 3 # "

" " " 3 + ' 3 " "

$ + # 8 # 1 " % " # "

" 1 " % ' 5 G7 " " &,

" " G7 "

0 - " " " 2BH

" " + " # + "

"" " # $ 3 + '

5 3 - " > 3 9 8 # " + " 3

: " - " %. " / 0

/ !" !" #"$ " % &

" " " $ ' " "

" #

" =- " > %. " / * > "

" # + + " . " "

# " " A " " "

" @ 3 "

$% ) " #

' # + " " 5

# # 2 " - "

" + ? K P 2 # + # "

" " > ## " % 8 ;

J 8; ; ; K +

< @ " + " " ! 4(

%(& " " # " # " " " "

" + " " " " " # "

+ " " . # ) " " 4 "

@ 3 ; " 2 " " H "

" ; 3 " " "

%(& + " 2 + +

; G " ; +

: %(& " #" " +

$ " ; ' # ; 8 ) "

" @ " " 3 3 # " @ 3 2 K #

+ 2K # # @ " # " 9 3 " " 3

%(& " # " # " "

" " # " 3 "

" 3 " &, # 1 " #

" " : + " " # " 3

" " # " # " "

" 3 " . 6 " I ?

1 H " "

3 ; # " + " + # " C " " " 3

# 3 @ + 3 # ; 5 " + 3

6 " ; 5 5 ; 1 : " #

" 3 " 3 1 " "3

!" !" #"$ " % & //

@ " " # " " 4" #

" # 1 6 " + " 3 " +

" " " " @

%(& # 3 " +

" # ; # " 3

" " " 3 8 " " # " "

"" " " + " " " " " # # "

# " & % " " 3 " C " #" . #

3 C " " # " ) " " ; 4"

,73 # $ " ; ' # " "

# :"" > " 3 C " #

" + " " # " > #

" "

%(& "

$ @ " + " " " # < + "

" # " " " " " '

$0 " " ; " " # #

'

+ " " #

+ " + # + + "

" + . " 9 " #

# " ## " " + " % #

" " " " #" 3 ; # # " 3

" " ) " 3

+ 3 # " #

; " 5 # 3 " + @ # "

" " + "

& % " " 3 "" . # +

% ; 3 5 "+ 8 ;

# + " &, " " 8 ;

" " &, 3 + " # " " + "

" + 3 + &, " "

" / "; 4 "

" " # " " " " " -%/0 4 "

/ !" !" #"$ " % &

@ 3 # " " " " + "

@ + 3 &,

# " 3 # $+ " ' " +

" " " @

"

. " " #" # # " 5#

" " " # " 1 + 2 3 6 "

223 &, " " " # " " 8 # " 0

6 " 2? # 223 "

" # # & " #

" 3 # " " # "

4 " " ; 3 . " " # +

" " " " # 3 E

: # - " %. " / 0

" " 4 " + " + "

+ " @ "" + $ " 3 + '

4 " " " " + # % "

@ " " " # @ " + " 5

" ; " - " + " 3

" # . + 5 + + +

" " + " " # - " + " A "

; + # " 3

" + 3 " C ; # "

- " " + " #"

+ ; # 3 " ) #

-%/0 " " + " " # " "

$

0 " %(& $% # " " & " ' 3 $* "

+ " # &, " '3 " ; ! 4( $!

" # " " ' 23

!" !" #"$ " % & /(

" %(& # + " # " "

0 " " %(& " # "

" " # " " " 3 %(&

" ) ) # " #

" ! 4( " # " " " %(& "

" # " # " "

0

" # "" #

&0- " # " " & 0 "; - " :" + 0

, + " % * %

:"" 3 7 2B? 3 $: ## # # " '3 #

3 E ICEH 3 " " :"" N % + ; 2B B 2H C2 I

:"" 3 7 2B?23 2B E 3 $% + + '3 0

2BB 3 $( # / " " # Q

% ; " " 2BIBC2BB '3 5%9< B ??2IC CB3 1 1 -

/ " 3 / "

2 3 $! " 4 " ( " '3 # " 3

& " N 3 5%9<CB CI H HCEC?

E 3 $G + # # '3 C E2 E G " E E

? 3 R( # / 6 " " # - & R3

3 ( , + " -"

3 R: * # &@ R3 & " N

3 5%9< B ICB CI H HCHC?

3 $! " # " " '3 3

& " N 35%9< B ICB CI H HC?C

/) !" !" #"$ " % &

3 $5 " 3 9 " + : # 9 "

4 @ - 4" " '3

" C E 3 " " + #" "

2BBB3 2 3 $ & - ;3 : # " 1 '3

% + 3 &*CBBC I 3 5%9< B CI H HC2CB

6 : 3 < 7 8 9 # H 3 $% " + " @ S ";

" '3 9" " # - % EH ? CC H3 " "

+ 3 # " 5 # 3 % % 0 "; - "

22EE

A " 2B 23 2BB 3 $ # '3 G@# "

7 " 3 0 D 3 $G - " > . " C" '3

" "C C ; - " #3 " " + #"

"

* 3 ( 3 $( + # " # & " , '3

- 22E HE TH H3 " ; HB 3 " " + #" "

% # " " & " 3 $* " + " # &, "

'3 " "C C ; C " 3 " " +

#" " '

% 8 ; 3 0 7 J 8; ; 3 $- " + . '

3 "@ + " 3 / " + #" "

!" !" #"$ " % & /*