#### EU2000

```    After looking at a text book "Combinatorial Analysis",
by Fred Roberts I implemented a method of computing the Banzhaf
index by use of polynomial generating functions. This gives exact
answers for the USA in an hour or so on my Olivetti Laptop which
runs a low end pentium. The generating function method does not
work on the Tompkins County example because the number of votes are
so high.

Voting coalitions are enumerated by terms in the polynomial
expansion P=(1+x^v[0])(1+x^v[1]).....(1+x^v[n-1]). With n states
the product expands to 1+Sum{ v[i]: i=0 to n-1} terms.

Given an individual state with v[i] votes, the critical
coalition count is obtained by looking at terms of P/(1+x^v[i]).
These are coefficients of the set x^(q-i) where q is the majority
figure and i=,1,2,v[i]. These polynomial products are re-evaluated
for each distinct number of votes.

Source data was taken from an article inThe Guardian.

The German government complained about their poor power for their
population. The Banzhaf power ratio is computed by dividing the voting
power of a block by the square root of its population. The voting power
of a block is the number of swing coalitions in which a block may
participate. For example with three blocks each having 50,50 and 1 vote
respectively the power of all blocks is equal. For each block there is
a choice of two partners for a critical coalition where defecting would
change the result of the vote.

Total =       87, majority =   44, max =   10
Block         Population  Weight           Power  BPR
-----         ----------  ------           -----  ------
Luxembourg          0.42       2            1002  2.6214
Portugal            9.90       5            2558  1.3784
Belgium            10.21       5            2558  1.3573
Greece             10.53       5            2558  1.3365
Ireland             3.74       3            1514  1.3273
Austria             8.08       4            2134  1.2728
Sweden              8.85       4            2134  1.2162
Italy              57.61      10            5342  1.1933
France             58.96      10            5342  1.1795
United_Kingdom     59.24      10            5342  1.1767
Finland             5.16       3            1514  1.1300
Spain              39.39       8            4166  1.1254
Denmark             5.31       3            1514  1.1139
Netherlands        15.76       5            2558  1.0925
Germany            82.03      10            5342  1.0000

EU VOTING WITH NEW ARRIVALS
This was a point of contention at the EU summit in Nice.

The Nice conference decided on new values for the voting weights.

As can bee seen Germany benefits, but despite its extra vote the
Netherlands remains near the bottom of the list.

Total =      346, majority =  173, max =   29
Block         Population  Weight           Power  BPR
-----         ----------  ------           -----  ------
Luxembourg              0.42   4         2579379  2.4054
Malta                   0.37   3         1934279  1.9218
Cyprus                  0.75   4         2579379  1.8000
Poland                 38.66  27        18068565  1.7562
Spain                  39.39  27        18068565  1.7399
Bulgaria                6.23  10         6476963  1.5683
Italy                  57.61  29        19536933  1.5556
France                 58.96  29        19536933  1.5377
United_Kingdom         59.24  29        19536933  1.5340
Portugal                9.90  12         7787735  1.4958
Hungary                10.09  12         7787735  1.4817
Belgium                10.21  12         7787735  1.4729
Czech_Rep              10.29  12         7787735  1.4672
Greece                 10.53  12         7787735  1.4504
Lithuania               3.70   7         4522771  1.4210
Ireland                 3.74   7         4522771  1.4134
Austria                 8.08  10         6476963  1.3771
Sweden                  8.85  10         6476963  1.3158
Germany                82.03  29        19536933  1.3036
Estonia                 1.44   4         2579379  1.2990
Netherlands            15.76  13         8445609  1.2857
Romania                22.48  15         9773539  1.2458
Finland                 5.16   7         4522771  1.2033
Denmark                 5.31   7         4522771  1.1862
Slovakia                5.39   7         4522771  1.1773
Slovenia                1.97   4         2579379  1.1106
Latvia                  2.43   4         2579379  1.0000
```

#### SOURCE CODE

```
Z<-POWER.D4F
|class description
|Coalition voting systems
SQRT2PI<-#sqrt 8 * _3 o 1

"POWER.D4S"
edpower|WW.ED"power.d4f ", ARGS
ldpower|#copy"power.d4f" & POWER.D4F
testp|BZ<-POWER_INDEX ARGS,(0=rARGS)/"test.dat"

"test.dat">>EOF
Sheepshire    12 12
Richfolk       9  9
Candlewick     7  7
Fiddlesex      3  3
Slurrey        1  1
Porkney_Isles  1  1
EOF

"tompkins.dat">>EOF
Lansing          8317 404
Dryden_East      7604 333
Enfield_Newfield 6776 306
Ithaca_3         6550 298
Ithaca_4         6002 274
Ithaca_SE        5932 270
Ithaca_1         5630 261
Ithaca_2         5378 246
Ithaca_NE        5235 241
Groton           5213 240
Caroline_Danby   5203 240
Ithaca_5         5172 238
Ithaca_West      4855 224
Ulysses          4666 214
Dryden_West      4552 210
EOF

"eu.2000">>EUF
Luxembourg          0.42       2
Ireland             3.74       3
Finland             5.16       3
Denmark             5.31       3
Sweden              8.85       4
Austria             8.08       4
Portugal            9.90       5
Netherlands        15.76       5
Greece             10.53       5
Belgium            10.21       5
Spain              39.39       8
United_Kingdom     59.24      10
Italy              57.61      10
Germany            82.03      10
France             58.96      10
EUF

"eu.2020">>EOF
Austria             8.08       4
Belgium            10.21       5
Denmark             5.31       3
Finland             5.16       3
France             58.96      10
Germany            82.03      10
Greece             10.53       5
Ireland             3.74       3
Italy              57.61      10
Luxembourg          0.42       2
Netherlands        15.76       5
Portugal            9.90       5
Spain              39.39       8
Sweden              8.85       4
United_Kingdom     59.24      10
Bulgaria            6.23       4
Cyprus              0.75       2
Czech_Rep          10.29       5
Estonia             1.44       3
Hungary            10.09       5
Latvia              2.43       3
Lithuania           3.70       3
Malta               0.37       2
Poland             38.66       8
Romania            22.48       6
Slovakia            5.39       4
Slovenia            1.97       3
Turkey             64.38      10
EOF

"usa.dat">>EOF
CA 29760021 54
NY 17990455 33
TX 16986510 32
FL 12937926 25
PA 11881643 23
IL 11430602 22
OH 10847115 21
MI  9295297 18
NJ  7730188 15
NC  6628637 14
GA  6478216 13
VA  6187358 13
MA  6016425 12
IN  5544159 12
MO  5117073 11
WI  4891769 11
TN  4877185 11
WA  4866692 11
MD  4781468 10
MN  4375099 10
LA  4219973  9
AL  4040587  9
KY  3685296  8
AZ  3665228  8
SC  3486703  8
CO  3294394  8
CT  3287116  8
OK  3145585  8
OR  2842321  7
IA  2776755  7
MS  2573216  7
KS  2477574  6
AR  2350725  6
WV  1793477  5
UT  1722850  5
NE  1578385  5
NM  1515069  5
ME  1227928  4
NV  1201833  4
NH  1109252  4
HI  1108229  4
ID  1006749  4
RI  1003464  4
MT   799065  3
SD   696004  3
DE   666168  3
ND   638800  3
DC   606900  3
VT   562758  3
AK   550043  3
WY   453588  3
EOF

Z<-CJ EXT_COLS NN
|Get columns from variable
CJ<-0 #ifndef "CJ"
Z<-0 #sed "X:d;\$",NN
Z<-AWK"\$<-\$\$[CJ]"
Z<-#sed":cm"

Z<-X CV Y;I;J;N;T
|Convolution or polynomial multiplication
Z<-iI<-0 & N<-1+c/J<-,(irX) #outer+ irY & T<-,X #outer* Y
AA: Z<-Z,+/(I=J)/T & I++ & ->(I(0=rIX)/0
->(10
AA: |Split
N<-(rIX)%2
Z<-(CV_PRODUCT NtIX) CV CV_PRODUCT NdIX

Z<-CO_COUNT VEC;GF;Q;V;VL;X
|Count swing coalitions for each value
Q<-(1++/VEC)%2
X<-(1+c/VEC)r 0 & X[VEC]++ & VL<-(X>0)/irX & Z<-i0
AA: ->(0=rVL)/XX & V<-1tVL & VL<-1dVL
GF<-CV_PRODUCT ((VECiV) #ne irVEC)/VEC
Z<-Z,+/GF[Q-1+iV]
->AA
XX: VL<-(X>0)/irX & Z<-Z[VLiVEC]

Z<-POWER_INDEX NN;DV;GF;H1;MAX;N;PR;Q;V;VS;VL;VOTE;XI;ZP
|Banzhaf power index in weighted majority voting patterns
|Use generating functions to check _all_ coalitions
N<-1tr:V<- #fi ," ",2 EXT_COLS NN
MAX<-c/V & XI<-(1+MAX)r0 & XI[V]++ & Q<-(1++/V)%2
H1<-"Total = ########, majority = ####, max = ####" z (+/V),Q,MAX
H1
VS<-VL<-(XI>0)/irXI & ZP<-i0
AA: ->(0=rVL)/XX
VOTE<-1tVL & VL<-1dVL
DV<- ((V i VOTE)#ne irV)/V
"testing ", (zVOTE), " < ", zDV
GF<-CV_PRODUCT ((V i VOTE)#ne irV)/V
ZP<-ZP,X<-+/GF[Q-1+iVOTE]
7 #print "V= ",(6zVOTE),"  ",16 0 z X
->AA
XX:| Use the table
0 r 7 #sed"X:d;\$",NN
PR<-i0
0 r AWK "\$<-(32t\$),16 0 z X<-ZP[VS i #fi \$2] & PR<-PR, X % SQRT2PI * #sqrt #fi \$1"
PR<-PR%f/PR
0 r AWK "\$<-(48t\$),8 4 z PR[LC]"
Z<- 7 #sed ":cm"
Z<-Z[ m #sort PR]
0 r 7 #sed ":d"
#print H1
#print BANZHAF.FMT
#print Z
0 r WW.FSCR 7
```

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