# [EM] Democratic Lottery Enhancement

raphfrk at netscape.net raphfrk at netscape.net
Fri Jun 9 02:59:30 PDT 2006

```Jobst Heitzig <heitzig-j at web.de> wrote:
>
>  Hello Raphael again!
>
>> If so, there is an odd number of winners in that case, which might
>> prove problematic.
>
>For example:
>
>2 abc
>2 bca
>3 cab
>
>Starting with a uniform distribution, we get:
>
>           no. of
>           papers for
>iteration    a    b    c
>0            1    1    1
>1            7    6    8
>2           43   38   66
>3          217  274  538
>4          727 2286 4190
>5                ?
>
>Algebraically, one finds that the following is an equilibrium:
>
>           no. of
>           papers for
>iteration    a    b    c
>0            1    2    2
>1            7   14   14
>

Interesting, it doesn't have equilibrum in the same proportion

It looks like the system is unstable.  It seems to give all the ballots
to one of the 3 candidates and then to the next.  It follows

B -> A -> C -> B etc.

(at each stage the candidate gets more than 20 of 21 ballots.

It thus follows the condorcet tie circle.

In the 2 candidate case, the relationship between each round is:

Ba(n) = ballots for A (normalised from 0 to 2)
Bb(n) = 2 - Ba(n) = ballots for B

P =  support for A as a fraction

assume Ba(n) >= 1

<A's supporters> + <B's supporters>

Ba(n+1) = P*(2) + (1-P)*(2 - (2 * Bb(n)))
= P*(2) + (1-P)*(2 - (2 * (2-Ba(n))))
= P*(2) + (1-P)*(2 - 4 + 2*Ba(n))
= P*(2) + (1-P)*(2*Ba(n)-2)

It is the weighted average between 2 and (2*Ba(n) - 2).

If P = 0.5, then the proportions never change.

If P> 0.5, then the value must increase as the weighting
given to the higher term is increased.

Thus, in a 2 candidate race, the majority supported candidate
gets 100% of the ballots in the limit.  This explains the oscillation
as the system trying to pick the majority winner but there is
a condorcet tie, so it goes in circles.

Also my previous post is wrong (again .. not going well with this

>> If there are 10 candidates with 9 from party B and 1 from party A
>> all voters return 10 papers not 5 as I thought.

>> 'A' supporters return 4 A's and 6 from B
>> 'B' supporters return 10 B's

I multiplied by 4 instead of 2 ... I'm not going well
getting this right :)

However, it does show a possible solution to the oscillation problem.

The method could be changed to:

Each voter is given 3 sets of the initial ballots and returns 1 set
worth of ballots.

This leads to a stable end point.

Candidate A: 0
candidate B: 12
candidate C: 9

The results are

with doubling (renormalised after each step)

(1.000000 1.000000 1.000000)->(7.000000 6.000000 8.000000)
(1.000000 0.857143 1.142857)->(6.142857 5.428571 9.428571)
(0.877551 0.775510 1.346939)->(4.428571 5.591837 10.979592)
(0.632653 0.798834 1.568513)->(2.530612 6.390671 12.078717)
(0.361516 0.912953 1.725531)->(1.446064 7.303623 12.250312)
(0.206581 1.043375 1.750045)->(0.826322 8.346998 11.826679)
(0.118046 1.192428 1.689526)->(0.472184 9.539427 10.988389)
(0.067455 1.362775 1.569770)->(0.269820 10.902202 9.827979)
(0.038546 1.557457 1.403997)->(0.385456 12.190562 8.423982)
(0.055065 1.741509 1.203426)->(0.550652 13.228792 7.220556)
(0.078665 1.889827 1.031508)->(0.786646 14.024306 6.189048)
(0.112378 2.003472 0.884150)->(1.123780 14.571322 5.304898)
(0.160540 2.081617 0.757843)->(1.605400 14.847545 4.547055)
(0.229343 2.121078 0.649579)->(2.293428 14.809096 3.897476)
(0.327633 2.115585 0.556782)->(3.276326 14.382980 3.340694)
(0.468047 2.054711 0.477242)->(4.680466 13.456082 2.863452)
(0.668638 1.922297 0.409065)->(6.686380 11.859233 2.454387)
(0.955197 1.694176 0.350627)->(9.551971 9.344268 2.103761)
(1.364567 1.334895 0.300537)->(12.655046 5.881312 2.463642)
(1.807864 0.840187 0.351949)->(14.119762 3.360750 3.519488)
(2.017109 0.480107 0.502784)->(14.051732 1.920429 5.027840)
(2.007390 0.274347 0.718263)->(12.719984 1.097388 7.182628)
(1.817141 0.156770 1.026090)->(10.112024 0.627079 10.260898)
(1.444575 0.089583 1.465843)->(5.983244 0.580032 14.436725)
(0.854749 0.082862 2.062389)->(3.418997 0.662893 16.918110)
(0.488428 0.094699 2.416873)->(1.953712 0.757592 18.288695)
(0.279102 0.108227 2.612671)->(1.116407 0.865820 19.017773)
(0.159487 0.123689 2.716825)->(0.637947 0.989508 19.372545)
(0.091135 0.141358 2.767506)->(0.364541 1.130866 19.504592)
(0.052077 0.161552 2.786370)->(0.208309 1.292419 19.499272)
(0.029758 0.184631 2.785610)->(0.119034 1.477050 19.403916)
(0.017005 0.211007 2.771988)->(0.068019 1.688057 19.243923)
(0.009717 0.241151 2.749132)->(0.038868 1.929208 19.031924)
(0.005553 0.275601 2.718846)->(0.022210 2.204809 18.772980)
(0.003173 0.314973 2.681854)->(0.012692 2.519782 18.467526)
(0.001813 0.359969 2.638218)->(0.007252 2.879751 18.112997)
(0.001036 0.411393 2.587571)->(0.004144 3.291144 17.704712)
(0.000592 0.470163 2.529245)->(0.002368 3.761308 17.236324)
(0.000338 0.537330 2.462332)->(0.001353 4.298637 16.700010)
(0.000193 0.614091 2.385716)->(0.000773 4.912728 16.086498)
(0.000110 0.701818 2.298071)->(0.000442 5.614547 15.385012)
(0.000063 0.802078 2.197859)->(0.000252 6.416625 14.583123)
(0.000036 0.916661 2.083303)->(0.000144 7.333285 13.666570)
(0.000021 1.047612 1.952367)->(0.000082 8.380898 12.619020)
(0.000012 1.197271 1.802717)->(0.000047 9.578169 11.421784)
(0.000007 1.368310 1.631683)->(0.000027 10.946478 10.053495)
(0.000004 1.563783 1.436214)->(0.000038 12.382680 8.617281)
(0.000005 1.768954 1.231040)->(0.000055 13.613704 7.386241)
(0.000008 1.944815 1.055177)->(0.000078 14.668858 6.331064)
(0.000011 2.095551 0.904438)->(0.000112 15.573262 5.426626)
(0.000016 2.224752 0.775232)->(0.000160 16.348446 4.651394)
(0.000023 2.335492 0.664485)->(0.000229 17.012862 3.986909)
(0.000033 2.430409 0.569558)->(0.000327 17.582323 3.417350)
(0.000047 2.511760 0.488193)->(0.000467 18.070375 2.929158)
(0.000067 2.581482 0.418451)->(0.000667 18.488626 2.510706)
(0.000095 2.641232 0.358672)->(0.000953 18.847013 2.152034)
(0.000136 2.692430 0.307433)->(0.001362 19.154038 1.844601)
(0.000195 2.736291 0.263514)->(0.001945 19.416969 1.581086)
(0.000278 2.773853 0.225869)->(0.002779 19.642005 1.355217)
(0.000397 2.806001 0.193602)->(0.003969 19.834416 1.161614)
(0.000567 2.833488 0.165945)->(0.005671 19.998660 0.995670)
(0.000810 2.856951 0.142239)->(0.008101 20.138468 0.853431)
(0.001157 2.876924 0.121919)->(0.011573 20.256915 0.731512)
(0.001653 2.893845 0.104502)->(0.016532 20.356457 0.627011)
(0.002362 2.908065 0.089573)->(0.023618 20.438945 0.537438)
(0.003374 2.919849 0.076777)->(0.033740 20.505600 0.460661)
(0.004820 2.929371 0.065809)->(0.048199 20.556949 0.394852)
(0.006886 2.936707 0.056407)->(0.068856 20.592699 0.338445)
(0.009837 2.941814 0.048349)->(0.098366 20.611539 0.290095)
(0.014052 2.944506 0.041442)->(0.140523 20.610824 0.248653)
(0.020075 2.944403 0.035522)->(0.200747 20.586122 0.213131)
(0.028678 2.940875 0.030447)->(0.286781 20.530535 0.182684)
(0.040969 2.932934 0.026098)->(0.409687 20.433726 0.156586)
(0.058527 2.919104 0.022369)->(0.585268 20.280516 0.134217)
(0.083610 2.897217 0.019174)->(0.836097 20.048860 0.115043)
(0.119442 2.864123 0.016435)->(1.194424 19.706968 0.098608)
(0.170632 2.815281 0.014087)->(1.706320 19.209159 0.084521)
(0.243760 2.744166 0.012074)->(2.437600 18.489954 0.072447)
(0.348229 2.641422 0.010350)->(3.482285 17.455618 0.062097)
(0.497469 2.493660 0.008871)->(4.974693 15.972081 0.053226)
(0.710670 2.281726 0.007604)->(7.106704 13.847673 0.045623)
(1.015243 1.978239 0.006518)->(10.152435 10.808460 0.039105)
(1.450348 1.544066 0.005586)->(14.503478 6.463003 0.033519)
(2.071925 0.923286 0.004788)->(17.258971 3.693145 0.047884)
(2.465567 0.527592 0.006841)->(18.821226 2.110368 0.068405)
(2.688747 0.301481 0.009772)->(19.696353 1.205925 0.097722)
(2.813765 0.172275 0.013960)->(20.171297 0.689100 0.139603)
(2.881614 0.098443 0.019943)->(20.406796 0.393771 0.199432)
(2.915257 0.056253 0.028490)->(20.490084 0.225012 0.284903)
(2.927155 0.032145 0.040700)->(20.464417 0.128578 0.407005)
(2.923488 0.018368 0.058144)->(20.345091 0.073473 0.581435)
(2.906442 0.010496 0.083062)->(20.127393 0.041985 0.830622)
(2.875342 0.005998 0.118660)->(19.789406 0.023991 1.186603)
(2.827058 0.003427 0.169515)->(19.291144 0.013709 1.695147)
(2.755878 0.001958 0.242164)->(18.570527 0.007834 2.421639)
(2.652932 0.001119 0.345948)->(17.536040 0.004477 3.459484)
(2.505149 0.000640 0.494212)->(16.055322 0.002558 4.942120)
(2.293617 0.000365 0.706017)->(13.938367 0.001462 7.060171)
(1.991195 0.000209 1.008596)->(10.913206 0.000835 10.085959)
(1.559029 0.000119 1.440851)->(6.591010 0.000477 14.408513)
(0.941573 0.000068 2.058359)->(3.766291 0.000545 17.233163)
(0.538042 0.000078 2.461880)->(2.152167 0.000623 18.847210)
(0.307452 0.000089 2.692459)->(1.229809 0.000712 19.769478)
(0.175687 0.000102 2.824211)->(0.702748 0.000814 20.296437)
(0.100393 0.000116 2.899491)->(0.401570 0.000931 20.597499)
(0.057367 0.000133 2.942500)->(0.229469 0.001064 20.769468)
(0.032781 0.000152 2.967067)->(0.131125 0.001215 20.867660)
(0.018732 0.000174 2.981094)->(0.074929 0.001389 20.923682)
(0.010704 0.000198 2.989097)->(0.042816 0.001588 20.955596)
(0.006117 0.000227 2.993657)->(0.024466 0.001814 20.973719)
(0.003495 0.000259 2.996246)->(0.013981 0.002073 20.983946)
(0.001997 0.000296 2.997707)->(0.007989 0.002370 20.989641)
(0.001141 0.000339 2.998520)->(0.004565 0.002708 20.992727)
(0.000652 0.000387 2.998961)->(0.002609 0.003095 20.994296)
(0.000373 0.000442 2.999185)->(0.001491 0.003537 20.994972)
(0.000213 0.000505 2.999282)->(0.000852 0.004043 20.995106)
(0.000122 0.000578 2.999301)->(0.000487 0.004620 20.994893)
(0.000070 0.000660 2.999270)->(0.000278 0.005280 20.994442)
(0.000040 0.000754 2.999206)->(0.000159 0.006034 20.993807)
(0.000023 0.000862 2.999115)->(0.000091 0.006896 20.993013)
(0.000013 0.000985 2.999002)->(0.000052 0.007882 20.992066)
(0.000007 0.001126 2.998867)->(0.000030 0.009008 20.990963)
(0.000004 0.001287 2.998709)->(0.000017 0.010294 20.989689)
(0.000002 0.001471 2.998527)->(0.000010 0.011765 20.988225)
(0.000001 0.001681 2.998318)->(0.000006 0.013446 20.986549)
(0.000001 0.001921 2.998078)->(0.000003 0.015367 20.984630)
(0.000000 0.002195 2.997804)->(0.000002 0.017562 20.982436)
(0.000000 0.002509 2.997491)->(0.000001 0.020071 20.979928)
(0.000000 0.002867 2.997133)->(0.000001 0.022938 20.977061)
(0.000000 0.003277 2.996723)->(0.000000 0.026215 20.973785)
(0.000000 0.003745 2.996255)->(0.000000 0.029960 20.970040)
(0.000000 0.004280 2.995720)->(0.000000 0.034240 20.965760)
(0.000000 0.004891 2.995109)->(0.000000 0.039131 20.960869)
(0.000000 0.005590 2.994410)->(0.000000 0.044721 20.955279)
(0.000000 0.006389 2.993611)->(0.000000 0.051110 20.948890)
(0.000000 0.007301 2.992699)->(0.000000 0.058411 20.941589)
(0.000000 0.008344 2.991656)->(0.000000 0.066756 20.933244)
(0.000000 0.009537 2.990463)->(0.000000 0.076292 20.923708)
(0.000000 0.010899 2.989101)->(0.000000 0.087191 20.912809)
(0.000000 0.012456 2.987544)->(0.000000 0.099647 20.900353)
(0.000000 0.014235 2.985765)->(0.000000 0.113883 20.886117)
(0.000000 0.016269 2.983731)->(0.000000 0.130152 20.869848)
(0.000000 0.018593 2.981407)->(0.000000 0.148745 20.851255)
(0.000000 0.021249 2.978751)->(0.000000 0.169994 20.830006)
(0.000000 0.024285 2.975715)->(0.000000 0.194279 20.805721)
(0.000000 0.027754 2.972246)->(0.000000 0.222033 20.777967)
(0.000000 0.031719 2.968281)->(0.000000 0.253752 20.746248)
(0.000000 0.036250 2.963750)->(0.000000 0.290002 20.709998)
(0.000000 0.041429 2.958571)->(0.000000 0.331431 20.668569)
(0.000000 0.047347 2.952653)->(0.000000 0.378778 20.621222)
(0.000000 0.054111 2.945889)->(0.000000 0.432889 20.567111)
(0.000000 0.061841 2.938159)->(0.000000 0.494731 20.505269)
(0.000000 0.070676 2.929324)->(0.000000 0.565406 20.434594)
(0.000000 0.080772 2.919228)->(0.000000 0.646179 20.353821)
(0.000000 0.092311 2.907689)->(0.000000 0.738490 20.261510)
(0.000000 0.105499 2.894501)->(0.000000 0.843989 20.156011)
(0.000000 0.120570 2.879430)->(0.000000 0.964558 20.035442)
(0.000000 0.137794 2.862206)->(0.000000 1.102352 19.897648)
(0.000000 0.157479 2.842521)->(0.000000 1.259831 19.740169)
(0.000000 0.179976 2.820024)->(0.000000 1.439807 19.560193)
(0.000000 0.205687 2.794313)->(0.000000 1.645494 19.354506)
(0.000000 0.235071 2.764929)->(0.000000 1.880565 19.119435)
(0.000000 0.268652 2.731348)->(0.000000 2.149217 18.850783)
(0.000000 0.307031 2.692969)->(0.000000 2.456248 18.543752)
(0.000000 0.350893 2.649107)->(0.000000 2.807140 18.192860)
(0.000000 0.401020 2.598980)->(0.000000 3.208160 17.791840)
(0.000000 0.458309 2.541691)->(0.000000 3.666469 17.333531)
(0.000000 0.523781 2.476219)->(0.000000 4.190250 16.809750)
(0.000000 0.598607 2.401393)->(0.000000 4.788857 16.211143)
(0.000000 0.684122 2.315878)->(0.000000 5.472980 15.527020)
(0.000000 0.781854 2.218146)->(0.000000 6.254834 14.745166)
(0.000000 0.893548 2.106452)->(0.000000 7.148382 13.851618)
(0.000000 1.021197 1.978803)->(0.000000 8.169579 12.830421)
(0.000000 1.167083 1.832917)->(0.000000 9.336662 11.663338)
(0.000000 1.333809 1.666191)->(0.000000 10.670471 10.329529)
(0.000000 1.524353 1.475647)->(0.000000 12.146118 8.853882)
(0.000000 1.735160 1.264840)->(0.000000 13.410958 7.589042)
(0.000000 1.915851 1.084149)->(0.000000 14.495107 6.504893)
(0.000000 2.070730 0.929270)->(0.000000 15.424377 5.575623)
(0.000000 2.203482 0.796518)->(0.000000 16.220895 4.779105)
(0.000000 2.317271 0.682729)->(0.000000 16.903624 4.096376)
(0.000000 2.414803 0.585197)->(0.000000 17.488821 3.511179)
(0.000000 2.498403 0.501597)->(0.000000 17.990418 3.009582)
(0.000000 2.570060 0.429940)->(0.000000 18.420358 2.579642)
(0.000000 2.631480 0.368520)->(0.000000 18.788878 2.211122)
(0.000000 2.684125 0.315875)->(0.000000 19.104753 1.895247)
(0.000000 2.729250 0.270750)->(0.000000 19.375502 1.624498)
(0.000000 2.767929 0.232071)->(0.000000 19.607574 1.392426)
(0.000000 2.801082 0.198918)->(0.000000 19.806492 1.193508)
(0.000000 2.829499 0.170501)->(0.000000 19.976993 1.023007)
(0.000000 2.853856 0.146144)->(0.000000 20.123137 0.876863)
(0.000000 2.874734 0.125266)->(0.000000 20.248403 0.751597)
(0.000000 2.892629 0.107371)->(0.000000 20.355774 0.644226)
(0.000000 2.907968 0.092032)->(0.000000 20.447806 0.552194)
(0.000000 2.921115 0.078885)->(0.000000 20.526691 0.473309)
(0.000000 2.932384 0.067616)->(0.000000 20.594307 0.405693)
(0.000000 2.942044 0.057956)->(0.000000 20.652263 0.347737)
(0.000000 2.950323 0.049677)->(0.000000 20.701940 0.298060)
(0.000000 2.957420 0.042580)->(0.000000 20.744520 0.255480)
(0.000000 2.963503 0.036497)->(0.000000 20.781017 0.218983)
(0.000000 2.968717 0.031283)->(0.000000 20.812300 0.187700)
(0.000000 2.973186 0.026814)->(0.000000 20.839114 0.160886)
(0.000000 2.977016 0.022984)->(0.000000 20.862098 0.137902)
(0.000000 2.980300 0.019700)->(0.000000 20.881798 0.118202)
(0.000000 2.983114 0.016886)->(0.000000 20.898684 0.101316)
(0.000000 2.985526 0.014474)->(0.000000 20.913158 0.086842)
(0.000000 2.987594 0.012406)->(0.000000 20.925564 0.074436)
(0.000000 2.989366 0.010634)->(0.000000 20.936198 0.063802)
(0.000000 2.990885 0.009115)->(0.000000 20.945312 0.054688)
(0.000000 2.992187 0.007813)->(0.000000 20.953125 0.046875)
(0.000000 2.993304 0.006696)->(0.000000 20.959821 0.040179)
(0.000000 2.994260 0.005740)->(0.000000 20.965561 0.034439)
(0.000000 2.995080 0.004920)->(0.000000 20.970481 0.029519)
(0.000000 2.995783 0.004217)->(0.000000 20.974698 0.025302)
(0.000000 2.996385 0.003615)->(0.000000 20.978313 0.021687)
(0.000000 2.996902 0.003098)->(0.000000 20.981411 0.018589)
(0.000000 2.997344 0.002656)->(0.000000 20.984066 0.015934)
(0.000000 2.997724 0.002276)->(0.000000 20.986343 0.013657)
(0.000000 2.998049 0.001951)->(0.000000 20.988294 0.011706)
(0.000000 2.998328 0.001672)->(0.000000 20.989966 0.010034)
(0.000000 2.998567 0.001433)->(0.000000 20.991399 0.008601)
(0.000000 2.998771 0.001229)->(0.000000 20.992628 0.007372)
(0.000000 2.998947 0.001053)->(0.000000 20.993681 0.006319)
(0.000000 2.999097 0.000903)->(0.000000 20.994584 0.005416)
(0.000000 2.999226 0.000774)->(0.000000 20.995358 0.004642)
(0.000000 2.999337 0.000663)->(0.000000 20.996021 0.003979)
(0.000000 2.999432 0.000568)->(0.000000 20.996589 0.003411)
(0.000000 2.999513 0.000487)->(0.000000 20.997077 0.002923)
(0.000000 2.999582 0.000418)->(0.000000 20.997494 0.002506)
(0.000000 2.999642 0.000358)->(0.000000 20.997852 0.002148)
(0.000000 2.999693 0.000307)->(0.000000 20.998159 0.001841)
(0.000000 2.999737 0.000263)->(0.000000 20.998422 0.001578)
(0.000000 2.999775 0.000225)->(0.000000 20.998647 0.001353)
(0.000000 2.999807 0.000193)->(0.000000 20.998841 0.001159)
(0.000000 2.999834 0.000166)->(0.000000 20.999006 0.000994)
(0.000000 2.999858 0.000142)->(0.000000 20.999148 0.000852)
(0.000000 2.999878 0.000122)->(0.000000 20.999270 0.000730)
(0.000000 2.999896 0.000104)->(0.000000 20.999374 0.000626)
(0.000000 2.999911 0.000089)->(0.000000 20.999464 0.000536)
(0.000000 2.999923 0.000077)->(0.000000 20.999540 0.000460)
(0.000000 2.999934 0.000066)->(0.000000 20.999606 0.000394)
(0.000000 2.999944 0.000056)->(0.000000 20.999662 0.000338)
(0.000000 2.999952 0.000048)->(0.000000 20.999710 0.000290)
(0.000000 2.999959 0.000041)->(0.000000 20.999752 0.000248)
(0.000000 2.999965 0.000035)->(0.000000 20.999787 0.000213)
(0.000000 2.999970 0.000030)->(0.000000 20.999817 0.000182)
(0.000000 2.999974 0.000026)->(0.000001 20.999843 0.000156)
(0.000000 2.999978 0.000022)->(0.000001 20.999865 0.000134)
(0.000000 2.999981 0.000019)->(0.000001 20.999884 0.000115)
(0.000000 2.999983 0.000016)->(0.000002 20.999900 0.000098)
(0.000000 2.999986 0.000014)->(0.000002 20.999913 0.000084)
(0.000000 2.999988 0.000012)->(0.000003 20.999924 0.000072)
(0.000000 2.999989 0.000010)->(0.000005 20.999933 0.000062)
(0.000001 2.999990 0.000009)->(0.000007 20.999940 0.000053)
(0.000001 2.999991 0.000008)->(0.000010 20.999945 0.000046)
(0.000001 2.999992 0.000007)->(0.000014 20.999947 0.000039)
(0.000002 2.999992 0.000006)->(0.000020 20.999947 0.000033)
(0.000003 2.999992 0.000005)->(0.000028 20.999943 0.000029)
(0.000004 2.999992 0.000004)->(0.000040 20.999935 0.000025)
(0.000006 2.999991 0.000004)->(0.000057 20.999922 0.000021)
(0.000008 2.999989 0.000003)->(0.000082 20.999900 0.000018)
(0.000012 2.999986 0.000003)->(0.000117 20.999868 0.000015)
(0.000017 2.999981 0.000002)->(0.000167 20.999820 0.000013)
(0.000024 2.999974 0.000002)->(0.000238 20.999750 0.000011)
(0.000034 2.999964 0.000002)->(0.000341 20.999650 0.000010)
(0.000049 2.999950 0.000001)->(0.000487 20.999505 0.000008)
(0.000070 2.999929 0.000001)->(0.000695 20.999298 0.000007)
(0.000099 2.999900 0.000001)->(0.000993 20.999001 0.000006)
(0.000142 2.999857 0.000001)->(0.001419 20.998576 0.000005)
(0.000203 2.999797 0.000001)->(0.002027 20.997969 0.000005)
(0.000290 2.999710 0.000001)->(0.002895 20.997101 0.000004)
(0.000414 2.999586 0.000001)->(0.004136 20.995861 0.000003)
(0.000591 2.999409 0.000000)->(0.005909 20.994088 0.000003)
(0.000844 2.999155 0.000000)->(0.008441 20.991557 0.000002)
(0.001206 2.998794 0.000000)->(0.012059 20.987939 0.000002)
(0.001723 2.998277 0.000000)->(0.017226 20.982772 0.000002)
(0.002461 2.997539 0.000000)->(0.024609 20.975389 0.000002)
(0.003516 2.996484 0.000000)->(0.035156 20.964843 0.000001)
(0.005022 2.994978 0.000000)->(0.050223 20.949776 0.000001)
(0.007175 2.992825 0.000000)->(0.071747 20.928252 0.000001)
(0.010250 2.989750 0.000000)->(0.102496 20.897503 0.000001)
(0.014642 2.985358 0.000000)->(0.146423 20.853577 0.000001)
(0.020918 2.979082 0.000000)->(0.209175 20.790824 0.000001)
(0.029882 2.970118 0.000000)->(0.298822 20.701178 0.000001)
(0.042689 2.957311 0.000000)->(0.426888 20.573112 0.000000)
(0.060984 2.939016 0.000000)->(0.609840 20.390160 0.000000)
(0.087120 2.912880 0.000000)->(0.871200 20.128800 0.000000)
(0.124457 2.875543 0.000000)->(1.244571 19.755428 0.000000)
(0.177796 2.822204 0.000000)->(1.777959 19.222041 0.000000)
(0.253994 2.746006 0.000000)->(2.539942 18.460058 0.000000)
(0.362849 2.637151 0.000000)->(3.628488 17.371512 0.000000)
(0.518355 2.481645 0.000000)->(5.183554 15.816446 0.000000)
(0.740508 2.259492 0.000000)->(7.405077 13.594923 0.000000)
(1.057868 1.942132 0.000000)->(10.578682 10.421318 0.000000)
(1.511240 1.488760 0.000000)->(15.044961 5.955039 0.000000)
(2.149280 0.850720 0.000000)->(17.597120 3.402879 0.000000)
(2.513874 0.486126 0.000000)->(19.055497 1.944502 0.000000)
(2.722214 0.277786 0.000000)->(19.888855 1.111144 0.000000)
(2.841265 0.158735 0.000000)->(20.365060 0.634940 0.000001)
(2.909294 0.090706 0.000000)->(20.637176 0.362823 0.000001)
(2.948168 0.051832 0.000000)->(20.792671 0.207327 0.000001)
(2.970382 0.029618 0.000000)->(20.881525 0.118473 0.000002)
(2.983075 0.016925 0.000000)->(20.932299 0.067699 0.000003)
(2.990328 0.009671 0.000000)->(20.961311 0.038685 0.000004)
(2.994473 0.005526 0.000001)->(20.977889 0.022106 0.000006)
(2.996841 0.003158 0.000001)->(20.987360 0.012632 0.000008)
(2.998194 0.001805 0.000001)->(20.992770 0.007218 0.000012)
(2.998967 0.001031 0.000002)->(20.995859 0.004125 0.000016)
(2.999408 0.000589 0.000002)->(20.997620 0.002357 0.000023)
(2.999660 0.000337 0.000003)->(20.998620 0.001347 0.000034)
(2.999803 0.000192 0.000005)->(20.999182 0.000770 0.000048)
(2.999883 0.000110 0.000007)->(20.999492 0.000440 0.000068)
(2.999927 0.000063 0.000010)->(20.999651 0.000251 0.000098)
(2.999950 0.000036 0.000014)->(20.999717 0.000144 0.000140)
(2.999960 0.000021 0.000020)->(20.999718 0.000082 0.000200)
(2.999960 0.000012 0.000029)->(20.999668 0.000047 0.000285)
(2.999953 0.000007 0.000041)->(20.999566 0.000027 0.000408)
(2.999938 0.000004 0.000058)->(20.999403 0.000015 0.000582)
(2.999915 0.000002 0.000083)->(20.999160 0.000009 0.000832)
(2.999880 0.000001 0.000119)->(20.998807 0.000005 0.001188)
(2.999830 0.000001 0.000170)->(20.998300 0.000003 0.001697)
(2.999757 0.000000 0.000242)->(20.997574 0.000002 0.002425)
(2.999653 0.000000 0.000346)->(20.996535 0.000001 0.003464)
(2.999505 0.000000 0.000495)->(20.995051 0.000001 0.004948)
(2.999293 0.000000 0.000707)->(20.992930 0.000000 0.007069)
(2.998990 0.000000 0.001010)->(20.989901 0.000000 0.010099)
(2.998557 0.000000 0.001443)->(20.985573 0.000000 0.014427)
(2.997939 0.000000 0.002061)->(20.979390 0.000000 0.020610)
(2.997056 0.000000 0.002944)->(20.970557 0.000000 0.029443)
(2.995794 0.000000 0.004206)->(20.957939 0.000000 0.042061)
(2.993991 0.000000 0.006009)->(20.939912 0.000000 0.060088)
(2.991416 0.000000 0.008584)->(20.914161 0.000000 0.085839)
(2.987737 0.000000 0.012263)->(20.877372 0.000000 0.122628)
(2.982482 0.000000 0.017518)->(20.824817 0.000000 0.175183)
(2.974974 0.000000 0.025026)->(20.749739 0.000000 0.250261)
(2.964248 0.000000 0.035752)->(20.642484 0.000000 0.357516)
(2.948926 0.000000 0.051074)->(20.489264 0.000000 0.510736)
(2.927038 0.000000 0.072962)->(20.270377 0.000000 0.729623)
(2.895768 0.000000 0.104232)->(19.957681 0.000000 1.042319)
(2.851097 0.000000 0.148903)->(19.510972 0.000000 1.489028)
(2.787282 0.000000 0.212718)->(18.872818 0.000000 2.127182)
(2.696117 0.000000 0.303883)->(17.961168 0.000000 3.038832)
(2.565881 0.000000 0.434119)->(16.658812 0.000000 4.341188)
(2.379830 0.000000 0.620170)->(14.798303 0.000000 6.201697)
(2.114043 0.000000 0.885957)->(12.140432 0.000000 8.859568)
(1.734347 0.000000 1.265653)->(8.343475 0.000000 12.656525)
(1.191925 0.000000 1.808075)->(4.767700 0.000000 16.232300)
(0.681100 0.000000 2.318900)->(2.724400 0.000000 18.275600)
(0.389200 0.000000 2.610800)->(1.556800 0.000000 19.443200)
(0.222400 0.000000 2.777600)->(0.889600 0.000000 20.110400)
(0.127086 0.000000 2.872914)->(0.508343 0.000000 20.491657)
(0.072620 0.000000 2.927380)->(0.290482 0.000000 20.709518)
(0.041497 0.000000 2.958503)->(0.165990 0.000000 20.834010)
(0.023713 0.000000 2.976287)->(0.094851 0.000000 20.905149)
(0.013550 0.000000 2.986450)->(0.054201 0.000000 20.945799)
(0.007743 0.000000 2.992257)->(0.030972 0.000000 20.969028)
(0.004425 0.000000 2.995575)->(0.017698 0.000000 20.982302)
(0.002528 0.000000 2.997472)->(0.010113 0.000000 20.989887)
(0.001445 0.000000 2.998555)->(0.005779 0.000000 20.994221)
(0.000826 0.000000 2.999174)->(0.003302 0.000000 20.996698)
(0.000472 0.000000 2.999528)->(0.001887 0.000000 20.998113)
(0.000270 0.000000 2.999730)->(0.001078 0.000000 20.998922)
(0.000154 0.000000 2.999846)->(0.000616 0.000000 20.999384)
(0.000088 0.000000 2.999912)->(0.000352 0.000000 20.999648)
(0.000050 0.000000 2.999950)->(0.000201 0.000000 20.999799)
(0.000029 0.000000 2.999971)->(0.000115 0.000000 20.999885)
(0.000016 0.000000 2.999984)->(0.000066 0.000000 20.999934)
(0.000009 0.000000 2.999991)->(0.000038 0.000000 20.999962)
(0.000005 0.000000 2.999995)->(0.000021 0.000000 20.999979)
(0.000003 0.000000 2.999997)->(0.000012 0.000000 20.999988)
(0.000002 0.000000 2.999998)->(0.000007 0.000000 20.999993)
(0.000001 0.000000 2.999999)->(0.000004 0.000000 20.999996)
(0.000001 0.000000 2.999999)->(0.000002 0.000000 20.999998)
(0.000000 0.000000 3.000000)->(0.000001 0.000000 20.999999)
(0.000000 0.000000 3.000000)->(0.000001 0.000000 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000000 21.000000)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000001 20.999999)
(0.000000 0.000000 3.000000)->(0.000000 0.000002 20.999998)
(0.000000 0.000000 3.000000)->(0.000000 0.000002 20.999998)
(0.000000 0.000000 3.000000)->(0.000000 0.000002 20.999998)
(0.000000 0.000000 3.000000)->(0.000000 0.000002 20.999998)
(0.000000 0.000000 3.000000)->(0.000000 0.000003 20.999997)
(0.000000 0.000000 3.000000)->(0.000000 0.000003 20.999997)
(0.000000 0.000000 3.000000)->(0.000000 0.000003 20.999997)
(0.000000 0.000000 3.000000)->(0.000000 0.000004 20.999996)
(0.000000 0.000001 2.999999)->(0.000000 0.000004 20.999996)
(0.000000 0.000001 2.999999)->(0.000000 0.000005 20.999995)
(0.000000 0.000001 2.999999)->(0.000000 0.000006 20.999994)
(0.000000 0.000001 2.999999)->(0.000000 0.000007 20.999993)
(0.000000 0.000001 2.999999)->(0.000000 0.000007 20.999993)
(0.000000 0.000001 2.999999)->(0.000000 0.000009 20.999991)
(0.000000 0.000001 2.999999)->(0.000000 0.000010 20.999990)
(0.000000 0.000001 2.999999)->(0.000000 0.000011 20.999989)
(0.000000 0.000002 2.999998)->(0.000000 0.000013 20.999987)
(0.000000 0.000002 2.999998)->(0.000000 0.000015 20.999985)
(0.000000 0.000002 2.999998)->(0.000000 0.000017 20.999983)
(0.000000 0.000002 2.999998)->(0.000000 0.000019 20.999981)
(0.000000 0.000003 2.999997)->(0.000000 0.000022 20.999978)
(0.000000 0.000003 2.999997)->(0.000000 0.000025 20.999975)
(0.000000 0.000004 2.999996)->(0.000000 0.000029 20.999971)
(0.000000 0.000004 2.999996)->(0.000000 0.000033 20.999967)
(0.000000 0.000005 2.999995)->(0.000000 0.000037 20.999963)
(0.000000 0.000005 2.999995)->(0.000000 0.000043 20.999957)
(0.000000 0.000006 2.999994)->(0.000000 0.000049 20.999951)
(0.000000 0.000007 2.999993)->(0.000000 0.000056 20.999944)
(0.000000 0.000008 2.999992)->(0.000000 0.000064 20.999936)
(0.000000 0.000009 2.999991)->(0.000000 0.000073 20.999927)
(0.000000 0.000010 2.999990)->(0.000000 0.000083 20.999917)
(0.000000 0.000012 2.999988)->(0.000000 0.000095 20.999905)
(0.000000 0.000014 2.999986)->(0.000000 0.000108 20.999892)
(0.000000 0.000015 2.999985)->(0.000000 0.000124 20.999876)
(0.000000 0.000018 2.999982)->(0.000000 0.000142 20.999858)
(0.000000 0.000020 2.999980)->(0.000000 0.000162 20.999838)
(0.000000 0.000023 2.999977)->(0.000000 0.000185 20.999815)
(0.000000 0.000026 2.999974)->(0.000000 0.000211 20.999789)
(0.000000 0.000030 2.999970)->(0.000000 0.000241 20.999759)
(0.000000 0.000034 2.999966)->(0.000000 0.000276 20.999724)
(0.000000 0.000039 2.999961)->(0.000000 0.000315 20.999685)
(0.000000 0.000045 2.999955)->(0.000000 0.000360 20.999640)
(0.000000 0.000051 2.999949)->(0.000000 0.000412 20.999588)
(0.000000 0.000059 2.999941)->(0.000000 0.000471 20.999529)
(0.000000 0.000067 2.999933)->(0.000000 0.000538 20.999462)
(0.000000 0.000077 2.999923)->(0.000000 0.000615 20.999385)
(0.000000 0.000088 2.999912)->(0.000000 0.000703 20.999297)
(0.000000 0.000100 2.999900)->(0.000000 0.000803 20.999197)
(0.000000 0.000115 2.999885)->(0.000000 0.000918 20.999082)
(0.000000 0.000131 2.999869)->(0.000000 0.001049 20.998951)
(0.000000 0.000150 2.999850)->(0.000000 0.001199 20.998801)
(0.000000 0.000171 2.999829)->(0.000000 0.001370 20.998630)
(0.000000 0.000196 2.999804)->(0.000000 0.001566 20.998434)
(0.000000 0.000224 2.999776)->(0.000000 0.001789 20.998211)
(0.000000 0.000256 2.999744)->(0.000000 0.002045 20.997955)

with tripling (renormalised after each step)

(1.000000 1.000000 1.000000)->(9.000000 8.000000 11.000000)
(0.964286 0.857143 1.178571)->(5.785714 7.285714 11.571429)
(0.704348 0.886957 1.408696)->(4.226087 8.504348 11.452174)
(0.524272 1.055016 1.420712)->(3.145631 9.902913 9.000000)
(0.428005 1.347424 1.224571)->(2.568032 10.287979 9.000000)
(0.352493 1.412149 1.235358)->(2.114959 10.590027 9.000000)
(0.292324 1.463723 1.243954)->(1.753941 10.830706 9.000000)
(0.243776 1.505335 1.250889)->(1.462657 11.024895 9.000000)
(0.204210 1.539249 1.256541)->(1.225260 11.183160 9.000000)
(0.171698 1.567116 1.261186)->(1.030187 11.313209 9.000000)
(0.144802 1.590170 1.265028)->(0.868810 11.420793 9.000000)
(0.122427 1.609348 1.268225)->(0.734564 11.510290 9.000000)
(0.103728 1.625376 1.270896)->(0.622370 11.585087 9.000000)
(0.088040 1.638823 1.273137)->(0.528242 11.647839 9.000000)
(0.074836 1.650141 1.275023)->(0.449014 11.700658 9.000000)
(0.063691 1.659694 1.276616)->(0.382145 11.745237 9.000000)
(0.054263 1.667775 1.277962)->(0.325578 11.782948 9.000000)
(0.046272 1.674624 1.279104)->(0.277632 11.814912 9.000000)
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