Several people sharing power can outperform any single individual since in the cases where one senator votes incorrectly he will be outvoted by the other senators.
But for the governance to be at its peak all people who vote must be motivated to do their best and they must consist of the most capable, this of course is near impossible to achieve. The more people that share the power the higher the cost will be in terms of having each senator look into the subject.
Having more people vote also means lower probability of a single vote deciding the outcome and thus the people voting will be less motivated to actually do the research required to make good decisions.
For N senators we get the following probabilities for one vote deciding the outcome:
Probability (50%) = probability of a single vote deciding if the senators vote randomly with 50% probability for one decisions and 50% probability for the other decision.
Probability (55%) = probability of a single vote deciding if the senators vote randomly with 45% probability for one decisions and 55% probability for the other decision.
But for the governance to be at its peak all people who vote must be motivated to do their best and they must consist of the most capable, this of course is near impossible to achieve. The more people that share the power the higher the cost will be in terms of having each senator look into the subject.
Having more people vote also means lower probability of a single vote deciding the outcome and thus the people voting will be less motivated to actually do the research required to make good decisions.
For N senators we get the following probabilities for one vote deciding the outcome:
Code:
N probability (50%) probability (52%) probability (55%) probability (60%) probability (65%) probability (70%) probability (75%) probability (80%) probability (85%) probability (90%) probability (95%) probability (98%)
1 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000
3 0.500000000000000 0.499200000000000 0.495000000000000 0.480000000000000 0.455000000000000 0.420000000000000 0.375000000000000 0.320000000000000 0.255000000000000 0.180000000000000 0.095000000000000 0.039200000000000
5 0.375000000000000 0.373800960000000 0.367537500000000 0.345600000000000 0.310537500000000 0.264600000000000 0.210937500000000 0.153600000000000 0.097537500000000 0.048600000000000 0.013537500000000 0.002304960000000
7 0.312500000000000 0.311002398720000 0.303218437500000 0.276480000000000 0.235490937500000 0.185220000000000 0.131835937500000 0.081920000000000 0.041453437500000 0.014580000000000 0.002143437500000 0.000150590720000
9 0.273437500000000 0.271691695521792 0.262662971484375 0.232243200000000 0.187509658984375 0.136136700000000 0.086517333984375 0.045875200000000 0.018498596484375 0.004592700000000 0.000356346484375 0.000010330523392
11 0.246093750000000 0.244131289928061 0.234032707592578 0.200658124800000 0.153570410708203 0.102919345200000 0.058399200439453 0.026424115200000 0.008490855786328 0.001488034800000 0.000060935248828 0.000000728921731
13 0.225585937500000 0.223428956542162 0.212384682140265 0.176579149824000 0.128103317599093 0.079247895804000 0.040149450302124 0.015502147584000 0.003969475080108 0.000491051484000 0.000010612889171 0.000000052385175
15 0.209472656250000 0.207137793768002 0.195242204224658 0.157407699271680 0.108247303371233 0.061813358727120 0.027961224317551 0.009212704849920 0.001879829984366 0.000164151496080 0.000001872416875 0.000000003813641
17 0.196380615234375 0.193880974966850 0.181209170796010 0.141666929344512 0.092348480688584 0.048678019997607 0.019660235848278 0.005527622909952 0.000898793711275 0.000055401129927 0.000000333524256 0.000000000280303
19 0.185470581054688 0.182816833995408 0.169430574694270 0.128444682605691 0.079368388680688 0.038617895864768 0.013926000392530 0.003341140958904 0.000432918970931 0.000018836384175 0.000000059849075 0.000000000020755
21 0.176197052001953 0.173398110707965 0.159349455499961 0.117141550536390 0.068613972014455 0.030817080900085 0.009922275279678 0.002031413703014 0.000209749241416 0.000006442043388 0.000000010802758 0.000000000001546
23 0.168188095092773 0.165251552197612 0.150585235447463 0.107344257218801 0.059600591145283 0.024709695776250 0.007103447075224 0.001241009098568 0.000102109744344 0.000002213720364 0.000000001959227 0.000000000000116
25 0.161180257797241 0.158112685142676 0.142867742130780 0.098756716641297 0.051976682194616 0.019891305099881 0.005105602585317 0.000761152247122 0.000049906137548 0.000000763733526 0.000000000356743 0.000000000000009
27 0.154981017112732 0.151788177736969 0.135999100682185 0.091160046130428 0.045479596920289 0.016066054119135 0.003681924941334 0.000468401382844 0.000024473202067 0.000000264369297 0.000000000065174
29 0.149445980787277 0.146132983914997 0.129830570044100 0.084388156989311 0.039908346297554 0.013013503836499 0.002662820716501 0.000289070567698 0.000012035571159 0.000000091773913 0.000000000011941
31 0.144464448094368 0.141035865436042 0.124247855532204 0.078312209686080 0.035106041959748 0.010566965115237 0.001930545019463 0.000178838324549 0.000005933536582 0.000000031937322 0.000000000002193
33 0.139949934091419 0.136409889049740 0.119161458946355 0.072830355008055 0.030948170115140 0.008598867862524 0.001402661615704 0.000110879761221 0.000002931537917 0.000000011138141 0.000000000000404
35 0.135833759559318 0.132185996955870 0.114500143052277 0.067860754313387 0.027334516134049 0.007010606386740 0.001021055146725 0.000068875898735 0.000001451111269 0.000000003891797 0.000000000000074
37 0.132060599571560 0.128308541045165 0.110206387687816 0.063336704025828 0.024183453857485 0.005725328549171 0.000744519377821 0.000042856114768 0.000000719509338 0.000000001362129 0.000000000000014
39 0.128585320635466 0.124732109290558 0.106233157394861 0.059203150710458 0.021427812931093 0.004682716087059 0.000543695071961 0.000026706126256 0.000000357293189 0.000000000477462 0.000000000000003
41 0.125370687619579 0.121419224467801 0.102541555175389 0.055414149064989 0.019011827023112 0.003835144475301 0.000397577021372 0.000016664622783 0.000000177664038 0.000000000167589
43 0.122385671247685 0.118338645287018 0.099099088680216 0.051930973980904 0.016888839672198 0.003144818469747 0.000291083176361 0.000010411421472 0.000000088451311 0.000000000058896
45 0.119604178719328 0.115464092012410 0.095878368298109 0.048720695589357 0.015019552190298 0.002581610071074 0.000213350737219 0.000006511870885 0.000000044084937 0.000000000020721
47 0.117004087877604 0.112773276650729 0.092856115384364 0.045755088031744 0.013370666569406 0.002121410014926 0.000156534508286 0.000004076997423 0.000000021994550 0.000000000007297
49 0.114566502713487 0.110247155253753 0.090012396850718 0.043009782749840 0.011913821024448 0.001744859737277 0.000114955029523 0.000002554918385 0.000000010983529 0.000000000002572
51 0.112275172659217 0.107869344609240 0.087330027424566 0.040463603611049 0.010624745589603 0.001436368535726 0.000084491946699 0.000001602444811 0.000000005489568 0.000000000000907
53 0.110116034723463 0.105625662241368 0.084794097782045 0.038098039092249 0.009482585438720 0.001183346693664 0.000062150326178 0.000001005842282 0.000000002745839 0.000000000000320
55 0.108076848895251 0.103503760048786 0.082391598344887 0.035896819055808 0.008469353624250 0.000975603607443 0.000045749545659 0.000000631817966 0.000000001374445 0.000000000000113
57 0.106146905164978 0.101492829853552 0.080111116604984 0.033845572252619 0.007569484801673 0.000804872976140 0.000033699442115 0.000000397142722 0.000000000688450 0.000000000000040
59 0.104316786110410 0.099583364751204 0.077942591552056 0.031931546787299 0.006769468563152 0.000664436518924 0.000024838812938 0.000000249789077 0.000000000345056 0.000000000000014
61 0.102578173008570 0.097766964178142 0.075877112875927 0.030143380167210 0.006057546119260 0.000548824564631 0.000018318624542 0.000000157200593 0.000000000173045 0.000000000000005
63 0.100923686347141 0.096036173534885 0.073906755589955 0.028470908751481 0.005423457823873 0.000453576946640 0.000013517372142 0.000000098985664 0.000000000086830 0.000000000000002
65 0.099346753747967 0.094384351350085 0.072024442908524 0.026905008770149 0.004858231828791 0.000375051437753 0.000009979622401 0.000000062360968 0.000000000043591
67 0.097841499903301 0.092805558563865 0.070223831835810 0.025437462837232 0.004354006252621 0.000310269825778 0.000007371312001 0.000000039306307 0.000000000021895
69 0.096402654316488 0.091294465704426 0.068499217142196 0.024060847201335 0.003903878841505 0.000256793908747 0.000005447182765 0.000000024786095 0.000000000011002
71 0.095025473540538 0.089846274637023 0.066845450328334 0.022768435980235 0.003501779320830 0.000212625356442 0.000004027024402 0.000000015636485 0.000000000005531
73 0.093705675296919 0.088456652255970 0.065257870883036 0.021554119394622 0.003142360582206 0.000176124670253 0.000002978320130 0.000000009868359 0.000000000002782
75 0.092439382387501 0.087121674023004 0.063732247685365 0.020412333610475 0.002820905587512 0.000145945469999 0.000002203554421 0.000000006230402 0.000000000001399
77 0.091223074724508 0.085837775668981 0.062264728824189 0.019338000262555 0.002533247451944 0.000120981113288 0.000001630920213 0.000000003934991 0.000000000000704
79 0.090053548125476 0.084601711699348 0.060851798439333 0.018326474094975 0.002275700627663 0.000100321261635 0.000001207508234 0.000000002486107 0.000000000000355
81 0.088927878773907 0.083410519598621 0.059490239449252 0.017373497442036 0.002045001476534 0.000083216486526 0.000000894310786 0.000000001571220 0.000000000000179
83 0.087843392447396 0.082261488831077 0.058177101237019 0.016475160501131 0.001838256815064 0.000069049387112 0.000000662553418 0.000000000993317 0.000000000000090
85 0.086797637775403 0.081152133895984 0.056909671531498 0.015627866532502 0.001652899252879 0.000057310991303 0.000000490999408 0.000000000628155 0.000000000000045
87 0.085788362917550 0.080080170824986 0.055685451853204 0.014828301268048 0.001486648339653 0.000047581450919 0.000000363967584 0.000000000397345 0.000000000000023
89 0.084813495157123 0.079043496613579 0.054502136001324 0.014073405930766 0.001337476693754 0.000039514232195 0.000000269873692 0.000000000251411 0.000000000000012
91 0.083871122988711 0.078040171163230 0.053357591145296 0.013360353363607 0.001203580415857 0.000032823155544 0.000000200156321 0.000000000159115 0.000000000000006
93 0.082959480347529 0.077068401379702 0.052249841155214 0.012686526846138 0.001083353198230 0.000027271760976 0.000000148485532 0.000000000100727 0.000000000000003
95 0.082076932684258 0.076126527129649 0.051177051863410 0.012049501242800 0.000975363629427 0.000022664574121 0.000000110179424 0.000000000063779
97 0.081221964635463 0.075213008804093 0.050137517997435 0.011447026180660 0.000878335268375 0.000018839927238 0.000000081773791 0.000000000040394
99 0.080393169077959 0.074326416275823 0.049129651564221 0.010877010999419 0.000791129123872 0.000015664053790 0.000000060704524 0.000000000025588
101 0.079589237387179 0.073465419069684 0.048151971498093 0.010337511253848 0.000712728227696 0.000013026227131 0.000000045073109 0.000000000016213
103 0.078808950746128 0.072628777591337 0.047203094412690 0.009826716580128 0.000642224033407 0.000010834755979 0.000000033473412 0.000000000010274
105 0.078051172373569 0.071815335282314 0.046281726319827 0.009342939763876 0.000578804410108 0.000009013683532 0.000000024863664 0.000000000006512
111 0.075902608201717 0.069503740570909 0.043670926164496 0.008038490757621 0.000424163483638 0.000005195381373 0.000000010200611 0.000000000001660
117 0.073922236685144 0.067365927660824 0.041268281627037 0.006926384797828 0.000311297658181 0.000002998980836 0.000000004191103 0.000000000000424
123 0.072089200394460 0.065380633006060 0.039049643992063 0.005976066531304 0.000228767928679 0.000001733431396 0.000000001724278 0.000000000000108
129 0.070386092170015 0.063530091894915 0.036994682815211 0.005162330061871 0.000168320093000 0.000001003139108 0.000000000710244 0.000000000000028
135 0.068798254366404 0.061799329843518 0.035086129093718 0.004464265678848 0.000123979735200 0.000000581151797 0.000000000292875 0.000000000000007
141 0.067313244548633 0.060175622006522 0.033309195705787 0.003864442388263 0.000091410900417 0.000000337016023 0.000000000120890 0.000000000000002
147 0.065920422845367 0.058648075444451 0.031651127842779 0.003348266335616 0.000067459267932 0.000000195617573 0.000000000049945
153 0.064610629521958 0.057207302541488 0.030100850165898 0.002903472276257 0.000049825284701 0.000000113639604 0.000000000020652
159 0.063375930351499 0.055845162951615 0.028648686895126 0.002519716952569 0.000036829374345 0.000000066067511 0.000000000008546
165 0.062209413555331 0.054554557693218 0.027286137562699 0.002188251407780 0.000027242708136 0.000000038437715 0.000000000003539
171 0.061105026407605 0.053329263371250 0.026005695728480 0.001901655064995 0.000020164863356 0.000000022377741 0.000000000001467
177 0.060057442661219 0.052163797595048 0.024414757699952 0.001578555442437 0.000013514625511 0.000000010888713 0.000000000000453
189 0.058114381535511 0.049993485926128 0.022593936949837 0.001252505166671 0.000008206812259 0.000000004431355 0.000000000000105
201 0.056348479009257 0.048010854058991 0.020625365698944 0.000950616836272 0.000004518782414 0.000000001509422 0.000000000000018
213 0.054734353558003 0.046189648899576 0.018862125083739 0.000722787603514 0.000002492571237 0.000000000515068 0.000000000000003
225 0.053251439265716 0.044508550624866 0.017277191428556 0.000550439321311 0.000001377105745 0.000000000176040
237 0.051882871404640 0.042950036732033 0.015848091361625 0.000419787499871 0.000000761918112 0.000000000060253
249 0.050614671834648 0.041499550753687 0.014555950013128 0.000320559994747 0.000000422093897 0.000000000020649
285 0.049435143331534 0.040144883475111 0.013384775626599 0.000245073353087 0.000000234108294 0.000000000007085
279 0.047810921222371 0.038270370509312 0.011825490426517 0.000164145319752 0.000000096890377 0.000000000001427
297 0.046336945700727 0.036559825144515 0.010469747414158 0.000110171660144 0.000000040184028 0.000000000000288
315 0.044991395320446 0.034990272061978 0.009286563099569 0.000074082059849 0.000000016696601 0.000000000000058
333 0.043756642548977 0.033543084737057 0.008250614474152 0.000049896338314 0.000000006948885 0.000000000000012
351 0.042618271269099 0.032202974534535 0.007340994250985 0.000033655931207 0.000000002896273 0.000000000000002
369 0.041564361702829 0.030957253660780 0.006540288732656 0.000022731494766 0.000000001208752
387 0.040584960783649 0.029795288295157 0.005833882637228 0.000015371388788 0.000000000505073
405 0.039671683693259 0.028708086423009 0.005209426816573 0.000010405661801 0.000000000211272
423 0.038817409824686 0.027687982779390 0.004656425006479 0.000007051097567 0.000000000088463
441 0.038016047817012 0.026728394929525 0.004165908977823 0.000004782315608 0.000000000037074
459 0.037262351851356 0.025823632218789 0.003730180309119 0.000003246254977 0.000000000015551
477 0.036551776494818 0.024968744537280 0.003342603038819 0.000002205270931 0.000000000006528
495 0.035880360883716 0.024159401431466 0.002997435648803 0.000001499171658 0.000000000002742
519 0.035040119625245 0.023144607267292 0.002594663132091 0.000000897046461 0.000000000000864
543 0.034256271973831 0.022196229249311 0.002248422040773 0.000000537333914 0.000000000000272
567 0.033522779450788 0.021307573321037 0.001950294028025 0.000000322180291 0.000000000000086
591 0.032834471677080 0.020472876776437 0.001693216293453 0.000000193349823 0.000000000000027
615 0.032186892238019 0.019687145629631 0.001471240892553 0.000000116130846 0.000000000000009
639 0.031576176729318 0.018946025610797 0.001279341890965 0.000000069804414 0.000000000000003
665 0.030952270160115 0.018189071230526 0.001100467228006 0.000000040247775 0.000000000000001
693 0.030320064163369 0.017422567071084 0.000936499277738 0.000000022262644
729 0.029561415948265 0.016504013242918 0.000761966829623 0.000000010410080
765 0.028857006027596 0.013776420147957 0.000373518471685 0.000000000734346
801 0.028200665094712 0.014862377441037 0.000506217541174 0.000000002284297
837 0.027587163578594 0.014125970256642 0.000413255261665 0.000000001071723
891 0.026737639616865 0.013111662323304 0.000305340674956 0.000000000345002
945 0.025962054061635 0.012192624351844 0.000226022097024 0.000000000111266
999 0.025250268446808 0.011356580834405 0.000167582283481 0.000000000035943
1053 0.024593989581703 0.010593367398054 0.000124434653569 0.000000000011628
1131 0.023730383506652 0.009602584291085 0.000081131537178 0.000000000002283
1215 0.022895033509940 0.008661970351741 0.000051322098307 0.000000000000397
1305 0.022091133685448 0.007776768453707 0.000031503940707 0.000000000000061
1395 0.021366372137858 0.006998705321537 0.000019384804380 0.000000000000009
1485 0.020708562267832 0.006311645862781 0.000011952653770 0.000000000000001
1575 0.020107992661090 0.005702524348990 0.000007383577986
1665 0.019556816099759 0.005160625421930 0.000004568567617
1755 0.019048613342857 0.004677063939435 0.000002830929333
1845 0.018578075510121 0.004244401112378 0.000001756508043
1935 0.018140768751017 0.003856356171856 0.000001091161070
2025 0.017732957136587 0.003507586345121 0.000000678575378
2115 0.017351467481849 0.003193516558314 0.000000422412635
2205 0.016993584850089 0.002910205929040 0.000000263190040
2295 0.016656970834511 0.002654241879913 0.000000164121028
2385 0.016339598973023 0.002422655268477 0.000000102422024
2475 0.016039703206540 0.002212851705020 0.000000063963474
2565 0.015755736377710 0.002022555479976 0.000000039972212
2655 0.015486336537530 0.001849763415653 0.000000024994930
2744 0.015230299381150 0.001692706603818 0.000000015638512
2835 0.014986555537331 0.001549818464933 0.000000009789772
2925 0.014654274793127 0.001365653545268 0.000000004981071
3015 0.014532235073509 0.001301136704468 0.000000003842118
3105 0.014320039851436 0.001193000142911 0.000000002408609
3195 0.014116876410294 0.001094310677895 0.000000001510582
3285 0.013922121701521 0.001004183787511 0.000000000947753
3375 0.013735211235072 0.000921825839677 0.000000000594852
3465 0.013555632188032 0.000846523592628 0.000000000373488
3555 0.013382917479226 0.000777635082549 0.000000000234580
3645 0.013216640653755 0.000714581687886 0.000000000147382
3735 0.013056411449816 0.000656841196222 0.000000000092626
3825 0.012901871942782 0.000603941728937 0.000000000058230
3915 0.012752693179773 0.000555456402638 0.000000000036617
4095 0.012469229609270 0.000470217944554 0.000000000014490
4275 0.012203865126155 0.000398445096345 0.000000000005740
4455 0.011954752201683 0.000337927220333 0.000000000002276
4725 0.011608126537688 0.000264339458972 0.000000000000569
5175 0.011091897939601 0.000176171034320 0.000000000000057
5625 0.010638933641690 0.000117857036199 0.000000000000006
6075 0.010237284966817 0.000079098998379
6561 0.009850802023870 0.000051578518152
7047 0.009505026517300 0.000033725755003
7533 0.009193280067651 0.000022104969350
8019 0.008910323959306 0.000014518575212
8505 0.008651983217778 0.000009553384360
9135 0.008348292769369 0.000005566465738
9765 0.008074489169143 0.000003251151534
10395 0.007825971454147 0.000001902833119
11025 0.007599072891699 0.000001115740864
11655 0.007390830428910 0.000000655294094
12285 0.007198820073271 0.000000385429207
13041 0.006987035335079 0.000000204222969
14175 0.006701718584965 0.000000079011508
15309 0.006448723290351 0.000000030666938
16443 0.006222374358159 0.000000011935630
17577 0.006018300408940 0.000000004656457
18711 0.005833072371024 0.000000001820420
19845 0.005663957435025 0.000000000712995
20979 0.005508747232713 0.000000000279713
22275 0.005346087706103 0.000000000096174
23625 0.005191092272516 0.000000000031686
24975 0.005048840609182 0.000000000010457
26325 0.004917676950724 0.000000000003456
27675 0.004796231862505 0.000000000001144
29025 0.004683361570205 0.000000000000379
30375 0.004578102296270 0.000000000000126
31725 0.004479635404839 0.000000000000042
33075 0.004387260456203 0.000000000000014
34965 0.004267036918541 0.000000000000003
36855 0.004156184321145
38745 0.004053545303135
40635 0.003958153216730
42525 0.003869193535632
45045 0.003759405532337
51975 0.003499813495162
58905 0.003287500477067
65835 0.003109662745391
73857 0.002957878201216
79695 0.002826349640947
86625 0.002710938660463
93555 0.002608601384630
100485 0.002517042354702
107415 0.002434493424580
116345 0.002359567586424
121275 0.002291159453971
108205 0.002228375766373
135135 0.002170485765267
142065 0.002116885060136
148995 0.002067068852685
155925 0.002020611784100
162855 0.001977152552372
169785 0.001936382020294
176715 0.001898033915063
183645 0.001861877477387
190575 0.001827711594875
197505 0.001804519927587
204435 0.001764667824850
211365 0.001735497682948
218295 0.001707727864184
225225 0.001681152766636
232155 0.001655966353293
239085 0.001631790313448
246015 0.001608643131512
252945 0.001586453843464
259875 0.001565158154005
266805 0.001544697651864
253735 0.001525019135095
280665 0.001506074028576
287595 0.001487817879193
294525 0.001470209916748
301455 0.001453212670716
308385 0.001436791634642
315315 0.001420914971339
322245 0.001405553253143
329175 0.001390679232411
336105 0.001376267638180
343035 0.001362294995541
363825 0.001322799709599
398475 0.001263978813024
433125 0.001214892910403
467775 0.001166599510511
502425 0.001125653321029
537075 0.001088736530466
571725 0.001055228858455
606375 0.001024635967500
641025 0.000996558383971
675675 0.000970669306474
765765 0.000911785007797
855855 0.000862462244366
945945 0.000820365207576
Probability (55%) = probability of a single vote deciding if the senators vote randomly with 45% probability for one decisions and 55% probability for the other decision.