The mathematics of voting, power and sharing

 !
" # $ %
&')( *,+.-(0/13254)(76
8:9<;>=?;A@CBEDGFIHJ9IKMLIHONGP>QINGNC;ALRNCSUTWV?@XHO9YDZFIHONXQI9IH[D3\IV^]J]_=?S?DZ;>@GN`FaV^Lb;>cEQaV?]dNeDfV?9ILgHJ9IKIhi`FgHJN`HJNj9IS?DXDG@GQI;kHO9
V?]O]_=lS^DZHJ9gKmNCHODZQIV^DZHOSl9INA\gV?NnNGFISpon9rqEBsDZFI;kTtSl]O]JSponHJ9IKu;wvgV?xzyI]J;>NA{
|~}€I,‚„ƒt…‡†?ˆ
‰ CN FaV?@G;>FISl]OLI;A@CNA{DZFI;>HJ@X=?S?DZ;.HJNnyI@CSlyŠSl@CDGHJSl9aV^]ŠDGSzDGFI;.9‡QgxUq_;>@nS?T„NCFaV?@C;ANnDZFI;wB‹FISl]OLdh
‰HOŒ LI]J;A;>9‘P>DZDGHJSlV?@Z]V?]P3V^Žj9IS?LI]JHJ]JLI;>KlV^;?DZ;^{—˜xmV?V?NU9‘VbB‹@GNC;ADZNCV^QIDZ;>]ODAN~\„@GNe;>DfcEV^QIDG;AHJ@CNu;’TtDGQIFa9gV^P>D~DZHOV?Sl]J9,]“DG]OFIHJ™^;A;MHO@n=?LIS?;ADZ]O;>;A@GK‘NuV^onDG;AH[NnDZF=lS?QIDG9I;”;ATtcESlQa@`V^DG]šFIoš;”;ANGHJV?K?xzFEDG;”NA\yIV^@C;A9IN–L •
DGF‡QgN›Qg9I;AcEQaV?]œHOxmyŠSl@CDZV?9IP>;0HJ9‹DZFg;’;>9ILR@C;ANCQI]ODAh
‰ DGŽjFISl;>QgB<9EV^D:Bu]J]“q_=?SlS?V?DZ;k@GLIDZN>Fg{;’NCSlNGV?xzxz;j;.DZSpojonV39IBlNC\IFIDGHJFIyIHJN„NnFaKlV3H[=l=l;A;`N`xzDGFISl;.@G;šLI@GHO;A¡_yg;A@G@C;A;ANC9‘;AD`9‘DfDZSpVžonDZHO=?9I;ANCNFIHJDZyIFIN›V?9ŸQI9IS?;>DZcEFgQa;AV?@GN>]œhy_ ~SpošNCNG;AQI@>xzh HJ9gK~DZFaVžD
7¢ Spo£P3V?9bSl9I;.xz;3V?NCQI@G;kDZFgHJN`yŠS¤oš;A@Z¥R8¦D›HJN`9IS?D›NCHJxzyI]OBmyI@GS?y_Sl@eDZHOSl9aV?]ŠDZSŸDZFI;.9EQIxUqŠ;A@nS?T=lS^DZ;AN>{
|~}€I,‚„ƒt…§ˆj8:9<VuNCFaV?@C;AFISl]OLI;A@CNA¨Ixz;A;wDZHO9IKI\EDZFI;>@G;kV?@C;k©uyaV^@CDZHOPAHOyaV?9‘DZN>hª«FaV?Nj¬E­l®S?T€DZFg;kNGFaV?@C;AN>\a¯
FaV?Nn¬‘°l®‹\§V^9ILR±DGFI;.@G;>xsV?HO9IHJ9gKM²l®Yhš ³xmV¤´CSl@CHOD:BsS?T²‡µ¤®¶HJN`9g;A;ALg;ALRDGSzyIV?NGN7V?9‘B‹xz;3V?NCQI@G;^hš 79EB
Kl@CSlQIybS?T˜·ŸP3V?9bTtSl@GP>;kDZFI;’xm;AV?NGQg@G;kDZSzyaV?NCN~Sp=l;>@›DGFI;”S?yIy_S?NGHODGHJSl9‹S^T„DZFg;’DGFIHJ@CLdhX¸‡Smª”\_¯m\§±¹FIVp=?;
;AcEQaV^]€y_Spoš;A@`•LI;ANCyIHODG;.DGFI;AHO@`QI9I;>c‡QIV?]œ9‡QgxUq_;>@nS?T„NCFaV?@C;ANAh
i`FI;>@G;’;wvºHJNeD7NC;>=?;A@ZV^]NCPfFI;Axz;ANnDZSMDZ@eBsDGSmxz;3V?NCQI@G;kDGFIHJNz»CyŠSpoj;>@G¼zS?TDZFI;.yaV?@eDZHOPAHJyIV?9‘DZNAh
½ 9I;nS?T“DGFI;nxmS?NCDonHJLg;A]OBŸV?PAP>;AygDG;AL5¾WV?xzSl9IK.y_;>SlyI]O;XonFIS”™E9ISpo¿V^q_SlQºDDGFIHJN„•9IS^DV?N˜xsV?9‘BMV^N˜DGFI;A@C;
NGFgSlQI]JL<qŠ;?ÀÂÁMHONXDZFI;’ÃXV^9IÄAFaVžTyŠS¤oš;A@nHO9ILI;wv_h
ųÆaǤÈtÉggǤÈtƊʐˆ‹Ë¿FI;>9ÌLIS)B?SlQ!FaV3=l;Ì»CyŠS¤oš;A@C¼E¥Ë¿FI;A9BlS?QI@MLI;AP>HJNGHOSl9xmV^DGDG;A@GN>hi`FIV^DMHJN>\˜onFg;A9
onFI;wDZFI;>@uBlSlQ=lS?DG;sSl9I;[email protected];mS?DZFg;A@’xsV?™^;ANuVrLIH[¡“;>@G;A9gPA;sHO9DGFI;mS?QgDZP>Slxm;^\„Sl@>\onFI;>9,BlS?QI@
=lS^DZ;.HJNnV»CNConHO9IKl¼Ÿ=lS^DZ;?h
¸ºSM]J;wDnQIN›LI;>Ía9g;.B?SlQI@›y_Spoš;A@nHJ9gLI;wvbV^NnDGFI;kTt@ZV?PwDZHOSl9
ΛQIxUqŠ;A@nS?TPAS‘V?]OHODGHJSl9INonFI;A@C;’B?SlQrV?@G;.VMNConHO9IKŸ=lS?DG;
iS?DZV?]€9EQIxUqŠ;A@nS?TPAS‘V^]JHODGHJSl9gN
µ
|~}€I,‚„ƒt…§ˆ
‰ 8:9<DZFI;.PAV?NG;’V?qŠSp=l;s¾Wªu{ ¬E­l®Y\a¯m{ ¬‘°l®‹\§±M{²l®MÁXDGFI;.y_S?NGNGHOqI]J;~PAS‘V?]OHODGHJSl9INXV?@G;
µ ª0¯U
± · ª0¯ ± © ª0± ¯ ¬ ¯u± ª
² ª ¯U± ¯ ª0± ­ ± ª0¯ ° sª0¯U±
8:9bP3V?NC;AN’µl\°º{9ISlqŠSºLgBYHJNnVŸNConHO9IKz=?S?DZ; 8:9bP3V?NC;AN~·º\ ­‡{Iª f¯ V?@C;’qŠS?DGFbNeonHJ9IKŸ=?S?DZ;>N 8¦DnTtSl]O]JSponNjDZFIV^Dnª f¯ V?9ILb±FaV3=?;’DGFI;.NZV?xz;
8:9bP3V?NC;AN7©g\ º{Iª f±V^@G;kqŠS?DZFbNConHO9IKŸ=lS?DG;AN yŠSpoj;>@7HO9ILI;wv ¬ ^° ²
8:9bP3V?NC;AN›¬I\ ²‡{a¯ f±V?@C;kq_S^DZFbNConHJ9gKM=?S?DZ;>N ‰ Ë¿V?]ONGFISM;wDZSlFI9‹;>on@`FaBlSlV^QrD›xmV?@GV¤;.´CSlVŸ@CHONeD:onBmHJ9IHJN`KŸ9I=l;>S^;ADZLI;.;>LrSl@`DZ9ISzS?@GDn;AV?LgPf;AFryŠ;AVM9ILgLI;AN›PA9IHONGS?HODnSl9dS?h 9I]OBmSl9‹BlSlQg@›9EQIxuq_;>@›S^TNCFaV?@C;ANA\§qIQºD
8¦TVuxz;3V?NCQI@G;~P3V?9‹qŠ;0yaV?NCNG;>LrHO9sDZFI;~;wvgV?xzyI]J;~V?q_Sp=?;kSl9I]OBMonFI;>9<HODXFIV?Nn²?©‘®S?T€DGFI;~=lS^DZ;AN`S?@
xzSl@G;^\gDZFIV?9‹DZFI;.NGH[DZQaVžDZHJS?9‹PfFaV?9IKl;>NU¾ Ía]J]“HJ9‹DZFI;’V?9gNCoš;A@GNZÁw{
²Y{
µz{ ÿNConHO9IKŸ=lS?DG;
·‹{
‹{
©<{
°‹­Y{{
¬b{
yŠSpoj;>@7HO9ILI; v‹S?Tª yŠSpoj;>@7HO9ILI; v‹S?T„¯
yŠSpoj;>@7HO9ILI; v‹S?T„±
rÆaÇpºÇ¤ÈWƊʘˆ
, {
Ec QISl@CQIx
9I;A;>LI;ALbDZSzyaV?NGN
VŸxm;AV?NGQg@G;
!" $# - ,/.
oš;AHOKlF‘DZNnS?T
DGFI;k=lS?DG;A@GN
&¾ %'(%(")%*!+ Á
- ,/.
yŠS¤oš;A@nHO9ILIHJP>;AN
S?TDZFg;.=?S?DZ;>@GN
8:9<DZFg;.D:ošSz;wvgV?xzyI]J;>NnV?q_Sp=?;?{ ²ºµ.{‘¬E­0Z¬‘°0 ² # ¾ ²0 ²0 ²lÁ
²?©Ÿ{‘¬E0­ Z¬‘0° ² # ¾ Á
·
~9b;wvgV?xzyI]O;0 onHODZF<¬Ÿ=lS^DZ;A@CNA{
|~}€I,‚„ƒt…§ˆ ²ºµ.{‘¬ f© 0 · 3µ # ˜¾ ¥ 7¥ ~ ¥ 0¥)Á
P>HJ@GP>]J;~=?S?DZ;>N ª
¯
±
DGFaV^D›V?@C;
¬
© ·
NeonHJ9IKŸ=lS^DZ;AN
©
Á
µ
°
²ºµ’{l¬0f© · 3µ
# ¾ µ
‡¸ DG;>=l;>9 Ãj@ZV^xr¨ N7qŠS‡Sl™§\›V^DZHOSl9aV?]Sl]OHODZHOPAN>\ŠP>Sl9‘DfV?HO9IN7V?9)HJ9‘DG;A@G;>NCDGHJ9IKmLIHJNCPAQINCNGHJS?9RS?TFgS¤o³DGFI;uyŠSpoj;>@
HJ9gLI;wv”S?TaDGFI;jNCxsV?]O]EPASlQI9‘DG@CB”S?T€Qºvº;AxUqŠSlQI@CK~oš;A9‘DDZFg@GSlQIK?FUNGSlxz;šB?S^•¦B?SkxmS^DZHJS?9INV?NDGFI; Œ QI@CSly_;AV?9
PAS?xmxUQg9IHOD:Bs;wvºyaV?9ILg;ALd{
jŽ SlQI9‘DG@CB
Ë ;AHOKlF‘D
a@ZV?9gPA;
¬
;A@GxmV?9‘B
¬
8¦DfV^]OB
¬
Ãj;>]JKlHOQIx
·
Î7;wDZFI;>@G]JV?9ILIN ·
€Qºvº;AxUqŠSlQI@CK
µ
Œ09I;A9gKl]JxsV?9IV?L@C™
8:@G;>] V?9gL
@G;A;>PA;
Spoš;A@
ºh©gµ3©
ºh©gµ3©
ºh©gµ3©
ºhOµp°l­
ºhOµp°l­
ºh Ë;AHOKlF‘D
µ
µ
µ
²
²
·
µ
©
©
Spoj;>@
gh[µ gh[µ gh[µ gh[
µ gh[
µ
gh l°
gh[µ gh ‘­?°
gh ‘­?°
Ë,;>HJKlF‘D
µ
µ
µ
²
²
·
µ
©
©
²
Spoj;>@
gh[µp° gh[µp° gh[µp° gh[
µ gh[
µ gh ?¬ gh[µp° gh ?¬ gh ?¬ gh[µ
79ILI;>@GNCPASl@C;AL=^V?]OQI;AN‹HO]J]OQINCDG@ZV^DG;rDZFg;5yaV?@ZV^LIS3v¿S?T”9I;>oxm;>xUq_;>@GN$¾%œQ‡vg;>xUqŠSlQI@GKg¨ NY=?S?DZHO9IK
yŠS¤oš;A@mHJ9IP>@G;AV?NG;>NYV?NmHODZNMyI@CSlyŠSl@CDGHJSl9!S?T~=?S?DZ;>NsTWV?]O]JNŸSl9!DZFI;ry_SlNeDC•Zµ ­?© V?9ILy_SlNeDC•Zµ °gµbPASlQg9IPAHO]JNfÁ
V?9IL¿DZFg;$yIV?@ZV?LgSpv¿DGFaV^DA\~Sl9¿DZFI; yŠSlNCDe•fµ °gµ PASlQI9gPAHJ] \~P>SlQI9‘DZ@CHJ;AN‹onH[DZF LIHO¡_;A@C;A9‘D<=lS?DGHJ9IKoš;AHJK?FEDGN
¾&€Qºvº;>xUq_S?QI@GKY=?;A@GNCQI'N 0;A9gxsV?@C™RV^9IL)8:@G;>] V?9gL§Á›xmV3BRFaV3=?;Ÿ;wvgV?P>DG]OBbDZFI;uNZV?xz;”=?S?DZHO9IKYyŠS¤oš;A@>h.i`FI;
LI;>PAHJNCHJSl9 @CQI]J;>Nz¾tDZFIV^D.HJN>\€c‡QIV?]JH[Ía;AL5xmV¤´CSl@CHODZHO;ANZÁnTtSl@0DGFI;ŸDZFI@C;A;MPASlQg9IPAHO]JNkV?@C;zV?N0TtSl]J]OSponNA{Uµ¤·sS?QgD.S?T
µ¤­<¾eµ ²?)° (е ­?©‘Á —g¬Iµ.SlQgDnS^T²?°b¾–µ ­?)© (§µ °gµ¤Á —IV?9gLb¬‘²ŸS?QgD›S?T ?©r¾–µ °g*µ (е °‘²?Áwh
"!$#
q_;wTtSl@G;’­^©z{ µ¤·U{‘¬*Z¬*G¬* ·0 · 3µ #
­?© °gµ.{ ¬Iµ.{aµ
3µ 3µ
f²0 ²0 · 3µ
f©0f©
°gµ °‘²U{ ¬E²U{aµ
3µ 3µ
f²0 ²0 · 3µ
f©0f© # ² #
½ 9I;)PAV?9 QING;)DGFI;5ÃXV^9IÄAFaVžT.y_Spoš;A@‹HJ9gLI;wvÌDZSV^9aV?]OB‡Ä>;)S?DZFg;A@YNeBºNeDZ;>xmN<V?Nmoj;>]J] \XonFg;A@G;5LgHO¡_;A@G;>9‘D
yI]JVpB?;A@CN›FIVp=?;”Qg9I;AcEQaV?]œHOxmyŠSl@CDZV?9IP>;?\gqgQgDnonFI;A@C;’DGFIHJN`xmV3B‹9IS?DnqŠ;.;wvºyI@G;>NGNC;AL)HO9<oš;AHOKlF‘DZN>h
|~}€I,‚„ƒt…‡†?ˆ
‰ ! mx V¤PA´CSlSlxz@CHOxzD:BmHODG@GDGQI;A]O;˜;kS?FIT*Sl7]JLgyŠNA;Ah SlyI]O;?\žonFI;A@C;šDZFg;jPfFaV?HO@A¨ N=?S?DZ;šHJNLI;AP>HJNCHO=l;onFI;A9”DZFI;>@G;jHON„VnDZHO;?—?S^DZFI;>@ConHJNC;
‰ «qŠ;.P>S¤Sl=?9I;AKl@C@C@G;AHJLgNCN`LI;AS?9bT²”q‘BryŠ;A· Sl?yI©Ÿ]O;kS?TV?9IDZFILR;.µkPAyIS?9I@G;>KlNG@GHJ;>LgNG;AN>9‘—ŠD39g\gSlon@GxmFg;AV?@G]J;0][BMDGFIDZFI;0;.yI@Cxm;ANCV¤HJ´CLISl;>@G9EH[D:DnBsPA@CV?QI9Y]J;k=l;wFgDZSlSI]J\gLIqIN>h QgD`FIHONš=?;>DZSMPAV?9
i@CB‹H[D›V?]JNCSUonHODGFRVŸPAS?9IKl@G;>NGN›S?T yŠ;AS?yI]J;^h
¬
u13/ 1 b1„1A+z2
|~}€I,‚„ƒt…‡†?ˆ
‰ NCyI]JH[DGDZHO9IKŸVŸP3V^™?;
‰ LIH[=‡HJLIHO9IKŸQIybV?9b;ANCDZV^DZ;’V?xzSl9IKŸDGFI;.FI;AHO@GN
‰ NCyI]JH[DGDZHO9IKUQIy<DZFg;”V^NGNG;wDZNnonFI;A9RVŸPASlxzyaV?9‘B‹qI@C;3V?™EN›QIy
Æ ”ƒWŠ…ž†m¾tLIHO=‡HONGHJS?9‹S?TV$»CPAV?™?;>¼sqŠ;>D:oš;A;>95·Uy_;>SlyI]O;pÁ
½ 9I;.PAQgDGNA\aDZFg;’S^DZFI;>@`PfFISºS?NG;AN>h
8:xzyI]JHOPAHODXV?NGNCQIxzygDZHOSl9INA{
‰ A;½ V? PfFronygH[] DZV3F‹Bl;>DZ@`FIV^HJN`D›V?yIqg] V3]J;~Bl;>DG@ SzLgHO=‡HJLI;~P3V?™^;’HO9‹NGQIPfFrVuojVpBsDZFaV^Dn;>HODGFI;A@`S?TœDGFI;0D:ojSMyIHO;AP>;AN`ojS?QI]JL<qŠ;
‰ KlH[=l;>9RV?9‘B‹LgHO=‡HJNCHJSl9‹S?TDZFg;’PAV?™?;^\§;3V?PfFbyI] V3B?;A@`ojS?QI]JL<Ía9ILrV^Dn]J;AV?NCDnS?9I;.yIHJ;>PA;’V?PAP>;AygDZV?qI]O;?h
¤…E…)ÆY,Ƥ…R‚ƒWŠ…ž†
€;ANCN7;AV?NCB5h>hAh
½ 9I;.y_S?NGNGHOqIHJ]OHOD:BŠ{„]JV?NCDnLgHJxzHJ9IHONGFI;>@Xxm;wDZFIS‡Ldhn¾t¬g¨ N ‹( ÃjV?9aV?PfF a9 V?NeDZ;A@ZÁ
‰ „HO@GNeD`yI] V3B?;A@”¾WS?TKl@CSlQIybS?TγyI]JV3Bl;A@CNfÁM»CP>QgDZNC¼sVŸyIHO;APA;kDGFaV^D›]JS‡Sl™ENjTWV?HO@jDZSMDZFaVžDnyI] V3Bl;>@Ah
‰ i`V?FIPf;.F)yIS^HO;AT„P>DZ;’FgK?;ANG;>;”DZNnyI; ] vIV3B?V^;Axm@GN7HO9IP3;AV?Lb9rq‘PfBYFIS‡DZFISl;.NG;.S?DZDGS FI;A»e@nDZ@CyIHJ]JxzV3Bl¼U;A@CDZNAFI\Š;’·UyIDZHJ;>FIPA@C;’SlQIHOTKlFbDZFIΒ;wBb\aNGDGQIFIP>HJ9gPA;A™YNCNGHOD›H[=lHJ;AN`][BlDGh SºSz] V?@CKl;kTtSl@›V
TWŒ V?HO@XNGFIV?@G;^h
‰ ›yI]JT V3DZBl;A;A@.@`;won=l;AFI@eSzB‡q_PfFIS‡SlLgNCB$;’FaDGV?SMN.LIHJHJ9gxzNGyŠHJ9I;AHOP>NGDGFY;ALH[D3\IHOD’S?@XV?9IDZSML$yIyŠ] SlV3NGB?NC;[email protected]][µ.BRHODZT@C9IHJxzSlqŠxmSº;>LgL5BsHOLIDAHJ\dL“DZh FI;zyIHJ;>PA;MKlS‡;ANkDZS‹DGFI;m]JV?NCD
‰ i`@C;AFIxm;’V?yIHJ9I@CSºHO9IP>KŸ;ALIÎ`QI•f@Cµ.;”yIPA]V?V39RB?;AqŠ@G;’N>h @G;AyŠ;3VžDZ;ALbonHODGFbDGFI;”@C;AxmV?HJ9gLI;A@`S?TDZFI;’P3V^™?;s¾ DG@GHJxzxzHJ9IKlNZÁTtSl@`DZFg;
²
i@CB‹H[DnSlQgD ¥
@GSlqg]J;Ax¶onHODZF‹DGFIHJN’¾WV?9ILbxsV?9‘BsS?DZFI;>@fÁXxz;>DZFgSºLIN>{HODnHON`9IS?Dn;A9‘=EB‘•¦Tt@C;A;?h
Ë¿FaV^D›HONnV?9<;A9‘=EBE• Tt@G;>;uNGS?]JQgDGHJSl9§¥
NCSl]JQgDGHJSl9rHJ9RonFgHJPfFd\€VžT DZ;A@~;>=l;>@CBRyI] V3Bl;>@kFaV?N0FgHJN žFI;A@7yIHJ;>PA;?\“9ISlqŠSºLgBbDZFgHJ9I™EN~DGFaV^DkNCSlxm;>Sl9I;U;>]JNC;
HJN`qŠ;>DCDZ;>@›S^¡„h
i`FIHONHON9IS?DKlQIV?@ZV?9‘DG;A;ALzHJ9UDGFI;›V?qŠS¤=?;nyI@GS‡PA;>LIQI@G;^{„onFI;>9mDZFg;XÍa@GNeDU»eyIHJ;>PA;A¼.HON˜V?]J]OSºPAV^DZ;>Ld\?DZFg;›yg] V3Bl;>@
onFISm@G;AP>;AH[=l;AN~HODnxmVpB<qŠ;’FaV?yIy‘B<onHODGFrH[D3\§qIQgD7FI;’xsV3B<PfFaV?9gKl;”FgHJN›xmHO9ILbonFg;A9)FI;’NG;>;AN›DZFaV^D7] VžDZ;A@
yI]JVpB?;A@CN›K?;>D›xUQIPfFbqIHJK?Kl;A@`yIHO;AP>;AN›V^T DZ;>@`FI;’FaV?N`]J;wT D`DZFI;.LIH[=‡HJNGHOSl9‹K‘V?xz;?h
RV?™EHO9IKŸTWV?HJ@XLgHO=‡HJNCHJSl9‹;>9E=EB‘•¦Tt@C;A;”HJN`xUQIPfF FaV?@CLI;A@>h
~9b;A9‘=EB‘•¦Tt@C;A;’LIHO=‡HJNCHJSl9YTtS?@XDZFI@C;A; yI] V3B?;A@GN>h
"!#%$ &(')')*!#,+ .- /0)1
‰ IyIy ]J]JV3V3BlBl;A;A@m@7·ºµYh PAQºDZNUPAV?™?;YHO9DGFI@G;>;YyIHJ;>PA;>NuDZFaVžDU]JS‡Sl™5;>c‡QIV?]DGSRDZFIV^DUyI]JV3Bl;A@>\V?9IL,FaV?9ILINuS¤=?;A@uDGS
‰ DGyISM]JV3DGBlFI;A;.@79I·U;wxmv‡DeVp•:B?] V^\I@GHOKlT;>NCNCFID3;k\aHOon9‹HJNGFIFg;A;A@nNAyŠ\a;ADG@G@CHJPAx¹;AyºDZDZFgHJS?;’9dygh˜HJi`;AP>FI;k;kDZDZFa@GV^HOxmDnNCxzFIHJ;k9gDZKUFIi£HO9I™EHON`NnNGHJ;>N`D7] V^V?@GNCKlHJLI;>NC;0D`TtS?NCSz@XDZDGFIFa;.V^DnxzHODXSlxmHJN`;>;A9‘cED3Qah V^]
‰ yI]JV3Bl;A@›©ŸPfFIS‡SlNG;>N›DGFI;.yIHJ;>PA;.FI;kDZFgHJ9I™ENnHJN`]JV?@GKl;>NCDAh
‰ DG9I@G; HJv‡xzDYxzyI;A]JLYVpB?yI;AHO@Y;AP>·$;?\IPfDZFIFgS‡;ASl95NG;>·UNAh xU8¦QgT’NCDn·$DfLgV?™^HJLÌ;kDZDGFI@GHO;kx DZ@CHJHJ9!xzxmDZFI;>;bL‹NGyI;>HJPA;>SlPA9I;^L h NeDZ;>yd\nV?9ILÌHOT.©$LIHJL!9IS?D DfV?™^;RDZFg;
‰ µ.Kl;wDZN`DZFI;.@C;AxmV?HJ9IHO9IKUyIHO;AP>;?h
¸ºSŸTWV?@>\IDZFI;wB<V?@G;’V?]O]“FaV^yIy‘Bl\ŠV?9IL‹DZFg;A@G;’HJN`9gSz;>9‘=‡BŠ{
‰ ©ŸPfFISlNC;’ÍI@GNCD
‰ ·UPfFISlNC;uV^9ILrK?S?DnSl9I;.S?TDZFI;kD:ošSmygHJ;AP>;ANnNGFg;”P>Sl9INGHOLI;A@C;AL<DZSzq_;.VŸDZHO;0TtSl@`] V^@GKl;>NCD
‰ ]Oµk;ANGKlN S^DnSl9I;kS?TœDZFg;’ygHJ;AP>;ANXDGFaV^DnFI;kP>QgD3\§V?9gLb;w=l;>@CB‡q_S‡LgB<;>]JNG;kK?S?D’¾WHO9YDZFI;>HJ@`;wBl;ANZÁjDZFI;.NZV^xm;0S?@
Î7SpoonFaV^DnDZSMLISMonHODGF<DGFI;kDZ@CHJxzxmHO9IKE¥
Ë¿FaV^DG;>=?;A@ŸFaV?ygy_;>9INuonHODGF HODA\„yI]JV3Bl;A@mµsonHJ]O]˜9I;>=?;A@Ÿ;A9‘=EB,DGFI;YyI]JV3Bl;A@’onFIS5@G;>PA;>HO=l;>L DZFI;mDZ@GHOxmxz;AL
yIHO;APA;uHJ9RDGFI;UÍa@GNeDk@GS?QI9ILd\dqŠ;APAV?QINC;?\dTtSl@0yg] V3Bl;>@Uµl\“DG@GHJxzxz;ALRygHJ;AP>; DZ@GHOxmxzHJ9gKzS?9I]OBbxmV?™?;UQgyV^N
xUQIPfFrV?NnFI;m¾eµpÁjKlS?DnHJ9‹DGFI;kÍa@GNeD7@CSlQI9ILrV?9‘BEoXV3B?h
€;>DA¨ N’P3V?]O]„DZFg;myI]JV3Bl;A@konFISrKlS?DkDGFI;zDG@GHOxmxz;AL5yIHO;AP>;zHO9$DGFI;ŸÍa@GNeDu@CSlQI9IL ¾ HJN.;AH[DZFI;>@”·<Sl@.©lÁ
V?9IL<DZFg;’S^DZFI;>@`Sl9I;m¾WS?T·MV?9ILb©‘Á Î7Spo onHJ]J]“PAQgD›DZFI;.DZ@CHJxzxmHO9IKUHO9EDGSzDGFI@G;>;”;>cEQaV?]yIHJ;>PA;ANu¾tTt@CSlxFIHON ^Fg;A@›y_SlHO9‘DnS?T„=‡HJ;wo7ÁwhXi`Fg;A9
DZFg;’S^DZFI;>@›yg] V3Bl;>@GNnPfFIS‡SlNG;^{Ía@CNCD \gDGFI;A9$µl\gDGFI;A9 DZV?™?;>NnDGFI;.] V?NeDnyIHJ;>PA;.S?TDZFI;kDG@GHJxzxzHJ9IKIh
n;ANGQg]OD3{
‰ HJN`FIV?yIy‘Bl\§V?9gLr;>9E=‡HO;AN›9ISMSl9I;?\aqŠ;APAV?QINC; PfFISlNC;’ÍI@GNCDAh
‰ µ.µ.LILIS‡S‡;A;ANnNn9I9IS?S?D›D›;>;>9E9E=E=EBB
¾WNC;Aq_;’;>V?P3V?qŠQISpNC=l;”;pÁ Fg;’PfFgSlNG;”V?FI;AV?LrS^T ‰ LISº;>Nn9IS?D›;A9‘=EBbV?9‘BlSl9g;?\§q_;>P3V?QgNG; LIHOL<DGFI;.PAQgDCDZHO9IKIh
i`FI;UTtQI9g9EBbDZFgHJ9IKYHON7DGFaV^D~DGFI;A@C;ŸHJN79ISY;3V^NCBRojV3BRDGSYKl;A9g;A@ZV^]JHJÄ>;”DGFIHJN›DZSY¬‹Sl@7xmS?@G;uyg] V3Bl;>@GNA'h #›9‘DZHJ]
µ ·º\I9ISM;A9‘=EB‘•¦Tt@C;A;’NGSl]OQgDZHOSl9YoXV^N7™E9ISpon9rTtS?@`¬zyg] V3Bl;>@GNA\aqgQgDnDZFI;>9rSl9g;.ojV?NnTtSlQI9IL<q‘BbÃj@GV?xmNnV^9IL
iVpB‡]OSl@’¾WNC;A;.DZFI;.]JHONCD`S^TÍa9IV?]€yI@GS¤´C;AP>DGNfÁ h
Y…‡, €†^ˆ
‰ B?SlQrPAV?9rV?]JNCSzQgNG;kDZFIHONXDZSzLIHO=‡HOLI;kQIybVz]OHJNeD`S?TPfFISl@G;>NAÀ
‰ DGFIHJN`PAV?9rqŠ;.;wv‡DZ;>9ILI;>LrDZSzxzSl@G;0P>SlxzyI]JHOP3V^DG;AL‹yI@CSlqI]J;>xmN>\INGQgPfF)V?NnLIH[=ºHOLIHJ9gKŸQIyRV^9b;>NCDZV^DZ;^h
-
-
-
- 1
-
-
-
-
-
-
1 b1A45132
z+z/!n(
-
-
-
5* u2 ( (£+z/ *5/˜+z*,(0/ „(A( (k2 1A2 451
<" ÈtɊÆ$#l…§ˆjQINCQaV?]J][BYSl9g]OBsD:ošSsyIV?@CDGHJ;AN>h
8¦D7HJNny_S?NGNGHOqI]J;kDGSm;>9IL)QIybonHODZFrVmNCHODGQaV^DZHOSl9‹onFI;A@C;u;AV?PfF)yaV?@eD:B<;A9ILgN~QIybonHODZF<onFaV^D›DZFg;>Bby_;>@GP>;AHO=?;
V?NnxzSl@G;0DGFaV?9‹DZFI;>HJ@XTWV?HO@XNGFIV?@G;^À
|~}€I,‚„ƒt…§ˆu 7]JHJP>;ŸV?9IL$ÃjS?qV^@G;ŸLIH[=lSl@CPAHJ9gKIhui`FI;wB FaV3=l;MSl9I][BRD:ošSrxmV¤´CSl@0V?NGNC;>DZN>\onFgHJPfF 9I;A;>L$DGS
qŠ;ULIHO=‡HOLI;ALdh „HJ@CNCD3\Š;3V^PfF$S^TDZFI;>x HON0V?NC™?;AL5DGS‹V?]J]OSºPAV^DZ;.yŠSlHJ9‘DGN~DZSsDGFI;uD:ošS<V?NGNC;>DGNA\€SlQgD~S?TVmDGS?DfV?]
­
?S T˜µ
g\gV^PAPAS?@GLIHO9IKuDZS”onFaV^DXDGFI;>Bz=^V?]JQI;7xmS?NCD3h 7]JHOPA;›HJNXV”PAHOD:BmyŠ;A@GNCSl9d\gV?9IL‹yI]JV?PA;>NXyI@G;>xmHOQIx=^V?]JQI;
Sl9)DGFI;UNGxmV?]O]Î’Ž V?yIyaV^@CDZxz;A9‘D0DGFaV^D7DZFI;UP>SlQIyI]O;USpon9INAh”ÃjSlqRHJN0@C;>DZHO@G;>L5V?9IL5]OHJ™?;>N~DZSsNGyŠ;A9IL FIHON
DZHOxm;.ÍaNCFIHJ9IKg—dFI;u=^V?]OQI;AN›DZFI;>HJ@~9gHJPA;uNGFISl@C;UFISlQgNG;ŸxUQIPfF5xzSl@C;’DGFaV?9)DZFg;ŸV?yaV?@eDZxz;A9‘D3h~i`FIHJN0@C;ANCQI]ODGN
HJ9‹DGFI;kTtSl]J]OS¤onHO9IKuDZV?qI]J;^{
~]OHJPA; jà Slq
NGFgSl@G;.FISlQgNG; © ­
ΒhUh Ž~hŠV?yºD3h ­ ©
8:9<DZFIHON`P3V?NC;?\§HOD`xmV?™^;ANnNG;>9ING;’DZSMKlHO=?;. ~]JHOPA;~DGFI;’V?yaV?@eDZxz;A9‘D3\IV?9ILrÃjSlq<DZFI;.NGFgSl@G;.FISlQgNG;?h
8:9byI@ZV?PwDZHOPA;?\IDGFI;.NGH[DZQaV^DGHJSl9‹HON`QINGQIV?]J][BsxzSl@G;kP>SlxzyI]JHOP3V^DG;ALd\‡onHODZFbxzSl@G;kV^NGNG;wDZN>{
|~}€I,‚„ƒt…§ˆ)µ µsi@GQIxzy LIH[=lSl@CPA;?hr8:9,DZFgHJNuPAV?NG;^\DZFI;my_S?HJ9‘DŸV?]O]JS‡P3V^DGHJSl9)DZV?qI]O;sHJN’9IS?Du™E9ISpon9d\S?T
PAS?QI@GNC;?hšÃXV?NG;>LrS?9<DGFI;.9I;AK?S?DZHJV^DZHOSl9INA\ºSl9I;.P3V^9rxmV?™?;kDGFI;kTtSl]J]OS¤onHO9IKUKlQg;ANGN>{
~NCNG;wD
0Sl9aV^]JL ¦8 =?V^9aV
ŽjSl9I9g;AP>DGHJP>QgD›;ANCDZV^DZ;
µ
l© °
„V^]JxÚ;3V?PfFbxsV^9INGHOSl9
¬ ·
i@GQIxzy ˜]JV?Ä3VMV?yIV?@CDGxm;>9ED µ
©
i@GQIxzy<iSpoš;A@nDZ@GHOyI]J; v
©l° µ
ŽXV?NCF ?;>oš;A]J@eB
·
·
†>Çp…E‚ ˆ OH =?;k;3V?PfFryaV?@CD:BsDZFI;.qIHOKzH[DZ;>xmNjDGFaV^DnDZFg;>B‹]OHJ™?;kxzSlNeD3h
0Sl9aV?]OLd{ „V?]OxÃj;AV?PfFrxmV?9INCHJSl9
i@CQIxzyriSpoj;>@›DG@GHJyg]J;wv
¬
©l°
­?°Ÿy_SlHO9‘DZN
8¦=?V^9aVg{ ŽjSl9I9I;>P>DGHJPAQºD~;>NCDfVžDZ;
©l°
i@CQIxzy ] V^Ä3VMV?yaV?@eDZxz;A9‘D ©
l°Ÿy_SlHO9‘DZN
>† Çp…E‚dˆ H[=l;kDZFI;’@G;>xsV^HJ9IHO9IK$»CNGxmV?]O]J¼UDZFgHJ9IKlN`DGSzDGFI;”yIV?@CD:B‹onFISmFaV?NnDZFI;.Tt;>oš;ANeD0yŠSlHJ9‘DGNA\IDZSz;>=?;A9
SlQgD.DGFI;m@C;ANCQI]OD”V?N’xUQIPfF,V?N.y_SlNCNGHOqI]J;^hm8:9$DZFIHON’PAV?NG;^\8¦=^V?9IVrKl;wDZNkDZFI;zP3V^NGF V^9IL<´C;woj;>]J@CB?\„V^9IL,FaV^N
9ISpo ­MyŠSlHO9EDGNAh
†>Çp…E‚ ˆ`i`FI;.NGHODGQaV^DGHJSl9‹HON`9IS?D›;>=?;A9dh˜Ë,;.9I;>;AL<DZSMDZ@GV?9INCTt;>@›VMqgHOD`Tt@GS?x 0Sl9aV^]JLYDZSM8¦=^V?9aVºh
°
º¸ HO9IPA;X8¦=^V?9IV7=^V?]JQI;>N„DZFg; „V?]OxÃj;3V?PfFŸxmV?9INCHJSl9uxUQIPfFŸxzSl@G;DZFaV?9”DZFI;jDG@GHOyI]J; v“\žonFIHJ]O; 0Sl9IV?]JL’=?V^]JQI;>N
DZFg;ANG;ŸD:ošSbV?q_S?QgDk;AcEQaV^]J]OB?\“H[DkxsV^™?;ANkNC;A9INC;ŸDZSYDZ@GV?9INeTt;A@0yaV^@CD.S?TDGFI; „V?]OxÚ;3V?PfF$xsV^9INGHOSl9dh.¢7Spo
xUQIPfFa¥
8¦Tdoš;~KlH[=l;`vs®S?TdDGFI; V?]Jx¹Ãš;3V?PfFYxmV?9INCHJSl9ŸDGSu8¦=^V?9aVg\EDZFgHJN˜]J;AVp=?;AN0¾eµ
¤• vIÁ˜®S?TdDZFg; V?]Jx¹Ãj;AV?PfF
xmV?9INGHOSl9YTtSl@ 0Sl9IV?]JLdh
¢7Spo£xsV?9‘B‹yŠSlHO9EDGN`LIS‡;ANn;3V?PfFbS?TDZFI;.yaV?@eDZHO;AN`FaV3=l;.DZFg;A9§¥
0Sl9IV?]JL
8¦=^V?9aV
iSmxmV?™^;0DZFIHO9IKlN`;>=?;A9d\§oš;’@C;AcEQIHO@G;
¸ºS?]O=‡HJ9IKUDGFIHJNXoš;.Ía9IL
©l° £­?°
¬ « ­­ « ­· µp© © ¬
· ­?° ·
¬
8:9byI@ZV?PwDZHOPA;?{
‰ 0Sl9IV?]JL‹LIHOL<Kl;>DnDZFg;’i@GQgxmy<iS¤oš;A@CN›DG@GHOyI]J; v
‰ 8¦=^V?9aVŸLIHOL<Kl;>DnDZFg;’i@GQgxmy ] V?ÄAVMV?yaV?@eDZxz;A9‘D3\IDGFI;”ŽjSl9I9I;>P>DZHOPAQgD›;>NCDfVžDZ;?\aPAV?NGF ´C;woj;>]J@CB
‰ 0Sl9IV?]JL‹KlS^D`DZFI; „V^]JxÚ;3V?PfFrxsV?9gNGHJS?95µlµ.xmSl9‘DGFIN žB?;3V?@
‰ 8¦=^V?9aVŸKlS^D`DZFI; „V^]JxÃj;AV?PfFbxsV?9gNGHJS?9$µ.xzSl9‘DZF ¤Bl;3V^@A\IonFIHOPfFrP>Sl@G@C;ANGyŠSl9ILgN›DGSm°ºh©?¬‘®‹h
"<ÈtÉ€È È Ê aÊ |k†>ÇpºÇp…
8:9zS?DZFI;>@˜;wvgV?xzyI]J;>NA\‘onFI;A@C;7xzSl@C;nDGFaV?9s·0y_;>SlyI]J;›V?@C;nHJ9‘=lSl][=l;>Ld\‘DZFI;›NGH[DZQaV^DGHJSl9ŸHONxmS?@G;nPASlxzyI]OHJP3VžDZ;AL
i`FI;0TtSl]O]JSponHJ9gK”; vIV^xmyI]O;0HJ]O]JQINeDZ@GV^DZ;>NšDZFg; 9aV?NCDG;A@`yI@CSºP>;ALIQg@G;?\ºonFIHJPfF‹ošSl@G™EN`HOTœyIV?@CDGHJ;ANjFaV3=l;.VUP3V^NGF
@G;>NG;>@C=l;uV?NXoš;A]J] h
|~}€I,‚„ƒt…§ˆš 7]JHOPA;?\ºÃjSlq“\aŽXV?@CSl]“V^9IL kV3=ºHOLsFaV3=?;0DZSUNCyI]JH[DjQIyYDGFI;AHO@šHO9IFI;>@GHODZV?9IP>;?\ºonFgHJPfFYP>Sl9INGHONCDGNXS?T
Sl9I;.qIHOKMV?NGNC;>DA{DGFI;.FISlQINC;.DGFaV^DnDZFg;AHJ@`yIV?@G;>9EDGNnS¤on9g;ALdh
iS”NeDfV?@eD3\‘DZFI;wBM;3V?PfFmyIQgDjV’NCQIq‡´C;>P>DZH[=l;›=?V^]JQI;`Sl9MDZFg;7FIS?QING;^\‡onFIHOPfFmDZV?™?;>NšHO9EDGS’V?PAP>SlQI9‘DšDGFI;›xsV?@C™?;wD
=^V?]JQg;?\aV?N`oš;A]O]œV?NXDZFI;>HJ@`Spon9RNC;A9‘DZHOxm;>9‘DfV?]œV^DCDfV?PfFgxm;>9EDAh
7]JHJP>; ÃjS?q ŽXV^@GSl] kVp=‡HOL
µ¤· g\ · º\ µp° º\ µ3¬ g\ ')
# º¸ HO9IPA;uÃjSlqR=^V?]OQI;ALbDZFI;”FISlQINC;UxmS?NCD7FIHJKlFg]OBl\ŠFI;”Kl;>DGN~HODAh›¢7;uDGFI;A9)ygQgDZN.·g\ Uv5© ž¬ µ¤²º\ HJ9<V »ey_S?DG¼I—ºDGFIHJNj@G;Ayg@G;ANC;A9‘DZN>\“V^PAPAS?@GLIHO9IKUDZSŸFIHONXSpon9<;>=^V?]JQIV^DZHOSl9d\EDZFI;kxzSl9I;wBsDZSŸqgQgB‹SlQgDXFIHJNjDGFI@G;>;
NGHOqI]JHO9IKlNjTtS?@XDZFI;>HJ@`NGFIV?@G;.S?TDZFI;.FIS?QING;^h
Î7Spo£;3V?PfFrS?TDGFI;.S?DZFg;A@XDZFg@G;A;’Kl;wDZN.µ ^¬MS?TDZFI;>HJ@`Spon9b;ANCDGHJxmV^DZ;kS^TDGFI;kDZS?DZV?]“=^V?]JQg;kS?TDGFI;.FISlQINC;?{
~]OHJP>;0DfV?™^;AN.µ ^¬Uv µ¤· g\ /© g\ uTt@CSlx DGFI;.y_S?DAh
ŽXV?@CSl]dDfV?™^;AN.µ ^¬Uv µ3° g\ / ¬E²‡\ uTt@GSlx DGFI;.y_S^D3h
kV3=‡HJL‹DfV?™^;AN’µ ^¬Uv µ3¬ g\ / ©‘²‡\ ŸTt@CSlx¶DGFI;.y_S^D3h
7T DG;A@›V?]O]_DZFgHJNA\ºDZFI;.yŠS?D›HJN`9IS?Dn;>xmyºD:B_—IDGFI;.@G;>xsV?HO9ILI;>@`HJNA{¬ g\ „HO9aV?]J][Bl\EDZFIHON`Kl;>DGNnLIHO=‡HJLg;AL<HJ9‘DZSM;>c‡QIV?]œNGFaV?@C;AN>\aNGSMDZFaV^D`DGFI;A@C;”HONk
µ g\ uTtSl@`;AV?PfF
ºÇsÈW†UÇ …RÇpÆaÇpaƒ ž…‡† ƒ Ç a@GSlx¶ ~]OHJPA;^¨ NnyŠSlHJ9‘DnS?T=‡HJ;wo.{šNCFI;”K?S?Dn¬ g\ Ÿ;>=?;A9bDZFISlQgKlFrNCFI;.DZFISlQgKlF‘D›DGFI;A@C;’ojV?N›Sl9I][B)µ¤· º\ DZSMLIH[=‡HJLI;^\IS?TonFIHJPfFbNGFI;’;wvºyŠ;APwDZ;ALR© g\ a@GSlx¹ÃšSlqd¨ NjyŠSlHJ9‘DS?Td=‡HJ;wo.{FI;7KlS?DDZFI;7FISlQINC;?\IV^9ILsNCy_;>9ED0µ3¬ g\ g—‡V?P>PASl@CLIHJ9gK”DZS”FIHJN;ANeDZHOxsV^DG;nFI;
DZFg;A@G;wTtSl@G;.KlS^D0· g\ ”•›µ3¬ º\ g\ Ÿ;>=?;A9bDZFISlQgKlFrFg;.DGFISlQIK?FEDnDGFI;A@C;’ojV?N›Sl9I]OB‹· g\ uDGS
LIH[=ºHOLI;?\gS?TonFgHJPfFbFI;.;wvºy_;>P>DG;AL ² º\ a@GSlx ŽXV?@CSl] ¨ N›y_SlHO9‘D~S?T=‡HO;>o.{nNGFI;uKlS?D.²?²º\ g\ŠV?]ODGFISlQIK?FRNGFg;U;wvºyŠ;AP>DG;AL µ ^¬zv,µ3° g\ ¬E²‡\ Sl9I][Blh
a@GSlx kVp=‡HOLd¨ N’yŠSlHJ9‘D.S?Tš=ºHO;>o.{”FI;mK?S?D’¬E²º\ g\V?][DZFISlQgKlFFg;m; vgyŠ;APwDZ;>L µ ^¬‹v!µ3¬ g\ ©‘²‡\ Sl9I][Blh
¢7Spoš;>=?;A@A\IDGFI;7onFgSl]J;7yI@GS‡PA;>LIQI@G;~HJNj9gS?DX;>9E=EB‘•¦Tt@C;A;^{˜V^T DZ;>@XV?]O] \ºŽXV?@GS?]_KlS^DjxzSl@G;›xzSl9I;>BzDZFIV?9 kV3=‡HJLd\
onFISzKlS?DnxzSl@G;0DGFaV?9< ~]OHJP>;?À
i`FIHON›yg@GS‡PA;ALgQI@G;.HJNnV^]JNGSMyŠSlNGNCHJqI]O;kSl9I][BYHOTÃjS?qrFaV^N7V‹µ3¬ g\ UP3V?NCFR@C;ANG;>@C=?;?À
µ