A cikin 2012, an ba da lambar yabo ta Nobel a fannin tattalin arziki ga Lloyd Shapley da Alvin Roth. "Don ka'idar barga rarraba da kuma al'adar shirya kasuwanni." Aleksey Savvateev a shekara ta 2012 ya yi ƙoƙari don sauƙaƙe da kuma bayyana ma'anar ma'anar ilimin lissafi. Ina gabatar muku da taƙaitaccen bayani .
A yau za a yi lacca na ka'idar. Game da gwaje-gwaje , musamman tare da gudummawa, ba zan fada ba.
Lokacin da aka sanar da cewa ya sami kyautar Nobel, akwai madaidaicin tambaya: “Ta yaya!? Har yanzu yana raye!?!?” Mafi shahararsa sakamakon da aka samu a 1953.
A bisa ka'ida, an ba da kari don wani abu dabam. Don takardarsa ta 1962 akan "ka'idar zaman lafiyar aure": "Jagora Shiga Jami'a da Kwanciyar Aure."
Game da aure mai dorewa
matching (matching) - aikin nemo wasika.
Akwai wani ƙauye keɓe. Akwai samari "m" da "w" 'yan mata. Muna bukatar mu aurar da su da juna. (Ba lallai ne lamba ɗaya ba, wataƙila a ƙarshe za a bar wani shi kaɗai.)
Wadanne zato ya kamata a yi a cikin samfurin? Cewa ba shi da sauƙi a sake yin aure ba da gangan ba. Ana ɗaukar wani mataki na zaɓi na kyauta. A ce akwai aksakal mai hikima da ke son kara aure don kada bayan mutuwarsa a fara saki. (Saki wani yanayi ne da miji ke son mace ta uku a matsayin matarsa fiye da matarsa).
Wannan ka'idar tana cikin ruhin tattalin arziki na zamani. Ba ta da mutunci. Tattalin arziki a al'ada ya kasance rashin mutuntaka. A fannin tattalin arziki, ana maye gurbin mutum da na'ura don haɓaka riba. Abin da zan gaya muku shi ne kwata-kwata abubuwan hauka daga mahangar ɗabi'a. Kar a dauke shi a zuciya.
Masana tattalin arziki suna kallon aure haka.
m1, m2,… mk - maza.
w1, w2,... wL - mata.
An gano wani mutum da yadda yake "umarni" 'yan mata. Har ila yau, akwai "matakin sifili", wanda a ƙarƙashinsa ba za a iya ba da mata a matsayin mata ba kwata-kwata, koda kuwa babu wasu.
Komai yana faruwa a bangarorin biyu, iri ɗaya ga 'yan mata.
Bayanan farko na sabani ne. Iyakance kawai zato / iyakance shine ba mu canza abubuwan da muke so ba.
Theorem: Ba tare da la’akari da rabon da kuma matakin sifili ba, a ko da yaushe akwai hanyar da za a kafa wasiƙun kai-tsaye tsakanin wasu mazan da wasu mata ta yadda za su yi ƙarfi ga kowane irin rarrabuwar kawuna (ba kawai saki ba).
Wadanne barazana za a iya samu?
Akwai ma'aurata (m,w) da ba su yi aure ba. Amma ga w miji na yanzu ya fi m, kuma ga m matar yanzu ta fi w. Wannan lamari ne mara dorewa.
Akwai kuma zaɓin cewa wani ya auri wanda ba shi da “ƙasa da sifili”, a wannan yanayin, auren ma zai wargaje.
Idan mace ta yi aure, amma ta fi son namiji marar aure, wanda ta wuce sifili.
Idan mutane biyu ba su da aure, kuma dukansu suna "sama da sifili" ga juna.
Ana jayayya cewa ga kowane bayanan farko irin wannan tsarin aure yana wanzuwa, mai jure wa kowane irin barazana. Abu na biyu, algorithm don gano irin wannan ma'auni yana da sauƙi. Bari mu kwatanta da M*N.
Wannan samfurin ya kasance gabaɗaya kuma an faɗaɗa shi zuwa "aurin auren mata fiye da ɗaya" kuma an yi amfani da shi a wurare da yawa.
Hanyar Gale-Shapley
Idan dukan maza da dukan mata sun bi "rubutun," tsarin aure da zai kasance mai dorewa.
Takardun magani.
Muna ɗaukar ƴan kwanaki kamar yadda ake buƙata. Muna raba kowace rana gida biyu (safe da yamma).
Washe gari da safe kowa yaje wajen fitacciyar macensa ya buga taga yana neman aurenta.
Da yammacin wannan rana, juyowa ya koma ga mata, me mace zata iya ganowa? Cewa akwai jama'a a ƙarƙashin taganta, ko dai namiji ɗaya ko babu. Wadanda ba su da kowa a yau sun tsallake juzu'in su jira. Sauran, waɗanda suke da aƙalla ɗaya, suna duba mutanen da suka zo don ganin cewa sun kasance “sama da sifili.” Don samun aƙalla ɗaya. Idan kun yi rashin sa'a kwata-kwata kuma komai yana kasa da sifili, to a aika kowa da kowa. Matar ta zaɓi mafi girma a cikin waɗanda suka zo, ta ce masa ya jira, ta aika sauran.
Kafin rana ta biyu, lamarin shi ne: wasu matan suna da namiji ɗaya, wasu ba su da.
A rana ta biyu, duk "kyauta" (aiko) maza suna buƙatar zuwa ga mace mai fifiko na biyu. Idan babu irin wannan mutumin, to an bayyana cewa mutumin bai yi aure ba. Wadancan mazan da suka riga sun zauna da mata ba su yi komai ba tukuna.
Da yamma, mata suna kallon halin da ake ciki. Idan wanda ya riga ya zauna ya kasance tare da fifiko mafi girma, to, mafi ƙarancin fifikon an aika. Idan waɗanda suka zo sun yi ƙasa da abin da aka riga aka samu, kowa sai a sallame su. Mata suna zaɓar mafi girman kashi kowane lokaci.
Muna maimaitawa.
A sakamakon haka, kowane mutum ya shiga jerin sunayen matansa gaba ɗaya, ko dai an bar shi shi kaɗai ko kuma ya haɗu da wata mace. Sannan za mu aurar da kowa.
Shin zai yiwu a gudanar da wannan tsari gaba ɗaya, amma mata su ruga da maza? Hanyar yana da ma'ana, amma mafita na iya bambanta. Amma abin tambaya a nan shi ne, wa ya fi wannan?
Theorem. Bari mu yi la'akari ba kawai waɗannan mafita guda biyu masu ma'ana ba, amma saitin duk tsayayyen tsarin aure. Tsarin da aka tsara na asali (maza suna gudu kuma mata sun yarda / ƙi) yana haifar da tsarin aure wanda ya fi kowane namiji kyau kuma mafi muni ga kowace mace.
Yin auren jima'i
Ka yi la’akari da yanayin “auren jinsi.” Mu yi la'akari da sakamakon lissafin da ke jefa shakku kan buƙatar halatta su. Misalin kuskure a akida.
Yi la'akari da 'yan luwadi huɗu a, b, c, d.
fifiko ga: bcd
fifiko ga b:cad
abubuwan fifiko ga c: abd
don d ba komai ya sanya sauran ukun da suka rage ba.
Sanarwa: Babu tsarin aure mai dorewa a wannan tsarin.
Nawa tsarin mutane hudu ne? Uku. ab cd, ac bd, ad bc. Ma'aurata za su rabu kuma tsarin zai tafi cikin hawan keke.
Tsarin “jinsi uku”.
Wannan ita ce tambaya mafi mahimmanci da ta buɗe dukkan fannin ilimin lissafi. Abokin aikina a Moscow, Vladimir Ivanovich Danilov ya yi hakan. Ya ɗauki “aure” a matsayin shan vodka kuma ayyukan sun kasance kamar haka: “mai zubo,” “mai magana da gasa,” da kuma “mai yanka tsiran alade.” A cikin halin da ake ciki inda akwai wakilai 4 ko fiye na kowane rawar, ba shi yiwuwa a warware ta da karfi. Tambayar tsarin dorewa shine bude daya.
Shapley vector
A cikin cottage village suka yanke shawarar kwalta hanya. Bukatar shiga. yaya?
Shapley ya ba da shawarar magance wannan matsala a cikin 1953. Bari mu ɗauka yanayin rikici tare da ƙungiyar mutane N={1,2…n}. Ana buƙatar raba farashi/ fa'idodi. A ce mutane tare sun yi wani abu mai amfani, sun sayar da shi kuma yadda za a raba riba?
Shapley ya ba da shawarar cewa lokacin rarraba, ya kamata a yi mana ja-gora da nawa wasu rukunin waɗannan mutanen za su iya samu. Kudi nawa ne duk 2N da ba komai ba zai iya samu? Kuma bisa ga wannan bayanin, Shapley ya rubuta wata dabara ta duniya.
Misali. Mawaƙin soloist, guitarist da mai buga ganga suna wasa a cikin hanyar ƙasa a Moscow. Su uku daga cikinsu suna samun 1000 rubles a kowace awa. Yadda za a raba shi? Yiwuwa daidai.
V(1,2,3)=1000
Bari mu yi kamar haka
V(1,2)=600
V(1,3)=450
V(2,3)=400
V(1)=300
V(2)=200
V(3)=100
Ba za a iya tantance rabo mai adalci ba har sai mun san irin ribar da ke jiran kamfani da aka ba shi idan ya balle ya yi da kansa. Kuma lokacin da muka ƙaddara lambobi (saita wasan haɗin gwiwa a cikin sigar siffa).
Superadditivity shine lokacin da suke samun fiye da na daban, lokacin da ya fi riba don haɗin kai, amma ba a bayyana yadda za a raba abubuwan da aka samu ba. An karya kwafi da yawa game da wannan.
Akwai wasa. Wasu 'yan kasuwa uku a lokaci guda sun sami ajiya mai darajar dala miliyan 1. Idan su ukun suka yarda to akwai miliyan daya daga cikinsu. Kowane ma'aurata na iya kashe (cire daga shari'ar) kuma su sami duka miliyan don kansu. Kuma ba wanda zai iya yin komai shi kaɗai. Wannan wasan haɗin gwiwa ne mai ban tsoro ba tare da mafita ba. Za a sami mutane biyu da za su iya kawar da na uku ... Ka'idar wasan hadin gwiwa ta fara da misalin da ba shi da mafita.
Muna son irin wannan mafita ta yadda babu wata kungiya da za ta so ta toshe mafita ta bai daya. Saitin duk sassan da ba za a iya toshe shi ba shine kernel. Yana faruwa cewa ainihin fanko. Amma ko da ba fanko ba, yaya za a raba?
Shapley ya ba da shawarar raba wannan hanyar. Jefa tsabar kudi da n! gefuna. Mun rubuta duk 'yan wasan a cikin wannan tsari. Bari mu ce mai ganga na farko. Ya shigo ya dauki nasa 100. Sai “na biyu” ya shigo, bari mu ce soloist. (Tare da mai ganga za su iya samun 450, mai ganga ya riga ya dauki 100) The soloist daukan 350. The guitarist shiga (tare 1000, -450), daukan 550. Na karshe daya a quite sau da yawa nasara. (Supermodularity)
Idan muka rubuta don duk umarni:
GSB - (nasara C) - (nasara D) - (nasara B)
SGB - (nasara C) - (nasara D) - (nasara B)
SBG - (nasara C) - (nasara D) - (nasara B)
BSG - (nasara C) - (nasara D) - (nasara B)
BGS - (saba C) - (saba D) - (saba B)
GBS - (nasara C) - (nasara D) - (nasara B)
Kuma ga kowane ginshiƙi muna ƙarawa da rarraba ta 6 - matsakaici akan duk umarni - Wannan shine Shapley vector.
Shapley ya tabbatar da ka'idar (kimanin): Akwai nau'ikan wasanni (supermodular), wanda mutum na gaba da zai shiga babbar ƙungiya ya kawo babbar nasara gare ta. Kwayar kwaya koyaushe ba ta da komai kuma ita ce haɗuwar maki (a cikin yanayinmu, maki 6). Siffar Shapley tana kwance a tsakiyar tsakiya. Kullum ana iya ba da ita a matsayin mafita, babu wanda zai yi adawa da shi.
A cikin 1973, an tabbatar da cewa matsala tare da cottages ne supermodular.
Duk n mutane suna raba hanyar zuwa gidan farko. Har zuwa na biyu - n-1 mutane. Da dai sauransu.
Filin jirgin saman yana da titin jirgin sama. Kamfanoni daban-daban suna buƙatar tsayi daban-daban. Haka matsalar ta taso.
Ina tsammanin waɗanda suka ba da lambar yabo ta Nobel suna da wannan cancantar a zuciya, ba kawai aikin gefe ba.
Na gode!
Nuna karin
- Channel "Math - Sauƙi":
- Tashar "Savvateev ba tare da iyakoki ba": edusex.ru, brainsex.ru, studfuck.ru
- Jama'a "Lissafi mai sauƙi ne":
- Jama'a "Masu ilmin lissafi wargi":
- Yanar Gizo, duk laccoci a can +100 darussa da ƙari: savvateev.xyz
source: www.habr.com