Munhu wese angaroora sei (wanano imwe-, mbiri- uye katatu-pabonde) kubva pamaonero esvomhu uye nei varume vachigara vachikunda

Muna 2012, mubairo weNobel weEconomics wakapihwa Lloyd Shapley naAlvin Roth. "Nekuda kwedzidziso yekugovera kwakagadzikana uye tsika yekuronga misika." Aleksey Savvateev muna 2012 akaedza nyore uye zvakajeka kutsanangura musimboti wezvakanakira masvomhu. Ndinokupai pfupiso hurukuro dzevhidhiyo.

Nhasi pachava nehurukuro yedzidziso. Nezvekuedza Ela Rota, kunyanya nemupiro, handichatauri.

Pazvakaziviswa izvozvo Lloyd Shepley (1923-2016) akagamuchira Mubairo weNobel, pakanga paine mubvunzo wakajairika: “Sei!? Achiri mupenyu!?!?” Mhedzisiro yake yakakurumbira yakawanikwa muna 1953.

Pakarepo, bhonasi yakapiwa kune chimwe chinhu. Kune bepa rake ra1962 pamusoro pe "chato kugadzikana theorem": "Kupinda Kukoreji uye Kugadzikana Kwewanano."

Nezvewanano yakasimba

Matching (matching) - basa rekutsvaga tsamba.

Kune mumwe musha uri woga. Kune "m" majaya uye "w" vasikana. Tinofanira kuvaroorerana. (Kwete nhamba imwe chete, pamwe pakupedzisira mumwe munhu achasiiwa ega.)

Ndedzipi fungidziro dzinofanirwa kuitwa mumuenzaniso? Kuti hazvisi nyore kuroora kana kuroorwazve. Rimwe danho riri kutorwa pakusarudza zvakasununguka. Ngatitii kune aksakal akachenjera anoda kuroorazve kuti mushure mekufa kwake kurambana kurege kutanga. (Kurambana imamiriro ezvinhu apo murume anoda mukadzi wechitatu semukadzi wake kupfuura mukadzi wake.)

Iyi theorem iri mumweya wehupfumi hwemazuva ano. Haana hunhu hunoshamisa. Economics yagara isina hunhu. Mune zvehupfumi, munhu anotsiviwa nemuchina kuti uwedzere purofiti. Zvandichakuudza zvinhu zvinopenga zvachose kubva pakuona kwehunhu. Usazviisa mumwoyo.

Nyanzvi munyaya dzehupfumi dzinotarisa muchato nenzira iyi.
m1, m2,… mk - varume.
w1, w2,... wL - vakadzi.

Murume anozivikanwa nematauriro aanoita vasikana. Kune zvakare "zero level", pazasi iyo vakadzi havagone kupihwa sevakadzi zvachose, kunyangwe pasina vamwe.

Zvose zvinoitika kumativi maviri, zvakafanana kuvasikana.

Yekutanga data ndeyekuda. Kungofungidzira chete/kupikiswa ndekwekuti hatishandure zvatinoda.

Theorem: Pasinei nekugovera uye chiyero che zero, pane nguva dzose nzira yekumisikidza tsamba-kune-imwe pakati pevamwe varume nevamwe vakadzi kuitira kuti ive yakasimba kumarudzi ese ekuparadzana (kwete kungorambana).

Zvii zvingava tyisidziro?

Pane vaviri (m,w) vasina kuroora. Asi kune murume wazvino uno akaipa kupfuura ini, uye kune wazvino uno akaipa kupfuura w. Aya ndiwo mamiriro asina kugadzikana.

Pane zvakare sarudzo yekuti mumwe munhu akaroora mumwe munhu ari "pasi zero"; mumamiriro ezvinhu aya, muchato unoparara.

Kana mukadzi akaroorwa, asi anoda murume asina kuroorwa, uyo ari pamusoro pe zero.

Kana vanhu vaviri vese vasina kuroora, uye vese vari "pamusoro zero" kune mumwe nemumwe.

Zvinopokana kuti kune chero data rekutanga hurongwa hwemuchato huripo, hunopesana nemhando dzese dzekutyisidzira. Chechipiri, iyo algorithm yekuwana iyo yakaenzana iri nyore kwazvo. Ngatienzanise neM*N.

Iyi modhi yakagadziriswa uye yakawedzerwa ku "barika" uye yakashandiswa munzvimbo dzakawanda.

Gale-Shapley maitiro

Kana varume vose nevakadzi vose vakatevera “zvakatemwa,” hurongwa hwewanano hunozobuda huchava hwakasimba.

Prescriptions.
Tinotora mazuva mashoma sezvinodiwa. Isu tinokamura zuva rega rega muzvikamu zviviri (mangwanani nemanheru).

Mangwanani ekutanga, murume wese anoenda kumukadzi wake wepamusoro uye anogogodza pahwindo, achikumbira kuti amuroore.

Manheru ezuva rimwe chetero, chijana chinotendeukira kuvakadzi.Ko mukadzi angawanei? Kuti paive nechaunga pasi pehwindo rake, angava murume kana pasina. Avo vasina munhu nhasi vanosvetuka nguva yavo vomirira. Vamwe vose, vanenge vaine mumwe chete, tarisai varume vauya kuzoona kuti “vari pamusoro pezinga razero.” Kuva nechimwe chete. Kana iwe usina rombo rakanaka uye zvese zviri pazasi zero, saka munhu wese anofanira kutumirwa. Mukadzi wacho anosarudza mukuru pane vakauya, anomuudza kuti amirire, uye anotumira vamwe vose.

Pamberi pezuva rechipiri, mamiriro ezvinhu ndeaya: vamwe vakadzi vane murume mumwe chete, vamwe havana.

Pazuva rechipiri, varume vose "vakasununguka" (vakatumirwa) vanofanira kuenda kumukadzi wechipiri wekutanga. Kana pasina munhu akadaro, murume wacho anobva anzi haana murume. Vaya varume vakatogara nevakadzi havasati vaita chinhu.

Manheru, vakadzi vanotarisa mamiriro acho ezvinhu. Kana mumwe munhu akanga agara akabatanidzwa nechepamusoro-soro, ipapo iyo yakaderera inotumirwa kure. Kana avo vanouya vari vakaderera pane izvo zvagara zviripo, munhu wese anodzingwa. Vakadzi vanosarudza iyo yakanyanya element nguva dzese.

Tinodzokorora.

Izvi zvakaita kuti murume mumwe nemumwe apinde mugwaro rose revakadzi vake uye akasara ari ega kana kupfimbwa nemumwe mukadzi. Tobva taroora munhu wese.

Zvinoita here kuita hurongwa hwese uhwu, asi kuti vakadzi vamhanyire kuvarume? Iyo nzira ndeye symmetrical, asi mhinduro inogona kunge yakasiyana. Asi mubvunzo ndewokuti, ndiani ari nani kubva pane izvi?

Theorem. Ngatitarisei kwete chete aya maviri esymmetrical mhinduro, asi seti yeese yakagadzikana hurongwa hwewanano. Nzira yepakutanga yakarongwa (varume vanomhanya uye vakadzi vanobvuma / vanoramba) inoguma muhurongwa hwewanano huri nani kune chero murume kupfuura chero upi zvake uye yakaipa kupfuura chero ipi zvayo kune chero mukadzi.

Kuroora Kwakafanana

Funga nezvemamiriro ezvinhu e“kuroorana kwengochani.” Ngatitarisei mhedzisiro yemasvomhu inokanda kupokana pakudikanwa kwekuinyoresa zviri pamutemo. Muenzaniso une pfungwa usina kururama.

Funga nezvengochani ina a, b, c, d.

zvinonyanya kukosha kune a: bcd
zvekutanga zve b:cad
zvinonyanya kukosha kune c: abd
nokuti d hazvina basa kuti anobata sei vamwe vatatu.

Chirevo: Hakuna hurongwa hwewanano inosimba muhurongwa uhu.

Masystem mangani aripo kuvanhu vana? Tatu. ab cd, ac bd, ad bc. Vakaroorana vachaparara uye maitiro achaenda mumatenderedzwa.

"Vatatu-vakadzi" masisitimu.
Uyu ndiwo mubvunzo wakanyanya kukosha unovhura ndima yese yemasvomhu. Izvi zvakaitwa nemumwe wangu muMoscow, Vladimir Ivanovich Danilov. Aiona “wanano” sokunwa vodka uye mabasa acho aiva akadai: “uya anodira,” “uyo anotaura toast,” uye “anocheka masoseji.” Mumamiriro ezvinhu apo kune 4 kana kupfuura vamiririri vebasa rimwe nerimwe, hazvibviri kugadzirisa nechisimba chechisimba. Mubvunzo wegadziriro yakasimba ndeye yakavhurika.

Shapley vector

Mumusha wekottage vakafunga kuisa asphalt mugwagwa. Kuda kupinda mukati. Sei?

Shapley akakurudzira mhinduro kudambudziko iri muna 1953. Ngatitorei mamiriro ekukonana neboka revanhu N={1,2…n}. Mari / mabhenefiti anofanira kugoverwa. Ngatitii vanhu vakaita chimwe chinhu chinobatsira pamwe chete, vakachitengesa uye kuti vangagovana sei purofiti?

Shapley akakurudzira kuti kana tichipatsanura, isu tinofanirwa kutungamirwa neakawanda sei mamwe madiki evanhu ava anogona kugamuchira. Imarii yaigona kuwana ese 2N asina-empty subsets? Uye zvichibva paruzivo urwu, Shapley akanyora fomula yepasirese.

Muenzaniso. Muimbi ari ega, gitare uye mutambi anoridza mugwara repasi pevhu muMoscow. Vatatu vacho vanowana 1000 rubles paawa. Nzira yekuiparadzanisa sei? Pamwe zvakaenzana.
V(1,2,3)=1000

Ngatiedzei kuti
V(1,2)=600
V(1,3)=450
V(2,3)=400
V(1)=300
V(2)=200
V(3)=100

Kupatsanurwa kwakanaka hakugone kutsanangurwa kudzamara taziva kuti ndezvipi zvakamirira kambani yakapihwa kana ikapanduka uye kuita yega. Uye patakaona nhamba (seta mutambo wemubatanidzwa mune chimiro chechimiro).

Superadditivity ndeye apo pamwe chete vanowana mari yakawanda kupfuura yakaparadzana, kana inobatsira zvikuru kubatanidza, asi hazvisi pachena kugovera mawinnings. Makopi akawanda akatyorwa pamusoro peizvi.

Pane mutambo. Vanamuzvinabhizinesi vatatu panguva imwe chete vakawana dhipoziti inokosha $1 miriyoni. Kana vari vatatu vakabvumirana, saka kune mamiriyoni avo. Chero vakaroorana vanogona kuuraya (kubvisa kubva munyaya) uye kuwana mamiriyoni ose ega. Uye hapana anogona kuita chinhu ari ega. Uyu mutambo unotyisa we-co-op usina mhinduro. Pachava nevanhu vaviri vanogona kubvisa chechitatu ... Cooperative game theory inotanga nemuenzaniso usina mhinduro.

Tinoda mhinduro yakadai zvekuti hapana mubatanidzwa uchada kuvharisa mhinduro yakajairika. Seti yezvikamu zvese zvisingakwanisi kuvharika ndiyo kernel. Zvinoitika kuti musimboti hauna chinhu. Asi kunyange kana isina chinhu, sei kupatsanura?

Shapley anokurudzira kupatsanura nenzira iyi. Kanda coin ne n! kumativi. Isu tinonyora vatambi vese nenzira iyi. Ngatitii mutambi wengoma wekutanga. Anouya otora 100 yake. Ipapo "wechipiri" anopinda, ngatitii muimbi ari oga. (Pamwe chete nemuridzi wengoma vanogona kuwana 450, mutambi wengoma atotora zana) Muridzi wega anotora 100. Gitare anopinda (pamwe chete 350, -1000), anotora 450. Wekupedzisira mune kazhinji kazhinji anokunda. (Supermodularity)

Kana tikanyora kune ese maodha:
GSB - (win C) - (win D) - (win B)
SGB ​​- (win C) - (win D) - (win B)
SBG - (win C) - (win D) - (win B)
BSG - (win C) - (win D) - (win B)
BGS - (kuwana C) - (kuwana D) - (kuwana B)
GBS - (win C) - (win D) - (win B)

Uye kune imwe neimwe column isu tinowedzera uye tinogovanisa ne6 - avhareji pamusoro peese maodha - iyi iShapley vector.

Shapley akaratidza theorem (inenge): Kune kirasi yemitambo (supermodular), umo munhu anotevera kujoina timu hombe anounza kuhwina kwakakura kwairi. Iyo kernel inogara isiri-isina uye ndeye convex musanganiswa wemapoinzi (munyaya yedu, 6 mapoinzi). Iyo Shapley vector iri pakati chaipo peiyo nucleus. Inogona kupiwa nguva dzose semhinduro, hapana achapikisana nayo.

Muna 1973, zvakaratidzwa kuti dambudziko remakottage ndiro supermodular.

All n people share the road to the first cottage. Kusvikira kune yechipiri - n-1 vanhu. Etc.

Nhandare yendege ine njanji. Makambani akasiyana anoda hurefu hwakasiyana. Dambudziko rimwe chetero rinomuka.

Ini ndinofunga kuti avo vakapa Mubairo weNobel vaive nehunhu uhu mupfungwa, uye kwete basa rekugumira.

Ndinokutendai!

Zvakadaro

• Channel "Math - Nyore": youtube.com/punkmathematics
• Channel "Savvateev isina miganhu": edusex.ru, brainsex.ru, studfuck.ru
• Veruzhinji "Mathematics iri nyore": vk.com/alexei_savvateev
• Veruzhinji "Mathematicians joke": vk.com/bsu_mmf_jokes
• Webhusaiti, hurukuro dzese ipapo +100 zvidzidzo uye nezvimwe: savvateev.xyz

Source: www.habr.com