CIN WASA DA ƘARANCIN MAKI

A wani jawabi ga ɗaliban da ke kammala karatu a Dartmouth College a watan Yuni 2024, Roger Federer ya ba da lambobi biyu: ya ci kusan 80% na wasannin singles (ɗan wasa ɗaya da ɗaya) 1,526 da ya buga a rayuwar wasansa, amma 54% kawai na maki. Darasin da jawabin ya ɗauko game da juriya ne — har mafi ƙwararru suna rasa kusan maki ɗaya cikin biyu. Lambobin kuma suna nuna wata siffa ta asali ta tsarin ƙidayar maki na tennis. Maki ne ke haɗa game (ƙaramin zagaye), game ne ke haɗa set, set ne ke haɗa wasan gaba ɗaya (match), kuma ƙaramin fifiko a maki yana ƙara girma a kowane mataki.

Andrés Martín, Johann H. Martínez da Javier M. Buldú, na Complex Systems Group a King Juan Carlos University da ke Madrid, sun warware wannan siffa gunduwa-gunduwa, sannan suka gwada ta a kan shekaru 35 na tennis na ƙwararru maza.

Tsawon wasan ne ke yin aikin

A ce ɗan wasa yana cin kowane maki da yiwuwa (probability) ɗaya, p. Mene ne yiwuwar ya ci wasan? Ƙungiyar ta lissafa hakan daidai-wa-daida ga tennis, badminton, ƙwallon raga (volleyball) da tennis na tebur (table tennis). A p = 50%, kowa yana a 50%. A kewayen wannan wuri, lanƙwan (curve) yana da tsauri sosai:

• ɗan wasan da ke cin 52.5% na maki yana cin 80% na wasannin tennis na set biyar;
• a tennis na tebur, wanda ake bugawa a mafi kyawun game biyar (best of five), sai an kai 54.6% na maki kafin a kai wannan kaso.

Tsaurin yana bin wata doka mai sauƙi: yana girma kamar tushen murabba’i (square root) na yawan maki a cikin wasa, kusan √(2N/π) ga maki N. Maki ɗaya yana yanke hukuncin gasa ne kawai idan sauran an raba su daidai, kuma hakan yana ƙara wuya yayin da wasanni ke ƙara tsawo. Dukan tsarin matakai na game da set yana sauya wannan da ’yan kaso kaɗan cikin ɗari kawai.

Daga maki zuwa wasanni: duk yadda wasa ya fi tsawo, lanƙwan yana ƙara tsauri (hagu); gangaren a 50% yana bin tushen murabba’i na yawan maki (dama). — Hoto na 1, Martín, Martínez & Buldú (2026), arXiv:2611.03702.

A 54% na maki, wannan samfurin (model) yana hasashen cin 86% na wasannin set uku da 91% na wasannin set biyar — fiye da 82% na Federer. Marubutan sun bayyana dalilin: 54% matsakaici ne (average) a kan abokan hamayya masu bambanci sosai, kuma ɗaukar matsakaici a kan lanƙwan da ke lanƙwasa ta wannan hanya yana ba da sakamako ƙasa da karanta lanƙwan a wurin matsakaicin.

Maki masu ƙarin nauyi

Amma ba duk maki ne ɗaya ba. Ƙungiyar ta raba su gida biyu: maki na yau da kullum, waɗanda ake ci da yiwuwa p₁, da maki masu muhimmanci (crucial points) — duk makin da wanda ya ci shi, ko wane ne, ke rufe game, set ko wasan — waɗanda ake ci da yiwuwa p₂.

Kullum akwai wani ƙunƙuntaccen yanki inda ɗan wasa ke cin yawancin wasanni yayin da yake cin ƙasa da rabin maki. Yana wanzuwa ne saboda maki masu muhimmanci suna da nauyi a kan sakamakon fiye da kasonsu a cikin maki, kuma yana wanzuwa a dukan wasannin nan huɗu. Amma siririn yanki ne mai siffar ƙusa (wedge), wanda ke buɗewa da digiri 0.3 zuwa 1.6 kawai. A tennis, ɗan wasan da ke cin 49% na maki na yau da kullum dole ne ya ci kusan 52.4% na makin game (game points) kafin ya kai can. Makin game suna yawan faruwa sosai, kusan 29% na dukan maki, har cin ƙarin daga cikinsu yana saurin ɗaga kason gaba ɗaya. Makin set da makin wasa (match points) ba kasafai ake samun su ba, don haka suna karkatar da wasan ne kawai da babban fifiko, kusan 70%.

Shekaru talatin da biyar na wasanni

Tsakanin 1991 da 2025, 4,530 cikin 96,826 na wasannin ATP na maza (4.68%) ɗan wasan da ya ci ƙarancin maki ne ya ci su. Maki masu zaman kansu (independent points) suna hasashen 3,892: ainihin adadin ya fi da 16%, kuma har yanzu ya fi da 5% bayan an yi la’akari da serve (bugun buɗe maki). A wasannin Grand Slam 166 da aka ci da ƙarancin maki, waɗanda suka ci sun ɗauki 49.1% na maki na yau da kullum, amma 51.1% na makin set da 61.9% na makin wasa. Natsuwar ɗan wasa a kan makin break (break points), wadda aka auna a wasu kakannin wasa, ita ma tana hasashen sakamako: fifikon maki ɗaya na kaso cikin ɗari (percentage point) a kan abokin hamayya yana ninka damar (odds) cin nasara da kusan 1.08, a daidai kason maki iri ɗaya.

Sake ƙidaya saɓanin Federer (Federer paradox)

Federer yana riƙe da wani baƙon tarihi (record): wasanni 36 da ya sha kaye a cikinsu bayan ya ci maki fiye da abokin hamayya, fiye da kowa. Amma Federer ya ci maki fiye da abokin hamayya sau da yawa har akwai damammaki da yawa na rasa irin wannan wasa fiye da cin ɗaya. Kwatancen adalci shi ne da abin da maki masu zaman kansu ke hasashe: kaye 25.6 irin wannan maimakon 36, da nasara 15.6 da ƙarancin maki maimakon 9. Don haka tarihin 36 wani ɓangare tasirin ƙidaya ne, wani ɓangare kuma ɗan ƙaramin gibi: gaba ɗaya, kusan wasanni masu kusanci 17 ƙasa da abin da maki ke nuni, wanda marubutan suka kira alama ba hujja ba. Pete Sampras, Andre Agassi, Andy Murray da Rafael Nadal duk sun ci wasanni masu kusanci fiye da yadda ake tsammani.

Saɓanin Fognini

A ɗaya gefen kuma akwai Fabio Fognini. A wasanni 774, Fognini ya ci maki 61,624 cikin 124,623, 49.4% — amma duk da haka wasanni 396, 51.2%. Maki masu zaman kansu da sun ba da 47.5%. Sirrin ba makin break ba ne, sai dai yadda maki suka rarrabu a cikin set: cin 56.6% na set masu kusanci, kuma set ɗin da aka rasa sau da yawa an yi masa rinjaye mai yawa a cikinsu — 18% sun ƙare 6–0 ko 6–1, idan aka kwatanta da 5% ga Federer. Makin da aka rasa a set da ya riga ya tafi ba sa jawo asara ko kaɗan. Ashirin da uku daga cikin ’yan wasa 328 da aka yi nazari a kansu sun ci fiye da rabin wasanninsu da ƙasa da rabin maki.

Wasanni masu kusanci da aka ci sama ko ƙasa da yadda ake tsammani: Fognini a sama, Federer a ƙasa (hagu); ’yan wasan da suka ci yawancin wasanni da ƙasa da rabin maki (sama a dama). — Hoto na 6, Martín, Martínez & Buldú (2026), arXiv:2611.03702.

Samfurin an yi shi da gangan ya zama mafi sauƙi: babu serve, yiwuwoyi tsayayyu a duk tsawon wasa, dukan maki masu muhimmanci na mataki ɗaya ana ɗaukar su iri ɗaya, kuma tennis na maza kaɗai ke cikin bayanan. Layinsa na ƙarshe yana aron kalmomin Federer — “maki ɗaya ne kawai” — kuma yana ƙarawa: “Amma ba duk maki ne ɗaya ba.”