Teburin Abubuwan Ciki
1. Gabatarwa
Fasahar Blockchain, musamman hanyar yarda ta Bitcoin ta Hujjar Aiki (PoW), tana wakiltar sauyi a tsarin tsarin da ba na tsakiya ba. Wannan takarda tana amfani da ka'idar Wasan Filin Matsakaici (MFG) don ƙirƙirar samfurin mu'amalar dabaru tsakanin masu hakar Bitcoin—ɗimbin jama'a masu gasa don warware wasanin sirri. Babban manufa ita ce siffanta halayen daidaiton ƙarfin lissafi gaba ɗaya (hashrate) da aka keɓe don hakar ma'adinai da kuma abubuwan da suka shafi tsaron blockchain. Fahimtar wannan tushen ka'idar wasa yana da mahimmanci, saboda tsaron ƙa'idar ya dogara gaba ɗaya akan daidaitattun abubuwan ƙarfafawa a cikin yanayin rashin amana.
2. Tsarin Ka'idar
2.1 Tushen Wasan Filin Matsakaici
Ka'idar Wasan Filin Matsakaici, wadda Lasry da Lions suka fara gabatar da ita, tana ba da tsarin lissafi don nazarin yanke shawara na dabaru a cikin tsarin da ke da adadi mai yawa na jama'a masu mu'amala. Maimakon bin kowane mutum, jama'a suna amsa ga rarraba ƙididdiga ("filin matsakaici") na yanayi da ayyukan dukkan al'ummar. Wannan ya dace musamman ga hakar Bitcoin, inda dubban masu hakar ma'adinai suka kafa hujjar su na saka hannun jari da ayyukan aiki akan jimillar hashrate na cibiyar sadarwa.
2.2 Aiwatarwa ga Wasan Hakar Ma'adinai
Ana ƙirƙirar tsarin hakar PoW a matsayin wasa mai ci gaba da lokaci, wanda ba na haɗin kai ba. Kowane mai hakar ma'adinai $i$ yana sarrafa ƙarfin lissafinsa $q_i(t)$, yana haifar da farashin makamashi $C(q_i)$. Yiwuwar samun nasarar hakar toshe yana daidai da rabonsu na jimillar hashrate $Q(t) = \sum_i q_i(t)$. Ladan toshe $R(t)$, wanda aka ƙidaya a cikin kudin sirri na asali, yana ba da ƙarfafawa. Daidaitawar matsalar hakar $D(t)$ tana tabbatar da tsayayyen lokacin da ake tsammanin toshe, yana haɗa ayyukan mutum ɗaya zuwa yanayin duniya.
3. Tsarin Samfurin
3.1 Matsalar Inganta Mai Hakar Ma'adinai
Mai hakar ma'adinai ɗaya yana neman haɓaka ƙimar yanzu mai tsabta na ladan da ake tsammani a nan gaba ban da farashi. Ana iya tsara aikin manufarsa kamar haka:
$$ \max_{q_i(\cdot)} \mathbb{E} \left[ \int_0^{\infty} e^{-\rho t} \left( \frac{q_i(t)}{Q(t)} \cdot \frac{R(t)}{\tau} - C(q_i(t), \theta(t)) \right) dt \right] $$ inda $\rho$ shine ƙimar rangwame, $\tau$ shine manufar lokacin toshe, kuma $\theta(t)$ yana wakiltar jihohin waje kamar farashin makamashi ko ci gaban fasaha.
3.2 Samun Ma'auni Babba
Ana siffanta daidaito ta hanyar Ma'auni Babba—ma'auni na bambance-bambance wanda ke bayyana juyin halittar aikin ƙima $V(m, t)$ ga wakilin mai hakar ma'adinai, idan aka yi la'akari da rarraba $m$ na yanayin duk masu hakar ma'adinai. Ma'aunin ya haɗa da sharadin inganci na Hamilton-Jacobi-Bellman (HJB) da ma'aunin gaba na Kolmogorov (Fokker-Planck) don juyin halittar rarraba:
$$ \partial_t V + H(m, \partial_m V) + \langle \partial_m V, b(m) \rangle + \frac{\sigma^2}{2} \text{tr}(\partial_{mm} V) = \rho V $$ Warware wannan yana ba da sarrafa daidaito $q^*(t)$ da kuma hanyar filin matsakaici da ta biyo baya.
4. Binciken Matsayin Daidaito
4.1 Matsayin Tsayayye Mai Kayyadewa
A cikin yanayin kayyadewa tare da tsayayyen ƙimar ci gaban fasaha $g$, samfurin yana hasashen cewa jimillar hashrate $Q(t)$ za ta karkata zuwa hanyar girma mai tsayayye. A daidaitaccen yanayi, hashrate yana girma daidai da yadda fasaha ta inganta: $Q(t) \sim e^{g t}$. Wannan ya yi daidai da yanayin dogon lokaci da aka gani a tarihin Bitcoin, inda hashrate ya ƙaru da yawa duk da sauye-sauyen farashi.
4.2 Manufar Hashrate Mai Ban Tsoro
Lokacin haɗa girgiza mai ban tsoro (misali, farashin kuɗin sirri bazuwar $S_t$), binciken ya bayyana "manufar hashrate" $Q^*(S_t)$ ga kowane yanayi na duniya. Tsarin yana nuna halayen komawa ga matsakaici: idan ainihin hashrate ya karkata daga $Q^*$, abubuwan ƙarfafawa na tattalin arziki suna motsa masu hakar ma'adinai su shiga ko fita, suna tura shi komawa ga manufa. Wannan yana ba da kwanciyar hankali ga cibiyar sadarwa.
5. Abubuwan Da Suka Shafi Tsaro
5.1 Dangantakar Hashrate-Tsaro
Babban ma'aunin tsaro na blockchain na PoW shine farashin da ake buƙata don aiwatar da harin kashi 51%, wanda kusan yayi daidai da jimillar hashrate. Samfurin MFG ya nuna cewa a daidaitaccen yanayi, wannan matakin tsaro ko dai yana dawwama ko kuma yana ƙaruwa tare da ainihin buƙatar kuɗin sirri. Wannan sakamako ne mai ƙarfi: yana nuna cewa ƙirar ƙa'idar tana haifar da tsaro daidai da ƙimar tattalin arzikin tsarin.
5.2 Jurewa Kai Hari
Samfurin yana nuna cewa faɗuwar farashin ɗan gajeren lokaci bazai yi barazana ga tsaro nan da nan ba. Domin hashrate yana daidaitawa zuwa manufa $Q^*(S_t)$, kuma kayan aikin hakar ma'adinai suna da farashin da aka nutse, hashrate—don haka tsaro—zai iya raguwa a hankali fiye da farashi. Duk da haka, ci gaba da faɗuwar ƙimar tattalin arziki a ƙarshe zai ja da manufar hashrate da farashin kai hari.
6. Sakamako & Tattaunawa
6.1 Tabbatarwa Ta Gwaji
Duk da yake takardar ta kasance ta ka'ida, hasashenta sun yi daidai da abubuwan da aka gani ta hanyar gwaji. Babban hasashen samfurin—cewa hashrate yana bin yanayin dogon lokaci wanda ya dace da ci gaban fasaha ($g$) yayin da yake jujjuyawa a kusa da manufa mai ban tsoro—ya yi daidai da tarihin tafiyar hashrate na Bitcoin (duba hoto mai nuni na 1: Hashrate na Bitcoin a ma'aunin log). Lokutan ƙimar farashi mai sauri suna ganin hashrate ya haura sama da yanayi, yayin da kasuwannin beyan suke ganin jinkirin girma ko raguwa na ɗan lokaci, sannan kuma a koma.
6.2 Binciken Hashrate na Bitcoin
Hoton da aka bayar (Hashrate na Bitcoin a cikin tera hashes a kowace dakika, ma'aunin log) zai nuna haɓaka mai yawa akan lokaci tare da sauyin yanayi mai mahimmanci. Tsarin MFG ya bayyana wannan a matsayin mu'amala tsakanin: 1) yanayin kayyadewa wanda ingancin kayan aiki (Dokar Moore) ya motsa, da 2) karkatacciyar hanya ta ban tsoro wanda sauye-sauyen farashin Bitcoin ya haifar, wanda ya canza ladan nan take $R(t)$. Tsarin daidaitawa na wahala shine maɓallin haɗin gwiwa wanda ke fassara waɗannan ƙarfin tattalin arziki zuwa ma'aunin lissafi.
Mahimman Fahimtar Samfurin
- Tsaro na Ciki: Hashrate na daidaito, don haka tsaro, yana da alaƙa da ƙimar kuɗin sirri.
- Manufar Hashrate: Ra'ayin daidaito mai ban tsoro yana daidaita cibiyar sadarwa.
- Daidaitawar Wahala: Ita ce mahimmin tsarin mayar da martani wanda ke haɗa tattalin arziki da lissafi.
- Daidaitawar Ƙarfafawa: MFG ta tsara ƙirar ƙarfafawa ta asali ta Nakamoto.
7. Cikakkun Bayanai na Fasaha
Ginshiƙin lissafi yana zaune a cikin Ma'auni Babba. Hamiltonian $H$ don matsala sarrafa mafi kyau na mai hakar ma'adinai shine:
$$ H(m, p) = \max_q \left\{ \frac{q}{\int z dm(z)} \cdot \frac{R}{\tau} - C(q) + p \cdot (\beta(q, m) - \delta q) \right\} $$ inda $p$ shine maɓalli mai farashi, $\beta$ yana wakiltar tasirin mu'amalar filin matsakaici, kuma $\delta$ shine ƙimar rage darajar kayan aiki. Ana ƙirƙirar daidaitawar wahala a matsayin $D(t) \propto Q(t)$, yana tabbatar da $\mathbb{E}[\text{Lokacin Toshi}] = \tau$. Wannan yana haifar da madauki na mayar da martani: mafi girma $Q$ → mafi girma $D$ → ƙananan ladan nan take kowace hash → yana rinjayar $Q$ na gaba.
8. Misalin Tsarin Bincike
Nazarin Lamari: Binciken Lamarin Rabawa Biyu
Yi la'akari da amfani da tsarin MFG ga "rabawa biyu" na Bitcoin, inda ladan toshe $R$ ya ragu da rabi. Samfurin yana ba da tsarin bincike:
- Girgiza: Aikin ladan $R(t)$ ya faɗi ba zato ba tsammani a lokacin $T$.
- Tasiri Nan Take: Manufar hashrate $Q^*$ ta koma ƙasa, yayin da ɓangaren kudin shiga na lissafin ribar masu hakar ma'adinai ya raunana.
- Daidaitawar Halitta: Masu hakar ma'adinai tare da mafi girman farashin aiki ($C(q)$) sun zama marasa riba kuma sun rufe, suna rage $Q(t)$.
- Sabon Matsayin Daidaito: Cibiyar sadarwa ta karkata zuwa sabuwar, ƙananan hanyar girma mai tsayayye na hashrate, duk abin da yake daidai. Duk da haka, idan rabawa biyu ya zo daidai ko ya haifar da ƙarin buƙatu (farashin $S_t$ ya tashi), sabon $Q^*$ na iya zama mafi girma, yana rama yankewar ladan.
Wannan misalin ya nuna yadda tsarin ke raba tasirin injiniyoyi na ƙa'idar ƙa'idar daga amsawar tattalin arzikin ciki.
9. Aiwatarwa & Jagororin Gaba
Hanyar MFG tana buɗe hanyoyin bincike da aiki da yawa:
- Madadin Hanyoyin Yardawa: Aiwatar da MFG ga Hujjar Hatsari (PoS) don kwatanta kaddarorin tsaro na daidaito da kwanciyar hankali.
- Samfurin Tasirin Tsari: Kwaikwayon tasirin harajin makamashi ko haramcin hakar ma'adinai ta hanyar haɗa su azaman girgiza farashi $\theta(t)$ a cikin samfurin.
- Gasar Multi-Blockchain: Ƙara zuwa MFG mai kuɗi da yawa inda masu hakar ma'adinai ke rarraba ƙarfin hash a cikin sarkunan PoW daban-daban, kama da samfura a cikin wasannin cunkoso.
- Ma'auni na Haɗari na Ainihin Lokaci: Haɓaka allunan bayanai waɗanda ke ƙididdige nisan hashrate na yanzu daga manufar da samfurin ya nuna $Q^*$ a matsayin ma'aunin damuwa na cibiyar sadarwa ko ƙarin tsaro.
- Binciken Haɗuwa & Saye: Amfani da tsarin don ƙima tafkunan hakar ma'adinai ta hanyar tantance ikonsu na yin tasiri ko daidaitawa da filin matsakaici.
10. Nassoshi
- Bertucci, C., Bertucci, L., Lasry, J., & Lions, P. (2020). Hanyar Wasan Filin Matsakaici zuwa Hakar Bitcoin. arXiv:2004.08167.
- Nakamoto, S. (2008). Bitcoin: Tsarin Kuɗin Lantarki na Peer-to-Peer.
- Garay, J., Kiayias, A., & Leonardos, N. (2015). Ƙa'idar Ƙarshen Bitcoin: Bincike da Aiwatarwa. EUROCRYPT.
- Lasry, J., & Lions, P. (2007). Wasannin filin matsakaici. Jaridar Lissafin Jafananci.
- Huang, M., Malhamé, R., & Caines, P. (2006). Manyan wasannin motsa jiki na ban tsoro na yawan jama'a: rufaffiyar tsarin McKean-Vlasov da ƙa'idar daidaiton Nash. Sadarwa a cikin Bayanai & Tsarin.
- Biais, B., Bisière, C., Bouvard, M., & Casamatta, C. (2019). Ka'idar jama'a ta blockchain. Binciken Nazarin Kuɗi.
11. Bincike Mai Mahimmanci & Fahimtar Masana'antu
Fahimta ta Asali: Wannan takarda ba wasa ne na lissafi kawai ba; shine tabbataccen hujja ta farko cewa kasafin tsaro na Bitcoin ana ƙayyade shi ta ciki kuma yana da ma'ana ta tattalin arziki. Tsarin MFG ya bayyana cewa "hashrate" da aka yi magana da shi da yawa ba kawai fitarwar fasaha ba ce amma shine matsakaicin ma'auni na wasan rarraba jari na duniya, na ainihin lokaci. Ma'auni babba yana ɗaukar madauki na mayar da martani tsakanin farashi, wahala, da saka hannun jari wanda wasu samfura ke bi da su ta hanyar rarrabuwa.
Kwararar Hankali & Ƙarfi: Ci gaban hankali na marubutan daga samfurin kayyadewa mai sauƙi zuwa mai ban tsoro mai wadata yana da ƙwarewa. Ta fara da matsayi mai tsayayye inda hashrate ke girma tare da ci gaban fasaha ($g$), sun kafa tushe wanda ya dace da yanayin dogon lokaci na gwaji. Gabatar da farashin ban tsoro don samun "manufar hashrate" $Q^*(S_t)$ shine fahimtar mai kashe takardar. Ya bayyana abubuwan kasuwa kamar jinkiri tsakanin faɗuwar farashi da raguwar hashrate—masu hakar ma'adinai ba sa barin nan da nan; suna aiki har sai farashinsu ya wuce sabon, ƙananan ƙimar da ake tsammani. Ƙarfin yana cikin amfani da ingantaccen tsari daga kuɗin lissafi (MFG) don warware matsala a cikin kimiyyar kwamfuta (yarda), yana ba da fahimtar tattalin arziki inda a da akwai tunani na dabara kawai.
Kurakurai & Haɗin da Ya ɓace: Kyawun samfurin shi ma iyakarsa ce. Yana ɗauka cewa ci gaba da ƙananan masu hakar ma'adinai, yana ɗauke da gaskiyar tsakiyar hakar ma'adinai da rinjayen tafki. Ayyukan ƴan manyan tafkuna (kamar Foundry USA ko AntPool) na iya yin tasiri a kan filin matsakaici ta hanyar dabaru, yanayin da ya fi dacewa da samfurin MFG mai haɗaka tare da manyan ƴan wasa. Bugu da ƙari, maganin ci gaban fasaha $g$ a matsayin waje shine kuskure mai mahimmanci. A zahiri, $g$ da kansa ana motsa shi ta hanyar ribar da ake tsammani na hakar ma'adinai—bege na ladan yana haifar da R&D a cikin ƙirar ASIC. Wannan yana haifar da wani madauki na mayar da martani da samfurin ya rasa. A ƙarshe, yayin da yake ambaton ayyukan mahimmanci kamar Lasry & Lions (2007), za a iya ƙarfafa shi ta hanyar haɗawa da wallafe-wallafen kusa da tasirin cibiyar sadarwa da kasuwanni masu bangare biyu, kamar yadda aka gani a dandamali kamar Ethereum.
Fahimta Mai Aiki: Ga mahalarta masana'antu, wannan takarda tana ba da tabarau mai ƙima. Masu Zuba Jari: Samfurin yana ba da shawarar sa ido kan rabon girma hashrate zuwa girma farashi a matsayin ma'aunin lafiyar cibiyar sadarwa. Tsawon lokaci inda hashrate ke girma da sauri fiye da farashi na iya nuna alamar saka hannun jari da yawa da kuma shiri na masu hakar ma'adinai. Masu Haɓaka Ƙa'idar: Binciken ya jaddada cewa duk wani canji ga tsarin ladan (misali, kone kuɗin EIP-1559) dole ne a bincika shi ta wannan tabarau na MFG don hasashen sauye-sauyen daidaiton tsaro. Masu Tsari: Ƙoƙarin hana hakar ma'adinai ta hanyar manufofin makamashi ba za su rage tsaro a layi daya ba; samfurin ya yi hasashen cewa masu hakar ma'adinai za su yi ƙaura (suna canza $\theta(t)$) har sai an sami sabon daidaito na duniya, mai yiwuwa kawai yana motsa tasirin muhalli. Babban abin da za a ɗauka shi ne cewa tsaron Bitcoin ba saiti ne mai ƙayyadadden saiti ba amma daidaito ne mai ƙarfi, wanda tattalin arziki ke motsa shi. Yin magani da shi in ba haka ba—ko don saka hannun jari, ci gaba, ko manufa—kuskure ne na asali.