1. Gabatarwa & Bayyani
Wannan takarda ta gabatar da sabon aikace-aikacen ka'idar Wasan Matsakaicin Matsakaici (MFG) don ƙirƙirar halayen gasa na hakar kuɗin sirri, musamman ma magance sabani na tsakaita na lada da ƙarfin lissafi a cikin tsarin da ba a tsakaita kamar Bitcoin. Babban tambayoyin bincike suna bincika abubuwan ƙarfafawa da ke motsa halayen ma'adinai, hanyoyin da ke bayan tattara arziki da ƙarfi, da tasirin abubuwa kamar rarraba arzikin farko, lada na haƙa, da ingantaccen farashi (misali, samun wutar lantarki mai arha).
Tsarin ya ɗauki ainihin haƙar tabbatar da aiki: masu haƙa suna amfani da ƙoƙarin lissafi (hash rate) akan farashi, suna gasa don lada mai ban mamaki. Haɗaɗɗun dabarun mutum ɗaya yana haifar da bayyani mai girma na juyin halittar yanayin haƙa.
2. Tsarin Asali & Hanyoyin Aiki
2.1. Tsarin Wasan Matsakaicin Matsakaici
Tsarin ya tsara gasar haƙa a matsayin wasan matsakaicin matsakaici na tsayawa mafi kyau ko sarrafa ƙarfin tsalle. Ana la'akari da ci gaba da masu haƙa. Matsayin kowane mai haƙa shine arzikinsu $X_t$. Suna sarrafa ƙarfin hash rate ɗinsu $\lambda_t$, wanda ke tasiri akan yuwuwar cin nasarar toshe na gaba da farashin aiki.
2.2. Matsalar Inganta Mai Haƙa
Mai haƙa ɗaya yana nufin haɓaka amfanin da ake tsammani na arzikinsa na ƙarshe $X_T$. Halayen arziki suna motsa ta hanyar lada na haƙa (tsalle-tsalle) da farashin ƙoƙari:
$dX_t = -c(\lambda_t)dt + R \, dN_t^{\lambda_t}$
inda $c(\lambda)$ shine aikin farashi don kiyaye hash rate $\lambda$, $R$ shine lada na toshe da aka kayyade, kuma $N_t^{\lambda}$ shine tsarin Poisson da aka sarrafa tare da ƙarfi $\lambda_t$ wanda ke wakiltar abubuwan nasarar haƙar toshe.
2.3. Sarrafa Ƙarfin Tsalle
Babban ma'aunin sarrafawa shine ƙarfin $\lambda_t$ na tsarin Poisson. Zaɓin $\lambda$ mafi girma yana ƙara damar samun lada $R$ amma yana haifar da farashi mai ci gaba $c(\lambda)dt$. Haɗin kai na matsakaicin filin yana tasowa saboda yuwuwar cin nasara kuma ya dogara da jimillar hash rate na duk sauran masu haƙa, yana haɗa dabarun mutum ɗaya zuwa rarraba yawan jama'a.
3. Sakamakon Bincike & Na Lambobi
3.1. Yanayin Amfani na Exponential (Mafita Bayyananne)
Ga masu haƙa tare da amfani na exponential $U(x) = -e^{-\gamma x}$ (matsakaicin ƙin haɗarin cikakke), tsarin ya yarda da mafita bayyananne. An samo dabarar hash rate mafi kyau $\lambda^*$ a cikin sigar martani, yana nuna yadda ya dogara da arzikin yanzu, ƙin haɗari $\gamma$, ma'auni na farashi, da matsakaicin filin.
3.2. Yanayin Amfani na Ƙarfi (Mafita ta Lambobi)
Ga amfani na ƙarfi mafi dacewa da gaskiya $U(x) = \frac{x^{1-\eta}}{1-\eta}$ (matsakaicin ƙin haɗarin dangi), an warware ma'auni na Hamilton-Jacobi-Bellman (HJB) tare da ma'auni na Kolmogorov Gaba (KF) don rarraba arziki ta lambobi. Wannan yana bayyana halayen ƙarƙashin raguwar ƙin haɗarin dangi.
3.3. Babban Bincike & Masu Haɓaka Tsakaita
- Arziki Yana Haifar da Arziki: Bambance-bambancen rarraba arzikin farko yana haifar da ƙarin rashin daidaito a tsawon lokaci ("masu arziki suna samun ƙarin arziki"). Masu haƙa masu arziki za su iya ci gaba da hash rate mafi girma, suna cin nasara da yawa.
- Tasirin Girman Lada: Babban lada na Bitcoin $R$ yana haɓaka tsakaita ta hanyar haɓaka ribar sikelin ga manyan masu haƙa.
- Matsayin Gasa Biyu: Yayin da ƙarin masu haƙa ke ƙara jimillar hash rate, tsarin ya nuna gasa kawai a hankali yana rage—amma baya juyar da—yanayin tsakaita.
- Ingantaccen Farashi a matsayin Fa'ida mai Ƙarfi: Mai haƙa tare da ƙaramin aikin farashi $c(\lambda)$ (misali, daga wutar lantarki mai arha) yana ba da gudummawar babban rabo na ƙarfin hash a cikin ma'auni, galibi ba shi da lahani daga gasa daga abokan hamayya marasa inganci. Wannan kai tsaye yana ƙirƙira haɓakar ƙungiyoyi kamar Bitmain.
4. Cikakkun Bayanai na Fasaha & Tsarin Lissafi
Gindin MFG shine tsarin haɗin ma'auni na bambance-bambancen:
- Ma'auni na HJB (Sarrafa Mafi Kyau): $\partial_t v + H(t, x, \partial_x v, m) = 0$ tare da sharadi na ƙarshe $v(T,x)=U(x)$. Hamiltonian $H$ ya haɗa da haɓaka akan $\lambda$: $H = \sup_{\lambda \geq 0} \{ \lambda [v(t, x+R) - v(t,x)] - c(\lambda) \partial_x v \}$.
- Ma'auni na KF (Juyin Halittar Rarraba): $\partial_t m + \partial_x (b^* m) = 0$, inda drift $b^* = -c(\lambda^*) + \lambda^* [\delta_{x+R} - \delta_x]$ aka samo daga sarrafa mafi kyau $\lambda^*$ kuma ya ƙunshi lokacin tsalle. Sharadin farko shine rarraba arziki da aka bayar $m(0,x)=m_0(x)$.
Ma'auni shine madaidaicin batu inda sarrafa mafi kyau $\lambda^*$ daga ma'auni na HJB, bayan rarraba $m$, yana haifar da juyin halittar rarraba ta hanyar ma'auni na KF wanda ke haifar da $m$ ɗaya.
5. Sakamako, Chati & Mahallin Ƙwaƙwalwar Ajiya
Sakamakon lambobi na takarda zai zama kamata ya kwatanta juyin halittar rarraba arziki $m(t,x)$ daga matsayin farko da aka watsar (misali, log-normal) zuwa rarraba mai karkata sosai, mai tattarawa a tsawon lokaci. Fitattun hotuna sun haɗa da:
- Rarraba Arziki A Tsawon Lokaci: Chati da ke nuna aikin yuwuwar yuwuwar arzikin mai haƙa ya zama mafi karkata dama, tare da wutsiya mai kiba.
- Hanyar Coefficient na Gini: Zane na coefficient na Gini (ma'auni na rashin daidaito) yana ƙaruwa akai-akai tare da lokaci, yana ƙididdige tasirin "masu arziki suna samun ƙarin arziki".
- Raba Hash Rate vs. Arzikin Farko/Farashi: Zane da ke nuna yadda rabon hash rate na ma'auni yake aiki mai haɓaka sosai na arzikin farko ko aikin raguwa na farashi na gefe.
- Haɗin Ƙwaƙwalwar Ajiya: Tsarin yana ba da tushen ka'idar don abubuwan lura na ƙwaƙwalwar ajiya kamar waɗanda ke cikin Kondor et al. (2014), wanda ya gano tattara Bitcoin ya ta'allaka ne ga ƴan adiresoshi, da rinjayen kasuwa na tafkunan da ke da fa'ida kamar Bitmain (wanda ke sarrafa kusan 33% na hash rate a cikin 2019).
6. Tsarin Bincike: Nazarin Shari'a Mai Sauƙi
Yanayi: Yi la'akari da nau'ikan masu haƙa guda biyu a cikin tsarin tsayayye mai sauƙi: "Babban" mai haƙa L tare da ƙaramin farashi na gefe $c_L$ da arzikin farko $W_L$, da "Ƙarami" mai haƙa S tare da babban farashi $c_S$ da arziki $W_S$, inda $W_L >> W_S$, $c_L < c_S$.
Dabarar Tsarin: Kowanne ya zaɓi hash rate $\lambda_i$ don haɓaka ribar da ake tsammani: $\pi_i = \lambda_i \cdot R / (\lambda_L + \lambda_S) - c_i \lambda_i$, inda ake raba lada daidai da hash rate.
Sakamakon Ma'auni: Warware sharuɗɗan mataki na farko yana haifar da $\lambda_L^* / \lambda_S^* = \sqrt{c_S / c_L}$. Tunda $c_S > c_L$, mai haƙa L mai fa'ida ya ba da gudummawar ƙarfin hash da yawa. Tazarar ribarsa ta fi girma, yana ba da damar sake saka hannun jari da ƙara faɗaɗa tazarar—wani ƙaramin duniya na sakamakon tsakaita na MFG. Wannan yana kwatanta yadda bambance-bambancen farashi, ba kawai arzikin farko ba, ke haifar da tsakaita.
7. Aikace-aikacen Gaba & Hanyoyin Bincike
- Hanyoyin Yarjejeniya Madadin: Yin amfani da tsarin MFG zuwa tsarin tabbatar da hannun jari (PoS) don bincika tsakaita mai tabbatarwa da matsalar "babu abin da ke cikin hannun jari".
- Tsari na Manufofi & Ƙa'idodi: Yin amfani da tsarin don gwada tasirin canje-canjen ƙa'idar da aka tsara (misali, lada na toshe mai canzawa, tsarin kuɗi daban-daban) akan ma'auni na tsakaita.
- Halayen Tafkin Haƙa: Ƙaddamar da tsarin don haɗa ƙirƙira dabarun da gasa tsakanin tafkunan haƙa, haɗa kuɗin tafki da abubuwan aminci.
- Haƙa Kaya Da Yawa/Tsallake Silsila: Ƙirƙirar masu haƙa suna rarraba ƙarfin hash a cikin kuɗin sirri da yawa, nazarin hulɗar yanayin muhalli.
- Haɗawa da Bayanan Ƙwaƙwalwar Ajiya: Daidaita ma'auni na tsarin (aikin farashi, ƙin haɗari) tare da bayanan haƙa na ainihi don hasashen bakin kofa na tsakaita.
8. Nassoshi
- Li, Z., Reppen, A. M., & Sircar, R. (2022). Tsarin Wasan Matsakaicin Matsakaici don Hakar Kuɗin Sirri. arXiv:1912.01952v2 [math.OC].
- Nakamoto, S. (2008). Bitcoin: Tsarin Kuɗin Lantarki na Peer-to-Peer.
- Kondor, D., Pósfai, M., Csabai, I., & Vattay, G. (2014). Shin Masu Arziki Suna Samun Ƙarin Arziki? Nazarin Ƙwaƙwalwar Ajiya na Cibiyar Ma'amala ta Bitcoin. PLOS ONE.
- Lasry, J.-M., & Lions, P.-L. (2007). Wasannin matsakaicin filin. Jaridar lissafin Jafananci.
- Carmona, R., & Delarue, F. (2018). Ka'idar Yiwuwar Wasannin Matsakaicin Filin tare da Aikace-aikace. Springer.
9. Ra'ayin Masanin Masana'antu
Fahimta ta Asali: Wannan takarda tana ba da hukunci mai ban tsoro amma mai kyau ta hanyar lissafi: injiniyancin tattalin arzikin haƙar tabbatar da aiki yana da tsakaita a cikin sa. Tsakaita ba ma'auni mai ƙarfi ba ne amma matsayi na wucin gadi da tattalin arziki na sikelin, fa'idodin farashi, da haɗakar arziki suka lalata. Tsarin ya tsara abin da masu lura da masana'antu suka daɗe da zargin—cewa "tsakaita" na Bitcoin labari ne da ke ƙara cin karo da ka'idar wasansa ta asali.
Kwararar Ma'ana: Hujja tana da ban sha'awa. Fara da masu hankali, masu haɓaka riba. Ƙara tsarin lada wanda ba shi da tabbas amma daidai da babban jarin (hash rate). Gabatar da farashi daban-daban (wutar lantarki, ingancin kayan aiki). Sa'an nan injinan MFG yana ci gaba ba tare da katsewa ba, yana nuna yadda bambance-bambancen farko—ko a cikin arziki ko ingancin aiki—suke haɓakawa, ba ragewa ba ta hanyar gasa. Mafita bayyananne don amfani na exponential dabarar tsafta ce, amma sakamakon lambobi na amfani da ƙarfi shine ainihin bayanin, yana taswira kai tsaye zuwa halayen mai haƙa na ainihi.
Ƙarfi & Kurakurai: Ƙarfinsa shine tsaurin ƙa'idarsa—tsarin tattalin arziki ne daidai, ba kawai hannun hannu ba. Ya yi nasarar haɗa ƙarfafawa na ƙanana zuwa sakamako mai girma (tsakaita). Duk da haka, kuskurensa shine zato. Ya yi watsi da mahimman gogayya: dabarun tsalle-tsalle na tafki, rawar masu kera ASIC (kamar Bitmain kanta) a matsayin ɗan wasa kuma alkalin wasa, haɗarin ƙa'ida/wurare na siyasa, da yuwuwar rassa masu wuya don mayar da martani ga tsakaita mai tsanani. Kamar yadda yawancin aikace-aikacen MFG suke, zato na "matsakaicin filin"—cewa masu haƙa suna hulɗa kawai tare da jimillar—na iya sauƙaƙa ƙawance na dabarun da siyasar tafki.
Fahimta Mai Aiki: Ga masu haɓaka ƙa'idodi, wannan bincike gargaɗi ne mai tsanani. Yin gyare-gyare da lada na toshe kadai ba zai gyara tsakaita ba; an toshe shi cikin lissafin lada-farashi. Dole ne mayar da hankali kan ƙirƙira hanyoyin yarjejeniya waɗanda ke hukunta sikelin ko lada rarraba, ko kuma karɓar rawar shiga tsakani na ƙa'ida akan abubuwan farashi (misali, harajin carbon akan haƙa). Ga masu saka hannun jari, yana jaddada cewa ƙimar dogon lokaci na kuɗin sirri tana da alaƙa ba kawai da karɓuwa ba amma da dorewa na tsakaitarsa. Cibiyar sadarwa da ƴan ƙungiyoyi masu fa'ida ke sarrafawa haɗari ne na tsarin. Wannan takarda tana ba da tsarin ƙididdiga don fara auna wannan haɗarin.