Dis­tri­bu­tion Over­push­ing

Guide writ­ten by Playspout. Con­tri­bu­tions from the Amaz­ing Com­munity.

Feel free to use the gloss­ary as needed.

What is dis­tri­bu­tion over­push­ing? #

A greedy way to push a the­ory dis­tri­bu­tion is to push all 8 ori­ginal the­or­ies un­til they have sim­ilar \(τ\)/​hour gains. However, this is not ne­ces­sar­ily op­timal for long term \(τ\) gains. Dis­tri­bu­tion over­push­ing is when you push cer­tain the­or­ies such that their \(τ\)/​hours are less than oth­ers. This seems to con­tra­dict op­timal play, as gen­er­ally you’d gain more stu­dents quicker if you push all the­or­ies to equal rates. However, some the­or­ies do not be­ne­fit as much from the 3rd level of the 9th Re­search in the Stu­dents tab (3R9). There­fore it makes more sense to push the­or­ies that don’t be­ne­fit much from ad­di­tional stu­dents first. This al­lows us to use the ad­di­tional stu­dents to ac­cel­er­ate the rates of the­or­ies that do be­ne­fit sig­ni­fic­antly from 3R9.

How the­or­ies are af­fected by ad­di­tional stu­dents #

We will com­pare the ef­fect of stu­dents on \(\dot{τ}\), vs the ef­fect of time on \(\dot{τ}\) for each the­ory. For the equa­tions be­low, ‘K’ and ‘A’ are just con­stants, ‘S’ is the mul­ti­plier gained from hav­ing stu­dents (3R9 mul­ti­plier).

The­ory 1 #

We know that in The­ory 1, the \(c_4\) term dom­in­ates late game:

\(\dot{\rho} = KS\rho^{0.3}\)

\(\rho = (KSt)^{0.7}\)

Des­pite the 0.7 ex­po­nent, we can con­clude that σ and time af­fect \(\rho\) equally.

The­ory 2 #

\(\dot{q_4}, \dot{r_4} = A; q_4, r_4 ~= At\)

By the same lo­gic:

\(\dot{q_3}, \dot{r_3} = At; q_3, r_3 ~= At^2\)

\(\dot{q_2}, \dot{r_2} = At^2; q_2, r_2 ~= At^3\)

\(\dot{q_1}, \dot{r_1} = At^3; q_1, r_1 ~= At^4\)

\(\dot{\rho} ~= (At^{4}At^{4})^{1.15} ~= At^{9.2}\)

\(\rho ~= KSt^{10.2}\)

Here we see that time af­fects \(\rho\) 10.2 mag­nitudes more than ex­tra σ.

The­ory 3 #

\(\dot{\rho_n} ~= A, \rho ~= KSt \)

Sigma and time af­fect \(\rho\) equally.

The­ory 4 #

Late game \(c_3\) term dom­in­ates so:

\(\dot{\rho} ~= Ac_3q \)

\(\dot{q} ~= \frac{A}{1+q} \)

Solv­ing the dif­fer­en­tial equa­tion yields $q$ is pro­por­tional to \(\sqrt{t} \)

\(\dot{\rho} ~= Ac_3\sqrt{t}\)

\(\rho ~= KSt^{1.5}\)

Time af­fects \(\rho\) 1.5 mag­nitudes more than σ.

The­ory 5 #

Late game we can treat \(c_{2}\) gains as in­stant­an­eous, so $q$ is treated as a con­stant.

\(\dot{\rho} ~= A \) only

\(\rho ~= KSt\)

Sigma and time af­fect \(\rho\) equally.

The­ory 6 #

Late game \(c_5\) term dom­in­ates.

\(\rho ~= Ac_5qr^2 \) after in­teg­rat­ing

$\dot{r},\dot{q}\sim =A;r,q\sim =At$

\(\rho ~= KSt^3\)

Time af­fects \(\rho\) 3 mag­nitudes more than σ.

The­ory 7 #

Late game \(c_4\) term dom­in­ates.

\(\dot{\rho} ~= Ac_6\sqrt{\frac{\rho_2}{\rho_1}} \) after par­tial dif­fer­en­ti­ation

However, \(\frac{\rho_2}{\rho_1}\) is ef­fect­ively con­stant. This is be­cause if \(\rho_2\) is higher than \(\rho_1\), \(\rho_1\) will even­tu­ally catch up to \(\rho_2\) and vice versa.

There­fore: \(\dot{\rho} ~= Ac_6\)

\(\rho ~= KSt\)

Sigma and time af­fect \(\rho\) equally.

The­ory 8 #

The cost value scal­ings for \(c_3\), \(c_4\), \(c_5\) are the same. Since they are ad­dit­ive with each other, we can sim­plify the equa­tion as:

\(\dot{\rho} ~= A\sqrt{Bc_3}\)

\(\dot{\rho} ~= A\)

\(\rho ~= KSt \)

Sigma and time af­fect \(\rho\) equally.

From the equa­tions above, we can con­clude that for the­or­ies 1, 3, 5, 7, and 8, \(\dot{\rho}\) and σ are lin­early de­pend­ent on each other.

For The­ory 4, \(\dot{\rho}^{1.5}\) is pro­por­tional to σ.

For The­ory 6, \(\dot{\rho}^{3}\) is pro­por­tional to σ.

For The­ory 2, \(\dot{\rho}^{10.2}\) is pro­por­tional to σ.

These con­clu­sions im­ply that for The­ory 2 for ex­ample, in­creas­ing σ hardly af­fects \(\dot{\rho}\).

Over­push­ing Coef­fi­cients #

From pre­vi­ous work­ings, the over­push­ing coef­fi­cients are:

This means that for ‘per­fect’ over­push, we should push the max­imum of {\(\dot{τ}_1\), \(10.2*\dot{τ}_2\), \(\dot{τ}_3\), \(1.5*\dot{τ}_4\), \(\dot{τ}_5\), \(3*\dot{τ}_6\), \(\dot{τ}_7\), \(\dot{τ}_8\)}. This will max­im­ize long term \(τ\) gain, as­sum­ing everything else is equal.