Dis­tri­bu­tion Over­push­ing

Guide writ­ten by Playspout. Con­tri­bu­tions from the Amaz­ing Com­munity.

What is dis­tri­bu­tion over­push­ing? #

A greedy way to push a the­ory dis­tri­bu­tion is to push all 8 ori­ginal the­or­ies un­til they have sim­ilar $\tau$/hour gains. However, this is not ne­ces­sar­ily op­timal for long term $\tau$ gains. Dis­tri­bu­tion over­push­ing is when you push cer­tain the­or­ies such that their $\tau$/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 $\dot{\tau }$, vs the ef­fect of time on $\dot{\tau }$ 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:

$\dot{\rho }=KS{\rho }^{0.3}$

$\rho =(KSt{)}^{1/0.7}$

Des­pite the 1/​0.7 ex­po­nent, we can con­clude that $\sigma$ and time af­fect $\rho$ equally.

The­ory 2 #

$\dot{q_4},\dot{r_4}=A;q_4,r_4\approx At$

By the same lo­gic:

$\dot{q_3},\dot{r_3}=At;q_3,r_3\approx A{t}^2$

$\dot{q_2},\dot{r_2}=A{t}^2;q_2,r_2\approx A{t}^3$

$\dot{q_1},\dot{r_1}=A{t}^3;q_1,r_1\approx A{t}^4$

$\dot{\rho }\approx (A{t}^{4}A{t}^4{)}^{1.15}\approx A{t}^{9.2}$

$\rho \approx KS{t}^{10.2}$

Here we see that time af­fects $\rho$ 10.2 mag­nitudes more than ex­tra $\sigma$.

The­ory 3 #

$\dot{{\rho }_n}\approx A,\rho \approx KSt$

Sigma and time af­fect $\rho$ equally.

The­ory 4 #

Late game $c_3$ term dom­in­ates so:

$\dot{\rho }\approx A{c}_{3}q$

$\dot{q}\approx \frac{A}{1+q}$

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

$\dot{\rho }\approx A{c}_3\sqrt{t}$

$\rho \approx KS{t}^{1.5}$

Time af­fects $\rho$ 1.5 mag­nitudes more than $\sigma$.

The­ory 5 #

Late game we can treat $c_2$ gains as in­stant­an­eous, so $q$ is treated as a con­stant.

$\dot{\rho }\approx A$ only

$\rho \approx KSt$

Sigma and time af­fect $\rho$ equally.

The­ory 6 #

Late game $c_5$ term dom­in­ates.

$\rho \ap­prox Ac_5qr^2 $ after in­teg­rat­ing

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

$\rho \approx KS{t}^3$

Time af­fects $\rho$ 3 mag­nitudes more than $\sigma$.

The­ory 7 #

Late game $c_4$ term dom­in­ates.

$\dot{\rho }\approx A{c}_6\sqrt{\frac{{\rho }_2}{{\rho }_1}}$ 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: $\dot{\rho }\approx A{c}_6$

$\rho \approx KSt$

Sigma and time af­fect $\rho$ equally.

The­ory 8 #

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

$\dot{\rho }\approx A\sqrt{B{c}_3}$

$\dot{\rho }\approx A$

$\rho \approx KSt$

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 $\sigma$ are lin­early de­pend­ent on each other.

For The­ory 4, ${\dot{\rho }}^{1.5}$ is pro­por­tional to $\sigma$.

For The­ory 6, ${\dot{\rho }}^3$ is pro­por­tional to $\sigma$.

For The­ory 2, ${\dot{\rho }}^{10.2}$ is pro­por­tional to $\sigma$.

These con­clu­sions im­ply that for The­ory 2 for ex­ample, in­creas­ing $\sigma$ hardly af­fects $\dot{\rho }$.

Over­push­ing Coef­fi­cients #

From pre­vi­ous work­ings, the over­push­ing coef­fi­cients are:

Class: break­down;
Cap­tion: Over­push Coef­fi­cients;

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