Visu­al­iz­ing Vari­able Power Throughout the Game

Guide writ­ten by spqcey. Con­tri­bu­tions from the Amaz­ing Com­munity.

Feel free to use the gloss­ary as needed.

Ini­tial Pub­lic­a­tion on Novem­ber 19, 2022

In­tro­duc­tion #

Let’s start by de­fin­ing which vari­ables we’re dis­cuss­ing. We will dis­cuss the main-game vari­ables, (\(x-\xi\)), NOT the up­grades bought in the­or­ies.

We can split the pro­gres­sion through the game into three sec­tions:

• From 0 to ee4310, this sec­tion in­cludes the un­lock­ing of all of the vari­ables, fin­ish­ing with the un­lock­ing of \(\xi\) at ee791. It also in­cludes the pur­chase of \(y\) power up­grades, of which we will dis­cuss more later.
  
  
• From ee9160 to ee47362, this is the range when we buy su­prem­acy up­grades known as “psi3” up­grades.
  
  
• Fi­nally, the fi­nal sec­tion is after ee47362, where we see the title of strongest vari­able change hands a few more times with a fi­nal change at ~ee70000.

Pro­gram #

To visu­al­ize the power of each vari­able throughout the game, I’ve cre­ated a script in Py­thon that can com­pute the power of all vari­ables at any given \(f(t)\). It does this by sim­u­lat­ing the pur­chase of all of the vari­able levels it can af­ford. It then com­putes the power the vari­able has at its cur­rent level. It does this for every vari­able.

To cre­ate the graphs seen shortly, I re­peated this com­pu­ta­tion at many dif­fer­ent \(f(t)\) points, then graphed the power of each vari­able across an \(f(t)\) in­ter­val.

Something to note about the pro­gram is that it com­putes the base power, \(x\), not \(x_8\) which is used in the main equa­tion to com­pute \(f(t)\) after \(x_8\) is bought as an up­grade. Be­cause of how \(x_8\) is com­puted in-game, it can’t be well rep­res­en­ted in these graphs. However, this has no ef­fect on which vari­able is strongest, so it does­n’t mat­ter given the pur­pose of this guide ex­ten­sion.

Un­lock­ing Vari­ables (0 - ee791) #

The cost of the first levels of each vari­able range from Free to an ex­pens­ive ee791. Here’s a table with the cost of the first level of each vari­able:

F(t) val­ues cour­tesy of the Ex­po­nen­tial Idle Wiki.

Here’s a graph of vari­able power up to ee50 \(f(t)\). The ini­tial pur­chase of vari­ables \(y\) through \(\beta\) can be seen.

Com­puted every ee1 from ee1 to ee50.

And here’s a graph with the same \(f(t)\) range, but in­stead of the power of each vari­able, it graphs the per­cent­age of total power each vari­able has at any given \(f(t)\).

Com­puted every ee1 from ee1 to ee50.

And here’s a graph of vari­able power up to the pur­chase of the fi­nal vari­able at ee791:

Com­puted every ee1 from ee1 to ee850.

And the per­cent­age graph:

Com­puted every ee1 from ee1 to ee850. A bit more spiky ;)

\(y\) Power Up­grades (ee50 - ee4310) #

You may have no­ticed the ab­rupt jumps in the above graphs. We’ll now dis­cuss why that oc­curs.
After we su­prem­acy for the first time at ee50, we are given a cur­rency \(\psi\) (psi). With this new cur­rency, we can buy up­grades to in­crease the ex­po­nent on \(y\). Each up­grade raises the ex­po­nent by \(0.2\), so the ini­tial su­prem­acy at ee50 turns \(y\) into \(y^{1.2}\). These up­grades con­tinue all the way up to \(y^{9.0}\) at ~ee4310, for a total of \(40\) levels.

Be­cause the power each vari­able has is propag­ated down all of its lower vari­ables (ex. \(z\) → \(y\) → \(x\)), the change in ex­po­nent af­fects how strong all of the vari­ables are.

The cost of each up­grade from \(y^{1.2}\) through \(y^{1.8}\) is cal­cu­lated us­ing this for­mula, where x is the level you are buy­ing start­ing at 1:

\(\psi = 1.5 \times 2^{2(x-1)}\)

The cost model changes after this, and from \(y^{2.0}\) through \(y^{4.0}\) is this for­mula:

\(\psi = 0.488281 \times 2^{3(x-1)}\)

The cost model changes one fi­nal time, and from \(y^{4.2}\) to \(y^{9.0}\) is this for­mula:

\(\psi = 0.0000000794093 \times 2^{5(x-1)}\)

With all of these cal­cu­la­tions in place, here is the new vari­able power graph, now ran­ging up to ee5000:

Com­puted every ee1 from ee1 to ee5000.

Each small jump present in the graph marks the pur­chase of an ad­ded \(0.2\) to the ex­po­nent of \(y\).

And fi­nally, here is the per­cent­age of total power for each vari­able:

Com­puted every ee1 from ee1 to ee5000.

Very spiky!

Psi3 Up­grades (ee9160 - ee47362) #

After we buy \(y^{9.0}\) at around ee4310, we don’t have any­thing to buy with \(\psi\) un­til ee9160 \(f(t)\), when we can fi­nally af­ford the first Psi3 up­grade. This time, in­stead of 40 levels to buy, there’s only 24 up­grades. However, each up­grade is sep­ar­ated by e20\(\psi\), so the last level, bought with e570\(\psi\), is all the way up at ee47362 \(f(t)\).

These up­grades help delay the de­cay play­ers would oth­er­wise ex­per­i­ence from ee20k-ee50k as their the­or­ies slow down and they gain less \(\tau\).

The first psi3 up­grade in­creases \(z\)’s ex­po­nent to \(z^{1.04}\), and the second up­grade in­creases it fur­ther to \(z^{1.08}\). The third and fourth up­grades in­crease \(s\)’s ex­po­nent to \(1.08\), and so on un­til the fi­nal up­grade in­creases \(\eta\)’s ex­po­nent to \(1.08\).
To il­lus­trate the ef­fect these pur­chases have, let’s use an ex­ample.

Let’s say these are the cur­rent equa­tions for your first five vari­ables:

\(x = 4.7893e7y^9 + 2^3 \times 165.2\)
\(y = 19891z^{1.08} + 2^{419} \times 87522\)
\(z = 18082s^{1.04} + 2^{376} \times 70017\)
\(s = 16433u + 2^{341} \times 56007\)
\(u = 14933v + 2^{323} \times 48973\)

And let’s say that shortly af­ter­ward you pur­chased a new psi3 level:

\(x = 4.7893e7y^9 + 2^3 \times 165.2\)
\(y = 19891z^{1.08} + 2^{419} \times 87522\)
\(z = 18082s^{1.08} + 2^{376} \times 70017\)
\(s = 16433u + 2^{341} \times 56007\)
\(u = 14933v + 2^{323} \times 48973\)

With this new level, the power of \(s\) and \(u\) will in­crease, be­cause in their propaga­tion down to \(x\) they get boos­ted by the ad­ded \(0.04\) on \(s\)’s ex­po­nent. However, the power of \(x\), \(y\), and \(z\) will get no boost, be­cause they are down­stream of the ad­ded ex­po­nent.

But don’t just take my word for it. Let’s look at the data from the pro­gram around this up­grade.

Com­puted every ee1 from ee13000 to ee15000. For the pur­pose of this visu­al­iz­a­tion, only the four vari­ables dis­cussed are plot­ted.

As can be seen in the im­age, both \(y\) and \(z\) get no boost, as we ex­pec­ted. Fur­ther­more, both \(s\) and \(u\) get an equal boost from the up­grade, as will the sub­sequent vari­ables all the way down to \(\xi\).

This leads to an in­ter­est­ing ef­fect where every two psi3 up­grades one of the vari­able’s power stops get­ting boos­ted from the up­grade, so we see a line sep­ar­ate from the rest every two jumps:

Com­puted every ee1 from ee1 to ee50000. This took a while to run…

Com­puted every ee1 from ee1 to ee50000.

Post-Psi3 Vari­able Power (ee47362+) #

We’ve made it. The end of the su­prem­acy up­grades is upon us. What’s next?

Well, now is when \(f(t)\) de­cay really hits. Let’s take a look at an up­dated graph.

Com­puted every ee5 from ee35000 to ee80000.

Im­me­di­ately after the fi­nal su­prem­acy up­grade, we gain a lot less \(x\) per \(f(t)\). Dur­ing the psi3 up­grades, we need an av­er­age of \(17 f(t)\) for an \(e1\) in­crease in \(x\).

Shortly af­ter­ward, dur­ing the period up to ~ee52000, \(\xi\) is the most power­ful vari­able. Un­for­tu­nately for us, its scal­ing is ab­so­lutely ter­rible, and it takes \(77 f(t)\) for an \(e1\) in­crease in \(x\), around 4.5 times worse than dur­ing the psi3 up­grades.

For­tu­nately, \(\xi\) is quickly de­throned by \(\eta\), which is the most power­ful vari­able from ee52000 all the way un­til ee70000 \(f(t)\). For \(\eta\), it takes \(45 f(t)\) for an \(e1\) in­crease in \(x\). This is the sec­tion where all cur­rent en­dgame and top play­ers are.

Fi­nally, at ee70000, \(\eta\) is de­throned by \(y\), which will re­main the strongest vari­able for the rest of the game. \(y\) dom­in­ates the en­dgame be­cause for \(y\) it only takes \(22 f(t)\) for an \(e1\) in­crease in \(x\), which is in­triguingly close to the \(17 f(t)\) the psi3 up­grades offered long ago in our dis­tant past. Cur­rently, all top lead­er­board play­ers are in this range thanks to a sig­ni­fic­ant buff to Cus­tom The­or­ies.

A plot with only \(y\), \(\eta\), and \(\xi\) to il­lus­trate the dif­fer­ence in their scal­ing and where each vari­able be­comes dom­in­ant.

Why We Buy \(y\) Dur­ing En­dgame Gradu­ation Re­cov­ery #

Even be­fore \(y\) be­comes the most power­ful vari­able, as long as you are past ee48000, you should still buy \(y\) dur­ing gradu­ation re­cov­ery. Why? Well, it has to do with the su­prem­acy equa­tions we be­gin ad­opt­ing at ee48000.

At ee48000, the su­prem­acy equa­tion we re­com­mend us­ing skips most of the psi3 su­prem­acy up­grades. In fact, it only su­prem­a­cies at \(e52, e411, e511, e531, e551,\) and \(e571d\psi\). It does­n’t buy any of the psi3 up­grades un­til e410\(d\psi\). Let’s take a look at what that does to the power of the vari­ables:

Com­puted every ee10 from ee1 to ee50000

Very in­ter­est­ing! So it looks like we should be buy­ing \(y\) un­til the su­prem­acy at e410\(\psi\). When we were buy­ing every psi3 up­grade, it was­n’t worth buy­ing \(y\) be­cause the other vari­ables were stronger than it. However, these vari­ables were only stronger than \(y\) be­cause they had been boos­ted by the psi3 up­grades, none of which af­fect \(y\)’s power.

After we reach ee52000, we are re­com­men­ded to use a new su­prem­acy equa­tion. This equa­tion only su­prem­a­cies at e52, e511, and e571 \(d\psi\). Let’s see what that does to the power:

Com­puted every ee10 from ee1 to ee50000

Once again, it looks like we should be buy­ing \(y\) dur­ing gradu­ation re­cov­ery. This time, we should be buy­ing \(y\) un­til the e510\(\psi\) su­prem­acy.

Fi­nally, some­where between ee58000 and ee60000 (we aren’t en­tirely sure where within this range) we switch to our fi­nal su­prem­acy equa­tion. This su­prem­acy equa­tion only su­prem­a­cies at e52 \(d\psi\) and e571 \(d\psi\).

Com­puted every ee10 from ee1 to ee50000

There we go! Now we should buy \(y\) all the way up to e570\(\psi\).

Ad­di­tional Graphs #

Where I put all of the in­ter­est­ing plots that did­n’t fit well in any sec­tion.

Full Graph
The plots of vari­able power from ee1 to ee75000.

Com­puted every ee5 from ee1 to ee75000.

And the plot with per­cent­age of total power for each vari­able:

Com­puted every ee5 from ee1 to ee75000.

Log Per­cent­age Total Power Plots #

More than just the reg­u­lar per­cent­age total power plots, I also tried log­ging the per­cent­age so we can see more than just 0% power for al­most all of the vari­ables, cre­at­ing fas­cin­at­ing res­ults. Ig­nore the ini­tial xi/ \(\xi\) line in the first im­age; it’s a rem­nant that only shows up in these logged plots.

Com­puted every ee1 from ee1 to ee120.

Com­puted every ee1 from ee1 to ee1000.

Com­puted every ee1 from ee1 to ee3000.

Com­puted every ee1 from ee1 to ee20000.

Com­puted every ee10 from ee1 to ee75000.

So many stor­ies told in one plot!