Step­wise Vari­ables

Guide writ­ten by Eyland­ing. Con­tri­bu­tions from the Amaz­ing Com­munity.

Feel free to use the gloss­ary as needed.

What are Step­wise vari­ables? #

All vari­ables in Ex­po­nen­tial Idle The­or­ies can be clas­si­fied into 3 types: Dir­ect Mul­ti­plic­a­tion (a.k.a. Dir­ect Multi), Step­wise, and Lin­ear vari­ables.

Which type of vari­able a given vari­able is can be mostly de­term­ined simply by look­ing at it. If it is in the form $b^L$ it is a Dir­ect Mul­ti­plic­a­tion vari­able, but if it is just a num­ber by it­self, then it is a Step­wise or Lin­ear vari­able.

In or­der to tell the dif­fer­ence between Step­wise and Lin­ear vari­ables there are a few in­dic­at­ors:

• Step­wise vari­ables are al­most al­ways in­tegers.
• Lin­ear vari­ables are al­most al­ways very small (<1e6).
• Step­wise vari­ables of­ten have sub­script (e.g. $q_1$). In con­trast, Lin­ear vari­ables are of­ten by them­selves (e.g. $n$).
• Lin­ear vari­able in­crease, be­lieve it or not, lin­early. If it’s not grow­ing lin­early, it is not a Lin­ear vari­able.

However, there in­dic­at­ors are not per­fect. $c_1$ in FP grows lin­early up to 100, and even though it stays an in­teger, the lin­ear growth has a low value, grow­ing by 1 per level keep­ing it small. It is ac­tu­ally a Step­wise vari­able with an un­usu­ally high step length).

For ex­ample:

• $c_1=2^{52}$ - Dir­ect Multi
• $q_1=1.67e91$ - Prob­ably Step­wise
• $r_2={7.4}^{81}$ - Dir­ect Multi
• $s=0.25$ - Prob­ably Lin­ear

All vari­ables have the same cost model (ex­po­nen­tial). The dif­fer­ence between Step­wise and Dir­ect Multi vari­ables is what hap­pens when you buy an up­grade.

The simple vari­ables #

Dir­ect Multi vari­ables are very simple. When you buy an up­grade, what hap­pens is ex­actly what you ex­pect to hap­pen. The ex­po­nent is in­creased by one, rais­ing it’s value by a factor of it’s base (e.g. $3{}^4\to 3^5\to 3^6$ is $3$x in­crease per level).

Lin­ear vari­ables in­crease by a set amount per level (e.g. $2\to 3\to 4\to 5$ is $+1$ per level). This is usu­ally very weak, so they are gen­er­ally in ex­po­nents or other ex­po­nen­tial func­tions (or capped).

Step­wise vari­ables #

Step­wise vari­ables are more com­plic­ated than Dir­ect Multi vari­ables. It is im­possible to know how a Step­wise vari­able will in­crease just by look­ing at it. One needs to ex­am­ine the code in or­der to know how it will in­crease. Step­wise vari­ables can be de­scribed with 3 num­bers, their ini­tial value (I), their step length (S), and their base power (B).

Note: “ini­tial value (I)” is not com­monly used, as due to the nature of the game, it does not ac­tu­ally af­fect any strategy, so it is usu­ally ig­nored. Fur­ther­more, it does not have a name in code - “ini­tial value” is just a term I per­son­ally use.

A Step­wise vari­able starts with a value of 0. Every time you up­grade it, it will in­crease by I, it’s ini­tial in­cre­ment. Every S levels, the amount in­creased per level in­creases by a factor of B.

This is easi­est ex­plained with an ex­ample. Let’s cre­ate a vari­able $c_1$. This vari­ables will have an ini­tial (I) of 1, a base (B) of 2, and a step (S) of 10. At level 0, $c_1$ has a value of 0. Every up­grade, $c_1$ in­creases by I, which is 1. Thus, $c_1=1$ at level 1, $c_1=2$ at level 2, and so on. At level 10, $c_1=10$, but something else hap­pens. It’s step length (S) is 10. Now the amount $c_1$ in­creases per level in­creases by a factor of B, in this case doub­ling, so $c_1$ in­creases by 2 per level. At level 11, our vari­able now hold a value of 12, up to a value of 30 at level 20, where the in­cre­ment doubles again, in­creas­ing by 4 now per level.

First 30 levels (0-29) of a vari­able with ini­tial (I) of 1, base (B) of 2, and step (S) of 10

No­men­clature #

As be­fore, in an at­tempt for a bet­ter ex­plan­a­tion, I made my own no­men­clature (sorry about that). Here is the uni­ver­sally ac­cep­ted no­men­clature. When talk­ing about Step­wise vari­ables, one usu­ally uses 2 num­bers to de­scribe it: its base and its step. It is writ­ten like this (B, S). For ex­ample, the vari­able as de­scribed be­fore, $c_1$, is a (2, 10) Step­wise vari­able.

Whilst the ini­tial value of a Step­wise vari­able af­fects the the­ory and could af­fect vari­able com­pet­i­tion, when vari­ables are mul­ti­plied to­gether—as they are in most the­or­ies, the ini­tial value can be factored out and ig­nored. The ini­tial in­cre­ment value does not in­crease how much a vari­able in­creases per level as a per­cent­age. For ex­ample, (2,10) with an ini­tial of 1 fol­lows: 1, (+100%) 2, (+50%) 3; the same vari­able with an ini­tial of 10 fol­lows: 10, (+100%) 20, (+50%) 30. Dif­fer­ent num­bers, same per­cent­age dif­fer­ence.

What Step­wise vari­ables am I work­ing with? #

The easi­est way to de­term­ine what Step­wise vari­able you are work­ing with is to look into the code. The code of all main and of­fi­cial cus­tom the­or­ies are at Con­icGames’ Git­Hub. Main the­or­ies are found in the SDK here and of­fi­cial cus­tom the­or­ies are found here.

In ALL main the­or­ies, all Step­wise vari­ables are (2, 10) Step­wise vari­ables. Fur­ther­more, there are no lin­ear vari­ables.

Ba­sic Step­wise vari­able strategy #

(2, 10) Step­wise vari­ables in­crease by on av­er­age $7\mathrm{\%}$ per up­grade, and in all lis­ted strats in the The­ory Strategies Ex­plained Guide are bought at a 10x ra­tio to doub­ling vari­ables (Dir­ect Multi vari­ables with a base of 2). However, due to the nature of Step­wise vari­ables, some­times it is wise not to fol­low this ra­tio.

First of all, Step­wise vari­ables work dif­fer­ently at the very be­gin­ning. The first up­grade is usu­ally free and re­quired, as it moves it from 0 to 1. Very early up­grades are usu­ally un­usu­ally power­ful. For ex­ample, the level 3 -> 4 up­grade in­creases it by $33\mathrm{\%}$, much more than the $7\mathrm{\%}$ av­er­age in­crease. More im­port­antly is the non-smooth nature of Step­wise power func­tions.

Step­wise power func­tions have a jump at each step length. This jump means that just after get­ting the in­crease to their amount per level, they are stronger. For ex­ample, a (2, 10) Step­wise vari­able would be slightly stronger just after each mul­tiple of 10, level 10 -> level 11 is stronger than level 9 -> level 10.

$s(x)$ is the (2, 10) Step­wise power func­tion. Red: $s(x)$, Blue: $100(\frac{b(x+1)}{b\left(x\right)}-1)$, i.e. the amount s(x) in­creases when you buy an up­grade start­ing from up­grade x, as a per­cent­age.

In the above graph, the blue line rep­res­ents ap­prox­im­ately how much $s(x)$ in­creases after buy­ing a level, start from level $x$. As you can see, this spikes at mul­tiples of 10 (or more gen­er­ally mul­tiples of S) and goes down from there. From this, one can see that Step­wise vari­ables are more power right a mul­tiple of S and weak­est right be­fore it.

With this know­ledge, one can ad­just their strategy. For (2, 10) Step­wise vari­ables spe­cific­ally, the re­com­men­ded ra­tio to buy them (usu­ally 10) re­l­at­ive to a doub­ling vari­able is lower just after a mul­tiple of 10 and higher just be­fore one. For ex­ample, let $c_1$ and $c_2$ be Step­wise and doub­ling vari­ables re­spect­ively. If $c_1$ is level 341 and costs $4.86e94$ and $c_2$ is level 57 and costs $3.91e95$, we would usu­ally save for $c_2$, but in this case, be­cause $c_1$ is just after a mul­tiple of 10, one should ac­tu­ally buy $c_1$ first.

More spe­cific­ally, the ra­tio is 5x at a mul­tiple of 10, in­creas­ing to 10x at 5 more then a mul­tiple of 10, and 15x at 9 more than a mul­tiple of 10.

Mod10 strategies #

Many strategies make use of this un­der­stand­ing of Step­wise vari­ables by chan­ging the ra­tio bought at de­pend­ing on where one is in the cycle of these Step­wise vari­ables. This is done us­ing the mod­ulo op­er­ator, $\mathrm{\%}$. Take the above ex­ample. $c_1$ is a (2, 10) Step­wise and $c_2$ is a 2x Dir­ect Multi vari­able. The ra­tio we want to buy it at is 5x when $c_1$’s cur­rent level = 10, 20, 30 etc. and up to 15x when it’s cur­rent level is 19, 29, 39 etc. Thus, we define the strategy like this

This uses a new op­er­a­tion you may not have en­coun­ters be­fore, the mod­ulo op­er­ator $\mathrm{\%}$.

The mod­ulo op­er­ator re­turns the re­mainder of di­vid­ing the first num­ber by the second num­ber. For in­stance, take $11\mathrm{\%}3$. IF we do 11/​3, we get 3 re­mainder 2. This re­mainder is what the mod­ulo op­er­ator re­turns, so $11\mathrm{\%}3=2$. Thus, by tak­ing the level of the Step­wise vari­able mod ($\mathrm{\%}$) it’s step length (in this case 10), we get how far into the Step­wise cycle that vari­able is.