Exponential cost scaling

In most incremental games, each purchase multiplies the next price by a fixed factor $r$: after $n$ purchases the price is $b\cdot r^n$. Income grows linearly with the count owned, so purchases speed up, then slow down. $r$ sets when the turn comes and how steep it is. In 2016, Anthony Pecorella of Kongregate called exponential costs against linear or polynomial production the typical idle-game design, a seesaw between production and costs¹.

#How it works

Let $b$ be the base price (the first one) and $r>1$ the growth constant. With $n$ owned, the next one costs

On a log axis, the price is a straight line: $b$ sets its height, and $r$ sets its slope. The price doubles every $\ln 2/\ln r$ purchases: 4.96 at $r=1.15$, 10.2 at 1.07, and 3.1 at 1.25².

#Buying several at once

Buying $k$ more with $a$ owned sums a geometric series:

At $r=1.15$, “buy 10” costs 20.30× the next single price³ whatever the count owned: $(r^{10}-1)/(r-1)$. Inverting it gives the most you can buy with $c$, $\lfloor {\log }_r(c(r-1)/(b\cdot r^a)+1)\rfloor$⁴.

#Why the growth constant sets the pace

Let each one owned earn $p$ per second, as each building type does in Cookie Clicker before upgrades⁵. With $n$ owned and nothing saved, the next one takes

The ratio is below 1 while $n<1/(r-1)$, then above 1, tending to $r$⁶:

• Waits shrink until about $n=1/(r-1)$: 6.7 at $r=1.15$, 14.3 at 1.07, and 4 at 1.25⁶.
• Then each wait is about $r$ times the last, doubling every $\ln 2/\ln r$ purchases like the price^6,2.

For Cookie Clicker’s Grandma (base price 100 cookies⁷, 1 cookie/s⁸), the wait is 115 s for the 2nd, 38 s at its low for the 8th, and 36 minutes for the 51st⁹. In Pecorella’s account, multipliers keep production close until the wait “gets prohibitively long”, the point to prestige¹⁰.

With linear costs $b+s\cdot n$ and income $p\cdot n$, the wait tends to $s/p$ and never runs away.

#Implementations

#Cookie Clicker

The web version (2.058) uses one constant for all 20 buildings: each one owned multiplies the next price by 1.15¹¹. The next price is $\lceil \mathrm{basePrice}\cdot {1.15}^{\mathrm{owned}-\mathrm{free}}\rceil$¹². Free buildings (Starter kit, Starter kitchen) come off the exponent¹³. Discounts such as ×0.99 apply before the rounding, lowering the line but not its slope¹⁴.

var price=this.basePrice*Math.pow(Game.priceIncrease,Math.max(0,this.amount-this.free));

Only the Cursor keeps its written base price of 15¹⁵. The rest come from a formula on the building’s index¹⁶: 100, 1,100, 12,000, 130,000, and so on⁷. A sibling formula sets base production⁸.

The store shows bulk prices (1, 10, or 100¹⁷) as the exact geometric sum, rounded up once³. The purchase buys one at a time, each rounded up, ignoring the total. Ten Grandmas from zero show 2,031 but cost 2,035, and one cookie short still buys nine¹⁸. Selling pays 25% of the next price, about 15% more than its label¹⁹.

#Antimatter Dimensions

In Antimatter Dimensions, each of the eight dimensions keeps one price for 10 purchases, then multiplies it by its tier’s factor²⁰. Factors run from 10³ to 10¹⁵, about ×2.0 per purchase for tier 1 and ×32 for tier 8²¹. Starting prices run from 10 to 10²⁴²².

Once a price passes Number.MAX_VALUE ≈ 1.8 × 10³⁰⁸²³, the shared ExponentialCostScaling class grows the ratio between sets by another factor of 10²⁴ each set. With $e$ sets past the threshold, the price is $B\cdot C^n\cdot S^{e(e+1)/2}$, quadratic in log space²⁵. For the 1st Dimension, the extra scaling starts after 103 sets, or 1,030 bought²³.

The same amount unlocks Infinity, the first prestige reset²⁶. The Break Infinity upgrade “Reduce post-infinity Antimatter Dimension cost multiplier scaling” lowers S to 3²⁷. Tickspeed upgrades use the same class at ×10 per purchase²⁸.

“Buy until 10” is exact, because the price is flat within a set²⁹. Max-buy charges only the most expensive set it buys, about 0.1% under the true sum³⁰ at ×1,000 per set. A code comment says it “assumes you only have to pay for the most expensive thing”³⁰. Over a long run, the true sum approaches $r/(r-1)$ times the last price: 1.001 at ×1,000 (Antimatter Dimensions) and 7.7 at ×1.15 (Cookie Clicker)³¹.

#AdVenture Capitalist

Pecorella gives the Lemonade Stand as $3.738\cdot {1.07}^{\mathrm{owned}}$³² at 1.67 per second each. At $r=1.07$, the shortest wait comes with 15 stands owned⁶. The game’s code isn’t public.

#A Dark Room: linear, with a cap

A Dark Room prices huts at $100+50n$ wood³³ and traps at $10+10n$ wood³⁴. The build action refuses past the maximum, so the cap, not the price, ends each run³⁵. Housing is 4 villagers per hut³⁶.

#History

Cookie Clicker launched on 8 August 2013³⁷. Its earliest capture, five days later, set each new price to $\lceil \mathrm{price}\times 1.1\rceil$³⁸. The rounding compounded, so the 51st Cursor cost 2,283, not 15 × 1.1⁵⁰ ≈ 1,761³⁸. By v1.035, captured 10 September 2013, the rule was $\mathrm{base}\times {1.15}^n$ through one Game.priceIncrease³⁹.

A Dark Room, announced two months before Cookie Clicker⁴⁰, used linear prices. Antimatter Dimensions (2016⁴¹) used steps of 10 and a super-exponential stage.

#Variants and combinations

• Stepped: Antimatter Dimensions.
• Super-exponential past a threshold: Antimatter Dimensions’ post-Infinity scaling.
• Linear with a cap: A Dark Room.
• Bulk buying: the exact sum (Cookie Clicker’s label) or only the last price (Antimatter Dimensions’ max-buy).
• Discounts: Cookie Clicker.
• Prestige: restarts the curve; production multipliers carry across¹⁰.