Learn · Advanced incremental
An idle game hands out bonuses by the dozen: +50% here, ×2 there, 10% off. Whether they add or multiply decides how fast the numbers grow, and what the next one is worth. Here are ten bonuses at a time, stacked six ways.
Every bonus joins one sum, so each moves the total by the same amount: a shrinking share of a growing total, or, for discounts, a growing share of a shrinking one.
Also called flat percent, increased
Ten bonuses of +50% make +500%, so income is ×6. Each adds the same half of the base, but to an ever bigger total, so the tenth adds only 9% to what's there. Easy to read, and it keeps numbers tame.
Also called flat reduction
Ten price cuts of 10%, added together, make −100%: free. Each takes a bigger share of what's left than the last (the ninth halves the price, the tenth takes the rest), and an eleventh would pay the player to buy.
Each bonus multiplies whatever is already there, so it's worth the same share however many came before it. Buckets mix the two: bonuses add within a bucket, and the buckets multiply.
Also called more, compounding
Ten bonuses of ×1.5 make ×57.7. Each multiplies by the same 1.5, so the tenth is worth as much as the first, and the total grows exponentially with the count of bonuses.
Also called categories, multiplier groups
Bonuses come in kinds: within a kind they add, and the kinds multiply. Ten +50% bonuses split five and five make 3.5 × 3.5 = ×12.25, twice the additive total, and the best next buy is always the smaller bucket.
Also called stacking reductions, diminishing reductions
Ten cuts of ×0.9 leave 0.9¹⁰, about 35% of the price. Each takes the same tenth of what's left, so the price never reaches zero, however many stack.
The bonus raises income to a power, so it's worth more the bigger income already is.
Also called power boost, ^ upgrades
Each bonus raises income to the power 1.05. On 1,000 coins a second the first is worth ×1.41 and the tenth ×1.71, since each one grows the number the next is applied to. On bigger incomes they're worth far more.
With n bonuses of +b each, the total and what the nth one is worth are:
total = 1 + b × n · nth worth ×(1 + b ÷ (1 + b × (n − 1)))
The bonus is the same b every time, but it's divided by an ever bigger total, so its worth sinks towards ×1. The total grows in a straight line with the number of bonuses.
total = mn · nth worth ×m
Every bonus is worth the same ×m, and the total grows exponentially with the count. Ten ×2 upgrades are ×1,024; twenty are over a million.
log(a × b × c) = log a + log b + log c
That's why the pictures use a log scale: every multiplier becomes a block of height log m, and blocks simply stack. Equal multipliers make equal blocks; added bonuses make blocks that shrink; and the total's log, roughly its number of digits, is just the sum of its blocks.
Within a bucket bonuses add; across buckets they multiply. For a fixed number of bonuses the product is biggest when the buckets are equal, which is the AM–GM inequality: (1 + a)(1 + b) with a + b fixed peaks at a = b. So the best next bonus always goes in the smallest bucket.
Split ten +50% bonuses into k equal buckets and the total is (1 + 5 ÷ k)k, the same shape as compound interest. It climbs from ×6 at one bucket to ×57.7 at ten, and if bonuses could be split ever finer it would close in on e5, about ×148.
xp ÷ x = xp − 1 · log(xp) = p × log x
Raising income to the power 1.05 multiplies its log by 1.05: a 100-digit income gains about 5 more digits. So the same bonus is worth ×1.41 on 1,000, ×2 on 1,000,000, and ×100,000 on 10100 coins a second. Applied one after another, as in the tower above, exponents compound: ten of them make the power 1.0510 ≈ 1.63. Some games add the extra exponents instead, so ten +0.05s make ^1.5, which grows more gently.
Added cuts of c each reach zero after 1 ÷ c of them, and go negative after that. Multiplied, (1 − c)n halves again and again but never arrives: ten 10% cuts leave 35% of the price, and fifty leave half a percent.
The same ten +50% bonuses, spread as evenly as they'll go over one to ten buckets. Bonuses in a bucket add; buckets multiply. Pick how many.
×6 on income
One bucket: everything adds, the same as the additive tower.
1 + 0.5 × 10 = ×6
×12.25 on income
Two buckets: 2.04 times what one bucket gives, from the same ten bonuses.
(1 + 0.5 × 5)2 = ×12.25
×18.75 on income
Three buckets: 3.13 times what one bucket gives, from the same ten bonuses.
(1 + 0.5 × 4) × (1 + 0.5 × 3)2 = ×18.75
×32 on income
Five buckets: 5.33 times what one bucket gives, from the same ten bonuses.
(1 + 0.5 × 2)5 = ×32
×57.7 on income
A bucket each: every bonus multiplies, the same as the multiplicative tower.
(1 + 0.5 × 1)10 = ×57.7
More buckets always pay more, which is why games keep them few, and name them: when players can see that a new kind of bonus multiplies everything they have, they know where their next choice counts.
The total after one, five, and ten bonuses, and what the tenth was worth. Exponent bonuses are on 1,000 coins a second.
| Kind | Total after n | After 1 | After 5 | After 10 | The tenth |
|---|---|---|---|---|---|
| Additive | 1 + 0.5 × n | ×1.5 | ×3.5 | ×6 | ×1.09 |
| Added discounts | 1 − 0.1 × n | ×0.9 | ×0.5 | ×0 | ×0 |
| Multiplicative | 1.5ⁿ | ×1.5 | ×7.59 | ×57.7 | ×1.5 |
| Buckets | (1 + 0.5a) × (1 + 0.5b) | ×1.5 | ×5 | ×12.25 | ×1.17 |
| Multiplied discounts | 0.9ⁿ | ×0.9 | ×0.59 | ×0.349 | ×0.9 |
| Exponent | x^(1.05ⁿ − 1) | ×1.41 | ×6.74 | ×77 | ×1.71 |