In Clash of Clans (a super popular social building / defending game by Supercell) players can speed up actions such as building, upgrading or training troops with gems that can be bought for real money. They’re basically monetizing the player’s impatience. Every social building game I know of does it.
I set out to learn more about the hidden formulas at work. This is how CoC puts a price on time.
Let’s start off by comparing some graphs.
This line graph shows how many gems it costs to skip time. More time = more gems. As expected.
But there’s a catch. It looks like the price increases exponentially. This is misleading! It only looks that way because the difference between 60 seconds (1 minute) and 604,800 seconds (1 week) is so vast. The time is increasing exponentially and it’s distorting the data.
Switching from line graph to a scatter graph gives a more truthful picture. The horizontal time axis is scaled evenly. But it’s hard to tell what’s going now in the bottom-left corner now. The values are so bunched up you can’t tell them apart anymore.
Let’s try interpreting the data differently. Think of it in terms of cost in gems to skip 1 second. Skipping more time costs more money, but you get a better deal. Think of it as a set of increasing discounts.
My testing has shown that the pricing algorithm graph consists of a series of linear line segments. They are not simplified versions of curved graphs. I will explain why later.
Now let’s look at some formulas.
To start, all we need to do is create a neat little table like this with your desired prices, and then I will show you how to use linear interpolation to calculate any other price.
It’s a nice system. Breaking the pricing down into chunks of time that make sense to humans (hour/day/week) makes it easy to visualize. Everybody knows what it feels like to have to wait a day. It’s also very easy to tune. If you change your mind later and decide that 1 day should cost more, all you need to do is update 1 value.
The nice thing about using a linear segment graph is that it allows you to set and tweak specific price points.
So let’s turn it into a formula that we can turn into a graph.
We start simple enough with the basic formula to find the equation of a line:
From there we can work out the equation of a line between 2 points 
With this information, we’re ready to start substituting values. Using Clash of Clans’ data, let’s say for example you want 1hr (3,600 seconds) to cost 20 gems, and 1 day (86,400 seconds) to cost 260 gems. We can thus specify a function for such a line segment.
We can use this function to find the cost of any time between 1hr and 1 day. For example to find the cost of speeding up 4 hours, substitute 
All the formulas for the individual line segments can be combined into one very large piecewise function.
This is the CoC gems -> time conversion rate formula.
To enter it into excel, copy-paste this long formula into cell B1. If you then enter an amount of time in seconds in cell A1, the cost in gems will appear in B1.
=ROUND(IF($A$2=0,0,IF($A$2<=60,1,IF($A$2<=3600,((20-1)/(3600-60))*($A$2-60)+1,IF($A$2<=86400,((260-20)/(86400-3600))*($A$2-3600)+20,((1000-260)/(604800-86400))*($A$2-86400)+260)))),0)
Try a live demo. Double click the time to edit it and watch the cost in gems update.
Let’s talk about how all this affects real money.
So far I’ve just been talking in terms of CoC’s ingame currency “gems”. But gems are bought in the store for money, so we can put a price on them. This is how Supercell prices gems.
The cost per gem actually decreases the more you buy, but more about discounts later. For now I’m going to simplify things a little and use the average cost: £0.0055 per gem.
This is super useful because now we can use the formulas from before to set real-world prices for things. Someone at Supercell decided that in CoC a 1 day skip should cost on average £1.43 and 1 week £5.49. If we wanted to make a change, we could for example change the price of 1 day to £2.00 and have everything else update automatically. It’s a really convenient and logical way to tune the game’s monetization.
(Brief and mostly unrelated side note: have you ever wondered why companies like Supercell or Microsoft love using abstract virtual currencies with perplexing exchange rates? During my research I came across a great article on the psychology behind such virtual currencies. It’s too good not to share. Read part 1 here, and part 2 here.)
Discounts
Buying Gems
The gem pack sizes are fixed. You can’t buy 501 gems. You have to either get the 500 or the 1,200 pack. Regardless, I though it interesting to view it as a graph.
It’s hard to see, but the line actually gets slightly less steep the more you buy. Supercell is giving you a discount if you buy a larger quantity.
You save a 16.4% for buying a pack of 14,000 when compared to the price of the 500 pack.
Skipping Time
When you compare the discounts for buying gems to the discounts for skipping time, it becomes obvious that the discount for skipping a longer time-piece is far more generous than the discount for buying a larger gem pack.
You save a massive 90.1% buying a 1 week time skip when compared to the price of skipping 1 minute.
It’s worth noting that the discount difference between 1 minute and 1 hour is 66.7%, but the discount difference between 1 hour and 1 week is only 23.4%. Instead of thinking of 1+ days as highly discounted, we could think of it the opposite way: 1-59 minute skips are overly expensive. Why are they so overpriced? I don’t have a definitive answer… my hunch is that only the most impatient of players are willing to buy such short skips. They may also be the most willing to spend, so perhaps its a little bit of price discrimination at work? I’m not sure. I wish I had a better answer. Leave me a comment if you think you know better! 
Stacking Discounts
If you buy gems, and then use them to skip time, the discounts stack.
The worst deal you can get is to buy 500 gems for £2.99 and then use it to skip a 1 minute timer. You have just paid £0.000100 per second you skipped.
The best deal you can get is to buy 14,000 gems for £69.99 and then use it to skip a 1 week timer. You have just paid £0.000008 per second you skipped.
That’s a price difference of 91.7%.
Discount Power
The cost per second sounds like a tiny insignificant amount, but when you are skipping 604,800 seconds it adds up. If you were charged the same rate for skipping 1 week as for 1 minute, it would cost you £60 instead of £5!!
The generous discount for skipping longer stretches of time is essential. Without it prices would quickly become prohibitive and nobody would buy them.
That’s what I think Supercell must have realized. They needed a way to heavily discount long skips that is easy to tune. Linear segmented graphs are a great solution.
Lastly, let’s consider alternative solutions.
There are other ways Supercell could have done it. Considering alternative solutions teaches us more about the pros and cons of their method.
A Linear Graph
My first thought was to use a simple linear graph. Then there would just be a gems -> time conversion rate, like there is for $ -> £.
It’s possible, but the cost of skipping 1 day would fall by 61.5% from 260 gems to 100 gems. Or, if you wanted to keep the 1 day price at 260 gems, the line would have to rise more steeply and 1 week would become more expensive.
Basically, it would only allow us to tune 2 price points. If you want to tune more than that – as Supercell obviously felt it was necessary – you need multiple linear segments. Theoretically they could tune even more price points. If they wanted to charge more or less for a 2 day skip, they could easily introduce 1 more segment.
A Curved Graph
Why not a smooth curve? Perhaps some exponential, quadratic or bezier function?
It looks nice and smooth, but it becomes really difficult to tune… the formulas involved are just a major pain to work with. Having messed around with it for a little while, I don’t recommend it 
Conclusion
The Clash of Clans way of monetizing time skips is really elegant. I can’t think of a better system.
To summarize:
- Skipping more time costs more money, but you get a better deal.
- The pricing algorithm graph consists of a series of linear line segments.
- The nice thing about using a linear segment graph is that it allows you to set and tweak specific price points in a easy to understand way.
- The discount for skipping a longer time-piece is far more generous than the discount for buying a larger gem pack.
- The generous discount for skipping longer stretches of time is essential. Without it prices would quickly become prohibitive and nobody would buy them.
Feedback, questions, concerns? Leave me a comment!
Disclaimer: I compiled the information in this article by playing lots of CoC and doing some research. To the best of my knowledge, is it accurate and I have triple checked all figures and calculations. Nevertheless, it is always possible that I have made a mistake somewhere or that the game has changed since publication. Use at your own risk, I don’t provide any guarantees. Please get in touch if you find anything amiss.
A lot of nice work with the graphs to do with time, it does leave me wondering though, what does it look like for currency?
Having played the game for a while, I regularly see people buying their way to very powerful buildings while the entire rest of their base is low level (obviously spending).
Following your above findings, is it a better or equal deal to spend gems on 1 huge pile of cash / elixir for 1 building or several small bundles of cash / elixir for multiple other buildings?
If you just care about spending efficiently, definitely buying the biggest gem packs and then spending them on the longest timers. That way you will get a maximum amount of seconds skipped per dollar spent.
But is it the best strategy? If you want to dominate your friends, you would also need to take into account the usefulness of each building / upgrade. For example upgrading a lvl 8 gold mine is a more efficient spend than upgrading a lvl 1 cannon, but it’s not going to help you defend against the next attack!
I wasn’t talking about skipping timers, you’ve covered that pretty well.
I was talking about the fact that you can purchase gold, elixir and dark elixir with gems.
My question is therefore:
Do you get more gold / elixir / dark elixir per gem if you buy more of it in one go?
I think you need to think more carefully about the retail psychology of players. No one spend $100 immediately in-game, so small IAP purchases need to be useful, especially during the start of game when build times aren’t long.
And you don’t get build times of a 1 day until you’re days, if not weeks into playing.
His analysis was designed to perpysfully to disregard psychology and focus on the math. It wasn’t about what people do when faced with various buying options, but what makes logical sense based on the options and relevant data. It perfectly provided the analysis and conclusion I was looking seeking.
[…] Вольфганг Граебнер (Wolfgang Graebner) в своем блоге разобрал Clash of Clans на формулы. Мы предлагаем краткую версию […]
I buy small amounts of time because I am actually playing at that moment. For example, if I have 37 minutes till my builder is free but I have to go to work for 4 hours soon there is a chance that some wilk attack and take my gold that is alotted for my next building. If I spend a few gems I get my builder back and can proceed with my plan of action. I get 3 rewrds. One- my initial building is completed. Two- I’ve protected my gold by spending it. Three- I no longer have to keep in my mind the next step of building my village. Sometimes I forget while waiting for resources or a builder to become available and am forced to re-assess my village.
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supercell is just a bunch of greedy bastards who just want money
They produce a quality game, excellent play mechanics, employ dozens of people producing it, more marketing it, paying for ads on TV, designing events to promote it, and don’t put ads in the game. And then give it away for free. And you never need to spend a penny to advance. Free. Lots of their effort in it. Run server farms to keep it all going. Free.
The greedy people are the ones that object to Supercell’s desire to make some money by selling in game gems to speed their progress because they aren’t patient. You can take the surface roads, or pay for the toll road, only in this case the toll road was privately constructed on private lands, with private improvements and maintenance.
No doubt u r right …but u should also note that it takes hell lot of time to upgrade building …..even after one fucking year i cant max out my base and yes my 5 build are working continuously….so i just think they should reduce the timing a bit for not so rich players like me….
a coc player
No matter how great a game is, once it’s once you add Pay To Cheat, it’s RUINED. COC developers don’t deserve to be rewarded, they deserve to rot in prison or worse.
Lmao. Wow, that was dumb.
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So is it a better deal to buy time or gold/elixir in this game once you have the gems?
ty sooo much i been hitting my head against a wall trying figure out these numbers
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This price structure is due to hyperbolic discounting, which is basically what you guessed. Thanks for the article. very informative
not working in training troops and spells
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Hi,is it true if we upgrade the the dark drill or gold mine or elixir collector and cancel that we can boost it for another entire week?
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Seriously who the FUCK reads this and understands it all. Times adding BITCHES YOI DICK HEAD. You could be creating a formulae for cancer right now but no, you waste your whole life on COC. Fuck you nerd bitches. And you waste so much fucking money. HOLEY SHIT! 0.0 ;’(
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clash of clans is one of the best game ever.
just the only thing that i do not like in the game is that when you do apply for soldier you need to wait time that lead me to get out of the game.
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It took forever to save up 750,000 in gold to upgrade the main building from 5 to 6. As upgrades go it’s pretty minor payoff. A few more additions. Very limited choices unlike other strategy games. Then once I’d saved up and paid the gold…4 DAYS wait time. This does not incite me to buy gems, but to leave coc for a more satisfying experience. Fail.
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I would say shorter skips are worth less because they are taking advantage of human psychology. When you spend .01 cents you don’t care how good a deal you get. If you spend $100 you are more likely to keep them honest.
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