Hi, I'm Michael. After years of being a passive reader here on Gamasutra I've decided to stop "leeching" and contribute back to the community what little I can :)
This is a repost from my blog: lifeinagraph.shalyt.com. Feel free to visit and geek out with me on some mathematical analysis of games, people and other complex but awesome systems.
This is the longest (non-officially-scientific) article/post I've ever written so, as is customary in scientific articles, I'll start with a TL;DR of the main interesting points:
Is Clash Royale "pay 2 win"? Technically yes. But we'll derive a rough skill estimation formula - neutralizing (most of) the "wallet size" parameter.
Turns out the resource that limits players' ability to upgrade cards in the long run is gold - not amount of card copies. Each chest contains only ~60% of the gold required to use the cards contained in the same chest. Choose wisely which cards to upgrade and which to ignore.
Don't buy cards in the shop unless you really need that very specific card.
Always donate to clan members - it's in your own best interest.
Almost half the gold income originates from "free" and "crown" chests - which you can get even without winning a single game.
Supercell used a pseudo-random hard-coded sequence of chest drops that seems to be carefully engineered to feel more random and "fair" than actual random drops, and to dispense "wow" moments at precisely planned time intervals.
We'll calculate how much time does a free 2 play player need to reach each level. It's possible to get level 5 in a few days but reaching level 12 takes years. Literally.
Clash Royale who?
Even if you are not a gamer, you've probably heard of Clash of Clans. After all, not many mobile games can afford a super bowl commercial featuring Liam Neeson:
Don't let the cartoonish design fool you - in 2015 Clash of Clans produced an estimated 1.5 million dollars in revenue per day for Supercell, the game's creators. And now the Nordic company released a new blockbuster: Clash Royale.
The game was globally launched about 1.5 months ago and already has tens of millions of downloads, an active community and user-generated content (Youtube videos, streamers, play guide websites, wiki etc. etc.) - basically every game developer's dream. Part of the reason for its meteoric success is the popular Clash of Clans brand (Clash Royale uses the same lore, characters and visual design), part is the huge marketing budget Supercell are willing to invest in the game, but perhaps the biggest part is that the game is actually good.
The game's combat theory is quite deep so for now I'll only focus on a few strategic questions and try to provide answers using the power of MATH :).
A tale of 2 progression systems
I'm usually more of a hardcore strategy gamer, but something in the play-and-counterplay PVP dynamics of Clash got to me. Click here for an example of the ~3 minute matches that comprise the "game itself". The winner of the match gains (and the loser loses) ~30 trophies - the game's standings system (similar to the Hearthstone star system).
The battlefield of Clash. Deploy troops and watch them fight autonomously. |
This is the first progression system in Clash - basically the "better" you are the more trophies you have ("better" is not necessarily more skilled). Players are matched according to their trophy number - which is basically their MMR (match making rating) - accumulating or losing trophies until they reach a state of equilibrium: ~50% average win rate. The trophy progression system provides a goal for the players, an objective to strive for that supposedly proves their merit and provides bragging rights. Note that this progression scale is reversible: if you play poorly or choose a bad deck to bring into these duels then you will fall back down through the ranks. This is in contrast to the second progression system - a slow grind-like journey of upgrading the quality of your cards and gaining player XP and levels - where you can only progress.
As you play Clash, you get chests of various quality - containing cards and gold. You can sacrifice multiple copies of a card to upgrade it - making it 10% better and getting some player XP with each new card level. While the "merit" progress is measured in trophies, "grind" progress is roughly measured by player experience. Each player level makes your towers stronger by 7-8%, in addition to the stronger cards, so getting more chests gradually makes you objectively "better".
[Side note: cards increase their power slightly faster than towers, so on higher levels it's easier to destroy towers and the natural defender's advantage of fighting on your side of the map is less pronounced - putting even more emphasis on unit vs. unit combat as the player progresses. A nice design touch.]
This process of acquiring and upgrading cards is the main mechanism of pacing long-term game progress.
A sample deck. Note the card levels, the amount of cards needed for the next upgrade, the card rarity (indicated by the blue-orange-purple colors), the player level in the top left and the adorable artwork :) |
Unsurprisingly, you can buy chests and gold using real money. This brought about a lot of player backlash - it's not "fair" facing opponents with objectively stronger cards and towers just because they spent more money on the game - right? Which brings us to the first question:
Is Clash Royale "pay to win"?
Short answer: yes. You can make your cards and towers strictly stronger by paying money.
Longer answer: depends on what you call "winning".
Let's say I want to know how good am I at the game - what's my actual skill and competence - what's my "merit score". At the time of writing I am yet to spend any money on in game purchases during the 2-3 weeks of almost daily play. I am level 6 (which has nothing to do with my skill) and hang around ~1500 trophies (which does). Is 1500 a lot? Well, the very best players in the world reach ~4000, and anything above 3000 is considered "legendary" status. 1500 is a long way from that - but it's also a long way from the initial 0. Should I simply say then that my skill level is 1500 trophies?
No. My "strength" in a dual is roughly the sum of 3 main factors:
Power = player level + in-game execution + deck choice
My player level defines my tower strength and is also an approximate indication of my card level. Obviously, the stronger my cards and towers the higher my chance to win. This has nothing to do with my "true skill".
Deck choice is a vague measure of how good your deck functions together, its various cards complement each other, as well as how good is it against the most popular decks you encounter. Skill does play a role here but sometimes the lack of access to certain cards limits deck building.
In-game execution is the most obviously skill reliant factor - how good am I in piloting my deck and responding to my opponent's moves. Money can't buy that.
Since I'm at my equilibrium state - I win and lose about the same number of games - my opponents and I have approximately the same power score. My opponents' average level is 6.3-6.4 (about 30-40% are level 7, the rest are 6, with a rare 5 here and there). Since I have a 50% win rate against players with a slightly higher player level than me, from our equation above it follows that my execution and deck building - hence my skill level - is slightly higher than theirs (on average). If my average opponent level was 7 or even more then it would mean I'm a substantially better player than they are - since I manage to overcome my handicap of having worse towers and cards.
This is a more objective measure of player skill - neutralizing the effects of card quality to the best of our ability. And it has almost nothing to do with the actual number of trophies:
Skill ~ (Average opponent tower and card level) / (your tower and card level)
As with all simple models - it's only an approximation and has many inaccuracies, but gives a more "fair" measure of merit.
Should you buy cards?
In Clash, the gold that you get from chests is used mostly to pay for card upgrading, but can also be used to directly buy specific cards in the shop.
Cards available for direct purchase. The more you buy, the more expensive they become. |
At first glance the question seems silly - if you want the cards and have the money - why not? But the correct approach is to think long term: our goal is to eventually upgrade our cards, which requires both gold and multiple card copies. What is the limiting factor? In the long run will we run out of gold or out of cards first? Let the math commence...
Let's look at common and rare card upgrade quantities and prices:
Calculating the amount of gold per card needed for upgrade we get the following graphs:
Note that the price per card becomes a constant (10) after level 6 for commons and an almost constant (40-50) after level 3 for rares. Aha! Smells like hidden order :)
So, if we want to upgrade every card that accumulates the required number of copies, each chest has to contain, on average, about 10 gold per common and about 40 per rare. If the chests contain more gold than that - then we should buy a lot to spend the surplus gold, if they contain less - we can't upgrade all of our accumulated cards and have to "give up" on our least favorite/used cards.
Chest drop statistics in Clash are somewhat complex but fairly well documented on the game's wiki so we'll use their numbers and guesstimate the data they don't have. Most of the chests you get are of the "silver" and "free" variations, and they seem to produce approximately the same prizes. The chest content also depends on the number of trophies you had when you received it. Taking my current standings as an example, each chest contains 7 common cards - each has ~10% of being upgraded to a rare or epic [Side note: that means each 7 card pack has a roughly 50% chance of having at least 1 card better than common] and 46 - 58 gold (assuming an average of 52 gold
