Happy Friday, Archons! I’m ridiculously excited to be working with Alex to develop a series of articles we feel will help get to the fundamentals of Keyforge strategy and game design. There is no shortage of great content coming from all corners of this community, and it’s our hope that what we explore here will only build on what’s out there.



Welcome to THE BOOK OF LEQ.

*Author’s note: Special thanks to @Flibber from the Sanctumonius Discord for catching an error I made in Part 2!

Chapter 1: The Ingredients

Card games are simple.

~ Sorry, let me try that again ~

Card games are NOT simple (to play). But what makes them fun is something as simple as it is ancient.



A few years ago, the YouTube personality and fellow geek Tom Scott was featured in a video where he played one the oldest known games in the history of the world. Likely invented by the Sumerians over 4,500 years ago, the Royal Game of Ur game is a total gem. No seriously, watch how it’s played if you haven’t already (if only to witness the most unironically-British museum curator in action). Games like RGU are notorious, and we see them everywhere. From kid’s versions like Trouble and Sorry to the immortal Backgammon, we enjoy it when our games take us through the thrills and disappointments of chance. But at the end of the day, there’s not much you can do about how the dice rolls.



Right?

Well, let’s leave a mental bookmark here and return in a bit.



BUT FIRST! I challenge you to a game. Just three rounds. Care to play?

Excellent! The rules are simple, and you’re totally going to win. Trust me.

PART 1.



I’ll crack open a normal 52-card deck and give you the stack of 26 black cards in it. I’ll take the remaining 26 red ones as my stack. Starting with the first player, we’ll each take turns flipping over a card from our respective stacks. The first person to flip over 13 cards of the same color wins. Easy enough, right? You can go first…aaand congrats on the win. See, you’re awesome at this!



Ok now round two — tiny rule I forgot to mention: after every round, the winner takes a random card from the loser’s deck and shuffles it into their own. But don’t even worry! If I was a betting man, I would totally put my money on you. In fact, let’s just say this round is a $100 buy-in. Easiest money you’ve ever made. Game on….aaaand congrats on the win again!



Last round, same rules. How about a buy-in of $10,000 this time. You’re looking a little nervous, pal. Don’t tell me you’ve lost track of the math. It’s all still in your favor, trust me! Ready?…Game on!

*laughs in Mr. Burns*



Okay okay, thank you for indulging my campiness — I promise to bring it out only when necessary. Let’s break down our thought experiment a bit by asking a few questions.

Why was the first round of that game utterly boring despite you winning it?



Well, where should we start: the mundane task of flipping cards as your sole interaction with the game or the mindless determinism of who would win? Is something even really a “game” if a specific player is guaranteed to win every time? If that were the case, the real fun would actually be flipping the coin to see who’s first. Maybe that’s cool. Or maybe we can afford something more challenging than spectating random chance.



What changed in Rounds 2 and 3 that made things more interesting?



Very quickly the game jumped from one wholly pre-determined to one of chance. Anyone could tell you you’re 100% guaranteed to win your first round and probably pretty likely to win your second, but what about your third? And making the situation even more complex is an increasingly intense risk-reward trade-off to further distract your logic. Add a “decision” component such as either player choosing to stop at any time before they hit their first streak-breaking card or lose immediately upon finding it, and we got ourselves the next Vegas money-maker. A player’s interaction with chance is the bread and butter of any and all good strategy games. Maybe you noticed something familiar in our ancient board game too.



Is “chance” different than “random”?



In a well-designed game, yes. If a game gives players an equal chance of winning no matter how they interact within the rules, the results are as good as random. Save some time and money by challenging your friends to a good ol’ fashioned die-toss instead. However, if all players have an unequally distributed chance at winning (especially if the weights and imbalances are initially unknown to players), a much more diverse landscape of strategy and game design emerges. The ability to win may be somewhat dependent on a player’s “assigned chance”, but the decisions each player makes throughout the game can more than make up for these imbalances. I would even argue that players dismissing a game’s end result as fate are ignoring the degree to which their decisions actually impact the outcome. We’ll pull on this string a bit more over the next few weeks.



{Bonus question: What actually happened at the end of Round 3? Was I lying when I said you’d always win? Here’s a table of each player’s winning probabilities for the first three rounds if you really want to explore that question. If you’re curious to check the math, you can follow my Bayesian decision tree here as well as some Monte Carlo results of the first player’s winning probabilities here.}

Winning Player Game 1 Game 2 Game 3 First player 100.00% 51.85% 50.96% Second player 0.00% 48.15% 49.04%

PART 2.



Let’s have a quick chat about “chance”. As in, when you desperately need a lucky break during your favorite card game and still manage to draw exactly the wrong card causing you to curse angrily to the cosmos “what are the chances??”. The words “chance” and “probability” can be used interchangeably in this context, but if we want to get into semantics one might argue probability is the chance of an event occurring… again, largely interchangeable. In any case, nearly everyone these days with with an internet connection has the ability to learn basic statistics (whether through school, a job, or online resources) so I’ll not spend time discussing why — say– there’s a 13% chance of drawing six cards from a 52-card deck and catching exactly three spades. We’ll save that fun math for another time. Instead, I’m more interested in talking about the information gain inherent to games of chance.



Suppose you have a bag with 12 marbles: 6 green and 6 purple. You draw them one at a time and put them on the table in the order you drew them. What’s the most probable arrangement for your first six draws? There are actually quite a few! Here are two:





Why? Well a fundamental property of the universe is also a driving mechanic in statistics — entropy!



Let me define this a bit better: entropy relates to the number of indistinguishable “states” (or variations) in which any set of objects can exist. The more indistinguishable states there are for a particular distribution of items, the higher the entropy. So in the above example, imagine if each ball had a number — I could have switched the “#6” purple marble with the “#1” purple marble and it would not make a difference in the color distribution. Play a hundred more games pulling these marbles out of a bag and you’re likely to end up with a configuration with an equal number of green marbles as there are purple. Note: There are no favored configurations once you draw all 12 marbles (thanks @Flibber)



Entropy becomes especially helpful when you manage to lower it. Suppose you play this marble game one more time and in the first six pulls you get this distribution:

It doesn’t take more than intuition to see that you’ve gained some information about whether you’re more likely to draw a green marble or a purple marble next, but did you know that this configuration has also lowered the total entropy of the marble game? In fact, this loss in entropy is what we would mathematically say led to your understanding that a green marble is more likely to show up next.



I wonder if this applies to any other games…

CONCLUSION.



By now, I’m sure you’re beginning to smell some deliciousness on the horizon. I’m sure you’ve also noticed I’ve yet to mention Keyforge (until just now). There will be plenty of time for that in the next article after I leave you with some food for thought. Speaking of which, let’s go back to our mental bookmark from earlier…



The Royal Game of Ur, a game in which two players throw dice and move pieces down the board — the question is, how much of a player’s success is due to chance, and how much of it is knowing when chance may be on your side?



A good question we’ll continue to explore in our next installment of The Book of leQ.

Until then, fellow Archons. May you play, fight, and reap forth!

-Kav