King of Tokyo: Math, Mayhem, and Blown Minds
There is an regular comparison that gets batted around the gaming world surrounding the game King of Tokyo. It is common, in gaming circles, to hear King of Tokyo be described as similar to (if not the same as) a classic dice game dating back to before 1940. Both require rolling dice, holding the ones you want and re-rolling to try and get your perfect hand. Both are games of probability at their heart and both require you to play the odds to some degree. That classic game is, of course, Yahtzee.
I have to admit that I believe saying King of Tokyo is the same as Yahtzee is a bit unfair. It feels like saying, to use a random simile, a Kobe Steak is similar to supermarket value beef mince.
I thought, now hoping to get back into writing regularly for this blog, I would kick start the analysis articles with something that would be a quick look into the mathematics behind King of Tokyo as a game. There are whole papers that go into the various different types of roll in Yahtzee and how you should play. These go into the odds for each type of result and I believed (foolishly) the same kind of mentality could be used for King of Tokyo. Granted there is a difference in the number of dice (Yahtzee has five dice, whereas King of Tokyo has six to eight) but that is just a matter of scale, right?
After an evening and a half trying to wrap my head around the numbers, I can tell you that it does not work like that. Most of the time when playing Yahtzee, you are looking to optimise for very specific things, which makes the mathematics of working out the odds difficult, but the end goals are fixed. They remain the same. No matter how difficult the math, you know there are only 15 different ways of ultimately scoring your hand.
Now – when playing King of Tokyo we remove those fixed end goals as they become adaptive per roll. They are constantly changing and moving based on what you are trying to achieve as a player in that turn. This makes it a game with a ridiculous number of possible outcomes. Yes, you ultimately want to either win by victory points or by conquest (being the last monster standing) but how you get there has billions of combinations.
If you remove the restrictions around Yahtzee and keep in the same number of dice it becomes complicated enough, but try adding an additional die to the base five, constantly changing circumstantial goals (they are always changing based on the environment around the table) and add in the option to occasionally roll with 7 or 8 dice.
One of the countless reasons I like games is because probability is a good place for an amateur mathematics fan like myself to geek out.
That being said, King of Tokyo is a probability nightmare. There are too many moving parts and too many potential strategies to be able to map them out in an easy way. Instead, it is some psychoanalytical calculus bomb that makes it too difficult to even know where to begin.
Let me show you what I mean –
The Calculus Nightmare
So if we look at the odds of rolling, as an example, six hearts in King of Tokyo (because we have just come out of Tokyo and want a lot of health), then we are looking at a fairly easy calculation. What are our odds of getting one heart? We have a 1/6 chance. How many dice do we have? We have 6 dice. That makes our calculation:
1/6 x 1/6 x 1/6 x 1/6 x 1/6 x 1/6 = 1/46,656
Pretty low odds, but we have two re-rolls to get what we want. Let’s say we re-roll three, then our odds are:
1/6 x 1/6 x 1/6 = 1/216
Better odds of getting what we want. If we went on to re-roll one, we would get a 1/6 chance of getting what we want.
Now let’s say we only want five hearts, and the other one can be anything? Well, we calculate five, and then waste one dice, surely? But what if we want it to be something specific? Now let’s say we want two of one thing, two of another thing, and two of a third? What are the odds on getting what we want on re-rolls? What if we were to add in additional dice, or be able to manipulate them in accordance with cards? What if you need to roll constantly changing results due to cards that are coming out each turn?
Just working out the basic combinations is difficult as, unlike in Yahtzee, almost every single combination of dice is a valid hand in its entirety under constantly adaptive goals, and often in multiple different ways. There are a few exceptions, especially around victory points, but a miss in King of Tokyo feels more redeemable than a miss in Yahtzee due to the flexibility of the strategies at play. You are always able to get a Chance in Yahtzee, but in King of Tokyo those dice results matter more and can help influence any number of strategies across any number of turns. I don’t know, maybe it is me oversimplifying Yahtzee but, for me, King of Tokyo offers more options per roll.
There are, in fact, 252 unique combinations of results you can get in Yahtzee, when you have five dice, and that is when you take the duplicate results out of the 7,776 possible rolls (so 5,5,1 and 5,1,5 and 1,5,5 are counted as one unique set as they all have 2×5 and 1×1). If we compare that to King of Tokyo, when you have 6 dice, that goes up to 462 unique combinations out of 46,656 possible rolls. If you unlock the additional two dice, that goes up 1,287 unique combinations of dice to out of 279,936 possible rolls!
There are 66 Power Cards in the game. If you buy three of them that gives you 45,760 different combinations in what those three cards could be. If you buy four it gives you 720,720 different combinations in those four cards. You may buy 5 (unlikely, but possible), and so end up with 8,936,928 different combinations.
Add those odds and combinations to the dice and mind blown…and that is before any re-rolls.
So What Does This Mean?
I think what this means is that, by comparing King of Tokyo to Yahtzee (something I have been guilty of in the past) we run the risk of doing the game a massive injustice. Where there are, for all intents and purposes, 252 different unique ways a roll can go in Yahtzee, there are potentially billions of different ways a turn can go in King of Tokyo once additional dice and cards are taken into account.
What King of Tokyo is, is an evolution of some of the same mechanics as Yahtzee whist doing more towards both providing the player with options and making the game more replayable. It is this, this advanced nature, that makes it both interesting and highly fun to play. The two games share similarities, but they are also incredibly different. Plus, King of Tokyo has monsters, which, in my opinion, beats an abstract dice game any day of the week!
It is also this complexity, however, that makes it difficult to analyse for the board game beardy gamer. Analysis of a game like King of Tokyo is not simple and can easily (oh so easily!) get out of hand incredibly quickly.
So, there we have it. An article that was meant to be an analysis turning into a reason as to why the game can’t easily be analysed.
What’s your take on the subject? What’s your opinion of the King of Tokyo/Yahtzee debate? Let me know your thoughts in the comments below.