# Going for broke in King of Tokyo

## 23 July 2017 15:24 GMT

King of Tokyo is a lighthearted but engrossing board game that's based on the classic monster movie genre. You play a monster in the style of Godzilla or King Kong and your objective is to defeat the other monsters battling for control of Tokyo. Since my brother introduced me to the game last summer it has quickly become a family favourite. It's a game I can play with my seven year-old daughter and her 83 year-old grandad and everyone has a good time.

I won't run through all of the rules here. The only thing you need to understand to read this article is the game's central dice mechanism. Underneath the thematic flavour of giant monsters with superpowers, King of Tokyo is a dice rolling game with similar rules to Yahtzee. The dice are the traditional six-sided type, but with different faces. These are the digits 1, 2, and 3, a claw, a heart, and a lightning bolt.

On your turn you roll six dice up to three times. After each roll you can keep any of the dice that you like and continue rolling the remaining dice until you either have all the faces that you want, or you run out of rolls. The dice that are face up at the end of all your rolls are your hand for that turn, and you then resolve the outcomes.

Dice with numbers give you victory points if you have three or more of the same number. Claws deal damage to your opponents, hearts restore your health, and lightning gives you energy which you can use to buy power cards.

Quite often in the game you can benefit from getting dice of more than one type on the same turn: do some damage to an opponent *and* heal yourself a bit; collect some lightning for power cards *and* gain a few victory points.

But at other times you need to go for broke and try to get as many of one dice face as possible. This tends to happen at the most decisive moments in the game: when you need just a few more victory points to win outright, when you need to do enough damage to kill an opponent before they win, when you need to heal quickly to prevent your own impending death, or when you need to collect enough lightning to buy a vital power card before someone else.

You can model this strategy with a Python function that looks like this:

```
import numpy as np
def max_outcome_for_face(dice=6, rolls=3):
hits = 0
outcome = [0] * rolls
for roll in range(rolls):
# Get a random number from one to six for each dice
results = np.random.randint(1, 7, size=dice)
# Count the number of ones: a one is a hit
numhits = np.count_nonzero(results == 1)
# Add to hits and remove a dice for each hit
hits = hits + numhits
dice = dice - numhits
# Store the hits after each roll
outcome[roll] = hits
return outcome
```

This function simulates a turn in the game with the given number of dice and rolls — six dice with three rolls by default. After each roll, it counts the number of times the target face was rolled, adds that number to a running total for the number of dice with the target face, and removes those dice with the target face from subsequent rolls. After all the rolls have completed, it returns a list showing the cumulative number of dice with the target face after each roll.

If you call this function a large enough number of times and collect the results, you can obtain the probability distribution for the outcomes of this strategy for any given number of dice and rolls. The heatmap below shows the distribution of outcomes after ten million turns with six dice and up to four rolls.

In King of Tokyo you get three rolls with six dice by default, so while this chart shows the outcomes for up to four rolls, the third column is the most relevant. This shows that if you are trying to get as many dice as possible with one particular face, you have an 80% chance of getting two or more dice with the given face, and a 21% chance of getting four or more.

So why does the heatmap show probabilities for up to four rolls? Because some of the power cards you can buy in the game give you an extra roll of the dice, and the Giant Brain card in particular gives you an extra roll as a permanent effect. With a fourth roll of the dice the probability of getting two or more dice with the target face increases to 91%, and the probability of getting four or more rises to 38%.

There are also power cards in the game that give you an extra dice on each roll: the Extra Head card gives you this as a permanent effect. Here is the distrbution of outcomes for seven dice and up to four rolls.

This shows that an extra dice is worth less than an extra roll when you are going after a particular face. With seven dice and three rolls the probability of getting two or more of a given face is 87%, and the probability of getting four or more is 33%. That's better than six dice with three rolls, but not as good as six dice with four. Although it's worth noting that an extra dice confers other benefits — you get more stuff.

Of course, if you can get the combination of cards that gives you seven dice with four rolls, then getting four or more dice with the target face becomes more likely than not, with a 54% chance.

This analysis was done using numpy and pandas, and the heatmaps were produced with matplotlib and seaborn. The complete source code is available on GitHub.