How To Conquer Tokyo

Well, not really, but sort of ;)

Let’s briefly discuss how to best conquer Tokyo.

Not literally, of course. But in the rather fun board game called “King of Tokyo”:

The goal of the game is to defeat all the other monsters so that your monster can be the sole ruler of Tokyo!

This can be achieved in multiple ways, and one of them involves the card “Wings”, which I got to use in a recent game:

The card acts like a protective shield that prevents all damage from other players. But in order to function it requires a constant influx of energy.

So how does one get the required energy? The way to collect energy is by rolling dice and scoring flash symbols

• Each results in 1 energy for the player

• There are a total of 6 dice to be cast, each one has only 1 symbol

• The player can recast the dice up to two times if he so chooses and with each recast he can decide which dice he wants to recast.

In that game I decided to go all in on that card and try and win the game by first gathering up a large supply of energy, which would in turn then allow me to essentially become impervious to damage and therefore give me the time to slowly but calmly work on taking over Tokyo.

I therefore tried to maximize the amount of energy per turn by scoring as many as possible in my turn.

The process for this looks as follows:

1. Cast all 6 dice

2. Recast those dice which do not yet show a

3. Repeat the last step once more if there are still some dice left which do not yet show a

My intuitive expectation was that this should give me on average at least 2 energy per turn, which would then be enough to power the shield. This was of course only a feeling at the time, as one does not have time to think these things through in the heat of battle!

Well, to make a long story short, in the end it didn’t work out because I didn’t roll enough energy, so my strategy utterly failed and I was defeated.

But after the game I had some more time to think about this, and I realized that inside of 3 turns I only rolled a total of 3 energy (0, 2 and 1 energy respectively), and that this was way less than I had hoped for.

So now that the battle for Tokyo is over we can take some time and verify this. Did I misjudge the probability of generating energy or was I just unlucky?

To test this we simulate the dice casting according to the energy-maximizing strategy detailed above, which will then give us the empirical distribution of the number of per turn.

# initialize  list with casts
l <- numeric() 

# number of simulated turns
n <- 1000

# start the simulation
for (ii in 1:n)
{
  # initial cast
  test <- sample(0:1, 6, replace=TRUE, prob = c(5/6, 1/6)) #1: energy symbol, 0: not an energy symbol
  
  # recast 1: keep the energy symbols and reroll all other dice
  hits <- test[test==1]
  test <- c(hits, sample(0:1, 6-length(hits), replace=TRUE, prob = c(5/6, 1/6)))

  # recast 2: keep the energy symbols and reroll all other dice
  hits <- test[test==1]
  test <- c(hits, sample(0:1, 6-length(hits), replace=TRUE, prob = c(5/6, 1/6)))

  # add final cast to the list    
  l <- rbind(l, test)
}

The resulting probability distribution looks like this:

The probability to roll at least 2 energy in a single turn is about 85%, so it seems that my strategy was somewhat resonable and I really had just a stroke of bad luck in that game.

With that knowledge I am now well prepared for the next big battle for Tokyo and will surely dominate my enemies!!!