The Long Night Riddle

When humans die, the Knight King converts them into White Walkers. How hard it is to bet them? It seems impossible! I like any sort of riddles, especially this one created by my flatmate Ferran (Alet, 2019), which is GoT related, not trivial but simple enough to find an (incredibly beautiful) answer in less than 10 minutes. I am sure Tyrion Lannister would solve it, had he had the math tools we have today. Would you know when to fight and when to retire?

The problem setup is straightforward, with lots of simplifications but still a good approximation of the dynamics of the Long Night. Let's put it in bullet points:

• There are W White Walkers, fighting against H Humans.
• If a White Walker kills a Human, the Human becomes a White Walker.
• If a Human kills a White Walker, the White Walker disappears. 
• Humans and White Walkers are equally skilled. At every period of the battle, one warrior is picked randomly and he kills an opponent. 

The question is simple: when the army of White Walkers is large enough, what is the number of Humans needed so that the battle becomes even? That is, that the probability of winning is approximately 50% for both sides. 

The riddle was solved by several MiT Ph.D. students from different fields: Econ, Math and Computer Science. And it was fun to see how different the approaches were, while the solution was the same. You can use a computer if you want, but a pen and pencil approach also works.

Once you find the solution, you won't believe how beautiful it is. Can't wait for it? Here is my (the Economist, closed form) proposed solution.

Extra Credit: want to extend this model? How the solution changes if, when a Human Kills a White Walker, with probability p it is the Knight King and Humans win? (An easy solution is: if Arya Stark is in your team, Humans always win.)