Does Crime Pay in the Den of Thieves? Taking Puzzle Proposals!

Let me know if you can find any other good questions associated with the Den of Thieves game

Last week we were introduced to a game called Den of Thieves. It goes like this:

[…] we have n thieves that each have $1. The game proceeds in rounds where every thief simultaneously and anonymously selects a target. If a player has been selected by at least one thief, they lose the money they had at the start of the round (but they will still gain the money for their theft for the round). Each thief will gain the amount money of their victim unless other thieves also selected this target. In this case, all the thieves split the money evenly (in this game world, there is some honor among thieves). To clarify, all of the monetary losses occur before collecting your ill-gotten gains so even if you are stolen from, you will still end the round with whatever amount you stole from your victim.

I wanted to ask a series of follow up questions about this game but of the questions I came up didn’t feel satisfying. Some of the ideas I played around with were:

1. What is the optimal play when the wealth distributions are more complex? Say each player has wealth w_i . If they all play optimally, how should they choose their next victim?

2. Assume the game goes on for many rounds, what does the wealth distribution converge to for a given thief?

3. Can we make the math for either of these questions simpler if we tweak the game a little? What if, instead of splitting the victim’s wealth with all thieves that pick the same target, one lucky thief get’s all of it? What if players that have $0 at the end of one of these rounds is eliminated? What if they are allowed to continue playing. I thought here was perhaps the question could be How long on average until all the wealth is consolidated to a single player?

4. What if the starting distribution were w_i = 2*i/n (triangular distribution)?

For (1) there is an answer but it struck me as messy and overly technical. In my opinion, it lacked that je ne sais quoi.

I couldn’t find a way to adequately answer (2). The ‘piecewise’ nature of the distribution made it too hard for me to reason about in any analytic way as the rounds progress.

(3) had issues when you get to two players, at which point the game goes on infinitely. Adding an endgame condition for 2 players felt ugly and unsatisfying.

I didn’t get a chance to explore (4) but it struck me a little bit contrived.

Finally, I had to give up on this problem. I already spent too much time trying to make it work. Perhaps there is still an elegant or interesting result buried in the problem but I need some help to find it.

So I’m turning it to the reader: Can you come up with any good follow up questions for this game or interesting tweaks to the rules? If you have any ideas, leave a comment!