The dreidel game is a gambling game in which players take turns spinning a 4-sided top (the dreidel). Each player starts with a certain number of tokens. Whenever the pot is empty, each player puts one token into the pot (antes). The player whose turn it is spins the dreidel and acts according to the result of the spin. The four sides and their associated actions are as follows:
When a player must ante but has no more tokens, (s)he is "ruined" and must leave the game. The game can end either when one player has won everybody's money or when the first player is ruined.
Doron Zielberger conjectured that, for a game with two players, each with n tokens, the expected running time of the game is O(n^2) [HTML]. This conjecture, along with Cyril Banderier's conjecture that the same asymptotic running time applies to a multi-player game, was proven by Sujith Vijay and Thomas Robinson [PDF].
Cyril Banderier conjectures that, if the time until the first ruin is defined by random variable T and the number of players is k, then (E^2[T]) / V[T] = 2 / (k+1). This conjecture is still open. Is it true?
Also, the asymptotic result proven by Sujith Vijay and Thomas Robinson is far from exact. Through experimentation Zeilberger has found a quadratic equation that gives the expected running time for a two player game [PDF results] [Maple package]. Can this equation be proven?
If there are multiple players that start with different numbers of tokens, what is each player's probability of victory/ruin?
Special thanks to Cyril Banderier for his correspondence regarding this open problem.