A roulette tournament is played as follows. At the beginning of the tournament each participant pays a participation fee and receives in return a fixed number of chips. Now the participants play again and again the roulette, winning and losing chips along the way. Each player has to determine when to stop playing. When the last player stops, the house counts the number of chips each player has, and the winner(s) is the player(s) with the maximal number of chips.

Christian Seel from Bonn studied this problem. In his model, which is played in continuous time, each player i observes a process (X_t^i) which is a Brownian motion with a drift: X_t^i = x0 + mu * t + sigma^2 W_t^i, where (W_t^i) is a standard Bronian motion. Thus, x0 is the number of chips each player starts with, mu is the (usually negative) constant drift, and sigma^2 is the variance of the Brownian motions that influence the players’ payoffs. The quantity X_t^i represents the number of chips that player i has at time t.

One assumes that the Brownian motions of the various players are independent. Each player has to choose a stopping time with finite expectation, which is the date at which he stops gambling. Once bankrupt, the player must stop, so that when X_t^i is equal to 0 the player must stop. To determine the winner, denote by tau_i the stopping time of player i, and compare the values X_{tau_i}^i: that is, the winner is the player(s) who has the maximal number of chips when he stopped gambling.

Since the game is symmetric, one expects to find a symmetric equilibrium (if the drift mu is not too large; if it is large, the players will continue forever). It is more surprising to learn that there is a unique equilibrium payoff in this game, and Christian went on and characterized this payoff, and also provided an explicit formula for the symmetric equilibrium strategy. Surprisingly, the expected number of chips in equilibrium is not monotone in either the drift or the variance of the Brownian motion.

So, before you participate in a roulette tournament, look for Christian’s paper, and, if you use his strategy and win, thank him.

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