N-player Ruin Problem
The above described problem (2 players) is a special case of the so-called N-Player ruin problem. Here players with initial capital dollars, respectively, play a sequence of (arbitrary) independent games and win and lose certain amounts of dollars from/to each other according to fixed rules. The sequence of games ends as soon as at least one player is ruined. Standard Markov chain methods can be applied to solve in principle this more general problem, but the computations quickly become prohibitive as soon as the number of players or their initial capital increase. For and large initial capitals the solution can be well approximated by using two-dimensional Brownian motion. (For this is not possible.) In practice the true problem is to find the solution for the typical cases of and limited initial capital. Swan (2006) proposed an algorithm based on Matrix-analytic methods (Folding algorithm for ruin problems) which reduces, in such cases, the order of the computational task significantly.
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