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It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from Pascal to Fermat written in the 1650s (cf. this manuscript by Oystein Ore ).

I wanted to see some mathematical research where variants of the classical gambler's ruin were studied (e.g. by accomodating multiple gamblers and/or letting the bet sizes for the rounds to be deterministic or non-deterministic variables, changing the rules for winning/losing a round, etc.).

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Multi-dimensional generalizations (one player against $d$ other players) are explored by P. Lorek in Generalized Gambler's Ruin Problem: explicit formulas via Siegmund duality.

For the analogue on a ring or torus graph, see Generalized Gambler’s Ruin Problem by Chayes et al.

Further generalizations include The conditional gambler’s ruin problem with ties allowed.

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