Gambler's ruin problem on Erdős-Rényi graphs

A multiagent ruin-game is studied on Erdős-Rényi type graphs. Initially the players have the same wealth. At each time step a monopolist game is played on all active links (links that connect nodes with nonzero wealth). In such a game each player puts a unit wealth in the pot and the pot is won with equal probability by one of the players. The game ends when there are no connected players such that both of them have non-zero wealth. In order to characterize the final state for dense graphs a compact formula is given for the expected number of the remaining players with non-zero wealth and the wealth distribution among these players. Theoretical predictions are given for the expected duration of the ruin game. The dynamics of the number of active players is also investigated. Validity of the theoretical predictions is investigated by Monte Carlo experiments.