Because expectation is linear and the total amount of money in play never changes, we know that the sum of the expectations for each player over a given turn is

Because expectation is linear and the total amount of money in play never changes, we know that the sum of the expectations for each player over a given turn is equal to 0. If no players have run out of money, this means that the expected change for each player over a single turn is zero. If instead we have m players with money and b broke players, each player still has an equal expected number of dollars received, and the m players with money each expect to give 1 dollar. Summing this, we have a total expected change of (m + b)E(received) - m, which must equal zero, meaning E(Received) = m/(m + b), so players with money expect a change of -b/(m+b) and players without money expect a change of m/(m+b). This tells us that the expectation for a turn is basically always zero and never gets above zero in a way that allows accumulation of wealth for a single player. So over long periods of time we should expect this to look like a drunk walk with a weird distribution.