I was studying martingales and came across this example.
Let's say there is a bet-type game. A player bets 1€, if he lose he stops, otherwise he bets 2€. Again he bets 3€. A single player can play just three rounds. Every round a new player comes in.
Define $S_n$ as the number of total winning at round n. The example states that $S_n$ is a martingale and that's okay, but also that $|S_n-S_{n-1}|\leq 7$. I guess the difference should be the maximal winning variation but I can't figure out how it can be 7.
What I thought is the following: the worst situation is when at round n-1 (suppose n>3) we have player A playing his third and last round, player B playing his second, and player C playing his first. Let's say they all win. I'll say $S_{n-1}=6$. At round n player A is out and starts player D. Now to maximize $|S_n-S_{n-1}|$ I believe that player B,C,D should all lose. If so $S_n=-3$ and so i get $|S_n-S_{n-1}|=9$. It should be trivial, maybe I just misunderstood the game, but I don't know how to make it works.