PvP leagues with entry fees

This diagram models a player-versus-player game with leagues (i.e. skill tiers), in which there is an entry fee to play matches. As the player wins more games and levels up, they gain access to higher leagues, which in turn increases the rewards (but also the costs).

The goal of the diagram is to estimate how long it might take for players to reach each league, and how often they might run out of Gold and be temporarily blocked in their progression.