I am interested in the following scenario:
A online/mobile app game is played by lots (for example, thousands, or millions) of people.
Each player starts with \$$20.$ A game is between two players and each player bets \$$1$ to play. The winner gets \$$2$ and the loser gets nothing. The app pairs players randomly (i.e. you don't know who your next opponent is).
Once a player hits zero, they are not allowed to continue playing (as they have nothing to bet).
My question is: After everyone plays lots and lots of games, what does the distribution of winnings look like?
I think if we assume the % win rate is a Normal distribution among the population with mean $50$% and standard deviation $4$%.
The $x-$ axis is the winnings to the nearest dollar and the $y$ axis is the number of people who won that amount of money to the nearest dollar. [If we didn't round to the nearest dollar then there wouldn't be many people who won the exact same amount, so the graph would be meaningless...]
The graph is convex decreasing with asymptote $y=0$ and I think $x=0$ is an asymptote also but I'm not sure. What is the graph? Is it $\ y=Ae^{-kx}?\quad y=k/x^n$ or something else, and why?