Up five per cent a round, and broke by the hundredth
A fair coin. You stake some fraction of everything you have: heads and the stake comes back with half as much again (+50%), tails and two fifths of it is gone (−40%). Stake the lot and every round is worth five per cent to you on average — a better edge than anything anyone will sell you. Play it a hundred times and the average player holds — times the stake, while the player in the middle holds — of it. Both of those are right. They are about different people.
A hundred rounds of it
Nothing here is simulated. After a given number of rounds the whole of what can happen is one number — how many of the tosses were heads — so the outcomes can be written down exactly and weighed by how likely they are. The band is where the middle ninety per cent of players are, the darker band the middle half, and the line through them is the player in the middle. The dashed line is the average.
One drawing convention, said out loud because it is the sort of thing a figure hides: a share of the players is a smooth thing and a number of heads is not, so between the two outcomes either side of a share the lines are drawn straight across. Without it the bands come out as a sawtooth, which on a page that computes rather than simulates is a picture telling the reader something untrue about how it was made. The middle line still lands on the middle outcome exactly, at every even number of rounds, and the checks at the foot of the page require it.
Where the money actually ends up
Every outcome, laid out. Each bar is one possible number of heads, put where the money it leaves you lands and made as tall as its chance. The tosses are fair, so the bars are a symmetrical bell — in the money's logarithm, which is the scale across the bottom. That is the whole trick of the page: a symmetrical bell in the logarithm is a very lopsided thing in money, and the average is dragged off into the thin right-hand tail by outcomes almost nobody gets.
So how much should you stake?
The average is no use as an answer: it is five per cent a round times whatever fraction you staked, so it tells you to stake everything, and staking everything is what the figures above do to you. Ask instead what the middle player gets, which is the same as asking what a run of this game does round after round rather than over everybody at once. That is the hump. It rises to a peak, comes back down, and crosses zero at a stake where the coin is no longer worth playing at all.
The hump is the whole argument of this page and at that scale it is nearly a flat line, which is worth saying out loud rather than drawing around: the best this coin can do for a run is under one per cent a round, against an average that promises five. Here is the same blue curve with the amber one taken away and the axis pulled in to fit it.
The two marks on that figure are worth a sentence each, because both come out in whole fractions. The peak is at a quarter: a quarter of everything, every round, and it is the fastest this coin can be made to grow for the player in the middle. The crossing is at a half, where a run of the game neither grows nor shrinks in the long term, because winning and losing rounds multiply out to exactly one. And the second is exactly twice the first — not for these two numbers, but for any coin of this shape, which the checks below confirm on a grid of other ones.