Today on IRC someone claimed that there is a mathematical proven algorithm to win money
with poker in the long run. I can not believe this is true because of the following:
Assume we have the algorithm A that guarantees a certain win when playing
poker. Now imaging 4 poker players. Each starting to play with an
starting amount of X euro against each other in a room. Each player plays
according to algorithm A, because that will guarantee him a certain win of
money over time.
Given by the rules of poker: The amount that can be won in a round (W) by a player
is equal to the amount lost in a round by the other players (L).
After 0 rounds the total amount of money in the room will still be 4X euro. Trivial.
After n+1 rounds the amount of money in the room is the amount of money in the
room after round n + W - L. Which gives Mn+1 = Mn + W - L. Given that W = L we have that
Mn+1 = Mn.
So after n rounds with n a natural number >= 0 the total amount in the room is still 4X.
Assume that algorithm A works and g is the average amount gained by each player
per round. If A is true than g > 0. Hence after n rounds (with n > 0 and sufficiently large),
every player has on average X+gn. Which means the total amount in the room
would be 4X+4gn with n > 0 and g > 0 after n rounds. Earlier we have proven that
for every n the total amount in the room is 4X. Therefor we have that:
4X+4gn = 4X for every value of n. But 4X+4gn > 4X for every n > 0,
hence the assumption that algorithm A works is false.
