Designing a casino cashback program Let's say a casino is considering offering a cashback program whereby it would return 50% of player losses twice a month. The casino has a house edge of 1% on each game.
What steps could the casino take to ensure that they remain profitable?
One way would be to enforce a minimum number of plays per user per cashback period- what would that minimum number of plays be?
Let's say the casino can't ban or ID users.
 A: It depends on the mechanism. For instance, if they look at your total in/out for a month, and it is the case that over over that month almost every player makes a net loss (however small) then refunding half of the total amount lost they simply cut down on profits without ever going negative (ignoring overheads).
A: Suppose the casino offers only one game: you bet £1, then with probability 49% you win £2 (otherwise you lose your wager). Your expected net is -£0.02. Now if you know that half your total losses will be refunded, here's how it breaks down.
One bet: Win £1 with 49% probability, lose £0.50 with 51% probability, expected gains £0.23½.
Two bets: Win £2 with $.49^2$ probability, wash with $2\cdot.49\cdot.51$ probability, lose £1 with $.51^2$ probability. Expected gains £0.2201.
Three bets: Expected gains £0.33015.
...
47 bets: Expected gains £0.6824276...
It seems that for this simple game 47 is the best you can do. The specifics will vary, but probably the rough result will remain the same: you can expect to win back a fraction of a single bet.
