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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.

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Winning rate of casinos is so big that refunding half of it only cuts down profits, they still make money. –  MyUserIsThis Jul 17 '13 at 0:15
    
Maximum bets usually are fairly small. And it's the rare gambler that can keep themselves to only a few bets in a month. –  Nate Eldredge Jul 17 '13 at 0:41
    
Thanks MY and @Nate, I've updated my question a bit. –  Eagle Jul 17 '13 at 0:43
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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.

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Thanks! How would I calculate that expected value for large numbers, say 47? –  Eagle Jul 17 '13 at 3:56
    
Here's the program I used (PARI/GP). First I defined f(n)=my(g=x->if(x>0,x,x/2)); sum(i=0,n, binomial(n,i)*.49^i*.51^(n-i)*g(2*i-n)) which gives the expected value of n bets. Then I computed a table of records with r=0; for(n=1,1000, t=f(n); if(t>r,r=t;print(n" "t))) (you can see that in this case I used a limit of 1000, you could increase this if desired). –  Charles Jul 17 '13 at 4:01
    
Wow, thanks a lot. Would upvote but I dont have enough rep. –  Eagle Jul 17 '13 at 4:30
    
@Eagle: Glad to help. You may wish to see my answer math.stackexchange.com/a/141422/1778 to a question which I perceive to be related. –  Charles Jul 17 '13 at 4:31
    
Thanks again for your help. One more question- I'm sure this is simple but it's confusing me. I believe the answer you provided assumes the casino reimburses losses after each win. To make the game more favorable for the casino, it seems they could impose some minimum number of rounds that the player must play per "cashback period" (every week or so) before the user is reimbursed half their losses. How could one calculate the number of minimum plays required for an expected value of 0? –  Eagle Jul 17 '13 at 5:11
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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).

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Thanks. I've updated my question. –  Eagle Jul 17 '13 at 0:42
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