[BruceZ]

Your friend's 2% expectation is an average of two numbers: the amount above 2% that he wins on average when he wins over 2%, and the amount below 2% that he wins on average when he wins less than 2%. Note this last number is probably a negative number; that is he loses money on average if you only look at the times he wins less than 2%. These two numbers are equal distance from 2% in opposite directions. The size of these two numbers depends on his standard deviation (sigma) for each session. If you assume his session results follow a normal distribution, then of the times he wins over 2%, the average amount that he wins turns out to be 2% + .7734*sigma. The amount that he wins when he wins less than 2% is 2% - .7734*sigma which is probably negative or a loss. Now if he tips half of his winnings over 2% when he wins over 2%, then his average win when he wins will become 2% + .7734/2 sigma. So his overall expectation will be (2% -.7734 sigma + 2% + .7734/2 sigma)/2 = 2% -.7734/4 sigma. This is almost certainly a negative expectation as the following example shows:

Say he bets an average of $1250/hr for a 4 hour session, for a total of $5000/session. He expects to win 2% of this or $25/hr or $100/session. Let's give him an hourly standard deviation of $500 which would be as low as you could ever expect; it could be twice this much. Then his standard deviation for the session would be sqrt(4 hours)*500 = $1000. So when he wins more than $100, on average he would actually win 100 + .7734*(1000) = $873, and on average when he wins less than $100 he would actually win $100 - .7734*(1000) = -$673, that is he would lose $673. Now if he is going to tip (873-100)/2 when he wins, then his overall session expectation would be (100 + 773/2 - 673)/2 = -$93.25. So instead of winning $100 for the session, he loses almost that same amount! Note again that this will become an even larger negative expectation if his standard deviation is larger which it most likely is.

In order to avoid a negative expectation, your average tip cannot exceed your average win. If you only tip when you win an above average amount, then you can tip twice as much since you are only tipping half as often. This doesn't guarantee you won't go broke; you can go broke even if you don't tip at all depending on the size of your bankroll. This is just how to avoid a negative expectation. If you have a negative expectation you will go broke for sure.

You probably don't want to tip so much that you have a zero expectation; you probably want to make some money. Tipping 10% of your expected win would be more than generous (it's like a tithe) and you could accomplish this by tipping 20% of your AVERAGE SESSION EXPECTATION only when you win over average, NOT 20% OF YOUR WIN FOR THE SESSION. The problem is that if you only tip this amount when you win big, many dealers will be more insulted than if you didn't didn't tip at all. They only see that you won big, and they expect to be tipped like a waiter 15-20% of your win for the session or something. If that was the only blackjack you played all year this might make some sense, but they don't understand that you are counting on that big win to make up for all the smaller wins and losses that they don't see.