Quote Originally Posted by RollingStoned View Post
I've been reading through it -- takes a little while because I'm not familiar with the whole coupon thing. I'm still having a really tough time seeing how there's a difference in playing $20 cash on roulette and $10 cash + $10 coupon in BJ....compared to $10 cash + $10 coupon on roulette and $20 cash on roulette. [Well....other than the times when "if you surrender, you lose half your bet AND the entire coupon" or the "if you double-down you're triping your risk ($10 -> $30) to win twice as much ($20 -> $40). Either way, the win/loss amount is going to be the same if we lose BJ and win roulette or win BJ and lose roulette or any other combination.] I'll start over from the beginning again....but I don't think my brain wants to see the difference! >
The difference is you are far more likely to lose in roulette, so you'd rather lose $10 cash and the coupon there instead of $20 cash. The value of the coupon as a loss rebate only kicks in on a loss, and that happens 37 out of 38 times on a number at roulette. Simplifying BJ to a fair coin flip for the sake of argument, you get the following for the two scenarios you mentioned:

Bet $20 on roulette and $10 cash + $10 matchplay on BJ
EV = 1/38 * 700 + 37/38 * (-20) + 0.5 * 20 + 0.5 * (-10) = $3.95

Bet $10 cash + $10 matchplay on roulette and $20 cash on BJ
EV = 1/38 * 700 + 37/38 * (-10) + 0.5 * 20 + 0.5 * (-20) = $8.68

It's worth $4.73 more with the matchplay on roulette because you get the $10 discount 37/38 times instead of 1/2 of the time. $10 * (37/38 - 1/2) = $4.73

To take advantage of a loss rebate, you should aim for a payout distribution that maximizes the frequency of the loss (but with a large infrequent win to make sure the actual EV of that payout distribution is still reasonable).