vinylweatherman said:
Expected value is simply looking at the playthrough, and working out how much will be lost to the house in the long term due to the house edge.
Focusing primarily on expected value is pretty traditional approach. It's not a bad method, but it makes a couple of assumptions which don't apply to most bonus play--the worst of which is that you'll be able to play bonuses forever.
While I don't want to turn this into BW 101, and this is bound to be either an offensive or boring post for some, there are probably some useful ideas for non-bonus gambling as well.
The really important figure for bonus (ab)use is the Risk of Ruin (RoR). While the house edge figures prominently in RoR, there are other significant factors to be considered such as bet sizing and game variance.
RoR is essentially the odds of losing an entire starting bankroll. For games with a positive return, RoR is the chance of going broke over an infinite period of time. For games with a negative return (which of course make up the majority of casino games) it's the chance of going broke over a specific set of conditions: for example, playing the game a set number of times or stopping when a specified bankroll is met.
RoR is more important than expected value because few of us will play enough
sessions to approximate the expected long-term return on a per-session basis. Minimizing RoR usually maximizes the chances of getting a positive return from any given session.
In the following examples the player has a starting bankroll of $200, a profit is ending up over $100, and you have to place $2500 worth of bets before you can cash out. All of the figures are taken from simulations, but they should be accurate to within 1%.
Let's consider RTG Pontoon with $1, $5, and $25 bets. What bet size is best? What are the chances of coming out ahead?
Typical but incorrect answers are "$25 bets" and "excellent, Pontoon has a really low house edge". You're placing fewer bets so $25 is lots faster, plus your long-term return is slightly higher. So $25
must be the best option... right? And when you win, you can get some pretty impressive wins. (Those are actual arguments I've read.)
The right answer is $1 bets. The probability of winning any given session (ending up over $100) is .89 for $1 bets, .67 for $5 bets, and .39 for $25 bets. Worse, the probability of having 3 consecutive losing sessions with $25 bets is .23, versus only .001 for dollar bets. And, the majority of the time you'll lose both the bonus and deposit on $25 bets--while with dollar bets only rarely will you end up completely broke, and the remaining money can then be used for the next bonus.
As for the "increased return", the average long-term return for $25 bets is only about $6 higher than the return from dollar bets. Hardly worth it. Sure, you get some impressive wins... and some mighty impressive losing streaks.
Another very good option for this bonus is single hand 9/6 Jacks or Better with quarters. The probability of completing the bonus with a profit is about .84. (This figure is very different from what you'd expect by just looking at the house edge.) If you're really dedicated, nickels are extremely safe (99% chance of earning a profit) but 40,000 hands for an average $87 profit is a lot of work.
Other points:
Baccarat is a better option than Pontoon with $5 and above bets even though the house edge is much higher; the probability of completing the bonus with $5 Baccarat is .76, and it's just about 50/50 with $25 bets. (Assuming Baccarat is allowed, that is. Usually it isn't.)
House edge makes a significant difference in the JoB example. 6/5 JoB with $1 or below bets is a long-term loss with this bonus. With $5 bets you'll probably come out
slightly ahead in the long run, but the probability of winning any given session is only about .25--a total crapshoot, in other words. (It's safest to say that 6/5 JoB is a lose, period.)
Slots are rarely a good option for bonus hunting even with a 99.9% return and a smaller WR, because the odds of making a profit are surprisingly low even with penny bets.