Does setting win/loss limits modify expected results?
We all know that every bet has some expected win/loss ratio associated with it, and we can expect our actual win/loss ratio to approach the expected value over time.
But when I walk into a casino, I don't sit there until I hit the expected value and say, "look at that, proved the math right AGAIN!" and walk away satisfied. I do what most people do, set limits. I walk in with, say, $500, and leave when it's gone. Or, if I'm lucky, when I hit $750 or $1000. So does that change my expected results? I guess the crux of my question is this: Does setting win/loss limits increase my chances of walking away a loser since I put upper and lower bounds on the variance I'm willing to tolerate in any given session?
Intuitively it seems like I could be "cheating" myself out of long winning streaks by leaving when up a pre-determined amount, but of course I'm also limiting my losses as well. Do they cancel out, or does the fact that I'm more likely to lose to begin with come into play?
Quote: scudderI've been thinking about the question posed in the thread title for a long time, but I've never tried to work out an answer mathematically, primarily because I have no idea where to start.
Intuitively it seems like I could be "cheating" myself out of long winning streaks by leaving when up a pre-determined amount, but of course I'm also limiting my losses as well. Do they cancel out, or does the fact that I'm more likely to lose to begin with come into play?
Yes and no.
Setting a win limit definitely decreases the likelihood of a losing session, as some portion of the time that you would have kept playing, you would have wound up losing.
Setting a loss limit definitely increases the likelihood of a losing session, as some portion of the time that you would have kept playing, you would have wound up winning.
Therefore, since you are playing a negative EV game (I presume), and therefore are more likely to hit the loss limit than the win limit, setting those limits INCREASES the likelihood of a losing session. Likewise, in a positive EV game, setting those limits would increase the probability of a winning session.
Nonetheless, the answer to your question is probably best stated, "not really". First of all, "sessions" are an illusory concept; it's all one long session, in reality. Stopping when you're ahead $500 or down $500 or when you turn into a pumpkin is meaningless: all that does is artificially lengthen the time between your most recent bet and your next one. Therefore, a long pause in your betting, caused by some arbitrary condition being met, will not affect your results at all.
From a mathy standpoint, your bell curve will still look the same, with the highest/middle point being your expected value, but in the case of loss/win limits, the ends of the bell curve will be chopped off.
Quote: mkl654321Yes and no.
Setting a win limit definitely decreases the likelihood of a losing session, as some portion of the time that you would have kept playing, you would have wound up losing.
Setting a loss limit definitely increases the likelihood of a losing session, as some portion of the time that you would have kept playing, you would have wound up winning.
Therefore, since you are playing a negative EV game (I presume), and therefore are more likely to hit the loss limit than the win limit, setting those limits INCREASES the likelihood of a losing session. Likewise, in a positive EV game, setting those limits would increase the probability of a winning session.
Nonetheless, the answer to your question is probably best stated, "not really". First of all, "sessions" are an illusory concept; it's all one long session, in reality. Stopping when you're ahead $500 or down $500 or when you turn into a pumpkin is meaningless: all that does is artificially lengthen the time between your most recent bet and your next one. Therefore, a long pause in your betting, caused by some arbitrary condition being met, will not affect your results at all.
From a mathy standpoint, your bell curve will still look the same, with the highest/middle point being your expected value, but in the case of loss/win limits, the ends of the bell curve will be chopped off.
Right, but are they chopped off evenly, considering its a negative EV game to begin with?
Quote: scudderRight, but are they chopped off evenly, considering its a negative EV game to begin with?
No, given that the highest point on the bell curve--the expected/mean result--is already on the negative side of the graph, more of the negative portion of the curve would be chopped off, assuming equal absolute numbers for your win and your loss limits.
Quote: mkl654321No, given that the highest point on the bell curve--the expected/mean result--is already on the negative side of the graph, more of the negative portion of the curve would be chopped off, assuming equal absolute numbers for your win and your loss limits.
Right, so starting with $500 and limiting myself to a -$500/+$250 session would change my overall EV towards being more negative ("losing" more +EV from the bell curve), but going with a +/- $500 limit would simply chop off the ends of the bell curve in equal proportions, thus leaving my overall EV unchanged.
Quote: scudderRight, so starting with $500 and limiting myself to a -$500/+$250 session would change my overall EV towards being more negative ("losing" more +EV from the bell curve), but going with a +/- $500 limit would simply chop off the ends of the bell curve in equal proportions, thus leaving my overall EV unchanged.
No, win/loss limits don't change your overall EV, they change the distribution of session results. The variables to consider are:
a) bet size relative to bankroll
b) EV and variance of underlying wager
c) Win and loss limits relative to bankroll.
Bigger bets, higher variance, and lower win limits increase your chances of coming out ahead.
For example, if you start with $500 and make $50 bets, you'll have very different short-term results than if you start with $500 and make $5 bets. Similarly, if you start with $500 and set win/loss limits of +5 and -$500, you'll win that $5 a vast majority of the time. Your overall EV (as a percentage of action) doesn't change, but your actual average session dollar loss will be smaller because your average session length is shorter.
You can't go broke taking a profit. Sure anytime you take that profit and walk you can imagine that had you but stayed for one more hour you would have broken the bank, but in truth that one more hour would be just like the first hour: entirely up to Dame Fortune.Quote: scudderIntuitively it seems like I could be "cheating" myself out of long winning streaks by leaving when up a pre-determined amount, but of course I'm also limiting my losses as well.
Its not the advantage of walking away when your losses reach a certain limit, its the advantage of setting a limit that is less than your entire bankroll. That coming back for another bite the next day is what gives you the ability to stick to a loss limit.
Setting win/loss limits doesn't change expected value, it just insures that you will be in the game longer so as to experience whatever Dame Fortune has in store for you.
Quote: MathExtremistNo, win/loss limits don't change your overall EV, they change the distribution of session results. The variables to consider are:
a) bet size relative to bankroll
b) EV and variance of underlying wager
c) Win and loss limits relative to bankroll.
Bigger bets, higher variance, and lower win limits increase your chances of coming out ahead.
For example, if you start with $500 and make $50 bets, you'll have very different short-term results than if you start with $500 and make $5 bets. Similarly, if you start with $500 and set win/loss limits of +5 and -$500, you'll win that $5 a vast majority of the time. Your overall EV (as a percentage of action) doesn't change, but your actual average session dollar loss will be smaller because your average session length is shorter.
Thanks for the explanation, I had completely forgotten about variance vs. bet size, and the profound effect that could have on your results in the short term. Everything you said makes sense intuitively, I'm just trying to make it make sense in my head mathematically. My stats knowledge is pretty rusty, so I'm not even sure I'm thinking in the correct terms.
What I'm envisioning is a bell curve with expected win/loss along the X axis, and the ends bounded by the upper and lower win/loss limits. Am I correct to assume that the area under the curve would represent overall EV and thus have a peak just to the left of 0 (assuming I'm not playing a game with some absurd house edge)?
Then, for any given range along the X axis, regardless of where the limits were placed, the area under the curve would be the same, assuming the peak value was included? i.e., a range of -50 to +50 would have the same area, and thus EV, as -40 to +60?
So setting a limit of -500 to +5 means the distribution will favor the area of the curve near and just beyond 0, making it more likely you'll come out ahead on a session to session basis, but end up with the same EV long term since there will be times when you hit the -500 limit?
Am I on the right track here or am I missing something?
Quote: scudderSo setting a limit of -500 to +5 means the distribution will favor the area of the curve near and just beyond 0, making it more likely you'll come out ahead on a session to session basis, but end up with the same EV long term since there will be times when you hit the -500 limit?
Am I on the right track here or am I missing something?
Yeah, that's basically it - you're adjusting the distribution of session outcomes, but not the overall aggregate (or the individual results). One of the things I try to do is adjust my bankroll, limits, and betting patterns to roughly hit the session distributions I feel like shooting for. The craps system I usually play leads to a somewhat slow dribble punctuated by moderate losses but even more infrequent big wins. One of those big wins is usually when I'll color up. If you want a lot of little wins, the limits above will do the trick.
