Jean,
Sorry... missed it the first time.
You are certainly right to look at the "random walk" example to way you do. That was
what I had hoped for when I originially wrote it down. VP is basically a random walk
process, where our bankroll (or stake) is the "position". I will get to your question abou
the bonus step in a moment. First, though, I think it is important to look at the
"units" (and common meanings of) EV.
Assume for a moment that you are playing a game with 101% return, a positive game and
someone asks the questions "What's the EV?"
Probaly the most common answer is : "The EV is 101 %"
But there are other possible answers such as:
"In the long run, expect to earn 1 cent for ever dollar I bet"
Likewise, if you are playing a negative game (Ghast!), with a 99% return and someone asks
the same question, the answer might be the same "Th EV is 99%" or even "My EV is 0! In
the long run I will certainly lose all my money. (Hmm... why am I playing?)"
There is an interesting computation and interpretation of EV hiddin in that second answer.
The player has computed the following:
Long Term Results = (starting Bankroll) * (0.99)^(infinity) = 0
The interpretation of "EV" in this case is something like: "What is MY theoretical EV if I
would play forever".
The point I am trying to make here is that the question "What is the EV?" is not particularly
well posed. It would be better to ask something like: "What is the EV for the game?" or
"What is the EV for that hand ?" or "What is your EV after 100 hands?" and of coarse to
agree upn the units (as in percentage or dollars or bets or units, etc)
Armed with this information, let's take a look at the random walk (drunkards walk)
example again. Let's assume now we are playing a VP-like game. There are only 2
outcomes of for any hand in this game. Either we win $100 (yes!) or lose $1 (ugh). Let's
make this an exactly even game, so that the probabiliy of losing is 100 times greater than
winning. In in idealized world then, if we playing this game for 101 hands, we would win
$100 once, but lose $1 100 times. The net results are = $100 * 1 - $1 * 100 = 0. Hence,
overall, on average, we would neither win or lose. For this game we could write that "EV =
100%" or "EV = 1" and some might write "EV = 0" though the last one could be
misleading.
Now, you sit down and play this game, and amazingly, you win 10 hands in a row. You
now have 10 * $100 = $1000 more in your pocket than when you started, for a huge
return of 10,000% = ($1000/$10 * 100%). But, What is your long term EV relative to you
original bankroll? Well, given what we know about the game, in the long term you expext
to be up exactly $1000, no more and no less. So in dollar units we know the answer: "EV
= $1000". But what is the EV in percentage units? Its a bit more complicated and depends
now on the number of hands played. You won the first $1000 in 10 hands, but you
continue to play now for N more hands. What's your EV afer these N more hands ?
EV (%) = [ ($1000 / $10) * 100% + N * $1 / $1 * 100%] / (N+10)
so
EV (%) = ( 10,000% + N* 100 %) / (N+10)
Notice that as N get very big, N is almost the same as N+10 [ as in 10,000,000 is almost
the same as 10,000,010 ]. In other words,
Ev(%, large N) = 10,000% / N + 100% = 100%
because 1/N ->0 as N becomes large.
What does this result mean? Well, in percetnage units, it says that your long term return is
100%, not more or less. But what it hides is that in the long term, you EV in real units
(dollars) is $1000. Yes, Units do matter in how we think we see the world!
Now lets switch the situation around. Instead of winning $1000 out of the gate, you lose
the first 1000 hands and therefore $1000. Now you have a $1000 loss. Your long term
EV in real units is -$1000, but in percentage units is comes out to 100% again!
Interestingly, you may have noticed, when people are losing they tend to think of long
term results in percentages rather than real dollars( compare "that loss doesn't matter
much, 'cause in the long term the game is still 100%!" to "I hit a RF on the first hand I ever
played, but that was years ago, and since then my percentage return has been dropping
since I haven't been loosing or wining")
Recap: I've been trying to make two points here (1), that EV's are basically additive (with
the weighting) , so long as each is in the same correct units, and (2) that the units of EV
matters (chose the wrong ones, and well eveything appears to change)
Now back to your question:
Your question about given a "bonus step" is a very intresting one. I'm assuming this step
trully is a bonus-- it is entirely free, with not cost at all to you? Assuming that you get
this bonus hand with a known rate (that is, it occurs randomly or deterministicly at a
constant average rate), it can be easily factored into the mathematical description of the
game. First, we can compute what a trully free play (bonus step) is worth for the game.
As you might expect , a free play has a BIG Positive EV.. Now assume that ou get this free
play every say on average nFP hands (nFP could be 100 or 1000 or whataver). The rate
that the free play hand occurs is then 1/nFP = 1 per every NFP hands. The overall effect
on the EV of the game would be:
EV (overall) = (1-1/nFP) * EV (normal game) + EV (Bonus Hand) * (1/ nFP)
Likely, 1/nFP is a small number so = 1 - 1/nFP = (approx.) = 1.
Again,we need that the EV's are additive (with the correct weighting factor). By definition,
the EV of the bonus hand is higher than the EV of the plain game (since we don't pay for
the bonus hand). Therfore, the bonus hand increases our EV. Let's assume that the new
EV( (overall) is positve, EV (overall) = 101%
Or, In real dollars, EV (overall) = 1.01 * N * B , where N is the number of hands and B is
the bet per hand in real dollars
[Yup, in real dollars, our long term, large N EV is infinite. Woo-hoo!]
No assume that you lost the first bunch of hands. Then we are back to the same situation
as before, but this time with a positive game. And that's a good thing.
Using what we learned before, we know that the long term percentage return is still 101%,
so our loss seems to dissappear. Likewise, in real dollars , we have
EV(overall) = 1.01 *N * B + (-$Loss)
But as N gets large, 1.01N *B becomes bigger than $Loss, and so we are happy folks
again. The loss doesn't seem to matter! BTW, the rate is the 1.01 factor. A bigger
number is a faster rate.
Caveat Emptor: All of this "math" depends on N becoming large. It can take a very large N
(with a typicall positive VP game) to come back from a big loss. BUT, we all know that
isn't what generally happens. We tend to come back from a loss quickly (or not at all!),
and we don't have to wait it out for infinite N. That's because of the variance. You see, so
far I have only taked about EV. A more important effect for VP (in the short run. small N
case) is the variance. In fact, it is the variance, that (in the short term) drives what appears
to be our rate of comming back from a big loss, likewise it is the variance that creates that
big loss in the beginning, not the EV itself (even in a negative game).
The point: I like our question, but the precise answer that includes the variance (and the
entire PDF) is comlplicated. Yup, the higher the EV the faster the rate you win, on
average. But, in practice the fluctuations about that average are much larger than that
rate: Variance matters.
···
--- In vpFREE@yahoogroups.com, "Jean Scott" <QueenofComps@f...> wrote:
No one responded to this message below. >
<<Am I right in thinking this way about your "steps to the right and left"
example. What if steps to the right stands for profits and steps to the
left stands for losses. I can see that if you are playing at exactly 100%
that it might take more time than you will live, to get back to the original
starting place ("even") if you had to take 10 steps to the left on the first
10 coin tosses. But what if you were given a "bonus" step to the right
periodically, like every so many coin tosses. You would be more apt to get
back to the starting place faster, right? I consider EV "bonus" steps - and
the higher the EV the more bonus steps you would get, and the faster your
trip toward the original EV.
<<First, if you have a big loss, (1) your long
term EV is this loss + EV of the game (in the correct units of course)>>
Despite the limitation of my math skills and understanding, I do understand
this - I think!. So this does not negate the concept I drum on all the
time. In fact it reinforces it. The higher the EV of the game the faster
you get to the long-term. If you have a long severe losing streak at the
first part of your play, it will just take longer than if you start with a
big winning streak.
Considering the VERY complicated posts it has spawned. I'm not saying this is an unimportant thread, it isn't. But it will take a lot of studying on my part to try to boil these numbers down to a non-math general explanation. I don't have time right now to study it - knee-deep in tax business.