1)If I were starting a casino online or offline (not sure if it matters)
and I wanted make financial projections 3 years out. Would I be using the
Theoretical vs. Actual House Advantage in my calculation.
How would it work?

Making a financial projection would be based off the theoretical house
advantage (T.H.A). Although this T.H.A might be less that the A.H.A.,
because some players know how to 'count-cards' and proper betting
strategies for table games.

The T.H.A and A.H.A is static with regards to slot machines, because
there is no strategy, and all probabilities are fixed.

The Actual House Advantage could be computed by setting the limit as t
-> +inf as the T.H.A. In other words, the longer your time frame, the
closer your A.H.A will be to the T.H.A.

You could view your A.H.A as stochastic ( random but with direction ),
and this direction would be similar to (but likely greater than) the
T.H.A. I say greater than, because inexperienced players who do not
use optimum betting strategies, as well as players under the influence
of alchole give the house a slightly greater edge. Craps may be a
49.29% chance of winning for the player, but as soon as an
inexperienced player starts throwing bets down, the house will obtain
a greater edge.

Concluding, the A.H.A is a stochastic process which will have the same
direction as the T.H.A, but is likely to be slightly higher than the
T.H.A. As a financial model over 3 years, you would always want to use
your theoretical house advantage, as that would pose the
'worst-case-scenerio' ( every player betting the best ). As far as the
theory behind the house advange is, you would need to consider the
proportions of people playing slot machines, and the different table
games. T.H.A is larger if larger proportions of your clients are
playing slot machines.

The T.H.A. could be written as:
numberofpeopleincasino*P(X=slot_player)*T.H.A(for slots) +
numberofpeopleincasino*P(X=crapsplayer)*T.H.A(for craps) ...

Where {X} contains all types of games in casino, P(X=slot_player) +
P(X=crap_player) + ... + P(X=nongambler) = 1 (if they are all
independent events, but it is likely you would have to use some sort
of inclusion-exclusion formula).

With appropriate theory, the Theoretical House Advantage would be
close to the Actual House Advantage. For some periods of time the
Actual House Advantage might be much different from the Theoretical
House Advantage because it is a stochastic process, but a good
theoretical model would contain some sort of random walk, and could
explain all A.H.A. to company owners.

The advantage to using a theoretical model for the house advantage as
opposed to  an 'actual' is you only need to crunch the numbers once.
The actual house advantage would be ever changing ( stochastic ), but
close to the theoretical.

Hopefully this helps you make your choice, (theoretical).

Mr_F

But let me ask you how is that in vegas we see the hold for the house
is anywhere between 14-20 percent over extended period of time, when
we know and like you say that over time the AHA and THA get closer and
closer. Thanks

Joshua

Let me rephrase your statement and you tell me if I am answering the
right question here.

Over the long run, how come vegas casino's holds are between 14-20%
and not a fixed number close to the T.H.A?

This would be accounted for by the fact the A.H.A is a stochastic
process, as well as the realization that different casino's would host
different types of people. Some casinos might have alot more slot
machines (thus more slot players), so there A.H.A would be greater
than other casino's (or less than if they decide they like to pay out
more often... which is unlikely). The Theoretical House Advantage
would be casino specific, using proportions of people playing
different games combined with the probabilities of winning in those
events.

Also some casino's have slightly different 'table' rules. (Eg. Hit on
soft 17 in blackjack), some casinos might also take a different rake
from poker games, would would contribute to the house.

The theoretical house advantage is not universal for all casino's but
rather very specific to individual casinos.

If the data for the A.H.A is there and computed, you could figure out
all the important statistical variables (STD, VAR.. etc) and apply
that to a financial model, which would be much simpler.

The issue is, if your not modeling financial projection of an existing
casino, it would be incorrect to fudge data to get an A.H.A.. Unless
you knew the house edge on the slot machines, and all the different
table styles, and your casino did _EXACTLY_ the same thing, in the
same demograph, which is unlikely, and probably wont be as correct a
T.H.A model.

The best approach would be having a Theoretical House Advantage
_MODEL_, in which you adjusted it after the first month with regards
to the proportions of people playing what.

Or you could just use average of the daily house advantage. (which
would approach the true house advantage with a larger time period
because of the central limit theorem)

Mr_F