Steve, as always, interesting points and I'm always glad to see you
post. Let's see if I can answer your observations
> I'm sorry, but the equation "more ER = more royals" is simply not
true.
> The max-EV strategy focuses intensely on squeezing the last drop
> of EV from the game without any regard whatsoever for the impact
> on frequency of royals.
>
> Now let me ask you this -- how do you define "more royals"?
Now that is central to my whole argument - how do I define more
royals? While the various strategies you outline are mathematically
correct for the respective definitions you give of that term, I
would argue that none of those definitons is realistic, in terms of
how the average reader on this board conducts his daily play. Let me
offer this definition:
"More royals" = "maximum number of royals over an individual VP
career, playing slightly positive games with an adequate bankroll,
which career will be terminated prematurely if excess losses are
encountered, and whose play levels will increase to the extent that
accumulated profits build up"
Also, let me expand the definition of "royals" to include all types
of jackpots worth 50% or more of a royal � this way it would include
Aces with kicker in DDB, Quad Deuces for Double, Triple and Loose
Deuces, Quad Aces in Super Aces Bonus, Ordered Aces in Aces Bonus,
Bonus Quads in ShockWave, etc.
OK, more "big payoffs" rather than royals.
I think this is most people's working definition and their
objective, or at least most recreational advantage players who
frequent this board. If you ask them explicitly you may not get this
as a response, but if you evaluate their actions over time, this is
how they play.
But there's a problem here -- you can't really determine a player's
true objective by watching them play, especially if the player doesn't
know how to optimize their results.
Now, would you not agree that under this definition, more ER = more
royals (jackpots)?
No. In fact, if I'm understanding your definitions, then it would be
more accurate to say "less risk = more big payoffs". Perhaps the
key phrase is "which career will be terminated prematurely if excess
losses are encountered". This phrase describes a very real
concern with risk of ruin, and suggests to me that the player would
feel it was more important to avoid going broke rather than squeeze
each last drop of EV from the game. Player's who don't have the
luxury of playing with unlimited funds would generally be better off
forgetting about EV and worrying more about RoR.
I am aware of, and know a few of, the group of wealthy individuals
who don't mind loses and play for the thrill of the jackpot. But
even this group would be better served by at least looking at the
math of the max-ER approach.
You seem to be arguing that max-ER is a "one solution fits all"
strategy, and that is precisely what I'm arguing against. If it is really
ER that you want to maximize, then max-ER is correct. If it is ANYTHING
ELSE that you want to optimize, then it is very unlikely that max-ER
strategy will also optimize the true objective.
I recently met one such individual in
Tunica who early in our acquaintance bragged that he "had 20 royals
in the past year, and needed 20 more in a row to break even money-
wise." Later on he began asking me various questions about proper
play strategies and the costs of his royal-chasing moves. I told him
about comparisons I had done between such strategies and high
variance games, and in the end he was planning to buy WinPoker and
learn how to play Super Aces Bonus, Triple Deuces and ShockWave by
the book.
This is a good example of helping a clueless player become a more informed
player. I agree that a player who gives up a "seat of the pants" strategy
and adopts a max-EV strategy is likely to greatly improve their probability
of getting more big payoffs without going broke. But that does justify the
claim that "more ER = more big payoffs". Compared to SotP strategy, ANY
mathematically derived strategy, whether it is max-ER or min-risk or
best-shot, is likely to significantly improve the outcome for the player.
> Not necessarily. If your objective truly is a higher frequency of
> royals, then playing more agressively is the only way to achieve
> that objective.
I disagree. First off, nobody said anything about higher frequency,
they simply wanted MORE.
The bare word "more" is quite ambiguous. I think if you asked 100
random players in a casino what "more royals" means to them, they
would say "getting royals more often".
This could be done without higher
frequencies by playing more hands, which for most people can be
accomplished by winning more and losing less. If all you want is
higher frequencies, money be damned, you should go for the royal on
every hand. This would shorten the royal frequency in JOB to around
23,000 hands. The problem is, your EV drops to 3.41% and most of us
would run out of money real fast and have to stop playing.
To be more precise, the "most royals" strategy on JoB 9/6 has a royal
cycle of 23164.7 and an EV of 55.02%. Far worse than Keno, and
approaching lotteries in terms of EV, but much better than 3.41% 
Extreme aggression has a hefty price. But my point remains -- when
"more" means "more often" then an agressive strategy fits the bill.
Unless
you have a greater limitation than money on your play (free time,
lifespan, etc.), you'll get more jackpots in the long run if you
make a little money along the way.
Agreed. However, this does NOT imply that the max-ER strategy
will give you the *most* jackpots in the long run.
But even if you do want a higher frequency, I disagree that playing
aggressive, non-max-ER strategy in your regular game is the only way
to achieve that. If your regular game is Bonus Poker, you can get a
higher frequency simply by changing games. This goes back to a point
I made in my original post, which I see as the deciding factor here �
in Tunica, at least, some of the highest ER games are high variance
games that many people don't play, like Super Aces and Triple
Deuces. As long as that is the case, then why in the world would a
jackpot-hungry player take a negative pay schedule like BP or JB,
and then give up even more ER by playing incorrectly in pursuit of a
jackpot? Far better to learn a new game, one that offers higher
frequency jackpots AND a higher ER.
Fine, pick whatever game you want. Then, within the context of
whatever game you finally pick, the playing strategy that gives
"most jackpots" will almost certainly be different than the max-ER
strategy for that game.
Simply put, "most ER" and "most jackpots" are not equivalent
concepts. You can maximize one or the other, but in general
if you choose to maximize ER than you will achieve slightly
fewer jackpots, and if you choose to maximize jackpots you
will get slightly lower ER.
It is easy to fall into the trap of believing that max-ER strategies
will automatically give you the best of all worlds, but that is
just wishful thinking.
> Sorry, but the max-EV "book" is the wrong book to read if your
> objective is anything other than "maximize EV". Different
> objectives almost always lead to a different form of optimal
> strategy.
In sum, what I'm saying is that for a player whose play is
constrained by available funds (most of us on the board here), the
goal of maximum jackpots lies down the same path as the goal of
maximum ER.
That is a mistaken notion. I agree that most players are constrained
by available funds, and for that very reason max-ER is slightly
misguided for most players (note that I say "slightly" here). They'd
probably be better off playing min-risk strategy instead.
ER is about money and time. Get the most money in the least time.
Dollars per hour. Based on your comments, it sounds like you
want to maximize "long term chances for hitting jackpots" while
avoiding going broke. That isn't directly about money, and time
isn't the issue, so on both counts this is a different objective
than maximizing ER.
Assuming the chosen game is favorable, the path of the max-jackpots
strategy lies down a path that is closer to min-risk strategy, and both
to these paths are a little to the "right" (more conservative) of the
max-ER path.
> Paradoxically, if your objective is to have your bankroll survive
> longer in order to increase your chances of (eventually) hitting
> a royal, then the optimal strategy is best_shot(royal), which tries
> for royals less often compared to the max-EV strategy.
I have heard of this, but do not know the details (although I think
I can guess some of it). Can you explain more about how to calculate
this strategy for a given game, and if possible give us an example
strategy for a game where it differs substantially from EV for an
adequately bankrolled player? Is bankroll one of the inputs? Does
the assumption of unlimited bankroll make this strategy the same as
max-ER?
Computing best-shot strategies is similar to computing min-risk.
Min-risk means that we minimize the probability of going broke.
Optimal strategy always minimize one thing while maximizing
just the opposite. Max-ER minimizes the average number of
plays need to win another dollar. Min-risk works the same
way, but the opposite of ruin is "success". More specifically,
for a favorable game, minimizing RoR is the same thing as
maximizing the probability that the bankroll will survive forever.
The funny thing about this is you can never know if you actually
succeed at reaching your goal, because the goal is "play forever".
Best-shot strategies are similar to this, but they have more tangible
goal. With a best_shot(royal) strategy, the objective is to have the
bankroll survive until you hit a royal flush. The opposite of this is
going broke before hitting a royal. So, min-risk and best-shot are
both forms of "maximize the probability of success before going
broke" but with different definitions of "success". For min-risk,
success means "play forever" while for best-shot success means
"hit a royal".
One can also modify the definition of "success" to target different
payoffs. In fact, for a game like Double Bonus it might make sense
to play best_shot(quad aces) or even best_shot(any quad) since
quads. It is also possible to define success with several different
pays, perhaps combining royal, str-flush and high paying quads
into one generic goal. Based on your comments, I think this is
probably the kind of thing you are after for "most jackpots".
Now some math. To compute a min-risk strategy, you can use
VP software to find the optimal strategy by using "virtual payoffs"
that are different than the actuall payoffs for the game. For min-risk,
you need to know the RoR value of the strategy in order to compute
the virtual payoffs. If R is the risk of ruin, then a payoff of N units
has a virtual payoff of:
V = (1 - R^N) / (1 - R)
This formula can be interpreted in the following way: R is the
risk of ruin, which is the same thing as the probability of eventually
going broke. If you don't go broke, then you play forever and the
probability of this "success" is (1 - R). If you get a payoff of N
units, then your new bankroll of N will give to N more tries at the
game of "start with one coin and play forever or go broke". The
probability of losing all N games is R^N, and this is risk of ruin for
an N unit bankroll. The value (1 - R^N) is the probability of
surviving forever by starting with a bankroll of N units. Favorable
games always give a value of R less than one, so as N gets
larger (1 - R^N) gets closer and closer to 1. In other words, the
bigger your bankroll, the higher the probability of playing forever.
A value of 1 here is the same as "certainty of playing forever".
The (1 - R^N) terms represent the new probability of success
after getting a payoff of N units. The overall probability of success
is found by summing over all possible payoffs, and that is equal
to (1 - R) so we get
(1 - R) = p(1)*(1 - R) + p(2)*(1 - R^2) + ... + p(1000)(1 - R^1000)
where p(N) is the probability of a payoff of N units.
Now think about the virtual payoff of V = (1 - R^N) / (1 - R). This
value is the probability of success with a payoff of N, divided by
the overall probability of success. The (1 - R) in the denominator
just "normalizes" the success rate in terms of the overall success
rate.
Suppose for now that R = 0.999 so that our probability of playing
forever is (1 - 0.999) = 0.001 = 1 / 1000. That means we have
a one in 1000 shot at playing forever. Let's see what the virtual
payoffs tell us. If a royal is worth 1000 units, then the virtual
payoff for a royal is (1 - (0.999)^1000) / (0.001) = 632.30. This
means that hitting a royal flush increases our probability of
success by a factor of 632.30 over what it was with a single
unit bankroll. Let's see what the virtual payoffs look like for
other cases:
Payoff Virtual Payoff
···
On Thursday 21 October 2004 10:02 am, blaw57 wrote:
--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:
-------------------------
1000 632.30
500 393.62
200 181.35
100 95.21
50 48.79
25 24.70
15 14.896
10 9.955
9 8.964
6 5.985
4 3.9940
3 2.9970
2 1.9990
1 1.00000000
-----------------------------
To clarify, this is not a payoff table. This just illustrates how
actual payoff translates into "probability of playing forever"
for a game with RoR = 0.999. If you want to maximize the
average probability of playing forever, then you can think
of the virtual payoffs as "virtual dollars" that don't directly
represent money, but express the relative "value" in terms
of the probability of success. If you think of a one unit
bankroll as "one shot at playing forever" then a 1000 unit
bankroll for this game gives you 632.3 "shots" at playing
forever.
You're probably lost by now, but I'll just say a little bit
more. The min-risk strategy can be found using virtual
payoffs as described above. The best-shot strategy is
slightly different. If you want the best_shot(royal) strategy,
then hitting a royal corresponds to a 100% probability
of success. For best-shot, the R value stil represents
probability of going broke, but the alternative to going
broke is hitting the royal flush. For, for all payoffs
except the royal, we use the same formula for the
virtual payoff:
V = (1 - R^N) / (1 - R)
but for the royal flush, the probability of success
increases from (1 - R^N) to (1) so the royal uses
a virtual payoff of:
V = 1 / (1 - R)
and the overall game satisfies
(1 - R) = p(1)*(1 - R) + p(2)*(1 - R^2) + ... + p(royal)*(1)
I've left out a lot of details, but that is the basic math.
In a nutshell, best_shot(royal) is just like min-risk
except that the royal uses a different virtual payoff.
Similarly, the best_shot(big_payoff) uses V = 1 / (1 - R)
for all the payoffs that are included in the "big payoff"
category. The best_shot(big_payoff) strategy maximizes
the probability that your bankroll survives until hitting
one of the target payoffs.
Best_shot strategies remain the same independent of
bankroll. You can think of it as if each unit in the bankroll
gives one shot at success. Having an unlimited bankroll
will virtually guarantee that you will hit one of the targets
before going broke. But, when the bankroll is small, the
best_shot strategy gives the highest probability of
surving unil hitting a target.