vpFREE2 Forums

max-EV

#1

Steve Jacobs wrote:

Kelly play is log-optimal, meaning that it seeks to maximize the expected
logarithm of the players bankroll.

However, for blackjack the loss from using a suboptimal playing strategy
is truly miniscule, because the low variance makes "close call" strategy
changes less significant.

Actually, the *effect* of Kelly-optimal strategy changes on bankroll growth
in real-world BJ play, and the *effect* of Kelly-optimal strategy changes on
bankroll growth in real world VP play, are both insignificant. An EV-only
BJ player will usually double A4 versus a dealer 4, for example. And if the
bet is near Kelly-optimal for the bankroll, that play is not Kelly-optimal.
But that changes the expected rate of bankroll growth by such a miniscule
amount that it really doesn't matter.

In VP, the bets are usually very small relative to bankroll, so the
difference between Kelly-optimal play and EV-only optimal play are rare and
small, making the overall effect on bankroll growth tiny. But they do
exist. When playing 10-7 DB at a denomination which makes the game
full-Kelly with two-thirds of a percent cashback, it is correct to hold the
full house with Aces full, for example; EV-based strategy sheets indicate
the three aces should be drawn to.

Min-risk strategies can be applied to short term objectives, such
as growing a bankroll by some fixed factor,

It has been proven that the Kelly criterion achieves this, as long as the
factor is large enough so that the central limit theorem holds (that is, if
you want to maximize the chance of achieving a 10% gain on a 10,000-bet
bankroll, it may be correct to pursue a royal flush at all costs at first,
because the payoff distribution is pretty lumpy; that strategy is dominated
by Kelly-optimal play when the goal becomes to triple that bankroll).

Let's suppose for a moment that you are a Kelly player who wishes
to adapt Kelly principles to VP. The correct way to do this is to
just forget about EV and use expected log(bankroll) as the criteria
for choosing betting denomination based on your bankroll, and for
making playing decisions. The main effect of choosing a Kelly
objective is to devalue royal flushes by some amount. The specific
amount depends on your bankroll, but it might be in the range of
10% to 20%.

It is approximately correct to map each payoff of the paytable to

(2*Bankroll * Payoff) / (2*Bankroll + Payoff). This devaluation factor is
10% for a Payoff of 2/9ths of the Bankroll, and 20% for a payoff of 1/4th of
the bankroll.

The optimal betting denomination, assuming your chosen game is available
in many denominations, will be whichever denomination gives a higher
expected value for log(bankroll) after averaging over all possible
outcomes while using the log-optimal playing strategy.

This could be greater than the Kelly-optimal denomination; for example, if a
particular game is available in quarters or dollars, and the Kelly-optimal
wager would be 89 cents, then the dollar denomination will probably have a
higher expected bankroll growth rate than the quarter. If the player
doesn't do well, though, bankroll shrinkage will cause the bankroll growth
expectation of continued dollar play to fall much faster than the bankroll
growth expectation of quarter play.

using a max-EV strategy when your true objective is something else is
sub-optimal, except in the (very unusual) case where the optimal strategy
happens to be identical to the max-EV strategy. For VP, I believe this
would happen only for players who have no real concern for risk.

Because the strategy table is lumpy, it wouldn't surprise me if there were
games whose strategy tables show no differences between a max-EV strategy
and a max-utility strategy, for reasonable bankrolls.

···

--
Randy Hudson

#2

[Formatting of tables will probably get scrambled, but I've decided to
post this here anyway in case there is interest -- Steve]

                       Equivalent Games
                Copyright (C) 2003, Steven R. Jacobs
                       All rights reserved.

Suppose we use a biased coin as the basis for creating a game. All
game outcomes will be derived by flipping coins that are identically
biased , but we are only allowed to bet on "heads" and the coin is biased
in favor of the house. The player has a probability of winning equal
to p(win)=0.49974550421575. This probability wasn't pulled from thin
air, but carefully chosen, for reasons that will become apparent later
in this article.

Now we will make the game more complex by creating several "sub-games"
and adding a random spinner which is used to select which sub-game is
played. The spinner will be divided into sections of various sizes,
and a number printed on each section will tell the player how much they
must try to win in order to record an overall win for the sub-game.

The following targets and frequencies have been chosen for the subgames.
Of course, these numbers are not arbitrary, but were also carefully chosen,
for reasons that will become apparent later.

Target Frequency

···

----------------------
1000 4.762161445
   50 0.558915312
   25 5.962619841
    8 9.224490842
    5 5.522918550
    4 4.523411750
    3 22.284452726
    2 25.809295114
    1 21.351734420
----------------------

The sections of the spinner are divided up in sizes that are
proportional to the frequency column, so that random spins
will have a 4.762161445% chance of selecting 1000 as the
target, and a 21.351734420% chance of selecting 1 as the
target, and similarly for the other targets.

Each sub-game is played as follows: The player starts with
a single coin as the bankroll for the sub-game, and spins
to determine the target bankroll. The player then flips
coins repeatedly, and is paid one coin for each win while
losing a coin for each loss. If the bankroll reaches
the target, then the sub-game is declared a "win" and the
player allowed to cash out that bankroll. However, the
player is not allowed to simply stop play in the middle
of a sub-game, so that once a target has been selected
the player is required to keep flipping coins until the
bankroll either reaches the target or hits zero.

Note the following:

1) All coins are biased in exactly the same way, so it
   doesn't matter which coin(s) the player flips.

2) Payoffs are effectively N-for-1, since the initial coin
   is purchased from the casino (so that the casino can
   control the bias) at the beginning of the game.

3) If the player loses the first flip, the bankroll is gone
   and the sub-game is lost.

4) When the target is "1" the player immediately wins
   without flipping. This result is equivalent to a push.

Now that the rules are in place, we can examine the results.
But first, a little bit of math to explain how the results
were computed.

Definition: unit-cost for a coin flip is defined as the
probability of losing divided by the probability of winning.
This number represents the average number of coins paid
in losses for each coin returned in winnings. For a biased
coin that favors the casino, the unit-cost will be greater
than one, indicating that player losses are greater than
winnings. When the bias favors the player, the unit-cost
will be less than one, indicating that the player loses
less than one coin for each coin returned. When the bias
favors the player, the unit-cost is numerically equal to
risk of ruin. For this problem, the coins all have
p(win)=0.49974550421575, so the unit-cost for our coins
will be 0.50025449578425/0.49974550421575 = 1.001085015457.
These coins are slightly in favor of the casino.

The letter 'C' will be used to represent unit-cost. With
a unit-cost of C and a target bankroll of N, the probability
of reaching the target before going broke is given by:

p(win) = (1 - C) / (1 - C^N)

This formula was used to compute the "Sub-cycle" numbers in
the table below. The columns in the table are described as:

"Return" is the total number of coins that we attempt to win from
the randomly chosen subgame.

"% Tries" is how often each sub-game is played. This describes the
relative size of each section of the spinner. It is assumed that the
spinner is completely fair in the sense that the probability of landing
on each section is exactly proportional to the size.

"Sub-cycle" is the average number of subgames needed to produce
a winning result from that sub-game. This is simply the inverse
of the probability of turning the one-unit starting bankroll into
an N unit payback. In short, sub-cycle = 1/p(win) = (1-C^N)/(1-C).

"% Hit" is the overall probability of reaching each of the
possible outcomes, and is given by (% Tries) / (Sub-cycle).

"Cycle" is the average number of subgames played between
favorable outcomes, and is given by Cycle = 100 / (% Hit).

         <---- Coin Flip Subgame ----> <------ Overall Outcome ----->
Return % Tries Sub-cycle % Hit Cycle
--------------------------------------------------------------------------
1000 4.762161445 1735.498092607 0.002743974 36443.495514000
   50 0.558915312 51.268241981 0.010901784 9172.810426000
   25 5.962619841 25.307949781 0.235602642 424.443457000
    8 9.224490842 8.028576209 1.148957250 87.035440178
    5 5.522918550 5.010195394 1.102335960 90.716445466
    4 4.523411750 4.006115160 1.129126740 88.564017180
    3 22.284452726 3.003056542 7.420590460 13.476016570
    2 25.809295114 2.001018502 12.898079200 7.753092414
    1 21.351734420 1.000000000 21.351734420 4.683460277
--------------------------------------------------------------------------

The table above summarizes the results of the overall game, based on
the combined results from the coin-flip subgames. The Sub-cycle column
is particularly interesting, because it illustrates how the losses of
a single coin flip get amplified when trying to win multi-unit payoffs.
For example, a payback of 50 in a fair game would tend to happen in one
attempt out of every 50, but in this biased game we win less often than
what is "fair" because we get 50 coins back only once in each 51.2682...
attempts. When trying for a payback of 1000 coins, we only reach the
target once in each 1735.5 attempts.

Now compare those results to the table below, which gives the expected
results from using a max-EV strategy while playing video poker on a
"Jacks+ 8/5" machine. The "% Hit" and "Cycle" columns are virtually
identical to the same columns for the coin-flip game. This means that
the two games have identical probability distributions, and so the
games are mathematically equivalent. An outside observer who looks
only at the freqencies and sizes of the payoffs cannot tell any
difference between these two games, and any kind of statistical
computation such as EV or variance will be identical for the two
games, because they have the same probability distribution. The
differences that we "see" between playing video poker and playing
"spin and flip" are cosmetic in nature rather than mathematical.

  Jacks or Better 8/5
  Distribution of Final Hands
----------------------------------------------------------------
Final Hand Payoff % Hit Cycle % Return
----------------------------------------------------------------
Royal Flush 5000 0.00274397 36443.495514 2.74397389
Straight Flush 250 0.01090178 9172.810426 0.54508921
4/Kind 125 0.23560264 424.443457 5.89006606
Full House 40 1.14895725 87.035440 9.19165803
Flush 25 1.10233596 90.716446 5.51167980
Straight 20 1.12912674 88.564017 4.51650696
3/Kind 15 7.42059046 13.476017 22.26177139
Two Pair 10 12.89807920 7.753092 25.79615840
High Pair 5 21.35173442 4.683460 21.35173442
----------------------------------------------------------------
                         45.30007244 2.207502 97.80863815

The exact amount of bias in the coin, and the section sizes for the
spinner, were chosen so that the coin flip game would yield results
that are indistinguishable from this video poker game. Now think
about what is happening here -- the spinner does nothing more than
select how often the player "tries" for a specific payoff, but
"trying" entails nothing more than flipping the same biased coin
over and over. The actual fluctuations in the players bankroll
are controlled completely by the dynamics of the biased coin (plus
the frequency of pushes, which is the same for both games). The
spinner is nothing more than an amusing diversion between sessions
of coin flips. So, the complex probabilities associated with a VP
game can be reduced to the simplicity of a single biased coin!

The same process can be applied to any game which involves only
integer payoffs which are multiples of the original bet. I
believe the results can be extended to include games such as
blackjack, where the size of losses and wins can both be more
than a single unit. In short, I believe that any game based on
payoffs which are multiples of some minimum unit size can be
reduced to a series of flips of the corresponding unit-cost coin.
For games that favor the player, the cost ratio of the equivalent
coin is numerically equal to the risk of ruin of the overall game.
In fact, I stumbled across this "equivalent coin" relationship
while studying the characteristic equation which describes risk
of ruin.

#3

Steve,

Handicapped by a "short attention span" 4am mind, I'm sure I'll digest
your article more fully later. There is one part at this time that's
a little troubling, but one which I gather you expect to be a
stumbling block:

Now think
about what is happening here -- the spinner does nothing more than
select how often the player "tries" for a specific payoff, but
"trying" entails nothing more than flipping the same biased coin
over and over. The actual fluctuations in the players bankroll
are controlled completely by the dynamics of the biased coin (plus
the frequency of pushes, which is the same for both games). The
spinner is nothing more than an amusing diversion between sessions
of coin flips. So, the complex probabilities associated with a VP
game can be reduced to the simplicity of a single biased coin!

Isn't the frequency with which a player tries for a specific payoff,
which in this game is determined by the spinner, an integral part of
the expected fluctuations in the players bankroll?

It seems to me that the spinner is closely analogous to a player's
strategy in vp since both determine expected distribution of wins. To
dispense with it, if I understand your statements correctly, would be
the equivalent of disregarding player strategy in determining bankroll
requirements.

Trust me when I say that I fully expect I'm overlooking the obvious in
your article and I'll give it another read later today. However,
maybe you could shortcut my headscratching a little and point me to
the section that I might want to most carefully mull over.

- Harry

#4

Steve,

Handicapped by a "short attention span" 4am mind, I'm sure I'll digest
your article more fully later. There is one part at this time that's
a little troubling, but one which I gather you expect to be a

stumbling block:
> Now think
> about what is happening here -- the spinner does nothing more than
> select how often the player "tries" for a specific payoff, but
> "trying" entails nothing more than flipping the same biased coin
> over and over. The actual fluctuations in the players bankroll
> are controlled completely by the dynamics of the biased coin (plus
> the frequency of pushes, which is the same for both games). The
> spinner is nothing more than an amusing diversion between sessions
> of coin flips. So, the complex probabilities associated with a VP
> game can be reduced to the simplicity of a single biased coin!

Isn't the frequency with which a player tries for a specific payoff,
which in this game is determined by the spinner, an integral part of
the expected fluctuations in the players bankroll?

Surprisingly, no. The same biased coin is equivalent to an infinite
variety of games, with each game represented by different spinners.
For example, the simplest game using this same coin would have only
two sections, a "push" section with size 21.3517% and a "payback 2"
section with size 78.6483%. Now imagine that you are using the VP
spinner while I use this other spinner, and we hire someone else to
do the coin flipping, and we observer the same coin while we record
results from a long sequence of spins. Our two bankrolls will fluctuate
in exactly the same way, since each winning flip will increase both
bankrolls and each losing flip will decrease both bankrolls. I'll be doing
a lot more spinning, since I spin after every single coin flip (sometimes
more than once, when pushes occur) but you often get to take a rest
from spinning and just watch the coin flips. So, the difference in our
recorded result is that I'm effectively sampling/recording the bankroll
level after every flip, while you are more selective about when you
record results based on your spinner.

One caution is in order though, because this kind of comparison does
weird things with the notion of time. Each spin (or outcome) constitutes
one tick of the clock. For the VP game, the number of flips per clock
tick varies widely.

It seems to me that the spinner is closely analogous to a player's
strategy in vp since both determine expected distribution of wins. To
dispense with it, if I understand your statements correctly, would be
the equivalent of disregarding player strategy in determining bankroll
requirements.

Choosing a different strategy generally alters both the spinner and the
amount of bias in the coin, but it is coin bias alone that determines
bankroll requirements. For favorable games, the unit-cost of the coin
is numerically equal to risk of ruin, and risk of ruin is what matters
for bankroll requirements. For unfavorable games, the unit-cost of
the coin is equal to the inverse of the casino's risk of ruin, so
minimizing the unit-cost is the same as maximizing the casino's risk
of ruin.

This does raise another point however. Using the min-ROR strategy
is the same as finding the spinner/bias combination which uses the
very best coin possible, without regard to how large the spinner
"push" section becomes. For a favorable game, the max-EV strategy
uses a slightly worse coin for controlling bankroll fluctuation, but the
max-EV strategy will have fewer pushes than the min-ROR strategy,
so that measuring the "drift rate" of the bankroll as a function of time
will show better results from the max-EV strategy. Said in another
way, the min-ROR will have the lowest bankroll requirements for the
same probability of reaching a given fixed bankroll target, but the
max-EV strategy will tend to get there faster.

I think this equivalent coin model can help make it easier to understand
the tradeoff from choosing different playing strategies, but it takes some
getting used to.

Trust me when I say that I fully expect I'm overlooking the obvious in
your article and I'll give it another read later today. However,
maybe you could shortcut my headscratching a little and point me to
the section that I might want to most carefully mull over.

I don't think you're overlooking anything obvious. I belielve this result
may have been overlooked for hundreds of years. I've asked several
other math types if they have seen this kind of equivalence demonstrated
before, and so far nobody has said "yes, so-and-so did this." I've also
done Google searches and have found nothing. The closest attempt
that I know of was when Griffin used as equivalent coin based on EV
and variance to model blackjack games, but that approach is not exact
and cannot be used to exactly recreate the probability distribution of
the original game.

···

On Sunday 09 November 2003 02:38 am, Harry Porter wrote:

#5

Steve Jacobs wrote:

So, the complex probabilities associated with a VP
game can be reduced to the simplicity of a single biased coin!

Steve, this single quote is not intended to shortcut the logic of your
original post, or you comments in reply to my questions. It's simply
a quick point of reference for this post.

These days, it seems, I only spend more than 5 minutes digesting a
complex problem when monetarily motivated :). So I'm going to ask you
to post what essentially is a repeat of your original work, only in a
somewhat reverse sequence.

Step us through the translation of an appropriately biased coin to a
representation of 9/6 JB vp, if you will.

- Harry

#6

Steve, first of all, I think that your biased coin analogy, while
difficult to grasp, is quite interesting.

I assume, then, that the min-ROR strategy is not dependent on
bankroll size...that all bankrolls have the same min-ROR strategy.
So, given that this implies that there is only one min-risk strategy,
how can you find it? I would think that in many games the min-ROR
strategy would be the same as max-EV strategy, seeing there are only
discrete figures that the biased coin's value could be calculated
at. More improtantly, I believe that there would have to be a
different strategy for each game, based on the cash back and
percieved value of comps and bounce-back for a particular game.

#7

Steve, first of all, I think that your biased coin analogy, while
difficult to grasp, is quite interesting.

It certainly takes some getting used to. I'm sure that I don't understand
all of the consequences of this result, and that I'll be spending a lot
of time thinking about it more.

I assume, then, that the min-ROR strategy is not dependent on
bankroll size...that all bankrolls have the same min-ROR strategy.

That is true, but I don't think it is necessarily a consequence of this
equivalent coin result. The min-ROR strategy applies to each individual
wager. The overall probability of going broke is all that changes with
bankroll.

So, given that this implies that there is only one min-risk strategy,
how can you find it?

The fact that the min-ROR strategy doesn't vary with bankroll does
not imply that the min-ROR strategy is unique. It is possible to have
some plays where the two best alternatives contribute equally toward
risk.

As for finding it, that is something I haven't worked out yet. I've heard
that Yuri Sorkin has published an article on min-ROR strategies, but
I haven't seen the article.

I would think that in many games the min-ROR
strategy would be the same as max-EV strategy,

In general, min-ROR and max-EV will be different. There is one special
exception to this -- if the max-EV strategy happens to give a game that
is exactly breakeven, then the min-ROR strategy will be the same as
max-EV. The same will hold if the max-EV strategy is "close enough"
to breakeven, but this too would be unusual. I doubt that a game
like Jacks+ 9/6 has matching strategies, but DB 10/7 with an EV
if 100.17% might.

seeing there are only
discrete figures that the biased coin's value could be calculated
at. More improtantly, I believe that there would have to be a
different strategy for each game, based on the cash back and
percieved value of comps and bounce-back for a particular game.

Sure, different pay schedules will lead to different min-ROR strategies.

···

On Tuesday 11 November 2003 01:59 pm, tooncesthecatwhocoulddriveacar wrote:

#8

More improtantly, I believe that there would have to be a
> different strategy for each game, based on the cash back and
> percieved value of comps and bounce-back for a particular game.

Sure, different pay schedules will lead to different min-ROR

strategies.

I'm not sure if we are on the same track or not. This isn't a
trivial observation, seeing that a traditional Max-EV strategy does
not change based on the amount of cash back or comps associated with
it. However, a min-ROR strategy would have to change depending on
the Cash-back rate, bounce-back rate, and how much value you
associate to the comps.

#9

Steve Jacobs wrote:

Choosing a different strategy generally alters both the spinner and the
amount of bias in the coin, but it is coin bias alone that determines
bankroll requirements. For favorable games, the unit-cost of the coin
is numerically equal to risk of ruin, and risk of ruin is what matters
for bankroll requirements. For unfavorable games, the unit-cost of
the coin is equal to the inverse of the casino's risk of ruin, so
minimizing the unit-cost is the same as maximizing the casino's risk
of ruin.

I'm in the same boat as Harry. I haven't fully digested this, plan to
later, and suspect that there's an easy answer to my question, which
I'm not sure I'll phrase very well. Wouldn't the different numbers of
trials in the sub games have the same effect as a different game
would? In other words, even if the exact same coin can represent two
different games, wouldn't the risk-equivalent EV of different games
still be different because of the different numbers of spins in the
sub games?

#10

--- In vpFREE@yahoogroups.com, Tom Robertson <thomasrrobertson@e...>
wrote:

Steve Jacobs wrote:

>Choosing a different strategy generally alters both the spinner

and the

>amount of bias in the coin, but it is coin bias alone that

determines

>bankroll requirements. For favorable games, the unit-cost of the

coin

>is numerically equal to risk of ruin, and risk of ruin is what

matters

>for bankroll requirements. For unfavorable games, the unit-cost of
>the coin is equal to the inverse of the casino's risk of ruin, so
>minimizing the unit-cost is the same as maximizing the casino's

risk

>of ruin.

I'm in the same boat as Harry. I haven't fully digested this, plan

to

later, and suspect that there's an easy answer to my question, which
I'm not sure I'll phrase very well. Wouldn't the different numbers

of

trials in the sub games have the same effect as a different game
would? In other words, even if the exact same coin can represent

two

different games, wouldn't the risk-equivalent EV of different games
still be different because of the different numbers of spins in the
sub games?

I'm not sure what you are asking, but I do understand what Steve's
saying. If you are using the same coin, the values on the spinner
are truly irrelevant. That's the magic of the concept. Let's say
you spin an 8. That just means "Keep flipping the coin until you are
ahead 7 (8-1) or are behind 1". If you spin a 2, that means "Keep
flipping the coin until you are ahead 1 (2-1) or are behind 1". But
since every time you complete the subgame and do a new spin, you are
just back to the same flipping that you were doing before.

The cool part to me of this equivalence is the distinction between a
Max-EV strategy and a Min-ROR strategy. For example, a sample Min-
ROR coin is heads 51% of the time (2% advantage), but you get in 10
flips per subgame for an EV of .2 units/subgame. But in the sample
max-EV strategy (what we call basic strategy), the coin is only heads
50.5% of the time (a 1% advantage), but since we "go for the royal"
more often, and going for the royal takes more flips to resolve than
going for smaller numbers on the spinner, we can get in 30 flips per
subgame for an EV of 0.3 units/subgame. Since each subgame is
equivalent to a single hand of VP, you clearly make more money per
hand with a max EV strategy, but by using a less biased coin to do
so, you are more at risk for a big losing streak.

#11

I don't understand what you are asking, but I'll try to throw out some
information that might help. You can have games with different EV but
the same unit-cost (and thus the same equivalent coin). Also, it may be
possible to play the same game using different strategies that give different
EV while both strategies have the same unit-cost. It may also be possible
to have two different strategies with the same EV (but neither strategy being
the max-EV strategy) while the unit-cost is different for the two strategies.

···

On Wednesday 12 November 2003 11:37 am, Tom Robertson wrote:

Steve Jacobs wrote:
>Choosing a different strategy generally alters both the spinner and the
>amount of bias in the coin, but it is coin bias alone that determines
>bankroll requirements. For favorable games, the unit-cost of the coin
>is numerically equal to risk of ruin, and risk of ruin is what matters
>for bankroll requirements. For unfavorable games, the unit-cost of
>the coin is equal to the inverse of the casino's risk of ruin, so
>minimizing the unit-cost is the same as maximizing the casino's risk
>of ruin.

I'm in the same boat as Harry. I haven't fully digested this, plan to
later, and suspect that there's an easy answer to my question, which
I'm not sure I'll phrase very well. Wouldn't the different numbers of
trials in the sub games have the same effect as a different game
would? In other words, even if the exact same coin can represent two
different games, wouldn't the risk-equivalent EV of different games
still be different because of the different numbers of spins in the
sub games?

#12

That last part isn't quite right. The max-EV strategy uses a more biased
coin than the min-ROR strategy. More bias equates to more risk.

Another way to view this is to look at pushes. For favorable games, the
max-EV strategy uses a riskier coin and accepts fewer pushes, and the
greater number of pushes for the min-ROR strategy dilutes the edge of
the "better" coin. For unfavorable games, the max-EV strategy still uses
a riskier coin but dilutes the edge by accepting more pushes than the
min-ROR strategy. This way of looking at it is simpler because you don't
have to even think about all the complexities of the VP game, you just
worry about the coin. The effect of pushes is the alter the time perception.
Min-ROR isn't concerned with time at all, only with risk and efficiency.
This demonstrates the fundamental tradeoff between risk and EV. The
only games that allow risk and EV to be optimized simultaneously are
breakeven games.

···

On Wednesday 12 November 2003 01:13 pm, tooncesthecatwhocoulddriveacar wrote:

The cool part to me of this equivalence is the distinction between a
Max-EV strategy and a Min-ROR strategy. For example, a sample Min-
ROR coin is heads 51% of the time (2% advantage), but you get in 10
flips per subgame for an EV of .2 units/subgame. But in the sample
max-EV strategy (what we call basic strategy), the coin is only heads
50.5% of the time (a 1% advantage), but since we "go for the royal"
more often, and going for the royal takes more flips to resolve than
going for smaller numbers on the spinner, we can get in 30 flips per
subgame for an EV of 0.3 units/subgame. Since each subgame is
equivalent to a single hand of VP, you clearly make more money per
hand with a max EV strategy, but by using a less biased coin to do
so, you are more at risk for a big losing streak.

#13

I'm working on this, but it will probably be a few days before I'll have
time to post the full result. The equivalent coin for the 9/6 game has a
unit cost of about 1.000425 compared to 1.00185 for the 8/5 game.
This is because the 9/6 game is almost breakeven, which would give
a unit cost of exactly 1.0000000.

···

On Tuesday 11 November 2003 07:37 am, Harry Porter wrote:

Step us through the translation of an appropriately biased coin to a
representation of 9/6 JB vp, if you will.

#14

> Since each subgame is
> equivalent to a single hand of VP, you clearly make more money per
> hand with a max EV strategy, but by using a less biased coin to do
> so, you are more at risk for a big losing streak.

That last part isn't quite right. The max-EV strategy uses a more

biased

coin than the min-ROR strategy. More bias equates to more risk.

In this instance I am referring to a favorable game, which would
include games below 100% that offer enough CB or comps to
compensate. I would assume the bias on the coin in these games favor
the player and that a max-EV game would have a lower bias (i.e. a
bias closer to 1.0) then the min-ROR game.

On the other hand, if you are defining player biases as biases of
less than 1, than yes, a max-EV strategy increases the bias number.

As for thinking of ROR in an unfavorable game, I don't understand the
application. Especially if our goal is to not go broke, the
probability of eventually going broke in all non-favorable games is
100%. I guess you can think of it as hitting a win goal, than
quitting forever, in which it a useful construct, but not
exploitable, as only irrational risk seekers would be interested in
that game.

Another way to view this is to look at pushes. For favorable

games, the

max-EV strategy uses a riskier coin and accepts fewer pushes, and

the

greater number of pushes for the min-ROR strategy dilutes the edge

of

the "better" coin. For unfavorable games, the max-EV strategy

still uses

a riskier coin but dilutes the edge by accepting more pushes than

the

min-ROR strategy. This way of looking at it is simpler because you

don't

have to even think about all the complexities of the VP game, you

just

worry about the coin. The effect of pushes is the alter the time

perception.

Min-ROR isn't concerned with time at all, only with risk and

efficiency.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:

On Wednesday 12 November 2003 01:13 pm, tooncesthecatwhocoulddriveacar wrote:
This demonstrates the fundamental tradeoff between risk and EV. The
only games that allow risk and EV to be optimized simultaneously are
breakeven games.

#15

> > Since each subgame is
> > equivalent to a single hand of VP, you clearly make more money per
> > hand with a max EV strategy, but by using a less biased coin to do
> > so, you are more at risk for a big losing streak.
>
> That last part isn't quite right. The max-EV strategy uses a more

biased

> coin than the min-ROR strategy. More bias equates to more risk.

In this instance I am referring to a favorable game, which would
include games below 100% that offer enough CB or comps to
compensate. I would assume the bias on the coin in these games favor
the player and that a max-EV game would have a lower bias (i.e. a
bias closer to 1.0) then the min-ROR game.

Nope. Min-ROR is the same thing as minimum bias.

On the other hand, if you are defining player biases as biases of
less than 1, than yes, a max-EV strategy increases the bias number.

When I say "bias" I mean "deviation from 1" which is the same as
"deviation from a fair game".

As for thinking of ROR in an unfavorable game, I don't understand the
application. Especially if our goal is to not go broke, the
probability of eventually going broke in all non-favorable games is
100%.

True. That is why I try (but sometimes forget) to talk in terms of unit-cost
instead of ROR.

However, if the goal is to play not forever, but only until you reach some
fixed target bankroll, then the probability of failure is less than 100%, and
is controlled by unit-cost.

I guess you can think of it as hitting a win goal, than
quitting forever, in which it a useful construct, but not
exploitable, as only irrational risk seekers would be interested in
that game.

I don't think you have to be irrational to play unfavorable games. For
example, if you have $255 but need $256 to buy a plane ticket home,
and only unfavorable games are available, then playing a Martingale
will minimize your risk and maximize the probability of flying home
instead of walking.

Playing an unfavorable game indefinitely with an expectation of coming
out ahead would certainly qualify as irrational.

···

On Thursday 13 November 2003 09:29 am, tooncesthecatwhocoulddriveacar wrote:

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:
> On Wednesday 12 November 2003 01:13 pm, > > tooncesthecatwhocoulddriveacar wrote:

#16

Steve Jacobs wrote:

···

On Tuesday 11 November 2003 07:37 am, Harry Porter wrote:
> Step us through the translation of an appropriately biased coin to a
> representation of 9/6 JB vp, if you will.

I'm working on this, but it will probably be a few days before I'll
have time to post the full result. The equivalent coin for the 9/6
game has a unit cost of about 1.000425 compared to 1.00185 for the 8/5
game.
This is because the 9/6 game is almost breakeven, which would give
a unit cost of exactly 1.0000000.

Take you time, Steve. The enzymes are still churning away at
digesting the posts on this topic thus far.

By the way, running the translation on 8/5 would be just as useful. I
just need to reverse engineer this in my brain for it to have an easy
shot at penetrating.

- Harry