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What stakes are optimum

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stho...@armstronglaing.com

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Jun 4, 1999, 3:00:00 AM6/4/99
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hi BG'ers!

Has anyone done any analysis on what optimum stakes to use given:

1) account size (money saved allocated for backgammon)
2) skill estimation (for example, assuming head to head and a 53 - 47%
skill advantage)
3) head to head; or chouette play (say with 4 players)

any thoughts?


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Patti Beadles

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Jun 4, 1999, 3:00:00 AM6/4/99
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I don't know if such work has been done for backgammon in particular,
but this sort of thing is well understood for other games (e.g.
blackjack and poker.) Check out the gambling theory books on my books
page (http://www.gammon.com/books/) and you'll probably find some good
information.

If I had to guess, I'd say that you probably need a $1000 bankroll to
comfortably play heads-up at $5/point, and maybe $2500 for a
four-person chouette at those stakes. But that's just a SWAG.

-Patti
--
Patti Beadles |
pat...@netcom.com/pat...@gammon.com |
http://www.gammon.com/ | The deep end isn't a place
or just yell, "Hey, Patti!" | for dipping a toe.

Ftardieu

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Jun 5, 1999, 3:00:00 AM6/5/99
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hi,

you will find some mathematical responds and others gambling stuff in the
following books :

Inequalities for stochastic processes
how to gamble if yoou must

Lester E. DUBIN
Leonard J. SAVAGE

DOVERS BOOKS

The theory of gambling and statistical logic
Richard E. EPSTEIN

Academic PRESS

John Graas

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Jun 5, 1999, 3:00:00 AM6/5/99
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In Bell's "Backgammon: Winning with the Doubling Cube," he states you
should be able to lose 100 pts per player in a chouette for any one
session without it affecting your play.

jdg


pat...@netcom.com (Patti Beadles) wrote:

**** Remove _spamme_ from e-mail address to respond. ****

Patti Beadles

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Jun 5, 1999, 3:00:00 AM6/5/99
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In article <37598dbe...@news.thegrid.net>,

John Graas <jgraas_...@csi.com> wrote:
>In Bell's "Backgammon: Winning with the Doubling Cube," he states you
>should be able to lose 100 pts per player in a chouette for any one
>session without it affecting your play.

Interesting! That's "in one session", but doesn't really address the
long haul. It sounds like my 500 point bankroll for a four-player
chouette was low.

Does he support this number in any way?

-Patti
--
Patti Beadles |
pat...@netcom.com/pat...@gammon.com |

http://www.gammon.com/ | Try to relax
or just yell, "Hey, Patti!" | and enjoy the crisis

John Graas

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Jun 6, 1999, 3:00:00 AM6/6/99
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Support? Not really.

His guideline is built around the idea of being able to make the
correct take/double decision at any cube level within the session
without money pressure getting in the way. The idea being not to give
up any equity over the long run due to being under captitalized.

For any one session, your bankroll looks fine. For the long haul, I'd
at least triple it: if you drop 400 pts twice in a row, you are
clearly out of your league, and you need to re-evaluate. And you'll
still have two sessions' worth remaining when you half the stake
you're willing to play for (realising that you are out of your
league).

Going back to my 21 counting days, if you had a 1% advantage over the
house, you needed a stake of 200 average bets to give yourself a 95%
(2 std deviations) chance of doubling you money before losing it all.
Twice this -- 400 bets -- for a long-run bankroll. At no advantage
(counting for comps), you needed a bankroll of at least 1,000 bets to
be reasonably assured of avoiding "gambler's ruin."

I _guess_ that the "average bet" for a bg game is a 2-cube (anybody
have the real number). Then, for example, every 4th game of a 4-way
chou, you're averaging an effective 6-cube. Assuming 50% win rate in
the box, then you looking at two 6-cubes and three 2-cubes every five
games: an average of a 3.6 cube. :-)

So... are we talking 3,600 pts for a long-run bankroll? Assuming even
competition? Sounds a bit high. Are we talking a 1,440 pt long-run
bankroll if you have a slight advantage? Sounds reasonable, but maybe
a bit conservative.

As for the long-haul, Keynes (economist) is often quoted as saying,
"In the long-run we will all be dead."

jdg

pat...@netcom.com (Patti Beadles) wrote:

**** Remove _spamme_ from e-mail address to respond. ****

Chuck Bower

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Jun 9, 1999, 3:00:00 AM6/9/99
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In article <7j91hk$ida$1...@nnrp1.deja.com>, <stho...@armstronglaing.com> wrote:

>hi BG'ers!
>
>Has anyone done any analysis on what optimum stakes to use given:
>
>1) account size (money saved allocated for backgammon)
>2) skill estimation (for example, assuming head to head and a 53 - 47%
>skill advantage)
>3) head to head; or chouette play (say with 4 players)
>
>any thoughts?

Sure! (Any SOUND answers? That may be tougher. :) And since
I haven't posted in two and a half months, I need to compensate by
writing a LONG one.

This is an interesting (IMO) topic. I don't know of any detailed
work that has been done on this subject--money management for head-to-
head (and choutte) backgammon play. I can imagine that a computer
simulation could be developed which would help answer this question
in greater detail than what I am going to present BoE. (I believe
John Graas's post was along a similar vein. I'm just more verbose. :)

One money management technique which HAS been studied fairly
extensively is known as the "Kelly Criterion". There is a web page
(which appears to have originally been a newsgroup post):

http://www.primenet.com/~jaygee/KELLY.HTM

This page is definitely more mathematical than what I am going to
present here, but seems to make sense (on my initial, quick reading).

First some sketchy history (and, as usual, from memory so I
may have messed up some of the details): In the 1950's, a researcher(s)
for Ma Bell (probably Bell Labs) wrote journal article about signal
routing. I believe it was pub'ed in one of Ma Bell's own journals.
The author (or one of the authors) was named 'Kelly'. Someone who
read the article (sorry, no name here) realized that the paper's
contents actually applied well to money management in some gambling
situations (blackjack)? Starting in the early 60's, Edward O. Thorp
(of "Beat the Dealer" and "Thorp(e) Count" fame) made the Kelly method
popular as a money management technique for casino blackjack (assuming
intelligent card counting practice).

Although I don't know if it has ever been proven, Thorp (see
for example, his book "Mathematics of Gambling", 1984, Lyle Stuart
publisher) contends that the Kelly Criterion is an optimal technique
for making money as fast as possible under the condition that you are
guaranteed never to blow your entire bankroll. There are some
conditions on the Kelly method in it's strictest application:

1) player MUST have an edge!
2) player has the option of varying his bet amount at each new
opportunity (e.g. before each round of cards is dealt).
3) during a particular trial (e.g. during a particular hand of
cards) the amount bet does not change.
4) bets are made "even money"; that is, you are paid the same for
a win as you would have to shell out for a loss. (NOTE: there
is an expanded Kelly System which takes into account odds being
offered, e.g. for horse racing. Although the enhancement is
simple, I don't think it applies to my model for backgammon
below so I'm not including it. See the above WWW page for
details of this enhancement.)

Before getting specifically to backgammon, let's create an
(artificial) example. (Note: this example is a modification of
one in Thorp's "Mathematics of Gambling" book.)

Suppose an HONEST (but maybe not particularly bright) person approaches
you with the following proposition:

You place a bet (size your choice). You roll a fair die. If it comes
up 1,2,3, or 4, he gives you the amount of your bet. If it comes up
5 or 6, you lose the amount of your bet.

He will play for the next four hours (or less, if YOU decide to quit).
You have $90 in your pocket. How much should you bet on each roll?
(I should also say that YOU are honest, meaning you won't bet more than
is in your pocket!)

Hopefully you realize that betting EVERYTHING each turn is likely
to end in ruin for you. But by betting very conservatively you'll
blow a chance to make a lot of money. What is the best compromise
so that you won't go broke (and have to quit such a lucrative proposition)
but still will rake in a large profit?

The simple ("oddsless") Kelly Criterion says you should bet a
percentage of your current bankroll which is equal to your percentage
edge. In the above example, you are a 2::1 favorite on each roll.
On average you will win 2/3 of the tosses and lose 1/3. Take the
difference (2/3 - 1/3) and that is your 'edge'--1/3. So you should
wager 1/3 of your current bankroll at every opportunity.

Note that you must CHANGE the amount bet on every new opportunity.
Let's take a hypothetical sequnce:

Bankroll Bet result

90 30 lose
60 20 win
80 26.67 win
106.67 35.56 win
142.23 .......................

By lowering your bet as your bankroll decreases, you insure that you
never go broke. But by raising it as your bankroll increases, you
give yourself the maximum opportunity for growth.

OK, back to reality--backgammon. First we see that backgammon
doesn't fulfill the strict conditions stated above. For one, a game
starts off worth a point but usually ends up being worth more than
that. The value of any given (money) game is unknown before it
begins. Secondly it is not customary in a BG money session for the
stake to change, and certainly not every game! Typically two players
agree beforehand on a stake and it remains that way thoughout the
session (barring certain kinds of 'steaming', of course).

This 'unknown' value of each game enters my money management
technique in two ways. First, we can assign a typical value to a
game. This has been discussed on the newsgroup before, and a good
number to use is '3'. That is the standard deviation for money play
for Jellyfish, and "not too loose, not too tight" humans as well.

The second place the value uncertainty enters is what I will
call an 'escrow'. I'm borrowing this term from finance, but most
likely I'm abusing its accepted meaning. (Sorry, bankers among you.)
Consider the MAXIMUM amount you are likely to lose on a single game
over MANY sessions. Experience enters here. I'll take myself as
an example. (Note that I tend to be conservative compared to your
typical money player in handling the cube.) In my lifetime, I'd
guess I've played around 20,000 'money' games of backgammon, head-
to-head and choutte. I only recall the cube reaching 32 twice.
16 is a rarity. So in several money sessions, the worst I can
imagine is seeing a 16 cube accepted. I set my escrow at twice that
= 32. If my bankroll EVER gets less than 32 (my escrow), I must stop
playing (or go to the bank machine...).

Next I need to estimate my percentage edge. Thanks to the
BG ratings formulas and online servers, this is a lot easier than it used
to be. I just need to know (or estimate) my opponent's online rating
and my own. The differece tells me my edge:

(see Kevin Bastian's page: http://www.northcoast.com/~mccool/fibsrate.html)

Ratings difference: Edge in a single game

50 3%
100 6%
150 9%
200 11%

(I assumed 1-point matches. Note that the relationship between rating
difference and edge is close enough to linear that interpolation is
reasonable.)

I think we have enough info to now speculate on a bankroll size
given a known stake:

bankroll = (escrow + 300/edge) * stake

where 'edge' is in percent, 'escrow' in points, and 'stake' and 'bankroll
in some appropriate monetary units.

Let's take me as an example (so 'escrow' = 32 points). Say I want
to play $5 per point against someone I estimate to be 50 ratings points
weaker than myself:

bankroll = (32 + 300/3) * $5.
= $660.

OK, ready to try a chouette? Let's assume n total players so a
maximum (when in the box) of n-1 opponents. How does this affect your
escrow? You must multiply it by n-1. And how about your "base amount"?
1/n of the time you are playing for n-1 times the stake (per point)
and (n-1)/n of the time you are playing for a single stake. So your
'average' stake in a chouette is 1/n * (n-1) + (n-1)/n * 1 = 2(n-1)/n.
You must multiply the 300/edge term in the above equation to apply
the formula to choutte's:

chouette bankroll = [(n-1)*("1-on-1 escrow") + 600*(n-1)/(n*edge)] * stake

Again, suppose I get in a chouette where I am 50 ratings points better
than the BEST of my opponents (i.e. assume you are ALWAYS playing the
best player), for a $5/point session with a total of four players, I
should have:

chouette bankroll = [3*32 + 600*3/(4*3)] * $5
= $1230.

For those who have read this far, I suspect 95% are going to
say "you're nuts! I get into money games all the time with nowhere
near this much cushion." And I believe you. And maybe my numbers
are completely worthless. On the other hand, how often do you have
to resort to "IOU's" or writing checks, or getting out of the game
prematurely because of an uncomfortable losing streak? And have
you ever conciously (let alone unconciously) changed your doubling
strategy because the cube was getting too high for your (payability)
comfort?


Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS


Patti Beadles

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Jun 9, 1999, 3:00:00 AM6/9/99
to
Chuck, I applaud your work! You've done an excellent job of wrapping
backgammon and Kelly betting together.

But I do question one of your assumptions:

> This 'unknown' value of each game enters my money management
>technique in two ways. First, we can assign a typical value to a
>game. This has been discussed on the newsgroup before, and a good
>number to use is '3'. That is the standard deviation for money play
>for Jellyfish, and "not too loose, not too tight" humans as well.

I don't think it's that simple, unless you're inclined to settle any
time you get a four cube or higher. Otherwise, you aren't really
accountin for the swings that you'll get when you get gammoned on that
eight cube.

Of course, we could be discussing apples and oranges here. I'm
talking about total bankroll, and you seem to be talking about session
bankroll.

-Patti
--
Patti Beadles |
pat...@netcom.com/pat...@gammon.com | You are sick. It's the kind of
http://www.gammon.com/ | sick that we all like, mind you,
or just yell, "Hey, Patti!" | but it is sick.

David Montgomery

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Jun 10, 1999, 3:00:00 AM6/10/99
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I asked Danny Kleinman about applying the Kelly criterion to backgammon
in a letter to the Chicago Point in the January 1989 issue. Kleinman
wrote that Michelin Chabot had written two books applying "Kelly theory" to
backgammon, but Kleinman disparaged the books and didn't give any
references. Kleinman then admitted that he didn't know about the Kelly
criterion and went on to suggest "stakes low enough to absorb a 200
point loss without emotional ruin." This isn't horrible advice, but
with Kelly you can do much better.

Let r be a random variable, the result of a heads-up backgammon
game. Let's say we have the probability distribution for r.
Then the Kelly criterion is that we should set our stakes at
proportion p of our total bankroll each game, where p maximizes
E[log(1+pr)]. If we bet this way, we achieve the greatest expected
bankroll growth in the long run.

An approximation for p that is often used is E[r]/E[r^2].

For example, let's say you have a .1ppg advantage over your opponent,
and that the variance of the single game result is 10. Then this
approximation says that you should set the stakes at .1/10=.01, or
1% of your total bankroll. If your bankroll is $1000, you do best
to play the next game for dimes.

A few months back I wrote a program that both calculates the Kelly
approximation and explicitly maximizes E[log(1+pr)], given r's
probability distribution.

For modest edges and/or decent variances, such that the approximation
indicates p < .015, the approximation was quite accurate. So if you
have a good idea of your edge E[r] and the variance E[r^2] (neither is
too hard to estimate), and if E[r]/E[r^2] < .015, then you can fairly
easily get a good estimate of what your optimum stake size is.

When the approximation indicates a bigger p, it could be substantially
too high. Playing around with my program I saw approximation values over
.04, but the p that maximized E[log(1+pr)] was in those cases a little
over .02. The possibility of extreme results in backgammon, like 16
and 32 cubes, makes it inavisable to set the stake size as high as the
approximation suggests. Doing so would make it too likely that you
might suffer a big drawdown, after which you would not be able to earn
as much.

(All of this assumes that we can bet exactly proportion p every game,
which in fact we cannot. Because of the inability to fine-tune the
stakes after every game, p should almost certainly be somewhat lower
than predicted by the Kelly criterion.)

Chris Yep, Michael Klein, and Gary Wong gave me a lot of help in
figuring out the Kelly criterion and what it means for backgammon.
Thanks guys.

If anyone has a reference for the Chabot books Kleinman referred
to, please let me know. I would very much like to read them.

David Montgomery
mo...@cs.umd.edu
monty on FIBS and GG


Murat Kalinyaprak

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Jun 10, 1999, 3:00:00 AM6/10/99
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Chuck Bower wrote news:7jmd7g$s7e$1...@flotsam.uits.indiana.edu...

>OK, back to reality--backgammon. First we see that backgammon
>doesn't fulfill the strict conditions stated above. For one,
>a game starts off worth a point but usually ends up being worth

>more than that. The value of any given (money) game is.....

I enjoyed reading your above extensive comments. Lately
I had been seriously considering playing against JF and
SW for money but I have also been afraid that there may
be more to gambling than to just playing for fibs points
or even for nothing, against robots in one's own privacy.
A few times it crossed my mind to post an article in rgb
to ask if somebody would be willing to partner up or do
some hand-holding/coaching to ease me into the gambling
world. Of course, there may be an element of personality
required and being coached, reading books on the subject,
etc. may not help...

>Next I need to estimate my percentage edge. Thanks to the
>BG ratings formulas and online servers, this is a lot
>easier than it used to be.

I think you should have made this comment with a big "IF".
For all the reasons offered before by several people, such
ratings are mostly useless. As part of my "experiments":),
I deliberately lowered my FIBS rating to 1400's. The other
day I invited a 1640(?) rated player and was turned down as
well as put down at the same time. After an exchange of some
"friendly" :) words with that person, I was offered to play
for $5 per point. Despite how I feel about gambling playing
bg, if that person was in front of me with his cash money
on the table, I'm pretty sure I would have given in to the
temptation... :) I feel it's my duty :) to remind that on
top of lack of necessary control and policies, popular but
bogus formulas like FIBS' and their products (ratings) are
nothing but useles...

MK


Chuck Bower

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Jun 10, 1999, 3:00:00 AM6/10/99
to
In article <pattibFD...@netcom.com>,
Patti Beadles <pat...@netcom.com> wrote:

>Chuck, I applaud your work! You've done an excellent job of wrapping
>backgammon and Kelly betting together.

Thanks!

>But I do question one of your assumptions:
>

>> This 'unknown' value of each game enters my money management
>>technique in two ways. First, we can assign a typical value to a
>>game. This has been discussed on the newsgroup before, and a good
>>number to use is '3'. That is the standard deviation for money play
>>for Jellyfish, and "not too loose, not too tight" humans as well.
>

>I don't think it's that simple, unless you're inclined to settle any
>time you get a four cube or higher. Otherwise, you aren't really
>accountin for the swings that you'll get when you get gammoned on that
>eight cube.

I agree that setting a bankroll--stake realtionship is not as
simple as my model made it sound. However, I don't understand the
part about "inclined to settle any time you get a four cube or higher".
In the example I gave, my bankroll vs. a player rated 50 points below
me was 132 units (points). Thus assuming I was even on all other games,
I could handle FOUR 24-point losses before getting close to my escrow.
As I mentioned, for MY play just one of these would be extremely rare.
If you see these fairly often (e.g. maybe once per session), then
probably the S.D. for YOUR play (and, let's say, against that particular
opponent/chouette) is higher than 3 and you should adjust accordingly.
In addition, your escrow should be higher.

>Of course, we could be discussing apples and oranges here. I'm
>talking about total bankroll, and you seem to be talking about session
>bankroll.

Hmmm. I didn't realize there should even be a distinction. I'll
have to think about that one.

On another note, having now read David Montgomery's post, I see
that a lot more work has been done on this subject than I thought.
Appartently, though, very little has been published, or at least
made available in form that is readily accessible. I think David's
numbers and mine agree pretty closely, which does give me SOME
confidence that what I said wasn't totally off-base.

Probably the biggest impediment to using my simplified model of
applying the Kelly Criterion is that typically you don't (can't?) change
the stake after each game. This hurts in two ways: it makes you more
likely to go broke (when you are losing) and it doesn't allow you to
maximize your earnings (when you are winning). This makes me think
there is probably a better money management scheme.

And maybe this is where the "total vs. session bankrolls" idea enters.
If you have money in reserve (total > session) then you have a chance
to adjust your bet size on the NEXT session. As in the true Kelly
method, this helps both when you are winning and when you are losing.
Besides, if you are getting hammered, maybe you underestimated your
opponent....

Finally, using the Kelly Criterion (or some other money management
optimization method) is really related to maximizing profit. If you
are just sitting down to a friendly game, that might not be your
primary goal. In that case, the size of your bankroll (or stake) may
be determined by other factors.

David desJardins

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Jun 10, 1999, 3:00:00 AM6/10/99
to
Chuck Bower <bo...@bigbang.astro.indiana.edu> writes:
> I agree that setting a bankroll--stake realtionship is not as
> simple as my model made it sound. However, I don't understand the
> part about "inclined to settle any time you get a four cube or
> higher".

The central limit theorem says that if you add up a large enough sample
from a suitable distribution, then the distribution of the total will be
roughly normal. It's not clear that this applies to backgammon, because
you can have arbitrarily large payoffs and it's not even clear that the
expectation and variance exist. But even aside from that, it's fairly
tricky to tackle the question of what is "large enough". It's certainly
the case that, for a distribution with "thick tails" such as backgammon
payoffs (because paying 16 points for getting gammoned at 8 would be a
5-sigma event with probability less than one in a million, if the payoff
distribution were normal with standard deviation 3, the probability of
large payoffs at backgammon is much larger than in the normal
approximation), the "large enough" that you need for the sum to converge
to the normal distribution is increased.

> In the example I gave, my bankroll vs. a player rated 50 points below
> me was 132 units (points). Thus assuming I was even on all other games,
> I could handle FOUR 24-point losses before getting close to my escrow.
> As I mentioned, for MY play just one of these would be extremely rare.

However rare it is for you, it's a lot less so than the normal
approximation would say.

David desJardins

Jim Williams

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Jun 11, 1999, 3:00:00 AM6/11/99
to

David Montgomery wrote in message <7jnkdc$o...@krackle.cs.umd.edu>...

>Let r be a random variable, the result of a heads-up backgammon
>game. Let's say we have the probability distribution for r.
>Then the Kelly criterion is that we should set our stakes at
>proportion p of our total bankroll each game, where p maximizes
>E[log(1+pr)]. If we bet this way, we achieve the greatest expected
>bankroll growth in the long run.
>
>An approximation for p that is often used is E[r]/E[r^2].
>
>For example, let's say you have a .1ppg advantage over your opponent,
>and that the variance of the single game result is 10. Then this
>approximation says that you should set the stakes at .1/10=.01, or
>1% of your total bankroll. If your bankroll is $1000, you do best
>to play the next game for dimes.

This is interesting. I suppose that your cube decisions should be
based on maximizing E[log(1+pr)], not based on maximizing E[r]. This
would mean getting more cautious as the cube gets bigger (no doubt
a sensible idea for anyone on a finite bankroll). I suspect that if
doubling is based on maximizing E[r], then E[r^2] would diverge
to infinity.
An interesting side effect is that the person with the larger bankroll
actually gains some equity from this effect, but probably not too much
if the stakes are reasonable.

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