Forum Archive : Miscellaneous

Money management and the Kelly Criterion

From:   Stuart Thomson
Address:   sthomson@armstronglaing.com
Date:   4 June 1999
Subject:   What stakes are optimum
Forum:   rec.games.backgammon
Google:   7j91hk$ida$1@nnrp1.deja.com

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?

Chuck Bower  writes:

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".

     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:

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

     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)] *

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)

       c_ray on FIBS

David Montgomery  writes:

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 .1 ppg 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

Chuck Bower  writes:

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

     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  writes:

Chuck writes:
>      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.

Patti writes:
> 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
> accounting for the swings that you'll get when you get gammoned on that
> eight cube.

Chuck 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.
Did you find the information in this article useful?          

Do you have any comments you'd like to add?     



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