Kelly Criterion, sometimes known as the Kelly formula or
strategy can be used to help sports bettors determine the optimal size of bet
to make, in relation to long term bankroll management.
Part I -- Expected Value vs. Expected Growth
A question I'm often asked is how exactly expected value
differs from expected growth. The difference is somewhat subtle but
understanding it is essential to risk management in general and the Kelly
criterion in particular.
.jpg "How do you manage your bankroll?")
The question frequently arises in the context of the idea
that betting one's entire bankroll implies -100% bankroll growth. (That's 100%
bankroll shrinkage -- a bankroll that shrinks to $0.) What's more, if you bet
your entire bankroll in one go, 100% bankroll shrinkage is implied regardless
of both the probability of the bet winning (as long as it wins less than 100%
of time) and the odds paid out on the bet (as long as the betting odds are less than
infinity).
Think about that for a moment, because it's an important
point: If when you bet you wager your entire bankroll each time then you expect
your bankroll to eventually shrink to zero.
Well, at least it should be important, but the truth is it
doesn't really get us any closer to understanding what exactly bankroll growth
is and how it differs from expected value. We’ll get back to this later.
Let's start with a brief review of expected value.
The notion of expectation is central to probability and
statistics and may be thought of as an average with an extra syllable. If you
were to flip a coin 10 times then you could expect it would land on heads 5
times and you could expect it would land on tails 5 times. In reality of course
the coin’s not always going to land on heads exactly 5 times out of 10 (in fact
it would only do so about 24.6% of the time), but if you were to repeat the
experiment (flipping a coin ten times) many, many times over then on average it
would land on heads 5 times each trial.
The same thought process is also applicable to sports. If
the Yankees can be expected to win a particular game (say the first of their series with the Red Sox) 60% of the time, then this
would mean that if the exact same game were repeated under the exact same
conditions across many, many parallel universes, we would expect the Yankees to
win 60% of those encounters.
So let’s say you bet $1 straight up that the Yankees are
going to win that game. Now that’s quite obviously a good bet. But just how
“good” is it?
That’s where expectations come in with sports betting. If
you made the same bet in each of those parallel universes you’d win $1 60% of
the time, and lose $1 40% of the time. Now let’s say that there are actually
1,000,000 of these universes. Exactly how much money would you make? Well, in
600,000 of those universes you’d make $1 for a total of $600,000 dollars, and
in the remaining 400,000 of those universes you’d lose $1 in each game for a
total of $400,000 dollars. So you'd receive $600,000 and would pay out $400,000
meaning that your total profit would be $200,000. Winning $200,000 across
1,000,000 means on average you would have won $200,000 / 1,000,000 games = $.20
per game.
Now of course 1,000,000 is just a made up number in this
context. There aren’t really 999,999 other universes where we could make such a
bet. This bet can only be made once. But that doesn’t actually matter in the
world of statistics. Whether you can make this bet only one time or you can
make it multiple times the expectation per game is precisely the same, namely
20%.
So to summarize, the expected value of a bet is the amount
we would receive on average if we were to repeat the exact same bet a very
large number of times. As such, the expected value of a bet is a metric by
which one might judge the relative attractiveness of that bet. If one bet has
an expected value of 5% (meaning that for every $10,000 we bet we would expect
to win $500) and another has an expected value of 10% (meaning that for every
$10,000 we bet we would expect to win $1,0000), then we would tend to think
that one would prefer the latter bet to the former.
But there's a bit of a difficulty here -- namely, expected
value ignores any consideration of the relative likelihoods of given outcomes
alone. For example a $10,000 bet on a 0.0000000000000000000000000000000000001%
likelihood event paying out at +110,000,000,000,000,000,000,000,000,000 ,000,000,000,000
odds corresponds to an expected value of 10% (+$1,000). But who among us would
be willing to essentially throw away $10,000 on such a long shot? To put it in
perspective you'd be about 1,870 times more likely to win the New Jersey State Lottery
five times in a row, than you would be to win this particular bet. Does it
really matter that if by some fluke of nature you actually did win you'd have
an unfathomably huge amount of money? If you're like most people, the answer is
probably not.
So now here's the difficulty ... there's no way whatsoever
to account for this very real phenomenon of preferences by appealing to the
theory of expected value alone.
(Enter stage right, expected bankroll growth.)
One major problem with the proposed bet is that for most
people, $10,000 represents a rather large chunk of one’s bankroll to be
throwing away on a bet that’s nearly certain to lose. But while a $10,000 bet
is probably too large a quantity to risk on this bet, there’s still a
sufficiently small dollar amount that most people would be willing to risk to
make this bet. Granted, for most people that dollar amount would be somewhere
in the neighborhood of a tiny fraction of a penny, but it nevertheless would still
be a positive dollar amount.
The fundamental issue with bets such as these is that,
despite being positive EV, placing them is an excellent way to go broke. The
apparent contradiction is easily reconciled. If you were to repeat this bet
once in each of a gigantically huge number of parallel universes, in nearly all
of the universes you’d lose your bet, but in a tiny, tiny, tiny, tiny, tiny
fraction of those universes you’d have won the bet and that win quantity would
make up for all the losses plus an additional 10% of the amount risked.
The fact is that most people just aren’t willing to live
through billions of trillions worth of bets just to have a vanishingly
minuscule probability of winning a huge odds bet once. So while the bet may
have positive expected value, the expected outcome is for your bankroll to
shrink by $10,000 each time the bet’s made. If your bankroll were $1,000,000
and you made the bet 100 times, you could expect to be broke after the 100th
bet (even though your expected value would be 10% × $1,000,000 = +$100,000).
So let’s look at some more practical numbers. Assume you’re
considering a bet that wins with 50% probability and pays out at odds of +200.
Further assume your total bankroll is $100,000 and that you want to place 1% of
your bankroll on this wager.
Question: Where do you expect your bankroll to be after 2
wagers?
Answer: There are 4 possible outcomes after placing two
wagers:
1. Win both
bets.
2. Win 1st
bet, lose 2nd bet
3. Lose 1st
bet, win 2nd bet
4. Lose both
bets
Now because winning and losing the bet are both equally
likely, all 4 outcomes occur with equal probability, namely 25%. Recall that
you’d be betting 1% of your bankroll on each bet and would be paid off at odds
of +200. Therefore, your ending bankroll under each of the 4 outcomes would be:
1. B =
$100,000 × (1 + 2×1%) × (1 + 2×1%) = $104,040
2. B =
$100,000 × (1 + 2×1%) × (1 - 1%) = $100,980
3. B =
$100,000 × (1 - 1%) × (1 + 2×1%) = $100,980
4. B =
$100,000 × (1 - 1%) × (1 - 1%) = $98,010
(The derivation of these equations is simple. Every time you
win your bankroll would grow to 102% of its previous value, and every time you
lose your bankroll would shrink to 99%.) The expected value from betting in
this manner would be 25%×$104,040 + 25%×$100,980 + 25%×$100,980 + 25%×$98,010 =
$101,002.50. To calculate expected growth, we would first need to recognize
that given our 50% win probability, our expected outcome would be to win a bet
and to lose a bet (# of wins = 50% × 2 bets, # of losses = 50% × 2 bets). Therefore
our expected growth would be that associated with that outcome (with expected
growth, the relative ordering of wins/losses is irrelevant), namely $100,980.
Therefore, the expected value from the two bets is $1,002.50
or 1.0025%, and the expected growth is $980 or 0.9800%. Notice that expected
value is higher than expected growth -- this is what you’re always going to
see. Expected value will always be higher than expected growth (except for
probabilities of 0 or 100%, we’ll they’ll be equal) because a few relatively
large, relatively uncommon outcomes will increase EV. Another way to think
about this is by realizing that the worst case scenario is losing everything
one time over., while the best case scenario would be winning your bankroll
infinity times over – in other words you while your maximum possible profit is
unlimited, your maximum possible loss is limited to your bankroll.
So in this instance our expected outcome would be a bankroll
of:
B* = $100,000 × (1 + 2×1%)2×50% × (1 - 1%)2×50% = $100,980,
implying expected bankroll growth ofE(G) = $100,980/$100,000 = 0.9800%
It should be readily apparent our expected outcome after n
bets would be a bankroll of:
B* = $100,000 × (1 + 2×1%)n×50% × (1 - 1%)n×50% = $100,000 ×
(100.48881%)n, implying expected bankroll growth of E(G) = (100.48881%)n -1.
By extension, our expected outcome after just 1 bet would
be:
B* = $100,000 × (1 + 2×1%)50% × (1 - 1%)50% = $100,488.81
And our expected bankroll growth would be E(G)= (1 +
2×1%)50% × (1 - 1%)50% - 1 =
0.48881%
(This last result bears a little discussion. We can talk
about expected growth after only 1 bet in the same manner as we can talk about
expected value after just one bet. In the same way as we’d never see a real
result equal to our expected value, we’d never actually see growth after one
bet equal to expected growth. This should cause absolutely no concern.)
So let’s generalize our results with expected outcomes and
growth.
Given a starting bankroll of B0, decimal odds of O, a win
probability of p, and a bet size of X (as a percentage of starting bankroll,
B0), the bankroll associated with the expected outcome from placing the bet
would be:
B* = B0 * (1 + (O-1) * X)p * (1 - X)1-p
And expected growth would be:
E(G) = (1 + (O-1) * X)p * (1 - X)1-p-1
Expected value, you’ll recall, would be:
EV = p*(O-1)*X - (1-p)*X = (pO - 1)*X
Q: So let’s look at a concrete example: What are the
expected value and bankroll growth associated with a bet equal to 1% of
bankroll paying out at -110 and winning with probability 54%?
A: EV = 1% × (54% × 1.909091 - 1) = 0.03091% of bankroll
E(G) = (1 + 0.909091 × 1%)54% × (1 - 1%)46% - 1 = 0.02638%
bankroll growth.
Q: Now let’s consider the same terms, but in the case of a
player placing a bet 25% of bankroll. What would expected value and growth be
in this case?
A: EV = 25% × (54% × 1.909091 - 1) = 0.7727% of bankroll
E(G) = (1 + 0.909091 × 25%) 54% × (1 - 25%) 46% - 1 =
-2.1510% bankroll growth = 2.1510% bankroll shrinkage
So think about these results for a moment. We have a
positive expectation bet and hence, quite naturally, the more we bet on it the
more we expect to make. However, if we were to wager too much on this bet then
we’d expect our bankroll to shrink by 2.1510% per wager (were we to place this
positive expectation bet 32 times, for example, we’d expect our bankroll to
depreciate roughly a half).
So this should help elucidate the huge odds bet above. No
matter how positive EV a bet might be, if you bet too much on it then you
expect your bankroll to shrink. This is the concept to which people are
referring when they talk about "money management". Even if you could
pick NFL spreads at 75% (which you can’t), were you to bet too much, you'd
expect to head towards bankruptcy.
So as a limiting case let’s look at one more example, the
example of betting one’s entire bankroll mentioned at this start of this article:
win probability = p, bet size = 100% of bankroll.
EV = 100% × (pO - 1) = pO – 1 (EV > 0 for p > 1/O)
E(G) = (1 + (O - 1)) p × (0) 1-p - 1 = -100% (for p < 1
and O < 8)
So what does this tell us? Well for one thing it tells us
that even if you were the “best handicapper ever”, were you to risk your entire
bankroll on every bet then you would expect to go broke. More generally, it
illustrates the concept that looking solely at expected value as a metric for
the attractiveness of a given bet is not the proper way to maintain long term
growth.