Advantage is a bogus statistical term.  I've said it many times, and
many times Steve Jacobs has jumped in to defend this abomination of
gambling mathematics.  I recently reviewed Steve Jacobs' post from
a year ago, and it didn't do anything to change my mind.

Advantage sprang into being, I believe, in the analysis of games
with even-money payoffs and flat wagers or at least independent
trials.  For example, saying that the advantage of the pass-line
bet in craps is -1.4% makes sense.

Advantage is defined as E(x)/E(w), this is, the expected outcome
divided by the expected wager.  It should be intuitively obvious
why, though it works for even money payouts, advantage is an
arbitrary bogus term in general.  The problem is that the wager,
the amount placed in the betting square or wherever, has at best
a loose relationship with the expected outcome and the risk
in the general case.  I will illustrate this with two examples.

Example #1.  It's a classic coin-toss: you bet w, and I
pay you w if you call it correctly.  You know the coin
is biased so that it lands on heads 1.5% more often than tails.
Your advantage is E(x)/E(w)=0.515*w-0.485*w/w=3% for betting
on heads.  No problem here.

Example #2.  Frontier messes up its keno payouts again, allowing
a positive expected value bet.  You can play every possible
combination to reduce the variance to zero.  If you bet w,
you are guaranteed to win back w plus 2% of w profit.
Your advantage is E(x)/E(w)=2%w/w=2%.

The first example has a higher advantage, 3% compared to 2%, yet
there is no rational utility function that would prefer the
situation of the first example.  Some risk-loving (irrational)
utility functions would prefer the first case, if you
are not allowed to wager any amount of money.  If you can place
any size wager, then obviously, any utility function that did
not *dislike* money would favor the second case, as you could
simply bet an infinite amount (either all at once or over an
infinite number of games) and be guaranteed to win an infinite
amount.

Here's another set of examples.  For all these examples,
50% of the remaining cards are considered to be 10's.
The player has blackjack versus dealer ace.

Example A: Player does not insure his blackjack.  His advantage
for the round is E(x)/E(w)=0.5*1.5*w/w=75%

Example B: Player insures his blackjack.  His advantage for
the round is E(x)/E(w)=(0.5*(1.5*w-0.5*w)+0.5*(0.5*w*2))/(w+0.5w)
=1/(1.5)~=67%

Example C: Player takes even money for his blackjack.  His
advantage for the round is E(x)/E(w)=w/w=100%.

So the "best" choice is even money, followed by no insurance,
followed by insurance.  But wait, we know that even money and
insurance are equivalent.  Contradiction.  Bogus statistic.

As you can see, I'm directing my attacks at the fact that
the wager in the advantage term does not necessarily have
any risk associated with it.  Some of you may object to this,
saying that a wager with no risk should not be considered
a wager at all for the purposes of advantage.  That is a
valid point, but then what if the wager has a 
small risk?  For example, what if there is a 1 in 100,000
chance that Frontier will cheat and steal some of all of
your Keno wager.  Then we cannot ignore the wager totally,
but including the whole wager in the advantage term leads
to the general conclusions above.

In blackjack, when we card count, we place different sized
wagers in different situations with different distributions
of outcomes.  Advantage does not properly take this into
account, pretending as if we were just flat betting a game
with an even money payoff.

The other reason I do not like advantage is that it
encourages one to take bets in excess of any arbitrary multiple
of the proper Kelly bet.  If you bet more than 2 times Kelly,
your bankroll is doomed to repeatedly crash arbitrarily close to
zero.  In order to avoid this fate, you must have different
information than advantage.

The use of advantage in blackjack books is a travesty,
especially when it is used as a metric to compare different games.
What I prefer to see is expected value and variance.  I can then
compute certainty equivalent or return on investment, and those
who aren't believers in Kelly can compute whatever it is that
they compute based on that expected value and variance.  (More
precisely, one might like to see the exact distribution of
results and their probabilities, but for a game like blackjack
where you don't bet a huge chunk of your bankroll on one hand, it
doesn't make much difference.)  Two of the few books that do use
expected value and variance extensively are Wong's _Professional
Blackjack_ and Schlesinger's _Blackjack Attack_.  Kudos to them.

Unfortunately, advantage is so entrenched in the blackjack
community, that we almost cannot communicate without reference
to it.

--
Abdul Jalib          |
                     |                Jihad on advantage!
AbdulJ@earthlink.net |