Say you enter a
casino with $100. What do the bosses expect you to leave behind? The
whole $100, or less? And, if less, do they consider you a better, worse,
or equivalent patron betting $10 or $25 per coup? The devil is in the
detail of how you gamble.
Say you go for broke, playing until you win $900 or lose $100. Many
folks believe the casino would figure them for $100 profit minus a small
luck factor, ascribing the effect to house edge.
But, imagine a gamble
with no edge. The casino has 90 percent chance of getting your $100.
Your shot at winning $900 before this occurs is the other 10 percent.
The house breaks even.
Here's why. Suppose
that 100 people each buy-in at the no-edge game for $100. They play
until they win $900 or go broke. On the average, 10 will succeed --
winning $900 each for a total of $9,000. The other 90 will lose $100
each for a total of $9,000. That's $9,000 in and $9,000 out. Nothing
is left for the casino.
With an edge, the
casino gets a share as well. Envision it by pretending the game is blackjack,
with everyone following perfect Basic Strategy. The house advantage
is half a percent.
When bets are a
flat $10 per round, this edge cuts the chance of winning $900 before
losing $100 from 10 to under 7 percent. On the average, of 100 players,
seven will therefore win $900 and 93 will lose $100. Winners take away
$900 x 7 or $6,300 and losers leave $100 x 93 or $9,300. The bosses
earn the $3,000 difference, so each of the 100 bettors looks like $30
to the casino.
What if everything
remains the same except that bets are $25 per round? The chance of victory
becomes 9 percent and that of defeat 91 percent. On the average, $900
x 9 or $8,100 then flows out the door while 91 x $100 or $9,100 stays;
the joint keeps the $1,000 difference. Each of the 100 $25 bettors represents
a theoretical profit of $10 to the casino.
The disparity arises
because the solid citizens with $100 stakes need longer either to hit
their profit targets or go belly-up betting $10 than $25 per round.
And, taking longer, the $10 players will end up betting in more rounds
and having a larger gross wager or "handle" on which the edge
takes its toll. To explore this counter-intuitive phenomenon, $10 bettors
being worth more to the bosses than their $25 counterparts, work back
through the edge and casino "take" to the handle.
For the house to
average $3,000 at half a percent edge, the gross wager would have to
be $3,000/0.005 or $600,000. At $10 per round, this is 60,000 rounds
-- an average of 600 rounds and $6,000 handle for each of the 100 players.
For the casino to earn $1,000 at the same edge, the handle would have
to be $1,000/0.005 or $200,000. At $25 per round, this is 8,000 rounds
-- an average of 80 rounds and $2,000 handle for each of the 100 players.
A higher edge with
all else being equal would lower the likelihood of winning the $900
before losing $100 even further. It would also raise the expected earnings
from each player.
Rising from 0.5
to 1 percent edge, prospects of prosperity for $10 players fall from
7 to 4 percent. At 1 percent, an average of four winners would accordingly
grab $900 x 4 or $3,600 while 96 losers would kiss $100 x 96 or $9,600
goodbye. The casino would earn the $6,000 difference, $60 per person.
Similarly for the $25 bettors, probability of success drops from 9 to
8 percent. On the average, eight winners would emerge with $900 x 8
or $7,200 while 92 losers would be stripped of $100 x 92 or $9,200.
The $2,000 difference would be the casino's commission, $20 per player.
Alternate exit strategies
would yield other results. But the underlying theme would still be that
neither bankroll nor bet size alone are good indicators of a player's
monetary worth to a casino. They may even lead to pampering the wrong
people. As the songster, Sumner A Ingmark, surely suspected when he
scribbled:
Conclusions
falsely prejudicial,
Arise from data superficial.