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Best of Alan Krigman
Does Equal Edge Mean Equal Chance of Profit in Different Games?8 September 2006
These games are equivalent to the casinos. Their bean counters figure that, assuming enough action, earnings will be equal. For instance, after a million bets averaging $10 each, they'll keep close to 5 percent of $10 million or $500,000 either way.
But no single player will make a million $10 bets. Were it even humanly possible,
the overwhelming likelihood of a $500,000 loss would militate against such fecklessness.
Individuals are more apt to play short sessions, for reasons of time and endurance
and also because volatility then affords them a good shot at grabbing a profit
before edge takes its toll. And volatility is where the alternate ways the house
gains its edge in these games differ. The "standard deviation," picture
it as average bankroll change per round, is essentially $10 in 10-10 and $9.50
Computer simulation yields some answers. These numbers of players and rounds are statistically small. Variability can therefore be expected in multiple simulation runs under the random conditions prevailing in one session or another. Just as in real games. Still, the simulations show some illuminating trends.
With 50 rounds at $10 each, each bettor's gross wager is $500. The theoretical
average loss is 5 percent of $500 or $25. Five simulated sessions showed losses
from $23 to $26, no bias toward either game. However, numbers of solid citizens
finishing even or ahead diverged substantially. In 10-10, the even-money game
with a greater chance of losing than winning and higher volatility, from 396
to 430 bettors ended where they started or at a profit. In 9-10, the 50-50 game
with $9 payoffs on $10 bets and lower volatility, 325 to 345 players finished
50 rounds even or ahead.
The same patterns appear for 200-round sessions. The theoretical average loss due to edge was $100, and the simulation showed a span from $93 to $104 with no fundamental difference between the two games. Numbers of players concluding even or ahead were below those in the shorter sessions but continued to favor the higher- to the lower-volatility game. Here, the range was 245 to 287 out of 1,000 for 10-10 and 201 to 232 out of 1,000 for 9-10.
None of this suggests a secret the casino bosses don't want anyone to know. Players could and did win in both games despite the stiff 5 percent edge. More finished even or ahead in 10-10, with higher volatility but lesser probability of winning, than in 9-10 with the converse characteristics. But the averages were equal. This implies more folks but lower profit levels in the game with sessions that were easier to win. Illustrating two important precepts. First, you can earn money at a casino whether or not you know what you're doing. And second, you can make choices with trade-offs that tailor your gambling to your personal preferences. Both of which were contemplated by the Coleridge of the casinos, Sumner A Ingmark, when he composed:
While lucky is who lucky does,
Best of Alan Krigman