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Best of Alan Krigman
Equal Edge Doesn't Make Games Equivalent4 January 2006
Picture a simple even-money table game with chances of 49 percent to win and
51 percent to lose an equal amount. These values give the house a 2 percent
edge. For convenience, posit $1 bets. The edge then represents a hidden "fee"
of $0.02 per round. Play a hundred rounds and the casino expects to earn an
average of $2 from you. Assume you begin with $100. You're not at much risk
of busting out, under these conditions, but neither are you apt to encounter
a long enough run of wins to buy you bragging rights.
Guess what! This game gives the house a 2 percent edge, too. The little bandit is still expected to net $0.02 per pull, $2 every hundred rounds, on the average. Only, now, you're more liable to deplete your fanny pack. This, because the casino's take hasn't changed and someone's gotta fund the fortunate few who hit $100. So, it's not the bosses who send the luckless to the lockers; it's the rare birds who head for home with the bulging billfolds.
Here's another variation. Again, you'd know the possible payouts but not their
probabilities. Imagine a machine with chances and returns of 0.08 percent for
$500, 0.10 percent for $100, 0.11 percent for $50, and 39.70 percent for $1.
The biggies are indeed elusive. But the machine says you win almost 40 percent
of your rounds even though it means you just got your own money back. From where
this discussion has been headed, you shouldn't be shocked to learn that the
edge is 2 percent, so the casino is once more looking at $2 for your hundred
rounds. However, the danger you'll go belly-up has risen further. Your chances
of any real wins have decreased and your action is financing the $499, $99,
and $49 profits for the occasional individuals who get them.
The effect of varying volatility for a fixed edge can be envisioned in terms of equivalent even-money bets. These would be 1-to-1 wagers on which the house still earns its $0.02 per round and the characteristic bankroll swings are the same as in the prototype game. In the actual even-money configuration, the bet is the $1 and the chance of winning is the 49.00 percent. In the implementation with the $100 return, the even-money equivalent of the $1 bet is an $8.35 wager with 49.88 percent probability of winning. In the $499, $99, and $49 version, the even-money equivalent is a $14.99 bet with 49.93 percent chance of winning.
Consider what this suggests. The third example is like betting $15 in an even-money game on a $100 bankroll. You could grab a nice profit. Or you could go broke in no time flat. With a given stake and bet size, you just can't avoid the trade-off between the shot at a big payoff and the peril of a wipe-out. As the prominent purveyor of punting poetics, Sumner A Ingmark, put it:
For a gambler it's no small ability.
To be cognizant of volatility.
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