PLAYING IT SMART by ALAN KRIGMAN

How Does Edge Affect Expected Bankroll Highs and Lows

The house advantage or edge in casino gambling arises because the payoff on a bet is somewhat less than the odds which must be overcome to win. When you make a Place bet on the five at craps, for instance, you're fighting odds of 6-to-4 but only get paid in the proportion of 7-to-5. On the basis of 20, the lowest common denominator for these ratios, these are odds of 30-to-20 for a payoff of 28-to-20. Likewise, in double-zero roulette, the odds against hitting a single number are 37-to-1 and the payoff for a win is 35-to-1. The figures can get more complicated and are often less obvious in other games, but the effect is the same.

The edge can be pictured as the percentage of each bet the house levies as a theoretical fee for letting players gamble. Bet-by-bet, or even an entire session basis, players rarely notice what they're missing. It's more statistical fiction than financial fact. But, being inconspicuous doesn't mean it isn't there.

One consequence of edge is the amount casinos expect to earn -- how much patrons expect to lose -- from the action. Say the edge in a game is 2 percent. You play 100 or 1,000 rounds at $1 each. In the first case, you bet a total of $100 and the casino figures your business is good for 2 percent of this, or $2. In the second instance, your handle will be $1,000 and the casino rates you as worth $20. Not that you'll finish the respective sessions exactly $2 or $20 behind. Because of the volatility of the game, over the short span of either session, you may instead earn a big profit or sustain a heavier loss. The edge actually takes its toll in the chances and amounts of one or the other.

Most solid citizens think less about where they'll be after some number of rounds, than whether they'll exhaust their stakes or reach a profit that satisfies their hopes and dreams during their visits. Again, volatility is a factor. For any particular edge, bankroll swings may diverge greatly with bets of low versus high volatility such as Red versus single spots at roulette, respectively. But, edge still has an underlying impact.

Here's an illustration of how edge affects the bankroll high and low points you can expect in sessions of reasonable duration. Consider four imaginary games. All have 50 percent chance of winning. The edge arises because of different payoffs. Per dollar bet, these are $1 -- 0 percent edge, $0.98 -- 1 percent edge, $0.96 -- 2 percent edge, and $0.94 -- 3 percent edge. The volatilities vary slightly. However, they're all close enough to fluctuations of 1 unit up or down per decision so the distinctions are essentially negligible. Computer simulations of 100,000 virtual players yielded the results given in the accompanying table for sessions of 100 and 1,000 decisions.

Effect of edge in low-volatility games on average bankroll low and high points during sessions of representative duration; values assume $1 bet on each round

Edge	100 rounds	1000 rounds

	Expected Loss	Avg. Low Pt.	Avg. High Pt.	Expected Loss	Avg. Low Pt.	Avg. High Pt.
0.00%	$0.00	(-)$7.50	(+)$7.50	$0.00	(-)$24.73	(+)24.73
1.00%	$1.00	(-)$7.96	(+)$6.97	$10.00	(-)$29.93	(+)$19.94
2.00%	$2.00	(-)$8.44	(+)$6.44	$20.00	(-)$35.96	(+)$15.97
3.00%	$3.00	(-)$8.96	(+)$5.94	$30.00	(-)$42.77	(+)$12.97

The data show that with no edge, on the average, players could expect bankroll swings to be equal and opposite down and up. As edge increases, low points are exacerbated while high points are diminished. The offsets in magnitudes of the peaks and valleys essentially equal the expected losses due to edge. For example, after 1,000 rounds at 2 percent, expected loss for every dollar bet per round is $20; at the average low point in this game a player would be down $35.96 while at the average high point the bettor would be ahead by $15.97 -- the difference being $20.

Of course, you could get lucky and clean the casino's clock in a high-edge game. Or unlucky and hit the skids despite a big break from the bosses. But, the average highs and lows are telling. They not only help you understand that the losses you're willing to endure aren't independent of the gains you seek, but how house advantage influences the relative values. Here's how the beloved bucoliast, Sumner A Ingmark, broached such brain bending:

Gamblers who survive the cut'll,
Oft be those who know things subtle,
Such as what won't help and what'll.

(c) 2005, ICON/Information Concepts Inc

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