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When do losses from edge overwhelm what you can earn from volatility?

30 May 2011

Pretend you make a bet whose chances are 15 percent to win 5-to-1, and 85 percent to lose. The house’s edge or advantage is 10 percent. Say you bet $10 five times. You’ll finish between $50 down and $250 up. The edge would cost you $5.00, 10 percent of the $50 gross wager. In this small set of trials, though, you won’t notice it. The $5.00 isn’t deducted directly from your poke; it’s buried in the offset between 5.00-to-1 payoffs and 5.67-to-1 (85-to-15) odds against success.

Changes in fortune caused by actual wins and losses on individual bets greatly exceed what the bosses earn from edge on the same wagers. For the high-edge hypothetical case described, the $1.00 latent commission per bet gets lost in the real $10 losses and $50 wins. The upshot is that you may finish one or more sessions or casino visits profitably despite the house advantage, or deeper in the hole than it would imply. The house operates on the overall net of many patrons’ wins and losses – a sum biased toward players in the aggregate leaving their money behind.

Edge always carries cash away from players. The associated losses therefore accumulate steadily, in strict proportion to numbers of rounds. Volatility moves money toward as well as away from bettors. The effect is somewhat self-cancelling so consequent expected final ranges expand at a rate less than proportional to numbers of rounds. Both are similarly influenced by amounts bet.

Owing to these factors, volatility overwhelms edge initially. However, the impact of the latter grows faster, so the dominance eventually reverses. How long can you play before what you lose on edge eclipses what you may win on volatility? The answer depends on the characteristics of particular bets and the confidence you want that you can still win after some number of coups.

Say you’d like to be at least 16 percent sure that normal bankroll upswings swings will leave you with a profit. The accompanying table gives maximum numbers of rounds for some common wagers. Try the calculations yourself for other bets. The formula is: n = (P + P*e - e)/(e*e), where n is the number of rounds, P the payoff for a 1-unit bet (5 in the example cited previously), e the edge (expressed as a negative decimal), and “*” the symbol for multiplication.

Theoretical number of trials before the loss due to edge exceeds one standard deviation, such that the chance of earning a profit is 16 percent or less.

bet                                    edge      payoff      rounds
single spot at 00 roulette          -0.0526     35-to-1      11,989 
2-no split at 00 roulette           -0.0526     17-to-1       5,833 
3-spot row at 00 roulette           -0.0526     11-to-1       3,781 
4-spot corner at 00 roulette        -0.0526      8-to-1       2,755 
12-spot column at 00 roulette       -0.0526      2-to-1         703 
single spot at 0 roulette           -0.0270     35-to-1      46,657 
2-spot split at 0 roulette          -0.0270     17-to-1      22,681 
3-spot row at 0 roulette            -0.0270     11-to-1      14,689 
4-spot corner at 0 roulette         -0.0270      8-to-1      10,693 
12-spot column at 0 roulette        -0.0270      2-to-1       2,701 
Place the four or 10 at craps       -0.0667      9-to-5         393 
Place the five or nine at craps     -0.0400      7-to-5         865 
Place the six or eight at craps     -0.0151      7-to-6       5,071 
Pass at craps, no Odds              -0.0141      1-to-1       5,001 
Blackjack                           -0.0050      1-to-1      40,000
8/5 video poker                     -0.0270   2.26-to-1       3,050 

The table shows a wide spread in numbers of trials needed for edge to become more of a factor in bettors’ performance than volatility. In general, high payoffs and low edge tend to lengthen the duration of play before edge reigns supreme. You can see this phenomenon from the tabulated data in two ways. Compare nominally equivalent bets at double- and single-zero roulette, where the former give the casino almost twice the edge as the latter for the same payoffs. Then, compare bets with different payoffs at the same house advantage, where greater returns lead to more rounds. Note that changes in edge have more impact on the results than those in volatility.

The formula is only an estimate for slots because there are multiple possible payoffs. An approximate number of rounds in the table was found for video poker using the actual edge and the average payoff on winning hands for a typical configuration. Neither edge nor average payoff would be usually known for reel-type slots. However, figure that the edge and volatility are higher on these than on video poker machines. Doubling both edge and volatility would cut the number of spins to roughly 1,500. Tripling both would drop the number to about 1,000.

Wouldn’t it be great if this were all there were to beating the bosses? Alas, casino gambling isn’t that simple. Keep edge as low as possible? Well, yes, subject to the games and the bets you favor. But go for wagers with the highest payoffs you can find? It’s inherent in the nature of big payoffs to be few and far between, so this only works if your bankroll is sufficient to outride normal cold spells. A situation not unlike the good-news/bad-news compromises you often face in the real world. Proving once more that much, if not everything, you need to know in life you can learn in the casino. The punters’ poet, Sumner A Ingmark, put it in gambling parlance like this:

A wagering strategy grounded in trade-off,
Oft helps folks who want to be frequently paid off.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.