Equal Edge Doesn't Make Games Equivalent

4 January 2006

By Alan Krigman

Casino gambling is rife with misconceptions and myths about the edge the house has on every bet. These range from the notion that edge is why the joints always clean everyone out, to the idea that it's the only thing distinguishing one game from the next. On an outlying tangent there's also the conviction that cryptic combinations of bets in games like roulette or craps cancel the edge ... but, you surely don't believe that one, er, do you?

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.

Ratchet up the action with a slot machine offering two payout levels on a $1 bet. These are a $100 return a $99 profit, and a $1 return a "push" with no gain and no loss although the machine beguilingly announces you've won when it really only flips your bet back. Pretend you somehow discover the chances are 0.71 percent of getting the $100 in your tray or on your credit meter and 49.29 percent of receiving the $1 return. The other 50 percent is the chance you'll lose. You're undeterred by failing to hit the $100 because you're deluded into thinking you're winning nearly half the time, proving the machine is hot, right?

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 edge is constant in these three alternatives. Yet the games obviously differ from one another. Not to the casinos, which average identical profits based on money wagered in each case. But, certainly, to the solid citizens who play. The distinctions arise from the volatility, a statistical measure of the bankroll fluctuations a player is likely to experience during a session.

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: