Who says players can't have an edge at the slots?

Cash-back incentives for slot buffs are popular on both sides of the casino fence. The appeal to solid citizens seems obvious: extra dough to pump through the machines, moolah they didn't need to scrape together and never actually had, which translates into longer gambling time and more shots at a big payday on their own recreation budgets. The allure from the joint's standpoint also appears to be evident: most bettors lose whatever they load into the machines, so the establishment will recover the cash-back and not really give away anything on what amounts to "funny money."

Good reasoning. Except, it's not quite correct. Or, at least, it's neither the whole story nor the only way to view the effect.

Make believe your favorite machine has a fairly high 10 percent edge, usually stated in up-beat terms as the complementary 90 percent return. This means, on the average, the house earns a dime and pays out $0.90 for every dollar bet on the game.

The operative term is "average." For instance, a thousand people betting a buck apiece might split a $900 return. However, the split will be lopsided. In the extreme, 999 hopefuls might lose a dollar and one win $899. The casino would reap $999 and sow $899, retaining $100 – 10 percent of the $1,000 bet. This is its theoretical gain due to edge. Theoretical, not because the gain is a conjecture. Rather, because the 10 percent isn't milked from each person on every bet, but statistically over many wagers.

Pretend a casino sent the thousand patrons $1 each in cash-back, $1,000 in all. On the $1,000 wagered during the first round, the casino expects to keep $100 and return $900. Since everyone could walk, the casino is looking at a $900 loss on $1,000 bet. Players therefore had an edge of $900/$1000 or 90 percent in that round.

Say they all try again, putting $1,000 more at risk. Of this, the casino again holds $100 and disburses $900. The joint's two-round net deficit becomes $800 on wagers totaling $2,000. Players who bet twice accordingly have an edge of $800/$2,000 or 40 percent.

A third round lowers the casino's average overall loss to $700 on a $3,000 handle, cutting the players' edge to $700/$3000 or 23.3 percent. After 10 spins, the casino has recovered $1,000 and is at break-even. No edge either way. At round 11, the bosses begin to rake it in. With that 11th coup, the machines will have kept $1,100, for $100 profit on an $11,000 gross wager. This boils down to an edge for the house of $100/$11,000 or 0.909 percent.

As play continues, the impact of cash-back on edge diminishes and the casino's rate of earnings approaches – but never quite reaches – the nominal value. After 500 rounds, for instance, 500 times $1,000 or $500,000 has been bet. The casino has averaged $50,000 from the wagers, minus the $1,000 it distributed as cash-back. So its edge is $49,000/$500,000 or 9.8 percent. After 1,000 rounds at $1,000 each, the figures are $100,000 minus $1,000 or $99,000 locked-up on $1,000,000 bet for an edge of 9.9 percent.

More realistically, a frequent visitor who bets $1 per spin from a bankroll of a few hundred bucks might get $25 in cash-back. At 10 percent on $1 bets, $0.10, 250 rounds are necessary before the casino earns a theoretical $25 and starts depleting the person's fanny pack. On fewer rounds, the edge is with the hoi polloi.

For other situations, the number of rounds in which players have an advantage over the casino rises with the face value of the cash-back and falls with the nominal house edge and the bet size per round. The good news is that individuals can accordingly play the slots with an advantage. The bad news is that 250 rounds is under a half hour of action for most slot buffs. Even 500 rounds, were the nominal edge to be 5 percent, takes less than an hour. Most folks want more playing time than this. And the casinos aren't apt to send cash-back incentives to anyone who gambles for such short periods. Which doesn't mean having an edge isn't possible. As the wily wordster, Sumner A Ingmark, wisely wrote: