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# Would you rather have \$50 in match play or \$25 to bet in cash?

30 July 2012

Which is a stronger inducement for solid citizens to visit a casino? Two \$25 match play coupons or one \$25 cash token the dealer will exchange for chips at any table (shades of those rolls of quarters). Assuming you won’t or can’t violate the letter or spirit of the incentive by doing other than betting with one or the other, the objective answer depends on the monetary equivalents of the premiums. And, since you’re going to bet with them, those equivalents are what math mavens call “expected value” – the amount you’ll have after a win times the probability of success.

Finding the expected values involves recognizing these differences between the alternatives.

• Match play coupons can’t be used by themselves but only to double the amount of a cash bet. On a win, you’re paid according to the total face value of the combined wager. You also recover your cash outlay but the coupon is a one-shot and isn’t returned.
•The chips you receive for your cash token can be used as all or part of a total wager. On a win, you’re paid based on the face value of the wager and also get back the chips you bet.

Red at single-zero roulette can be used to find expected values under three conditions for use in comparing these premiums for even-money bets. These conditions are risking 1) your own \$50 by itself, 2) your own \$50 and \$50 in match play, and 3) \$25 the joint gave you for the token. The chance of winning is 18 out of 37 and the payoff is 1-to-1. The following results would change a bit for other even-money situations but the inferences drawn would be essentially the same.

Starting with \$50 from your fanny pack, the chance is 18/37 you’ll finish with \$100 , the \$50 payoff plus your initial \$50 . The expected value of the \$50 you put at risk is therefore (18/37)x\$100 , which equals \$48.65.

When you bet \$50 in real moolah and \$50 in match play, the chance is 18/37 you’ll finish with \$150 – the \$100 payoff plus your original \$50. The match play coupon will be history. Expected value is (18/37)x\$150, which equals \$72.97. The expected value of the \$50 match play coupon alone is that of the combined bet minus what it was with your \$50 by itself. That’s \$72.97 - \$48.65 or \$24.32.

With the third alternative, betting \$25 from the cash token, a win pays \$25 and you get your \$25 back, for a total \$50 return. The expected value is (18/37)x\$50, which equals \$24.32. On the even-money bet, the \$25 cash token is therefore worth as much as \$50 match play.

Inquiring minds want to know what would happen to these figures if you use the incentives to bet on longshots rather than even-money propositions. For this, assume you wager on a single spot at single-zero roulette. This has one chance out of 37 of winning and pays 35-to-1.

Were you to simply bet \$50 in legal tender, a win would pay 35x\$50=\$1,750 and you’d get back your \$50, leaving you with \$1,800. The expected value of the \$50 you took from the cash machine in the lobby is consequently (1/37)x\$1,800, which equals \$48.65.

What if you bet \$50 of your own dough and \$50 in match play? After a win, you’d have 35x\$100=\$3,500 plus your original \$50 , a total of \$3,550. The expected value of the combination would be (1/37)x\$3,550, which equals \$95.94. The expected value of the \$50 match play is accordingly the \$95.94 characteristic of the combined bet minus the \$48.65 your real \$50 would be worth by itself, which comes to \$47.30.

Here’s how the values work out betting just the \$25 from the cash token. A win would get you 35x\$25=\$875 and your \$25 would be refunded so you’d finish with \$900. Expected value would be (1/37)x\$900, which is \$24.32. With this longshot, the \$25 cash token is worth only \$24.32/\$47.30 or 51 percent – slightly over half – of the match play.

Pretend you may bet longshots with match play and do so. Then, \$50 in these coupons would be far superior to\$25 cash tokens. They’d also cost the bosses lots more – \$47.30 rather than \$24.32. This is why almost all casinos limit use of match play to even-money propositions. And, when the restriction to even-money bets applies, the expected value of match play equals that of half the nominal amount in cash tokens. If, of course, you can find an establishment whose honchos didn’t do their homework and let patrons use match play on any bets they like, well... there’s still the tradeoff between odds and return. But the high expected value on low probability results would be mighty tempting to the cognoscenti. The 1-to-1 versus long odds shift in expected value is what the bettors’ bard, Sumner A Ingmark, may have had in mind when he wrote:

A plan whose structure’s not well numbered,
May be with poor results encumbered.