Tuesday, September 01, 2009

A small mystery resolved

I occasionally get "match play" coupons. For any even-money table-game bet of $5 or more (that's the usual stipulation, anyway), they match the bet amount. For example, you can use it on odd/even or red/black at roulette, which is my usual approach. It's definitely a +EV (expected value) thing to do.

The question I have wrestled with, though, is exactly how much EV is there in, say, a $5 coupon? This should be a completely straightforward question, but I had difficulty with it. My instinct at first was to say, "Well, it adds $5 to the bet, but only pays half of the time, so the EV is $2.50." (For present purposes, I'm assuming it's exactly an even-money payout and odds, disregarding the small chance of 0 or 00 on roulette.)

But when I did the math, I kept coming up with it being a $5 EV. Suppose we have two such coupons, and we bet $5 plus the coupon on red both times. The ball lands on red once, black once. When it hits red, the payout on a $10 bet is $20. I.e., you get the $10 bet back plus $10, net gain of $15, after investing $5. When it hits black, you lose $5. The sum of these two events is +$10, so the EV of each coupon is $5--in conflict with my first conclusion.

Hence my conundrum.

Today I had lunch with Cardgrrl at the Orleans, where she is staying. She was given one of these coupons on check-in, so I explained my dilemma. She wasn't sure of the correct resolution, either.

So we took the coupon to a table, put a $5 chip and the coupon on black, and the ball fell on 22 black. The woman placed two more $5 chips on our winning spot, and we walked away.

It took us a bit of thought to realize what had happened. The payout was smaller than expected--$15 rather than $20. And, once it dawned on us, the reason was obvious: They don't replace the coupon with a $5 chip before spinning the wheel. That makes it different from making an actual $10 bet in this crucial way: With a $10 bet, you get your original $10 back plus the $10 win, or $20. With a $5 bet and a coupon, you get back your original $5 plus the $10 win, or $15.

In fairness, I had seen this happen a couple of other times that I used these coupons and won, but I left wondering why the payout was less than I had thought it would be, without ever stopping to think through it all the way.

When you win, your net gain is $10. When you lose, your net loss is $5. Those two things happen with equal frequency, so the EV for the coupon is $2.50.

On the rare occasions that my gestalt sense of what should happen in a gambling situation conflicts with what the math is telling me, it's usually the math that is right and my subjective sense that is wrong. Here, though, I had the math set up incorrectly, which led me astray, and my original thought about the value of the coupon was correct. I suppose that's reassuring in some way, but, on the other hand, it's a bit embarrassing and distressing that I made such a fundamental error in how to evaluate the situation mathematically.

Can you tell that, aside from poker, I really don't gamble much?


Anonymous said...

There are a few matchplays that actually "match" your bet. i.e. bet $10 win $20. Floor at Cannery...oops... not your favorite place...was telling me that if he could get 2 a day from enough places he would never have to work again. When you are out and around you should be on the lookout as they can definately boost a bankroll. The Las Vegas Advisor and American Casino Guide are well worth thier price in matchplays alone. And you are in casinos everyday so its almost a sin to pass up free money.

Oh, and you should check out "win cards". Lot of places you play have them. You get $30 in promo chips for $20 and keep the promo chips as long as you win your bet.

genomeboy said...


Perhaps you are overlooking something quite obvious for the sake of simplicity, but there is a glaring error in your model.

While the payout may be even money, at roulette (or the pass line at craps, for example) are not even odds. Recall that there are 36 numbers on the roulette wheel in addition to 0 and 00 (or just 0 on European roulette wheels). Thus, while the payout is even money (1-1), the true odds are more like (1.111-1). Thus, you're true EV is $2.25 ($2.50/1.111)

KenP said...

Reminds me of those old trick math jokes where $5 disappears in the exchanges.

Does EV have relevance beyond poker might be the real question.

It really a simple ratio decision just like 17 outs. You are a dog by 17 or 18 to 16. Do you invest with 3:1 odds?

EV has always seemed an after market accessory to me. It justifies your outs/pot decision while telling the bad beat story.

bellatrix78 said...

You didn't include the house edge into your calculations. Your matchplay coupon is worth less than 2.50$. I once gave an assignment that asked exactly that:

What is the expected value of a 5$ matchplay coupon placed on red in the game of roulette?

Assuming a double zero wheel there are 18 ouy of 38 combinations that favor us. However the thing with the matchplay coupon is that you have to match the coupon value, when placing a wager, but since you got the coupon for free, you only risk those 5 matching dollars.

EV = 10$ × 18/38 − 5 × 20/38 = 2.11$

You are getting back 42.11% of the coupon’s value. Clearly roulette in the U.S is not the best wager for a matchplay coupon.
Baccarat has the best odds at around 47%.
Blackjack is a bit difficult, since some casinos pay you 3:2 on your matchplay, while some don’t (correctly). Also, casinos have
different rules on splitting and doubling down, so it is difficult to get the value of your matchplay for blackjack.

For completeness with single zero wheel:

EV = 10$ × 18/37 − 5 × 19/37 = 2.30$

or about 45.95% coupon’s value.

Pat in MN said...

I know you like to be precise so don't forget about 0 and 00 when betting black or red in roulette. Black and red are not 50-50 propositions. Your actual EV for a $5 match play bet on black or red is $2.37 after rounding up. IT is still a good deal. I do prefer roulette for match plays if it is an option. You can usually walk up and make a bet and have that bet quickly resolved. Blackjack and craps take a little more time to get a bet down and have that bet settled. Plus, I don't usually feel like there is any hit-and-run scrutiny in roulette.

Rakewell said...

For reasons I cannot understand, several readers/commenters all seem to have overlooked this rather explicit sentence in the post:

(For present purposes, I'm assuming it's exactly an even-money payout and odds, disregarding the small chance of 0 or 00 on roulette.)

Pat in MN said...

After rethinking my calculations I realized that I forgot to take off the house edge on the original $5 bet of .26 leaving an EV of 2.11 on the entire bet. I also realize that you noted the 0 and 00 but it is important not to forget those. It really pays to take advantage of these match plays as they are easy to use and not a "gamble" at all.

Sean G said...

I might actually have to use one of those coupons if they hand me any during my upcoming stay in Vegas. I've always just tossed them out, but your red/black thing doesn't sound like a bad idea for a fellow non-gambler.