Casino Promotions and Rebates
Matchplay coupons, chips you cannot cash and refunds on losses move a little money from the casino to the player. Priced correctly, some of them are worth more than the wheel takes.
A coupon that doubles an even-money payout is worth about 46% of the bet on a European wheel. A one-use promotional chip is worth (36 − n)/37 of its face on a bet covering n numbers: 95% on a straight up, 49% on red. A refund of half your net loss makes a short session of red worth playing. None of this changes the wheel; it changes who pays.
Why a promotion can beat the edge
The house edgemathHouse edgeThe share of each stake the casino keeps on average: 2.70% in European roulette.Read full → is a price per unitmoneyUnitThe base stake a system or a player works in. Progressions are described in units so they apply at any table.Read full → bet: 2.70% on a European wheel. A promotion adds money to one side of a bet, and if it adds more than 2.70% of the stake, that bet is positive for as long as the promotion lasts. Casinos offer them anyway because they are limited: one coupon per visit, chipsmoneyChipA token representing a fixed amount, placed on the layout to make a bet. Also: check.Read full → that can be bet but not cashed, refunds only for players who lose large sums. The arithmetic is the same as everywhere else on this site: chance times payoutmoneyPayoutWhat a winning bet returns, quoted as “X to 1” on top of the returned stake.Read full →, added up over the outcomes.
Matchplay coupons
A matchplay coupon sits next to a cash bet and doubles what the bet pays if it wins; win or lose, the casino keeps the coupon. Bollman (2024) prices a Las Vegas coupon from 2022 that matched up to $50 on an even-money betbetEven-money betRed/black, odd/even, high/low. 18 numbers, pays 1:1.Read full →. On a European wheel, $50 on redbetRedEven-money bet on the 18 red numbers. French: rouge.Read full → with the coupon wins $100 with chance 18/37 and loses $50 with chance 19/37: an expected +$22.97, or 45.9% of the bet. On an American wheel it is +$21.05, 42.1%. On a straight upbetStraight upA single number, pays 35:1.Read full → the same coupon would be worth twice as much, which is why coupons are limited to even-money bets.
Chips you cannot cash
Promotional chips, also called non-negotiable chips, can be bet but not exchanged for money. The usual rule is one use: a winning bet is paid in ordinary chips and the promotional chip is taken; a losing one is taken too. A $100 chip on a bet covering n numbers wins (36 − n)/n × $100 with chance n/37, so it is worth (36 − n)/37 of its face. The chip’s own value never comes back, only the winnings, and a bet that wins rarely but pays a lot loses least that way: $94.59 on a straight up, $48.65 on red. Where the chip may stay on the layoutequipmentLayoutThe printed betting grid on the table. Also: table, felt, cloth.Read full → until it loses, an even-money bet is worth 18/19 of its face on a European wheel and 9/10 on an American one (Bollman, 2024).
Turning the chips into cash
A value is an average; a player who wants the money for certain can buy it with a hedge. Bollman’s example is from the old Vegas World casino: $1,000 of one-use chips on an American wheel, $500 on red and $500 on blackbetBlackEven-money bet on the 18 black numbers. French: noir.Read full →, with $20 in cash on each of 0 and 00. Red or black wins $500 and loses the $40 in cash: +$460. A zerowheelZeroThe single green pocket that no even-money, dozen or column bet covers; the source of the house edge. Also: single zero, 0.Read full → loses both chips and pays $700 on the cash: +$680. The player banks at least $460, and the average, $471.58, is 99.6% of what the chips are worth on red alone. Where the rules allow chips on single numbers, a European wheel offers a better hedge: $25 chips on all 36 numbers ($900) and $25 in cash on zero. Whatever number comes up pays $875, and the cash on zero is lost unless zero itself comes up, so the player keeps at least $850, 94% of the chips’ face, with no risk at all.
Refunds on losses
High-stakes players are often offered a rebate: a share of their net loss back at the end of an agreed session. A rebate pays only when you lose, so it is worth most when results swing widely and the session is short enough that the edge has not yet taken over. Epstein (2013) works the case for an American wheel; our figures for a European one follow. With half the net loss refunded, flat even-money betting is worth playing for up to 103 spins and is best at 25, worth +0.51 stakes on average. On a straight up the swings are so large that the session stays positive for about 3,600 spins and is best at 882, worth +17.25 stakes. With a more usual 10% refund, even money is positive only for a single spin, while a straight up is best at 20 spins, worth +0.62 stakes.
What the casino is counting on
Every promotion has a cap, and the cap protects the casino: one coupon a visit, chips issued in a fixed amount, rebates agreed only with players whose expected losses over a trip dwarf the refund. A player who uses the coupon and then stays for an evening of $50 bets on red gives the $22.97 back within 17 spins, at $1.35 a spin. The value is real for the player who takes it and leaves, or who was going to play that much anyway.
Formula
Questions
For the one bet it goes with, yes. Every other bet still costs 2.70% of the stake.
Few players think of it, the chips are limited, and the casino has already priced the promotion as advertising.
Rarely. They are a high-roller term, usually agreed with a casino host and paid on the net loss over a trip.
Fun with Math
Try these yourself first. Each solution is folded; open it when you are ready.
You are given a $25 promotional chipmoneyChipA token representing a fixed amount, placed on the layout to make a bet. Also: check.Read full → that is taken after one bet, win or lose. Where should it go: on redbetRedEven-money bet on the 18 red numbers. French: rouge.Read full →, on a dozenbetDozen1–12, 13–24 or 25–36, pays 2:1.Read full → or on a single number?
Show the solution
Answer On a single number. There it is worth $23.65 on average; on a dozen $16.22 and on red $12.16.
The chip itself never comes back, only what it wins. On red it wins $25 with chance 18/37: $12.16. On a dozen it wins $50 with chance 12/37: $16.22. On a single number it wins $875 with chance 1/37: $23.65.
In general a bet on n numbers is worth (36 − n)/37 of the chip, so the fewer the numbers, the better. With an ordinary chip this makes no difference, because the stake comes back with every win and every bet is worth 36/37 of it. Taking the stake away hurts the bets that win often and hardly touches the bet that wins once in 37.
A casino will refund half of your net loss after exactly one spin. Is $100 on red worth playing? And $100 on a single number?
Show the solution
Answer Both are: about +$22.97 on red and +$45.95 on the number.
On red you win $100 with chance 18/37 and, after the refund, lose $50 with chance 19/37: (1,800 − 950)/37 = +$22.97.
On a single number you win $3,500 with chance 1/37 and lose $50 with chance 36/37: (3,500 − 1,800)/37 = +$45.95.
These are the matchplay figures again. A $100 bet that loses only $50 wins $100 or loses $50, exactly like a $50 bet with a coupon that doubles its win. Over a longer session the refund is paid only on the net result, wins and losses cancel before it is paid, and the value shrinks; for red with half back, 25 spins is the best length.
The casino will refund half of your net loss after a fixed number of $100 bets on red, chosen in advance: 1, 2 or 3 spins. Which is worth most?
Show the solution
Answer Three spins: about +$31.45. One spin is worth +$22.97, and two only +$20.96.
One spin: win $100 with chance 18/37, or lose $100 and get $50 back with chance 19/37: +$22.97, as in the exercise above.
Two spins: both win (324 times in 1,369), +$200. One each (684 times), $0, with nothing to refund. Both lose (361 times), −$200, −$100 after the refund. Average: (200 × 324 − 100 × 361)/1,369 = +$20.96. The refund only pays on a loss, and two spins turn many single-spin losses into a draw.
Three spins can never draw. Three wins (5,832 times in 50,653): +$300. Two wins (18,468): +$100. One win (19,494): −$100, refunded to −$50. None (6,859): −$300, refunded to −$150. Average: +$31.45.
The refund is worth more the longer you commit, while the edge costs only $2.70 a spin, so the value rises to about +$51 at 25 spins (see the table above), and an odd number of spins always beats the even number after it.
A Las Vegas casino once handed out a coupon: bet $5 on 00 and, if it wins, you are paid $250 instead of the usual $175. What is the $5 bet worth with the coupon, as a share of the stake, on its double-zerowheelZeroThe single green pocket that no even-money, dozen or column bet covers; the source of the house edge. Also: single zero, 0.Read full → wheel?
Show the solution
Answer About +34%: the bet is worth +$1.71 on average. If the $250 includes the returned $5, it is +31.6%.
The 00 wins once in 38 spins. With the coupon: +$250 once, −$5 the other 37 times. Average: (250 − 37 × 5)/38 = 65/38 = +$1.71, which is 34.2% of $5. Without the coupon: (175 − 185)/38 = −$0.26, the usual −5.26%.
If $250 is the total paid back, the win is $245 and the average is (245 − 185)/38 = +$1.58, or 31.6%.
Bollman (2024) sets this coupon, from the Lady Luck casino (now the Downtown Grand), as an exercise; the answer printed with it, 7.89%, matches neither reading. Like a matchplay coupon, it is worth something once: play it and stop.
Comic relief
A promotion is worth most to the player who takes it and goes home.
Sources
- Bollman, M. Basic Gambling Mathematics: The Numbers Behind the Neon, 2nd ed.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.