If there's one common casino procedure with which I take exception, it is that of the, "Maximum Aggregate Payout," which essentially gives the casino the ability to take bets without being on the hook to pay based on the paytable of the game in question when a big pay is hit.
Imagine if there existed a Video Poker machine that had the usual graduated pays to the Royal based on coins bet, 250-500-750-1000-4000, but on that machine was a caveat saying, "The maximum payout of this machine is $300." Essentially, this is the same concept of a Maximum Aggregate Payout, the Video Poker paytable claims you will get 4000 quarters ($1000) on a bet of five credits ($1.25), however, that is clearly not the case as $300 is equivalent to 1200 coins.
This (hopefully) non-existent Video Poker game is similar to Maximum Aggregate Payouts in another way: If the paytable of the Video Poker game is otherwise perfectly graduated on lesser paying hands, then the Optimal bet is anywhere between 1-4 credits and the player gets a worse return by betting the Max. The reason is because 1200 credits would graduate perfectly with the rest of the game on a bet of 4.8 credits, anything else and the player is essentially giving the house a Loss Rebate.
Maximum Aggregate Payouts often exist for Carnival Games, and in rare instances, I've even noticed them on Roulette. One common misconception is that the Maximum Aggregate Payout will only have a negative impact on the highest of the high rollers, but sadly, that is not necessarily the case. Progressives, however, are largely player-banked and should not apply to any Maximum Aggregate Payout.
For the purposes of a case study, we're going to look at the game of Let It Ride, in this game players are dealt three initial cards and have to decide whether they want t leave all three bets out there or take one back. After making this decision, the fourth card is revealed and the player again gets to decide whether to leave any remaining bets out there or to take one unit back. This is certainly not the only game for which a Maximum Aggregate Payout has applied, but for the purposes of an example, it is probably mathematically the easiest to use.
We will use the Standard Paytable from WizardofOdds:
Further, the most recent Let It Ride game I have seen has a Min-Max bet range of $5-$50 and a Maximum Aggregate Payout of $25,000.
The highest roller might bet at $50 per spot, and assuming a player is going to be paid in full, the Optimal Strategy for Let It Ride would have us let ride any Three Royal Flush cards that we get initially. For instance, we would let Jc-Ac-Kc ride largely due to our chance of hitting the Royal.
Hitting a Royal Flush in Let it Ride is mathematically the same as getting a dealt Royal Flush, 1/649739 or 0.000153907%. In other words, it's an astoundingly unlikely event. Even with that being the case, it contributes 0.004617 or 0.4617% of the return to the game with a House Edge of 3.5057%.
Unfortunately, that's when the Royal Flush pays 1000 to 1 which would be a pay of $150,000 under the posted odds with a $50 bet per spot. However, due to the Maximum Aggregate Payout, this $150 bettor would only receive $25,000 (assuming no side bet and no other players playing and hitting winning hands) which is 1/6th of what he should receive. Therefore, the return of the Royal is 1/6th what it should be.
Essentially, it reduces the return of the Royal Flush to .0007695 or 0.07695%, which effectively adds 0.38475% to the House Edge resulting in an Effective House Edge of 3.89045%.
But, as every infomercial host this side of 2a.m. has said, "But, wait, that's not all!" You see, the casino would actually have to pay $30,000 on a Straight Flush holding all cards, which is a return of .004063 or 0.4063%. Because the casino is only paying 5/6ths of that, the return goes to .0033858 or 0.33858% costing the player an additional .0006772 or 0.06772% in return making the new Effective House Edge 3.95817%. In sum, our $50 bettor bucks an additional House Edge of nearly half a percent and the casino gets an effective loss rebate on any jackpot this player might win.
Figuring out the Optimal sized bet to ensure the player always gets paid according to the paytable is a relatively simple task. Simply take the $25,000 and divide by 3000 which is the overall return of the Royal (1000 to 1, per spot). Therefore, the player should never bet more than $8.333- per spot, such a bet is impossible, of course, so just call it $8 per spot.
Does $8 per spot sound like a high-roller to you?
First of all, Maximum Aggregate Payouts are patently indefensible, however, casinos institute them for a very simple reason: They want all of the benefits (House Edge = Expected Win) of players making large bets, but they don't want to take the risk of sustaining a tremendous loss.
That, in and of itself, is almost understandable...particularly for a small casino. The casino could very easily have a losing week (or month) on a Let It Ride table if a couple of Royals are dealt out to Max (or near Max) bettors.
The casinos want to gamble with as little Variance as possible given that they are at an advantage, but at the same time, they want to size their Max Bets to make as much money as possible without sustaining a loss that they cannot tolerate.
Many casinos probably could not deal Let It Ride if they were not willing to accept a Max Bet greater than $8.00, it's a bit of a dying game and would probably not be profitable even at a Min-Max spread of $5-$10. Therefore, there must be some sort of compromise.
My suggestion is that the casinos should treat Maximum Aggregate Payouts for any side bets and the Base Game separately to the extent that hitting the Maximum Aggregate Payout of one does not apply to the other in any way whatsoever. Secondly, in a game such as Let It Ride, if a Royal is hit, only one player could have possibly hit it. The worst case scenario is one hits a Royal and the other a straight Flush holding 7-8-9, suited, while the community cards are a suited 10-J, quite unlikely.
Therefore, Maximum Aggregate Payouts should only apply to the top pay of the game possible and any other players should be paid according to the paytable.
Finally, in simple English, at the bottom of the Maximum Aggregate Payout sign it should read, "The most you can bet while assuring the full return of the game is $8.00, per spot."
Only then will Maximum Aggregate Payouts even approach something that I can tolerate, or that any players should tolerate. In the meantime, my advice is to know the math and never make a bet for which you would not receive the full return in case you ever do hit that, "Miracle Hand."