Recently, I had an exchange with a game inventor while we were tweaking his game. The topic came up about the relatively common betting structure of Ante/Play, used in many games such as Three Card and Four Card Poker.
In this structure, the player makes an initial wager (the ante), sees part or all of his hand, and then must either make a Play wager or Fold. As we discussed using this structure for his game, he remarked how the player could now just make the Ante wager and only Play when he has a strong hand.
This would not generally be the proper strategy. Players don’t fold all that often. In Three Card Poker, it is a fairly high fold rate, about 30%. You certainly don’t want any higher than this. It really isn’t in the casino’s best interest to have too much folding. A player’s interest might wane while he is waiting for the rest of the hand to play out and he might choose to leave. At the same time, every strategy decision is a possible point of human error that increases the profits to the casino. So, a careful balance must be struck.
So, the player isn’t likely only going to play when he has a strong hand. He’s going to play when he doesn’t have a lousy hand is probably more accurate. When we look at the math of the game, we realize the player has two choices. The first is to fold. This will cost him his Ante wager – one unit. If he chooses to play, he will wager a second unit. If his hand still loses, he will lose two units, which is the worst case scenario. So, the player should only make the Play wager if, on average, he can expect to win back at least one unit, which will make his net loss less than folding.
Assuming for the moment there is no dealer qualifying, the player would have returned to him four units if he wins the hand. We’ll also assume all wagers pay even money for our simplified example. I hope you’ll all forgive me for using a very tiny amount of algebra here. We know we want his average return to be one unit and we know he wins four units per win. Thus, we have a relatively simple equation to solve for probability of winning times 4 > 1.
We divide both sides by 4 and find the probability of winning must be equal to 0.25 or higher (i.e. 25%). That’s right. If the player’s hand is strong enough to win just 25% of the time, on average, it is worthy of playing. That’s not a very high amount. It also tells us the player will play hands where he is likely to win 25%-50% of the time. These are hands he will lose more often than win!
If we take a look at Three Card Poker, I can explain better. Now, with Three Card Poker the equation is a bit trickier as there is dealer qualifying. Thus, sometimes, the player only has three units returned instead of four. This makes the calculation a bit more complex. It puts the win frequency just a tiny bit higher, but still well below 50%. If the player has a hand of Q-6-4 or better he should play. This is the lowest hand that, on average, will have at least one unit returned to the player, making his average total loss less than one and better than folding.
But, it is still a losing hand on average.
In fact, all hands Queen High AND King High are losers on average. But, the loss will be less than folding, even though we might only win 35% or 42% of the hands. It is only when we get to Ace High hands that the player can expect to win more than lose.
This is the essence of a Play or Fold decision. You are NOT deciding to play strong hands you will likely win. Instead, you are playing hands that aren’t horrible that would cause you to lose almost all the time. This is very different than a Play-Check decision, which exists in some games. For example, in Let It Ride, when you decide to pull back one of the first two wagers, you do not have to fold the hand. While Let It Ride works in reverse in that you pull back your wager, mathematically, this is the same as checking as you simply don’t make a wager at that point and keep going without folding.
In a game like Ultimate Texas Hold’em, both situations occur. Early on, you have the choice to Play or Check and at the end, you have to choose between playing 1x or folding (if you haven’t made a wager so far). UTH adds the extra wrinkle that if you bet early, you can’t bet again, but that impacts those decisions. If you get to the 1x decision and haven’t wagered, you are facing the very same type of calculation as in Three Card Poker. You will be forced to make the wager even with only so-so hands.