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Preflop Controversy — Ace-King

by Reid Young |  Published: Mar 19, 2014

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Reid YoungAce-king carries different weights in different poker circles. There are big reasons, more like logical fallacies, that explain why this is the case and what is to be considered and understood to correctly play the hand preflop. The reasons extend to understanding much of what is going on preflop, so let’s take a look.

Blockers

Blocking cards remove possible value combinations from an opponent’s range. For example, if our opponent makes a preflop bet, and his distribution of holdings for this particular raise are pocket queens, pocket kings, pocket aces, ace-king, and a few bluffs, then it seems like matching ace-king against those hands is an incredibly bad idea. It appears as if there are two premium hands that annihilate ace-king and one against which we race (our opponent’s ace-king). But there’s a problem with that line of thinking that holds many players back from making the correct decision: the combinations of aces and kings that our opponent may possibly hold are tremendously altered by our own hole cards of one ace and one king.

To see what I mean, consider that normally there are six ways to make any pair, like pocket kings, and 16 ways to make an unpaired hand, like ace-king. If one king is removed from the deck, as is the effective case when that card is dealt to another player, there are only three possible ways to make pocket kings. When cards are “removed” from the deck by being dealt to another player or by being face up on the board, unpaired hand combinations are also blocked. With two such cards removed from the deck, hands like ace-king have only nine possible ways to make the hand instead of the unblocked usual 16 possible combinations. That means without holding an ace or a king in our hand that our opponent has six pocket queens combinations, six pocket kings combinations, six pocket aces combinations and 16 ace-king combinations, for a total of 34 hand combinations. However, when we hold ace-king, our opponent holds six combinations of pocket queens, three combinations of pocket kings, three combinations of pocket aces and nine combinations of ace-king, for a total of only 21 combinations.

This information is incredibly valuable to us because we have the ability to use what is called hidden information, information unavailable to our opponents. In this case, the information that our opponent’s effective value range is quite limited (there are 62 percent as many combinations, in fact) means that he is bluffing far out of proportion. Our opponent believes that he’s balancing his unblocked value bets (34 combinations of Q-Q plus and A-K) with a few bluffs, but in reality, we know that those few bluffs are a much higher percentage of his total range as only 21 value combinations remain unblocked. We’ll quantify this value shortly.

Weighted Equity

Realizing that nearly half of the strongest hands our opponent can hold are no longer possible changes our equity a bit. We can calculate that here by using what is called weighted equity. Weighted equity is the total of all conditional and possible equities for a particular situation. For example, if our opponent’s distribution were one hand against which we have 25 percent equity and one hand against which we have 75 percent equity, our weighted equity is simply:

[ (1 hand combination) * (25 percent equity) plus (1 hand combination) * (75 percent equity) ] / (2 total hand combinations) = 50 percent equity

With ace-king versus a more complex distribution, it’s helpful to use computer software to perform a Monte Carlo simulation, a simulation that mimics hands facing off all-in hundreds of thousands of times. You can find such a calculator at ProPokerTools.com, or just take my word for it for now that:

A-K versus A-A — A-K has 8.16 percent equity
A-K versus K-K — A-K has 31.12 percent equity
A-K versus Q-Q — A-K has 43.94 percent equity
A-K versus A-K — AK has 50 percent equity

Before we account for card removal, the equity against ace-king seems to be calculated with all combinations present for the relevant hands, meaning pocket aces and pocket kings match up against our ace-king six times each, which is simply not true. This yields a lower than actual equity for ace-king, which is part of the reason less learned players fear moving all-in preflop with ace-king in several situations.

Total weighted equity = [(6 combinations * 0.0816) plus (6 combinations * 0.3112) plus (6 combinations * 0.4394) plus (16 combinations * 0.50)] / (34 combinations)

Total weighted equity equals 38.22 percent

With our knowledge of the way blockers work, let’s adjust the weighted equity calculation to describe the true match up of ace-king versus our opponent’s distribution. The result is different. The point being that intuition can be incorrect once we drill down to the specifics of the mathematics and that card removal affects weighted equity.

Total weighted equity equals [(3 combinations * 0.0816) plus (3 combinations * 0.3112) plus (6 combinations * 0.4394) plus (9 combinations * 0.50)] / (21 combinations)
Total weighted equity = 39.59 percent

Dead Money

39.59 percent equity is still not an impressive number. Such a small amount of equity discourages a lot of players from playing a hand in a particular instance. Before we write off moving all-in preflop with ace-king, let’s account for the dead money in the pot. Dead money is money that remains from the blinds and previous bets in the pot, which effectively lays us odds to make a particular play since we are “only” risking our stack size, but have the potential to win even more money. These odds laid to us means that we don’t need to win the hand the majority of the time to make a profitable play. While that concept may seem alien to some players, its importance is paramount. To push thin edges often that add up to being the best player at the table, you need to understand the concept of playing with the odds offered to you.

For our example hand, let’s pretend that everyone at the table starts the hand with $1,000. Our opponent and other players (including us) have already put $400 into the pot. With only two players remaining, him and us, we need to decide if that $400 buffer to our risk warrants moving all-in with ace-king. How do we do that with math?

Expected Value

Let’s say $100 of that money is the amount we reraised preflop (now it’s in the pot and independent of our future decisions — we’ll see why shortly). That means we have $900 remaining to risk by moving all-in. In order to determine the expected value of moving all-in preflop with ace-king and calculate the long term outcome as if we made the play constantly and the results were averaged (we would not always win or always lose any all-in confrontation in the long run), we need to account for the equity we have when called and the dead money in the pot. We earlier found that our equity when called against a distribution of pocket queens, pocket kings, pocket aces and ace-king is actually 39.59 percent, not the smaller 38.22 percent. We also learned that there is $400 of dead money in the pot, $100 of which belonged to us and constituted a preflop reraise. There is also $45 dollars in the pot from other players who have folded. Our opponent reraised our reraise from $100 to $255, so now we are facing a decision.

There are two possible outcomes once we are all-in and called: we win the pot 39.59 percent of the time, or we lose the pot 100 percent minus 39.59 percent of the time or 60.41 percent. Each outcome corresponds to a particular dollar amount. Once those two conditional outcomes are added together, we have the total expected value of moving all-in preflop with ace-king. Because the value of folding is zero (we neither win nor lose money from further betting), we know that the expected value (EV) only needs to be positive to justify moving in with our ace-king, meaning the value of moving all-in must be better than folding, or we would simply fold to cut our losses. The math looks like this:

Expected value equals p(win) * (amount won) plus p(lose) * (amount risked)
Expected value equals (0.3959 * $1,045) plus (0.6041 * (-$900))
Expected value equals $413.72 minus $543.69
Expected value equals minus $129.97

So far, the play seems pretty crappy! But what if there were another possible outcome? What if some of the time, our opponent is folding a bluff?

Folding Equity

The more aggressive and skilled players we are against should be bluffing with their preflop reraise some percentage of the time, which means when we move all-in that their hand is too weak to call off the rest of their stack. So there is a third possible outcome: our opponent folds and we win the pot immediately. We can factor that outcome into our EV formula to see how it affects the value of our preflop all-in. First, let’s break up the expected value formula into two parts, when we are called and when we are not called (our opponent folds). The times our opponent folds are equivalent to the times he is bluffing, so let’s use the value (x) to mark bluffing percentage. Since he calls the rest of the time that he is not bluffing, calling percentage is then (1-x).

Expected value = (1-x) * [Stuff that happens when he calls] plus x * [Stuff that happens when he folds]

We just solved for what happens when our opponent calls us: we win some and lose some, but the overall outcome is minus $129.97. When our opponent folds, we win the $400 in the pot without risk, a great boon to our expected value.

Expected value equals (1-x) * [minus $1,29.97] plus x * [$400]

And finally, we can set the expected value formula equal to zero and solve for the value x because we know that the value of moving all-in with ace-king simply has to be zero or greater. X, then, is minimum bluffing percentage our opponent has to have in order to justify moving all-in pre-flop.

0 equals (1-x) * [-$1,29.97] plus x * [$400]

0 equals minus $129.97 plus 129.97x plus 400x

$129.97 equals 529.97x

x equals 129.97 / 529.97

x = 0.245

Opponent’s minimum bluffing percentage for a profitable all-in is therefore 24.5 percent. Any decent player undoubtedly meets this requirement, but by how much? Well let’s consider an example. If our opponent believes that he’s bluffing preflop 40 percent of the time, then how often is he folding?

Remember, we have hidden information our opponent can’t access. Our opponent believes that he’s bluffing 40 percent times 34 combinations and value raising with all his Q-Q plus and A-K combinations, what he believes are 34 combinations. So instead of bluffing with about 14 combinations and value betting 34 combinations (48 total combinations), our opponent is actually bluffing with 14 combinations and value betting 21 combinations (35 total combinations)! That means our opponent is going to fold to our all-in bet when we hold ace-king 14 combinations divided by 35 combinations, or 40 percent of the time! Forty percent is much greater than the minimum 25 percent folding percentage that we require, so we have an easy all-in with ace-king.

Controversial plays in poker often mask higher truths. It’s important to be critical and to examine outcomes and variables on your own so that you become a better player. In the case of playing ace-king preflop, we know that if we account for blockers, weighted equity, dead money, and folding equity that we find an easily positive expectation all-in bet preflop with our example stack sizes. That even holds true if our opponent does not bluff!

I invite you to try the same calculation with ace-queen suited and with pocket tens. Replace the necessary variables and see if you should move all-in preflop with these two hands against the same opponent. If not, then what type of opponent would you look for before moving all-in with those slightly weaker hands? The more answers you find like these, like how to play ace-king preflop, the better a player you shall be. ♠

Reid Young is a successful cash game player and poker coach. He is the founder of TransformPoker.com.