Poker and Rational Thinking - Part 2

In part two of this series we will analyse the relevance of our pot-equity in making our decisions. While it is typically considered a very important factor, our examination should lead us to understand that equity is of limited use in many situations.

Equity can be defined as - "The percentage of the time our hand will win at showdown if both players check down"

Imagine a situation where we call an all-in preflop for 100bb. Would we rather hold AK, 22, or 9Ts? The answer really depends on what our opponent is holding. If we knew our opponent held specifically AK we'd prefer the 22. If he held 22 we'd prefer the 9Ts. If he held the 9Ts we'd prefer the AK. This example illustrates that hand rankings are not absolute - it would be impossible to rank all starting hands in NLHE from the worst to the best.

This translates to postflop situations also. If we want to rank holdings from best to worst we'd need to know our opponents precise range. However this still assumes that we are all-in on the flop. The only time raw-equity is the most relevant concept is when we are all-in (or very close to it). Any time there are stacks left to play for, other factors take precedent.

Let's say we open from the BTN with 500bb effective stacks, and BB who is a huge nit decides to 3bet us for a large sizing. For the purposes of this example let's say we can define our opponents range clearly as AA/KK. We hold 22 which gives us around 18% equity, depending on the exact suits of our hand.

If we look purely at our equity here this is a pretty clear fold. We are crushed. However what does your instinct tell you is the best option here? It's a clear call. We have a strong idea of our opponents range which will allow us to play close to perfectly postflop. If we hit our set there is a reasonable chance we can extract a decent proportion of those 500bb effective stacks.

It should be a familiar concept to us, one which we refer to as implied odds. We don't even think about our equity here. In fact the weaker our opponent and the more accurately we can define his range the less it really matters how much equity we have when the stacks are sufficiently deep. We'd play 23o in this situation if we could see our opponents hole-cards and knew he had big postflop leaks.

This is the opposite situation to that described above. Let's imagine we open UTG and a very nitty player 3bets us from the BTN with a reasonably small sizing. We hold KQo. Depending on what we think of our opponents range, we may have a pot-odds call based on our equity. However what does your instinct tell you is the best option here?

It's probably better to fold here. The problem is that on a Kxx board we can lose a considerable amount to AK, AA, KK. Assuming our opponent holds something like QQ/AQ meaning our hand is good, we probably won't get a big payout. We refer to this kind of situation as suffering from reverse-implied-odds.

Although we may have the equity to continue initially, we realise there is a lot more to play for on later streets. We would in essence be putting our whole stack at risk to protect our hand preflop. As a result of the remaining stacks we are no longer getting the correct risk:reward for a profitable call, despite having the correct equity if we were to refer directly to our pot-odds.

To put this another way, despite our opponent not directly investing his entire stack preflop - we need to factor in these additional chips to our decision. The simple presence alone of these remaining chips make calling preflop significantly harder. This is a concept we can refer to as leverage. While those chips are currently out of play, the potential threat of them being used aggressively on later streets makes our effective preflop price significantly worse.

100bb effective stacks, MP opens to 3bb. Hero is on the BTN. Would we rather cold-call here with 8d9d or Ac8d. What does your instinct tell you?

It's likely that you feel 8d9d is a more profitable cold-call in this situation. Ac8d has more equity however (vs an average MP rfi range). So why is it that we feel that we should cold-call the lower equity hand as opposed to the higher equity Ac8d?

Essentially it has to do with the playability of our hand. Imagine the flop comes down Td5c2h. Ac8d actually still has the higher amount of equity but it's very tough to call the flop here facing a cbet. We may even feel reasonably comfortable calling the flop with 8d9d here however. The hand has much better playability on later streets and will be able to realise it's equity much more frequently than the Ac8d. What do we mean by this?

Firstly think about the additional cards that can help us when we hold the 8d9d compared to when we hold the Ac8d. There are 10 cards in the deck that can give us a flush draw. There are 8 cards in the deck that can give us an open-ender, the 7's and the J's. 2 of these are diamonds however so we will count 6. There are also 8 cards in the deck than can give us a gut-shot, again two of which are diamonds so we will count 6.

There are also 6 cards that can give us a pair and potential 2-pair/trips draw by the river (maybe even the best hand on the turn). This means that 28 turn cards can potentially improve our hand here, that's over half the remaining cards in the deck! Many of these cards will allow us to continue on the turn and see a river (hence fully realising our equity). We can also potentially make strong 5-cards hands by the river (straights/flushes) which we can feel very comfortable about calling a 3rd barrel with.

Let's now contrast this to the Ac8d which has better equity on the flop when compared to the 8d9d. How many turn cards improve us in this case? We have 6 cards only, the 3 Aces and the 3 8's. Even if we hit one of these cards it can still be pretty awkward to continue on the turn with the knowledge that facing a third barrel on the river is going to be extremely tough. We are going to be folding this hand much more frequently postflop which makes it very hard for us to realise our equity effectively. Ac8d may be a slight equity favourite in this example situation, but the 8d9d is a far superior hand.

As mentioned our raw equity is most relevant in all-in situations where no further betting takes place. This is because when we bet we a) have fold-equity on top of our pot equity b) manipulate the equity we have as a result of our opponents narrower continuing range.

Good poker play involves using our aggressive actions to our advantage not to our detriment. 

Let's imagine a way-ahead/way-behind situation. We have 95% equity vs our opponents range. 95% of his hands are way-behind while 5% of his hands are way-ahead and have us crushed. If we were to purely consider our equity in this situation we might think it's great to put out a large bet here. After all, we have such a huge amount of equity.

This would be a huge mistake however. Facing a large bet our opponent may fold all of the hands that are way behind and only continue with that 5% of his range that has us crushed. Despite having a huge amount of equity, it's clearly not the only factor here and we can't profitably bet large for value.

This is not to say equity is of zero relevance and can't help us. It is actually extremely important in some situations, just not all of them or even most of them.

Equity is most relevant in situations where we are either all-in or very close to it. For example the shorter the effective stacks, the more relevant our equity becomes. We are closer to an all-in when the stacks are shallow and we also cannot take advantage of some of the other discussed concepts such as implied odds. With shallower stacks we will realise our equity more frequently regardless of the action.