The Equity Delusion: How much does chasing cost you?

Take a look at the picture above and ask yourself, “Why are these men running?”

We can’t be sure, but the chances are that they aren’t running from some rabid beast. In fact they likely aren’t running from anything at all. We can infer this from the fact that not only are these gentlemen wearing inappropriate beast-fleeing attire but that the lead character is looking at his watch. All rather mundane, so why, dear reader, have I bothered bringing this to your attention?

For those of you hoping for a surprise beast-chomps-man denouement complete with a graphic YouTube video this blog post will be something of a disappointment. My readers with a poker-focused disposition will have a lot more fun and should read on.

The point is that these gentlemen are roused to action by anticipatory anxiety: the threat of something bad happening in the future. In this case our heroes are presumably at risk of missing a connection or event. We routinely account for future events in our life decisions yet it seems harder to do in poker. Why?

The answer lies in the uncertainty built into a poker game. For the suited sprinters in the picture the situation is very simple as there are only two, mutually exclusive outcomes- they either make the event or they miss it. Thus the future outcomes {Attend Event, Miss Event}, and their consequences {Payoff(Attend Event), Payoff(Miss Event)} are both known. It is thus easy to arrive at the correct decision- in this case to run- which increases the probability that they attend the event.

The reason contemplating future events is more difficult in poker is because the future events and their associated payoffs are conditional upon changes in the board texture, our opponent’s future actions and even our own future actions!

We must endeavor to negotiate this uncertainty if we want to play strong poker. The future events are just as consequential for our poker decisions as (more) predictable events are for our life decisions.

This brings us to the theme of this article. Below I will demonstrate a critical deficiency in a simplistic, equity-based approach to poker situations: failing to account for future bets.

To arrive at the situation displayed below we have called a raise from the BB and then check-called the flop and the turn for a pot-sized bet on each street. We then checked the river and now face a pot-size bet:

How much equity do we need against our opponent’s betting range for this to be a profitable call?

This is a simple calculation,

Minimum Equity = Price to call/Total Pot after we call

Minimum Equity = $300/($300 + $300 + $300) = 1/3 = 33.3%

So we arrive at the (hopefully familiar) conclusion that to profitably call a pot-sized river bet we must have at least 33.3% equity against our opponent’s river betting range. All that remains is to plug our opponent’s river betting range into Pokerjuice and check our hand vs range equity on the river.

So far we haven’t gone outside of the reader’s comfort zone, so it’s time to ramp up the complexity. Let’s roll back the river and consider a superficially similar decision on the turn.

Remember, so far we have called a raise from the BB and then check-called the flop. We now check the turn and face a pot-sized bet from our opponent.

How much equity do we need against our opponent’s betting range for this to be a profitable call?

You would be forgiven for saying 33.3% again, but you would be wrong. In the paragraphs that follow I will demonstrate to you why that’s a poor answer and why, in fact, asking, “How much equity do we need?” is usually a poor question.

We’ll begin by building a decision tree at the point where we face a bet from our opponent on the turn. In order to keep the size of the game manageable we’re going to make some limiting assumptions:

1. Hero’s choice of actions on each street is constrained to CHECK-CALL or CHECK-FOLD.
2. Villain’s choice of actions on each street is constrained to BET POT or CHECK BACK

These assumptions do not represent a major deviation from how most of you would play this hand presently. However they are quite limiting when we move to a range versus range approach and so we will revisit them in our concluding analysis.

Now we are ready to build our decision tree:

Each of the diamond shapes represents a decision point for either player and each of the hexagons represents an end point or ‘outcome’ of the game. Notice that, even with the constraints on each player’s action, we still have four possible outcomes to account for at ‘Decision 1’.

What follows is a ‘Proof by Counter-example’: we are going to create a greatly simplified range for Villain which is a 2-1 favorite against us on the turn. We’re then going to calculate the Expected Value (EV) of our Turn Check-Call against this range. If this is a losing play for even this single example then we will have shown that simply having 33.3% equity against our opponent’s betting range on the turn is not sufficient justification to call a pot-sized bet.

Consider opponent’s range composed of two distinct hands, each of which you have the correct equity to call against individually.

1. A♥A♠J♦3♦ is a 67.5-32.5 favorite against us on the turn.
2. A♣A♠Q♣K♦ is a 67.5-32.5 favorite against us on the turn.

Notice that because I have specified the hole cards exactly for each hand we can construct our opponent’s turn betting range as weighted equally between the two hands.

After we check-call the turn we arrive at the river and our equity against our opponent’s range is conditional upon the river card, as shown below:

• Since our actions are constrained to either check-call or check-fold the river we automatically check-fold on the 22 cards where our equity is zero (greyed out in the image)²
  Profit per card = -$100
• On 6 of the cards where we have 100% equity {A♣, T♠, T♥, T♦, 7♥, 7♦ } it makes little sense for our opponent to turn either hand into a bluff so he checks back and we win the pot.
  Profit per card = +$200

This leaves us 15 cards to consider, 14 of which we have around 50% equity versus his range and one, the Q♣, where we have 100% equity.

• We will first consider the 8 remaining flushing rivers. We’ll assume our opponent will bet every time when he hits his flush and will attempt to balance this by bluffing half the time with his non-flush hand. So on a flushing river he is firing 75% of the time and when he does bet we are indifferent between calling and folding on the river. When 6 of the the flush cards hit we (effectively) check-fold  75% of the time and win at showdown the other 25%. For each of those 6 cards our profit on the hand is:
  Profit per card = 0.75*($-100) + 0.25*(+$200) = -$25
• For the remaining Q♣ we generate implied odds as a result of our opponent’s strategy because he can only hold a bluff, which he bets on the river exactly half the time:
  Profit per card = 0.5*(+$500) + 0.5*(+$200) = +$350
• The other 8 cards are the three non-flush 9s, the two non-flush 8s and the three non-flush 4s. I will assume that our opponent will check back his entire range on the 9s and 8s, allowing us to realize our equity. On each of those 5 cards our profit is:
  Profit per card = 0.5*(-$100) + 0.5*(+$200) = +$50
• On the three non-flush 4s our opponent can make us indifferent to calling or folding with our two pair by betting half of the time with his bluffs, as before:
  Profit per card = 0.75*($-100) + 0.25*(+200) = -$25

We are now ready to construct an EV calculation for our turn decision. We’ll do this by finding the average profit: summing the results above for each river card and then dividing by the total number of possible river cards.

There is one additional complicating factor, which is that the weighting for the 6 river cards which we know to be in our opponent’s turn betting range is half that of the other 37 which are not.

So rather than divide by 43 cards, we are going to double-weight 37 cards and single-weight 6 cards and thus divide by 80 (37*2+76 = 80). Here we go…

EV (Turn Check-call) = (40*(-$100) + 11*(+$200) + 12*(-$25) + 1*(+$350) + 10*(+$50) + 6*(-$25))/80 = -$17.50

Under our assumptions then:

EV (Turn Check-call) = -$17.50

in a situation where we had 32.5% equity against our opponent’s turn betting range.

It’s important to understand exactly what I have and have not achieved above. I have not proven that check-calling a pot-sized turn bet with 9♣8♠7♠4♣ is a losing play against every conceivable turn betting range.

I have shown that it is possible to construct a turn betting range for our opponent which we have 33% equity against and yet the turn check-call is a $17.50 mistake. In so doing I have proved by counter-example:

With no threat of future bets the answer is once again obvious:

Minimum Equity = Price to call/Total Pot after we call

Minimum Equity = $100/($100 + $100 + $100) = 1/3 = 33.3%

An equity-based approach to the turn decision is maximally useful at a Stack-to-Pot Ratio (SPR) less than 1.  For higher SPRs a decision metric dominated by equity is fundamentally flawed. It is the future bets on the river which dictate whether a turn check-call is actually profitable whenever there is money left to play for after the turn decision.

This result should accord with your intuitive belief, likely developed from hundreds of thousands of Omaha hands, that this is an uncomfortable spot to check-call the turn. Still, the heuristics that most players use to make a decision in this spot are very crude.

One approach is to simply count ‘outs’ and estimate the equity we hold against our opponent’s range from that. A more sophisticated approach leads us to adjust for ‘implied odds’: those future bets we profit from hitting our flush and having our opponent pay us off with a poorer hand. We soon discover that there are unfortunate cases of ‘reverse implied odds’: those future bets we lose from improving to an inferior bluff-catching hand.

Before we finish, we’ll take a moment to consider the implications of this result on our strategy in this situation.

Let’s revisit the constraints we used to construct a decision tree for this situation:

1. Hero’s choice of actions on each street is constrained to CHECK-CALL or CHECK-FOLD.
2. Villain’s choice of actions on each street is constrained to BET POT or CHECK BACK

If we focus on our own actions we see that it is possible to increase our expected value by including either a river leading frequency or a check-raise frequency. This is because strategic options have non-negative value³. If we play optimally on the river, the addition of these options with a non-zero frequency would have to increase our EV for it to be a rational deviation from a strategy built only from the plays {Check-Fold, Check-Call} .

Now that I’ve shown you the limitations of equity it’s your turn to tell me some factors that would encourage you to check-call the turn.  To get started in the right direction, ask yourself why our EV would increase if we knew our opponent’s hand..

QUESTION(S) OF THE WEEK: What conditions are there on our opponent’s river betting range that would increase the Expected Value (EV) of our Turn Check-call? Look back carefully over the steps in the profit calculation for clues. If the community can find three different possible imbalances in our opponent’s river betting range then I’ll write a follow-up post on this topic. One imbalance only per commenter please!