Expected Value (EV) Calculations

Expected Value Calculations Part 4 - Implied Odds

3,984 Views 1 Comment on 24/4/15

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Our understanding Implied Oddsof Expected Value calculations is almost complete, but in part 3, Semi-Bluffing, we recognized there is a limitation we have been overlooking.

When calculating the expected-value of a call, we have only been facing all-in bets. In this scenario we can calculate our EV with a 100% certainty. When we are not calling an all-in bet this changes. We need to make estimates about what will happen on later streets.

This article assumes you are already familiar with the concept of implied-odds and pot-odds–it is recommended you check out the article on implied odds if you aren't sure. This article is not going to be a discussion of implied odds, but rather an examination of how various factors, such as implied odds, will affect our EV calculations.

There is $50 in the pot on the flop, with $200 effective stacks behind. You have a flush-draw with 36% equity vs your opponents range. Your opponent bets pot-size $50. What is the $EV of this call?

It would be very simple to just plug the relevant numbers into our EV formula. Those of you with a good understanding of pot-odds might recognize we are getting 2:1 on a call and therefore getting 3% more equity than we actually need. It should technically be a profitable call – if we really did have 36% equity. 

The problem is the $150 still left behind on the turn, exactly a pot-sized bet. If villain decides to shove the turn we can't call. We'd be getting 2:1 on a call, but this time only with 18% equity, which clearly will be non-profitable. The problem is not with the turn however, but the fact we didn't consider on the flop how later streets might play 

Let's imagine our opponent is going to shove the turn with 100% frequency. It would actually be incorrect for us to imagine we had 35% equity. Given that we will have to fold on the turn when we miss, we really only have 18% equity. If we ran the numbers on having 18% equity we'd see this is a non-profitable call.

This is still not the complete picture. If we did call, when we do hit, if villain is guaranteed to shove the turn we will make a lot more than the $100 that is already in the pot. In fact, if villain shoved the turn with 100% frequency we'd effectively be investing $50 to win $250. If we take into account our implied odds we are actually getting 5:1 on a call. We'd need only 16.667% equity to make the call, and we have 18%. 

Variables on Later Streets

The key to calculating EV of a call when there are stacks behind is to acknowledge the most important variables. By “variables” we mean values that can potentially change depending on which opponent we are facing or how the hand proceeds. 

1) How often will our opponent barrel on a later street?

We said our opponent is going to fire a turn barrel with 100% frequency. Why is this important? Firstly if our opponent were to barrel the turn less often it would effectively mean we had more equity. If our opponent never barrelled the turn, we could count all of our pot-equity. If we estimated he/she would barrel the turn 60% of the time we could discount our equity proportionally.

So if we expect our equity (36%) to be halved if we only see one card, rather than deducting the full half of our equity we could do -->

Full-Equity – (half-equity * barrelling-frequency)

0.36 – (0.18 * 0.6) 
0.36 – 0.108 =  .252    or  25.2% implied equity

2) How much will we make on a later street if our opponent bets?

In our simplified example we said that villain would shove all-in for his remaining $150 100% of the time. Sometimes villain will not necessarily give you everything that is remaining in his stack when you hit, especially if he is a bit deeper.

Perhaps he has $1000 left behind but you think he will only invest $350 of that with the second-best hand. In practice you will calculate an average because villain will pay you off varying amounts depending on his hand strength. So perhaps villain has $1000 behind when you make your flush and when he does pay you it'll work something like this →
  • 40% of the time villain pays you off $0  (check-folds)
  • 30% of the time villain pays you off $200
  • 15% of the time villain pays you off $300
  • 10% of the time villain pays you off $450
  • 5% of the time villain pays you off  $1000 
The most important figure to use in EV-calculations and implied-odds calculations is the average amount you will get paid. Since you get paid these amounts with varying frequencies it will also need to be a weighted-average. (Calculating averages was discussed in part 3). You could find the average amount you get paid by

(40 x $0) + (30 x $200) + (15 x $300) + (10 x $450) + (5 x $1000)     =  $200
                                   100

$200 is the average amount we'd make in this spot. While it's true we may sometimes get paid off $1000, if we wanted to calculate implied odds we should use $200, as this is the amount we will make in the long-run.

3) How often will our opponent fold on a later street?

Much of the time this is not considered as implied odds but is included in it's own category - “fold-equity”. It can still be considered a form of “implied-odds” however.

In our example with a flush draw, imagine villain will barrel the turn 50% of the time. The 50% of the time he doesn't barrel he will check-fold. If we would just check the hand down when we miss our draw, we couldn't technically make the flop-call. We don't have sufficient implied-odds because villain is not barrelling often enough when we hit our draw. 

However if we were to take advantage of your opponent's tendencies to check-fold the turn we would net ourself an extra $150, 50% of the time on the turn. Our additional equity through our chance to steal makes up for our lack of implied-odds on the flop. While “fold-equity” is not strictly the same as “implied-odds”, they counter-balance each other and both add to our (implied) equity.

Putting it Together

Let's continueimplied odds with the previous example in section 3) but run some numbers. Have a go at it yourself first. And don't worry, this is as tough as it will get in this series on EV calculations.

There is $50 in the pot on the flop, with $200 effective stacks behind. You have a flush-draw with 36% equity vs your opponent's range. Your opponent bets pot-size $50. On the turn you expect your opponent will shove 50% of the time on the turn and you will have to fold unless you hit. The other 50% he will check and fold to any bet. What is the $EV of this call?

We begin to see that EV calculations can start to become increasingly complex. In reality villain's turn play will not be this black and white. Sometimes he might check-call; sometimes he might bet an amount that isn't a shove. Maybe there is a 1/7 chance he might shove the turn with 100% frequency depending on whether it is his preferred night of the week for going out drinking or whether his wife is currently throwing objects at him. The permutations are endless. We have to draw the line somewhere and understand that we cannot calculate our EV exactly.

Let's break this problem down into stages. As discussed previously, since we need to fold on the turn if our opponent barrels we really have only 18% equity, not 36%. However, we expect to have some implied odds on our call. When we hit we expect to make an extra $150 50% of the time. We could say that on average we expect to make $75 every time we hit. We are therefore calling $50 to win $175. So to run the EV calculation purely on our call

(0.18 x $175) – (0.82 x $50) = 
($31.5)          -       ( $41 )       =     $-9.5

We can see that we don't quite have sufficient implied odds to make the call on the flop. We are not making enough on the turn.

However, when we miss, 50% of the time we expect our opponent to check, allowing us to take down the pot with a bluff to make $150. The EV of the turn situation assuming we've missed is therefore $75. We shouldn't even need the calculation for this

(0.5 x $150) – (0.0 x $0) = $75 

Unfortunately we can't simply add this $75 to our flop call EV. Everything needs to be weighted appropriately – we can't just “assume” we've missed” because we only miss 82% of the time. This 82% of the time we miss our hand, we don't really lose $50 because we immediately proceed to make an average of $75 on the turn.  We effectively make  (-50 + 75) $25 every time we “lose” (82%) and can write our EV formula --->

(0.18 x $175) + (0.82 x $25)
    31.5             +     20.5          =   $52

Note that instead of calculating the average amount we make on the turn we could have broken the calculation down into stages. 41% (50% of 82%) of the total time we will make $100 overall (we don't count hero's $50 flop call as winnings) while 41% of the time we will lose $50 (hero's flop call).

(0.18 x $175) + (0.41 x $100) – (0.41 x $50)  = $52

For those of you who are struggling to follow, remember that we can simplify everything by means of a flow chart. It does takes a little longer than just thinking logically about the situation, however it is a lot easier to understand, and you are less likely to make errors this way.

(0.09 x $250) + (0.09 x $100) + (0.41 x $100) – (0.41 x $50)

$22.5          +             $9           +   $41              -      $20.5                 =       $52     

In reality this is only one step more complicated than in our last article. We have 4 total outcomes rather than 3. We could easily have more than this. If villain had more realistic turn tendencies we might expect him to check-call sometimes rather than fold outright. If this were the case we'd need to factor in the size of hero's bet, because sometimes he'd lose it.

We'd also need to factor in hero's pot equity on the turn, because sometimes his bluff would fail but he'd still get there by the river. If you fancy doing further calculation try making a flow-chart for this. Imagine Villain folds 70% on the turn after he checks and hero shoves $150, but when he calls hero will have 18% pot-equity.

This is it for intense calculation. In part 5 we will be discussing various ways of measuring EV and their relevance to us. 

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w34z3l

I am of British nationality and go by the online alias w34z3l. I am considered one of the top consultants in the field for technical analysis (i.e. database work) and application of game theory concepts to various card games. I make a range of educational content ( ... Read More

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suertudojack

suertudojackon 16/6/16

For me, everything is easier with decision trees because I actually visualize every single action. I think is good to know how to do the maths manually before using the softwares available out there :)

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