The expected value in SNGs part 1

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The expected value in SNGs part 1

from Bobbs

To start with, you should review the definitions of ICM that can be found in the following article: To the article.

EV (expected Value): also called the chip-EV, it is the number of
chips that you expect to have on average after a certain action. An
action is +EV if it increases the chip count on average.

$EV: This is the estimated $ value of the EV. An action is +$EV if it increases the expected profit on average.

In the first part of the column I'll show you
how to calculate $EV with the formulas you already know from
cash games and the ICM. I will also show you how the necessary fold
equity for a resteal changes in the mid game when comparing cEV and $EV
or MTT and SNG, respectively.

As
the $ value of a chip changes over the course of a SNG, the odds change
as well. Chips lose value over the course of a SNG, which is why you
need better odds for drawing. Therefore I want to show you some
examples of how our $odds change in comparison to the chip odds in the
early, middle, and late stages of a SNG, where people like to argue
with cEV.

We will try to answer the following questions that arise in the different phases of a SNG:

Early Phase:
What odds do I need in order to be able to continue playing my flush draw?

Middle Phase:
My opponent bets 1/3 PS, how often do I have to be ahead to be able to call profitably?

Late Phase:
I get odds of 2:1 (in chips) for my call. How much equity do I really need to make +$EV call?

Lets start with some basics: Every chip that we win is worth less than a chip that we lose.

Example 1:
Every chip in our starting stack, purchased for 100$ is worth 0.05$. However, if we were to win the SNG,
we would have 20000 chips and win 500$ in a 10 player SNG where
all the chips will have lost half of their value. They are only worth 0.025$
now.

We would only make use of the $EV in the push or fold
phase though. When we are playing an SNG, we are trying to maximise our
$EV. In many cases, people overlook this fact and start to make use
of the cEV again.

Obviously there are cases in which the $EV and the
cEV are very similar, so that it makes no difference whether you use the $EV or the cEV in order to reach a decision. These cases are very rare in SNGs
though.

Many players justify their decision arguing that their action is +cEV. However, it is not
enough to make profit in the long run because the value of the chips in
a SNG depends on the number of players and the different stack sizes.
(In reality, their value is also dependent on your position and skill).

Example 2:
In this example I will compare
the cEV and the $EV for doubling our stack and determine how much equity
we need against the range of our opponent to call a push profitably.

No-Limit Holdem Tournament

Blinds: small
10 players

Stack sizes:
UTG: t2000
UTG+1: t2000
UTG+2: t2000
MP1: t2000
MP2: t2000
MP3: t2000
CO: t2000
Hero: t2000
SB: t2000
BB: t2000

Pre-flop: (10 players) Hero is Button with xy
UTG pushes all-in for t2000, 6 folds, Hero ?

Disregarding the profit from winning the blinds
and the possibility that the SB and the BB could have a very good hand behind hero, we are looking at the following question: How much
equity do I need to call profitably?

cEV:
We can easily figure out our cEV by determining our odds in order to conclude our equity from these:

cEV(Hero) > 0, if Hero has more chips after a call than when he folds on average. If this is true: Chips(Hero| Call) > Chips(Hero| Fold)

It is clear that hero would still have 2000 chips
if he folds and 4000 chips if he wins. We can easily calculate our
equity using these values:

P(Win)*Chips(Hero| Win| Call) > Chips(Hero| Fold), therefore
P(Win) > 2000/4000 = 0.5

We therefore have to win more than 50% to call with +cEV.

$EV:
So what is our $EV? By using the ICM-calculator (for example ICMCALC) we calculate our $Equity for folding, calling and winning 2000 chips.

We get the following results:
$Equity(Hero folds) = 0.1

Hero is entitled to 10% of the prize pool, which is equivalent to his buy-in.

$Equity(Hero | Hero wins| Hero calls) = 0.1844

Hero is entitled to 18.4% of the prize pool.

Notice the devaluation of the chips. Our first
2000 in chips are worth 10% of the prize pool, but the next 2000 chips
are only worth 8.4% of the prize pool.

We basically don't have 4000 chips of the same
value but rather 2000 starting stack and roughly 1680 new chips (the
additional 2000 chips are only worth~84% of the starting stack). Our "real" EV is therefore:

rEV(Hero) > 0 gdw P(Win) = Chips(Hero| Fold)/Chips(Hero| Win| Call) = 2000/3680 > 0.54.

We therefore need 4% more equity than in a cash game to come to the same expected value.

You don't have to determine your $EV the
complicated way. You could determine it by looking at your $Equity
rather than your chips.

$EV(Hero) > 0 gdw. P(Win) > $EQ(Hero/Fold)/$EQ(Hero/Win/Call) = 0.1/0.1844 = 0.54

We can reach a couple of conclusions through this example:

1. In general we need more equity for a call in a SNG than in an MTT or cash game.

We can conclude further:

2. We should avoid marginal calls in SNG, like draws, that are good in cash games.

3. It is +$EV to
stick around and fold in a SNG. In the example above, 15% of villains
buy-in are distributed among the eight other players who didn't
participate. These players therefore profit from our call without
having risked anything.

4. In turn we profit from players that risk their chips in the early phase of a SNG with marginal hands.

To summarise, we can say that it becomes very
obvious, in example two, that playing tight
in the beginning of a SNG to protect your chips is really important. Drawing expensively
should be avoided.

One ability a poker player should have, is to see
whether he made the right decisions while playing, or in his analysis
after playing. As soon as we reach the push or fold phase, most of this work is done for us in tools like the SNG wizard or the SNGPT.

However, some players are not able to calculate the expected value (in $) for certain situations because of this.

After I have carried over the formulas from cash
games to SNGs, I will show you how you can use this analysis for
resteals by using some examples.

How much equity do I need to profitably call a push?

If I push from the SB and the BB has me covered, the term EQ(Hero| Lose| Call) is removed and the following term is used in its place:

P(Win) = EQ(Hero folds)/EQ(Hero calls)

I often see the following sentence in our sample hand forum: "I think I have enough fold equity to push, is that right?"

Due to this I want to analyse a situation in the middle phase of a SNG to determine the difference in playing SNGs and MTTs.

Resteals are of very high importance in MTTs. Usually it is argued as following.

"I am holding 89s and CO raises with a wide range
here. I have the perfect stack to resteal with my 15 BB and even if he does
call I roughly have~40%(35%) equity.

Example 3:

Lets move on to example 3.

BB/SB = 100/50

Spieler 1: 1830
Spieler 2: 3420
Spieler 3: 2590
Spieler 4: 3030
Villain : 3880
Spieler 5: 1900
Hero : 1460
Spieler 6: 1890

Everyone folds to villain in the CO who raises to 300. Hero holds 9 8 and thinks that villain raises a wide range here. How often does villain need to fold for the move to be +EV:

a) in an MTT
b) in a SNG

a) I assume that we are far enough from the bubble so that we can
rely on the cEV to help us reach a decision. Furthermore I assume that
we have 40% equity after a call from villain. We then calculate the
necessary equity (for a reasteal with EV = 0) like so:

EV(Hero pushes) = EV(Hero folds)

H = Hero and V = Villain. The results are as follows:

Therefore this applies:

Therefore:

P(Fold) = [1410 - 1208]/[1860 - 1208] = 202/652 = 0.31

Villain needs to fold 31% of the time so that the push is
profitable. This is equivalent to him folding 31% of his range. If villain raises 40% of his hands and only
calls 20%, a push would be very profitable here. Lets look at the same
example in a SNG with the same stacks.

b) The same formula applies here if we substitute our chips with our $Equity.

Equity(Hero| Villain folds| Hero pushes) = 0.0968
Equity(Hero| Villain calls&loses| Hero pushes) = 0.1482
Equity(Hero| Hero folds) = 0.0753

P(Fold) = [Equity(H| H folds) - P(Win)*Equity(H| V calls&loses| H
pushes)]/[Equity(H| V folds| H pushes) - P(Win)*Equity(H| V
calls&loses| H pushes)]
P(Fold) = [0.0753 - 0.05928]/[0.0968 - 0.05928]
P(Fold) = 0.016/0.0375 = 0.42

Villain has to fold 42% of the time which is 11% more than in case a.
Calculating P(fold) for example a and b with the new P(Win)= 35% gives
a value of P(Fold) = 0.43 for a and P(Fold) = 0.52 for b.

Our opponents have to fold a lot more hands in a
SNG than in an MTT to make our resteal profitable. From this, it is clear that you have to be very careful with resteals in SNGs.
Without any reads, they should be avoided, as opposed to MTTs which
are about winning chips.

» SUMMARY

The analysis above shows some of the most
important concepts for playing an SNG. You should be trying to protect
your stack because its value (in $) automatically increases during the course of the
tournament which is not the case in an MTT or a cash game.
Many useful practises used in cash games and MTTs are not profitable in
SNGs.

You should be especially careful with
resteals. In most cases, you need a much higher equity against your opponent's range than you might expect. One further piece of wisdom to be gleaned from this examination is that coin flips in the early stage of a SNG should definitely be avoided.