## Looks At Whether Bets That Maximize Expectation or Bets Which Are More Likely

To Be Called Are Best On The River In SNG Tournaments.

The excellent 2+2 Book ‘**NL Holdem: Theory and Practice’** by David Sklanksy and Ed Miller – Has a list of important NL Holdem concepts. This Sit n Go Strategy article was inspired by ‘concept number 51’.

To summarize this: The authors suggest that in a *NL Holdem Poker Tournament one should prefer small river bets that will often be called to large ones that will seldom be called.* This refers to Heads-Up Pots and rests on the assumption that additional chips decrease in value in a poker tournament situation - and so the additional expectation for the seldom called large river bet is cancelled out by this decreasing chip value.

Our question is whether this applies to **SNG Strategy** too?

Here is the hypothetical scenario...

There are 6 players left, equal stacks of 2000 chips and BB=100. You checked your BB after 1 mid position limper and then called a 100 chip bet on the flop after flopping middle pair – the turn went check / check and you hit an unlikely gutshot straight on the river. There is 450 in the pot, both you and your opponent have 1800 behind and you put him on a medium strength hand (maybe top pair top kicker)... the question is - **how much should you bet?? **(assuming that you have the nuts / no shared cards etc).

You estimate the likelihood of being called for various bet sizes as follows...

- Bet = 200, you'll be called 100% of the time
- Bet = 400, you'll be called 50% of the time
- Bet = 1800, you'll be called 11% of the time.

As I am sure you noticed the expectation in terms of chips gained is the same for each bet (200 chips - ok a couple of chips different for 1800 but close enough!!). So what are the factors that could influence your decision as to which amount to bet??

We will break this down into 2 areas - mathematical (Independent Chip Model) factors and strategic factors. Let us start with the numbers...

At the start of the hand every players $ equity was equal, with a $100 prize pool you each have $16.66 (ignoring the blinds). If there were no betting on the river and you took the current 450 chips in the pot then the $ev for the table would look like this. $ev assumes a $100 pool split 50/30/20.

- You = 2250 chips = $18.34
- Villan = 1800 chips = $15.28
- Player A = 2000 chips = $16.68
- Player B = 2000 chips = $16.68
- Player C = 2000 chips = $16.68
- Player D (SB) = 1950 chips $16.34

Let us look at what happens in the 3 scenarios...

**1) Bet 200 chips and called 100% of the time;**

- You = 2450 chips = $19.63
- Villan = 1600 chips = $13.85
- Player A = 2000 chips = $16.72
- Player B = 2000 chips = $16.72
- Player C = 2000 chips = $16.72
- Player D (SB) = 1950 chips $16.37

You increase your $ equity by $1.29 over checking, Villan loses an additional $1.43.

**2) Bet 400 chips and get called 50% of the time. (equity where called)**

- You = 2650 chips = $20.89
- Villan = 1400 chips = $12.36
- Player A = 2000 chips = $16.77
- Player B = 2000 chips = $16.77
- Player C = 2000 chips = $16.77
- Player D (SB) = 1950 chips $16.43

So your equity is now $4.23 higher than before the hand and Villan's is $4.30 lower - but hang on this only happens half the time - it thus makes sense to take the '100% called' smaller bet on river as a baseline and compare the equity here with that... so $1.29 (the assured equity gain after smaller bet) is your risk.

Half the time the medium bet is called and your equity goes up from $19.63 to $20.89 - a gain of $1.26 divide this by 2 and we have a risk of $1.29 to win ($1.26/2) = 63c.

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**3) Bet 1800 Chips and Get Called 11% of the Time...** (will work with 10% to keep the numbers clear!!)

- You = 4050 chips = $28.90
- Villan = 0 chips = $0.0
- Player A = 2000 chips = $17.87
- Player B = 2000 chips = $17.87
- Player C = 2000 chips = $17.87
- Player D (SB) = 1950 chips $17.51

So your gain for the bigger bet when called compared to checking it down is $28.90 - $18.34 or $10.56... but this will only happen approx 10% of the time; so out of 10 tries you lose your assured 200 chips for the smallest bet 9 times (9*$1.29 = $11.61) and gain the difference between your assured 200 chip win and the bigger 1800 chip win once ($ 28.90 - $19.63 = $9.27).

So on average you actually give up equity by making the biggest bet - at a rate of $2.34 per try.

To conclude the mathematical part:

It seems as if Sklansky and Millers 'concept' holds up well when we look at the numbers. The key point here is that additional chips are worth less in terms of $ equity than previous ones (a key concept of ICM). Additionally, If you look from the baseline through the bet sizes you will see that the people not in the hand are steadily gaining $ev by doing nothing!!

**What about the strategic reasons?**

The question to ask here is whether there are any factors that mean having a 2650 stack as opposed to the assured 2450 stack would make the small equity risk worthwhile?

With such a small gain I can not honestly see it - you have more chips than your opponents in each case, but not enough so that getting into a confrontation with any of them would leave you unscathed (possible exception of the now short-stacked Villan in the above example).

Of course having the extra large 4050 stack would put you in a fantastic position to expolit the upcoming bubble - but giving up a potential $2.34 in $ev will not be compensated for enough. ie; you could argue that the times you do double up you can gain back more than $2.34 at the bubble by bullying the smaller stacks - but since this will only happen 1 time in 10 it seems clear that the smaller gain gives you more opportunity more often to get into the money...

It appears that ‘concept 51’ is good in terms of Sit n Go Strategy - and as ever in poker there is always an 'it depends' about things... hope to have provided some food for thought though!