Betting a draw for implied value

[_TKO_]

You may have read somewhere that betting with a draw builds the pot so you can make more when you hit. I think this applies particularly when you have little to no fold equity against a loose and passive opponent. I decided to check the math behind some of these assumptions using a common example for the foundation.

SB: $100
BB: $xx
UTG: $xx
CO: $xx
Hero (BTN): $100

Preflop: Hero is dealt K[img]/images/graemlins/diamond.gif[/img] 9[img]/images/graemlins/diamond.gif[/img] (5 Players)
2 folds, <font color="red">Hero raises to $4.00</font>, SB calls $3.50, BB folds

Flop: ($8.50) 2[img]/images/graemlins/spade.gif[/img] 3[img]/images/graemlins/diamond.gif[/img] A[img]/images/graemlins/diamond.gif[/img] (2 Players)
SB checks, <font color="red">Hero bets $6.00</font>, SB calls $6.00

Turn: ($20) J[img]/images/graemlins/heart.gif[/img] (2 Players)
SB checks, Hero ...

Okay, each of us has $90 left. We need to have some assumptions in place to illustrate the point.
1. Villain is passive.
2. Villain will pay off river for about a pot size bet.
3. Villain will check river when we miss, and we will never have the best hand.

So, is it better to bet or check this turn?

Well, if we check and hit, let's say we can get paid on $20. We are 20.46% to hit. So, 20.46% of the time we win $20, and 79.54% of the time, we make nothing.
EV = 20 * 0.2046 = $4.09.

Let's say we make a bet, b, and get called all the time, and the same conditions above apply for the river. By that time, the pot will be $20, plus 2b, since we each put the bet in the pot. When we hit (20.46% of the time), we will win the turn bet, plus an additional 20+2b on the river. When we lose, we lose b.
EV = 0.2046*(b+20+2b) - 0.7954*b
EV = 4.09 + 0.6138b - 0.7954b
EV = 4.09 - 0.1816b

For the non-math people, what this means is that every dollar of bet you put in the pot means you lose $0.1816 in EV. Once you hit a bet of about $23, you actually start to move into negative EV territory. So, with these assumptions, betting here is bad, and it’s not close. Or is it?

Since you only give up 18 cents of every dollar to your opponent, this leads me to believe that there might be some cases where betting is +EV. This shouldn't surprise you. What might is that I want to take a look at some of the possible areas of EV increase. Because a multi-variable problem such as this one can be complex to analyze all at once, I’ll try to look at the effects of individual variables first, and maybe see if I can combine the results afterwards.

In NL, the math behind EV determination is greatly simplified when it’s a linear problem. What this means is that an EV calculation is usually ignorant of bet sizing, save for its inevitable effect on the magnitude of the EV. Okay, I realize that might sound complicated and idealized, so I’ll try to explain it further. In the above example, we saw that by checking behind, we had $4.09 EV in the pot. When we bet, we found that we lost EV with every additional dollar put into the pot, even factoring in implied odds that increase with every dollar. In a linear problem, this effect is assumed to continue infinitely (or at least until we run out of money), when in reality, this may not be the case. This idealized assumption of linearity ignores some key factors – namely, the ones I would like to examine:

1. Pot equity.
2. Fold equity on the turn.
3. Fold equity on the river.

Pot Equity

In some cases, we will have additional outs to win. Sticking with the assumption that we can bet our hand as confidently on the river against a similarly passive opponent with a limited range of top pair, then there must be some pot equity e where the implied odds make it worth it to bet the turn for implied value. We would normally define an equity “edge” as having greater than 50% pot equity. Let’s see how our EV equation changes according to increased pot equity.

EV = 0.5*(b+20+2b) - 0.5*b
EV = 10 + 1.5b - 0.5b
EV = 10 + b

As expected, the EV is much better. Every additional dollar we bet increases our EV by a full dollar. But surely there must be some value of e where we break even, irrelevant of bet size.

EV = (3b+20)e – b(1 – e)
EV = 3be + 20e – b + be
EV = 20e + 4be – b
EV = 20e + (4e – 1)b

Notice now the two parts of the equation. The first part, 20e, represents our current pot equity, which is constant relative to the bet we choose to make. That is, it can only be affected (positively or negatively) by betting. In order to eliminate the effect of betting, the second part of the equation, (4e – 1)b, must be equal to 0.

(4e – 1)b = 0
4e – 1 = 0
4e = 1
e = 0.25

What we learn from this is that, if we have at least 25% equity in the pot and expect to make a full pot bet on the river, then it is +EV to build the pot for implied value. But that’s not all we learn.

Note that the required equity for implied value isn’t at all dependent on the current size of the pot, nor is it dependent on the size of the bet. In fact, all it depends on is the size of the bet you plan to win on the river relative to the bet you are making now.

Herein lies the true strength of implied value – the fact that implied odds matter not only when deciding whether or not to call, but also when deciding whether or not to bet.

I’ve already gone on long enough, so I think I’ll leave turn and river fold equity for another time.