Preflop raising decisions can be very complex. What stakes are we playing? Is it a live game or online? How strong is our hand? Which limpers will call? Will we be in position postflop?

For example, in a $1/$2 no-limit hold’em game, the small blind is $1 and the big blind is $2. The small blind is usually equal to half of the big blind. The blinds get passed around the table at the beginning of each hand in a clockwise direction one player at a time.
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Sometimes our decision is obvious: 'I have aces, so I'm going to raise for value here.' Sometimes it's subjective: 'I think I can force the limper to fold here.' But we should also consider what a mathematical analysis of preflop raising can teach us. When is a raise our most profitable action for the moment? When might it be the most profitable action for the hand?

Las Vegas $1/$2 NLH Stats

In order to calculate the expected value for our preflop actions accurately, we need to know certain key frequencies that are typical for our game.

Figure 1 below contains some of the most useful stats for these EV calculations for both online NL100 and $1/$2 NL live games in Las Vegas, most of them published here for the first time. (Check out Donkey Poker Volume 2: Postflop, Section 7.5 to see how these stats were generated.)

Here I define 'EP/MP' as the first four seats after the big blind on a nine-seat table. Since the vast majority of Vegas $1/$2 players are position-dumb, we can combine the results from these seats into a single stat in order to make the stat more reliable. The hijack and cutoff stats are similar to the EP/MP stats.

VPIP and PFR are the primary stats useful to determine an opponent's playing style. '35/6' means that the average Vegas player voluntarily invests in 35 percent of his dealt hands, while raising only 6 percent of them. This player would be considered a 'loose-passive' player in an online game, but he is average in a Vegas $1/$2 game.

Calling a Preflop Raise (CPFR) is a statistic that applies only when a player calls a raise after having already invested in the pot, which applies to limp-callers as well as to the blinds. This is not the same thing as Cold Calling Preflop (CCPF).

Compared to an online game, a Vegas preflop raise is fraught with more risk. Not only is our average raise size larger, our Vegas opponents are more likely to call it. Although their calling range is weaker, we often face multiple callers, which is rare online. Consequently, it is extremely important to have a solid idea of how these stats influence the profitability of our preflop action.

A Simple Example: A Big Blind Hero vs. a Single EP/MP Limper

Let's consider a situation which is relatively easy to calculate. Suppose an average player limps from one of the EP/MP positions. It folds around to Hero in the big blind who can check his option or raise.

Figure 1 indicates that that Mr. Average has a range of 35 percent, but he normally raises with 6.1 percent of this range. So we can assign his limping range as [6.1–35]. The specific card combos within his range depend on the ranking system we use. Although there is no single ranking system that is always correct, I will use the Flopzilla NLHE ranking in this column, since it is easy for anyone to access.

Suppose Hero has and decides to check. We can estimate Hero's 'Momentary Expected Value' (MEV) as.

MEV_Check = Pot₀ × EWC = $5 × 0.474 = $2.37

.where Pot₀ = $5 and EWC (Showdown Equity When Called) is 47.4 percent for Hero's pocket deuces when facing Villain's range. I call this his 'momentary' EV since it applies only at the moment he checks.

Suppose Hero decides to raise to $10 with the same . Figure 1 shows that an average villain will call a preflop raise after limping in EP/MP about 56 percent of the time (Row 8). This would be a Flopzilla range of [6.1–22.7].

Hero's Fold Equity is FE = (1 - 0.564) = 0.436. His showdown Equity-When-Called is EWC = 0.454. Notice that even though the villain has folded nearly half of his range, Hero's EWC is only slightly worse than when he checks. This suggests that the villain's actual limping and calling ranges are not critical in establishing Hero's MEV.

After deciding to raise, we can calculate Hero's MEV as.

MEV_Raise-$8 = (FE × Pot₀) + ((1-FE) × (EWC × Pot₁ - Raise)) = $2.27 + $0.28 = $2.55

The first term is just the average amount earned when the villain folds. The second term is the amount earned when the villain calls. ' Pot₁' is the size of the pot after he calls, minus the rake. 'Raise' is the size of our raise ($8 in this case). Since three-betting is so rare in Vegas $1/$2, and limp-three-betting is even rarer, we can ignore it for this calculation. Again, we consider this our momentary EV since it only applies at the end of the preflop action.

So Hero's raise appears to be more profitable (by $0.18) than checking. It gains nearly all of its profitability from fold equity and a just little from showdown equity. Add language digital download rosetta stone mac. This is generally the case for heads-up battles.

However, there is a serious issue with these estimates. https://aeyerl.weebly.com/blog/how-to-change-which-app-opens-a-file-mac. MEV uses showdown equity, which essentially assumes that the hand will be checked down postflop. This rarely happens in a real game, which makes its usefulness imperfect. We haven't considered such factors as skill and position, which can increase or decrease our expected profit in the hand. Hero's inferior position in this scenario suggests that his actual 'Hand EV' (HEV) will be lower than his calculated MEV. Even Hero's superior skill will not likely overcome his inferior position.

Since Hero's inferior postflop position makes it difficult to realize the full MEV of either action, it's difficult to say which action would have the higher HEV. Hero's best decision could depend on whether the villain appears more or less likely than the average villain to fold to Hero's raise. We might also consider that if both actions result in a similar HEV, the check does so with less risk.

Figure 2 shows the results of these calculations applied to some of the hands listed in the Donkey Poker starting hands chart. The solid symbols represent Hero's MEV when raising to $10, which is positive for each of these combos. 'JTs-' denotes 'suited connectors JTs and smaller.' Meanwhile the '+' symbols represent the difference in MEV between raising and checking.

When MEV is positive, raising is more profitable than checking, which is true for a range including approximately 22+, A2+, KT+, and QJ (i.e., any pocket pair, any ace-x hand, any two Broadway cards). This represents 27.6 percent of all starting hands.

Since we are out of position in the scenario, our HEV is probably lower than these values, so checking is probably preferable when the Delta MEV is close to zero.

Button Hero vs. One Limper

The previous scenario is rare in Vegas $1/$2 games, but it serves to illustrate the principals used for our analytical EV calculations. A more common and useful example is when the Hero is on the button with and facing a single EP/MP limper. In this case Hero can fold, limp, or raise, where folding has an EV_fold = 0.

This calculation has additional complexity because we have the blinds behind us, which means we must

Las Vegas $1/$2 NLH Stats

A Simple Example: A Big Blind Hero vs. a Single EP/MP Limper

MEVCheck = Pot0 × EWC = $5 × 0.474 = $2.37

MEVRaise-$8 = (FE × Pot0) + ((1-FE) × (EWC × Pot1 - Raise)) = $2.27 + $0.28 = $2.55

Button Hero vs. One Limper

MEV_Check = Pot₀ × EWC = $5 × 0.474 = $2.37

MEV_Raise-$8 = (FE × Pot₀) + ((1-FE) × (EWC × Pot₁ - Raise)) = $2.27 + $0.28 = $2.55

Texas Holdem 1 2 No Limit Strategy