Hero is dealt 
Folds to SB/Hero, who opens to 3bb.
BB/Villain calls.
Flop (6bb) 
SB checks.
BB bets 1.88bb. SB calls
Turn (9.76bb) 
SB checks.
BB bets 6.12bb. SB calls.
River (22bb) 
SB checks.
BB bets 10.45bb. SB calls.
We will analyze this line using our model solution, exploring various strategies on the same runout. Our focus is on guiding you through optimal bet sizing and pot commitment in each line, especially when it comes to bluff-catching.
Let’s delve into the flop strategy. The Small Blind checks roughly 60% of the time and bets around 40%, with bet sizes ranging from 50% to 150% of the pot. A large bet of 150% is polarizing, often made with overpairs, top pairs, two pairs, and sets for value, constituting about 35% of the range. Bluffs typically include hands with overcards, representing about 55%, along with numerous straight draws. Interestingly, Ace-high hands are quite rare in this bet size. It’s crucial to note that not all strong hands are bet; some overpairs are kept in reserve for a check-raise play. In this scenario, pocket-sevens are not part of this bet size, so we won’t analyze this line further.
Another prevalent bet size is 50%, which is less polarized. This range includes fewer overpairs and more top pairs, along with a significant presence of middle pair and Ace-high hands, particularly stronger combinations like Ace-Queen or Ace-King. These hands seek value from weaker Ace-high hands and dominate high card combinations like King-high and Queen-high. In this case, pocket-sevens are included in this bet size, so we’ll explore this as an alternative line. Facing a 50% bet, the Big Blind raises about 11% of the time, typically with top pair or stronger hands, using a small 50% raise size. However, the Small Blind aggressively three-bets with most overpairs to deny equity from draws, especially with hands like Nines through Queens, as an overcard on the turn could significantly weaken their relative hand strength. Pocket-sevens, however, do not three-bet but opt to call the raise. On a King-high turn, following a check, the Big Blind bets 50%, and we observe a fold from pocket-sevens. The betting range here leans more towards Seven-x bluffs than value hands.
Continuing our in-depth analysis of this intriguing hand, let’s examine the bet sizes on the flop within our model solution: 25%, 50%, and 100%. In this specific hand history, the closest bet size is 25%; the Small Blind checks the flop, and the Big Blind bets 25%, to which the Small Blind calls with pocket-sevens. Overpairs and two-pair hands are check-raised for value, along with a plethora of bluffs including overcards and draws. Intriguingly, in this simulation’s equilibrium, the 25% size is not preferred by the Big Blind, who opts for a 50% bet instead. This line will be the focus of our analysis.
Against a 50% bet, the Small Blind’s response includes raising overpairs, two-pairs, and top pair with a top kicker, along with bluffs. The raises are substantial, about 150% of the pot, with bluffs constituting over half of this range. Pocket-sevens are always calling in this situation. On the Jack of Diamonds turn, the Small Blind checks their entire range. The Big Blind bets with a frequency of 48%, choosing either a 50% or 100% bet size. The betting range includes second-pair and stronger hands for value. For example, with a 50% bet size, about 22.1% are Eight-x hands and 1% weaker pairs, with a significant presence of top pairs, two-pairs, and sets. Bluffing occurs around 40% of the time, mostly with overcards to the Eight, but also with hands like Jack-Seven suited and Queen-Seven suited. Since the value range includes hands like Ace-Six, calling with pocket-sevens is feasible, particularly with a diamond, as this negates the possibility of a flush on the board, albeit it’s a low expected value call.
With a 100% pot bet by the Big Blind, the range is more centered around middle pair or stronger, leaving no value overlap with pocket-sevens. The value range is heavily skewed towards top-pair and two-pair hands. From the perspective of pocket-sevens, it’s important to consider both the bluffs and the value that is blocked – here, pocket-sevens block the rare Jack-Seven value, but also block King-Seven, Queen-Seven, Nine-Seven, and Seven-Five bluffs, which comprise a significant portion of the bluff range. Against this bet, pocket-sevens with a diamond are mostly folding, though somewhat indifferently.
On the river, with a King of Hearts, the Small Blind has a small range for leading out but checks 95% of the time. The Big Blind bets half of the time, choosing between an overbet of 200% of the pot, as well as smaller bets of 50% and 75%. The overbet range primarily consists of two-pair and sets for value, while the 75% range is mostly top-pair King-x hands. The 50% bet mostly represents second-pair Jack-x hands for value. Against a 50% pot bet, pocket-sevens always fold, and calling is a significant mistake, costing about 0.5 to 1 big blind.
We have explored multiple lines for the same runout, demonstrating that pocket-sevens tend to commit to a maximum pot size of about 12 big blinds, then begin folding to larger bets. This concept, known as line invariance, suggests that the same hand on the same runout will contribute roughly the same amount to the pot across different lines.
For additional examples, consider the flop checking line. If the flop checks through, the Small Blind, in this line, has an option to overbet 300% of the pot, frequently taken with most overpairs and top-pair hands. Pocket-sevens always check on this turn. If the Big Blind bets 50% of the pot, pocket-sevens call, leading to a post-call pot size of 12 big blinds. However, if the Big Blind bets 100% of the pot, pocket-sevens are mostly folding, if not completely indifferent.
Overall, this hand could be optimized at equilibrium. In practical play, players often bluff small, especially when no straights are made, and are tempted to bluff on the river with unmade hands. Note that many players in the Big Blind will bet these kinds of flops too frequently, perceiving a range advantage. Therefore, check-raising an overpair on the flop can be a stronger play than betting out. With pocket-sevens, being a weaker blocker, it’s often wise to fold on the turn against players likely to bluff the flop and check back to draw, a common tendency with weaker hands.
In conclusion, this hand history offers a rich lesson in strategic poker play, emphasizing the importance of bet sizing, hand reading, and understanding player tendencies. By dissecting each street and considering the range of possible plays, we gain valuable insights into making more informed decisions at the poker table.