The 7-2 Game

A few nights ago I had the chance to play at friend's home game where we implemented the 7-2 game. For those of your not familiar, this is where anytime a player wins with 7-2, every other player at the table has to give them some amount of money. In our case, we were playing a deep-stacked 1/2 game with six players and when someone won with 7-2, they would get $10 (5 BB) from every other player. 25 BB total is not a bad score, especially when you're able to take it down preflop. Some people hate the game, others love it, and I certainly fall into the later category. Anything to drum up action and encourage bluffing is a win in my book.

At first, it no one was getting dealt 7-2. After at least four orbits the hand was not shown down and everyone said they hadn't seen the had once. This makes sense though- of the 1326 possible starting hand combos in NLHE, 7-2 comprises only 16 of them, for a little over 1% of total possible hands. After about an hour though of no one getting the hand, seemingly all at once, a very high proportion were getting dealt, and this continued for the rest of the night. There were at least 4x as many 7-2 combos dealt as what one would expect based on the odds (I certainly wasn't complaining about that!).

While the game is normally fun, somewhat loose, with a good amount of aggression, the 7-2 game transformed the table to have a preflop aggression frequency higher than the toughest online 6max games. It seemed like there was a 3bet every few hands with no one ever really choosing to back down with 7-2. On top of the standard 3 and 4bet bluffs with 7-2, there were also a few notable pots where 7-2 triple barreled on a scary board and got called down on all three streets and where a player opted to flat with 7-2 preflop and make a series of bluffs postflop to take it down. 

For the home game that this was played in, I think the 7-2 game makes a lot of sense. Everyone could afford to play these stakes so although the hyped up aggression left some people frustrated by the end of the night, it wasn't going to make anyone not come back. The only scenario in which I could see the 7-2 game not making sense for one's home game is if the stakes being played are meaningful to some, and the thought of losing 3 buyins or more in a friendly game is something that would discourage players from coming back (although in this type of case, my recommendation would be to lower the stakes, up the stack depth, and bring on the preflop aggression!). 

What I'm excited to further explore is not the merits of whether or not to play the 7-2 game sometimes -- unless you hate action and people bluffing more, it's worth at least trying for an hour or two. I want to look at how this game effects decisions so if you find yourself in a game where people are playing the 7-2 game, you know how to adjust. I think it's fairly obvious for those that have played the 7-2 game, most people over-adjust and bluff too much when holding 7-2. I'm going to look at how the reward of winning a hand wth 7-2 impacts one's EV and your frequencies. For the sake of simplicity, let's work with the assumption that the reward for winning with 7-2 is 30 BB - 5 BB at a 7 handed home game. 

Preflop

Let's say you normally open 3 BB to win 1.5 BB. Now with the 7-2 game in play the reward is 31.5 BB. So it's clear even in early position 7-2 is a slam-dunk open. Now what about a 3bet? Let's say you standardly 3bet to 10 BB over a 3 BB open. So now instead of risking 10 BB to win 4.5 BB, you're risking 10 to win 34.5 BB. At first glance it might seem like we should be 3betting 100% of the time with 7-2. I think in most games this is probably correct, but if you're in a really loose game where people rarely fold to 3bets, or up against a particularly sticky player, it might be best to just fold against those type of players. Because once called preflop, 7-2 has such poor equity against a calling range so without much fold equity postflop, best to just fold pre. Note in these games I would have a tiny or non-existent 3bet bluffing range without the 7-2 game.

Most players will have a frequency that they fold to 3bets, even in a loose, aggressive, and deep stacked game, so most of the time you should replace some of your 3bet bluffs with 7-2. The key when adjusting for this game is not completely throw off your relative frequencies - if you normally 3bet in late position with 9s+ AQ+ for value and A2s-A5s as a bluff, don't just add 7-2 to your 3betting range unless these players won't adjust to the 7-2 game - almost no one doesn't adjust when playing the 7-2 game, if anything, most players in my experience over-adjust and always "put you on 7-2". So against most players you should also add at least the proportionate amount of value combos to keep your ratio of value hands to bluffs the same, if not more value hands due to overadjustment. 

Now on to 4bet bluffing. If a standard 4bet to a 10 BB 3bet is 35 BB, you're normally risking 35 BB to win 11.5 BB, and with the 7-2 game to win 41.5 BB. As you can see, after more preflop betting occurs, you're starting to risk more to win relatively less. The same logic for when to 3bet bluff with 7-2 applies to 4betting, although because of the price we're laying ourselves, we need to be a little more conservative than with 3betting. Against a relatively balanced player, we should be 4bet bluffing all combos of 7-2. But against someone who only 3bets very good hands or is looking to gamble with a merged value range, best to fold all combos of 7-2 preflop. I imagine there aren't many opponents where it is correct to do anything but fold all combos or 4bet all combos. It would take a particular opponent who is somewhat balanced in their 3betting range but a little too loose to warrant a mixed strategy with 7-2.

Postflop

Barreling frequencies with 7-2 postflop are largely dependent on the size of the pot after the preflop betting. In a similar fashion to preflop, it's likely correct to cbet 100% in a single-raised pot heads up- if our cbet sizing is on average 1/2 pot, then one is risking 3.25 BB to win 37.5 BB. With multiple players in the pot, it still is likely correct to cbet 100% with 7-2 because of the price. Even if the 3.25 BB cbet only gets through 15% of the time in a 4way pot, it's still a really profitable cbet because you're risking 3.25 BB to win 43.5 BB (only needs to work about 7.5% of the time to break even). If you're at a table where it's so loose that cbets don't go through on the flop when playing the 7-2 game because everyone puts you on it, don't ever bluff postflop with 7-2 and please let me know if you ever need another player for the game.

In a 3bet pot, the same logic largely applies. In a heads up pot when cbetting the flop you're risking 10 BB to win 51.5 BB, so you only need the bet to work 18% of the time as opposed to the normal 33% without the 7-2 bonus. Note how much more of an attractive proposition cbetting is in a single-raised versus heads up pot: cbets only need to work 8.5% of the time versus 18% of the time. And for 4bet pots this then changes to 26.5% which while is better than the 33% that it would need to work without the 7-2 game, won't change your range as significantly. In a 4bet pot you should probably give up with some combos of 7-2 and replace your worst normal bluffing candidates with 7-2.

Don't be the guy that makes the hero triple barrel - on each street the extra 30 BB becomes much less of a factor. If it's a 3bet pot heads up pot with 200 BB stacks to start the hand, and you get to the river with 100 BB in the pot and 150 BB behind. You decide to overbet the river and risk 150 BB to win 100 + 30 BB because goddamnit if you'll lose with 7-2. Normally you would need this bluff to work 60%. But with the extra 30 BB, this bet still needs to work 53.5% of the time, not that significant of a difference. If you decide it makes sense to have an overbetting range on a particular river card, it will likely make sense to include at least a combo or two of 7-2, just not all 12 combos.

 

Accurately Constructing a Range

 

For many of you the above approximations will be enough actionable information for how to adjust your game for the 7-2 game. But let's say you wanted to do an in depth range construction of your own. Below is a formula for figuring out at what point 7-2 game becomes unprofitable compared to the normal ideal bluffing candidates: high equity draws. 

Equity when called + fold equity - bet when called and miss + bounty equity = 0

Equity is when called = x

% Opponent folds = y

7-2 Bounty = z

So let's say I bet 50 into 100 on a flop in a heads up pot.

So the base equation before knowing our exact hands, equities, and bounty is the following knowing the size of the bet:

x(1-y)*200 + y*100 - 50*(1-x)(1-y) + z = 0

The flop is Kc6h9c.

Which is a better c-bet bluffing candidate, 72o or J10c?

Let's approximate that 7-2 has about 5% equity against a continuing range and J10c has 35% equity. Your opponent will fold 33%, 8% more than optimal. In the home game I played, the 7-2 bounty was 50.

                             7-2

.05(1-.33)*200 + .33*100 - 50*(1-.05)(1-.33) + 50 = 57.875

                            J10c

.35(1-.33)*200 + .33*100 - 50*(1-.35)(1-.33) + 0 = 58.125

So in this case, we'd expect to profit about $7 (answer of equation - the bet) with our best bluffing candidate as well as 72o betting half pot in a medium sized pot for the stake, without much theoretical difference between the two hands.

Now let's look at what happens if this flop was bet called and a blank turn comes out.

Kc6h9c4s

Which is a better bluffing candidate now for betting 140 into 200? Let's adjust the base equation for this bet and pot size, how often your opponent folds (33%, a few % less than optimally against this bet size), and updated equities - 0% for 7-2 and 18% for J10c.

x(1-y)*480 + y*200 - 140*(1-x)(1-y) + z = 0

                             7-2

0(1-.33)*480 + .33*200 - 140*(1-0)(1-.33) + 50 = 117

                            J10c

.18(1-.33)*480 + .33*200 - 140*(1-.18)(1-.33) + 0 = 201.796

As you can see, as the pot gets bigger, 7-2 becomes significantly worse (EV of -$23 in this example) to bluff compared to good draws (one would expect to profit $61 semibluffing J10c here).

Now a note on river play - if you do get to the river with 7-2, then it becomes your best bluff because none of your bluffs have equity but you get the extra bounty with 7-2. This doesn't necessarily mean that you should always bluff with all combos of 7-2 you get to the river with, but you should defintely bluff all 7-2 combos before adding other bluffs.

Conclusion

The big takeaway is to still be quite aggressive with 7-2 - the extra 30 BB in most circumstances makes it an excellent bluffing candidate. This becomes less and less true on later streets, and in bloated pots. Just remember to not get too crazy and have it make your ratio of value bets to bluffs go out of whack - with the addition of 7-2 to a bluffing range, remember to value bet extra thinly.