Now let’s take a look at how our first equilibrium example changes as we change a few factors. Here we compare several situations at the same time, so I present this in a slightly different format for easier comparisons.
In the tables above, I’m comparing six different situations where we’re in the BB facing an opening raise from the same MP player. Some situations use 10,000 effective stacks as before; others use 6,000. The blinds could be 100/200/25 or just 100/200 to see the effect of adding an ante. In addition, the opening raise could be to 500 or 600.
The left-most list in the first table is the same situation we discussed in the previous section; the other five columns are new.
So, what are the obvious things that jump out at you from these tables? Some things that jumped out at me: • The solution 3-bets about 6% of the time in all cases.
• For 6,000 stacks with an ante, most raises are pushes.
• When looking at the “other-calling-hands” ranges, the presence of an ante makes no difference.
The solution 3-bets about 6% of the time in all cases. There’s some variation, such as raising more often when there is an ante, but it doesn’t make much of a difference. When he increases his raise size from 500 to 600, we tighten up our calling range, which is important, but our 3-bet range is almost unchanged. With 10,000 stacks, we 3-bet a bit less and with 6,000, we 3-bet a bit more, but for the most part, it doesn’t matter if he opens with 500 or 600. This makes sense, as his strong opening range hasn’t changed, only his raise size.
For 6,000 stacks with an ante, most raises are pushes. There are occasional smaller value 3-bets (and bluffs), but most of the time, a raise is a push. His raise is 8%-10% of our stack and we’re out of position. We’d hate to make a smaller raise, put in a good chunk of our stack, and still be out of position post-flop. It’s better to push all-in and eliminate our positional disadvantage, even though a push is sometimes more than a 4x-pot bet.
When looking at the “other calling hands” ranges, the presence of an ante makes no difference. This really surprised me. When we compare ante versus no ante:
• We are more willing to fight for a pot with an ante (makes sense). • Our minimum calling standards are almost unchanged (what?!?).
It didn’t really make sense to me at first, but a little investigation into the details of the solution hints at an answer.
Our intuition fails us, because we think, “When there’s an ante and I lose the pot, I lose the same amount, but when I win, it’s a bigger pot. Why wouldn’t I play more hands?” The answer is that our opponent will fight a lot more to win a pot that has an ante.
Now this only applies when we’re thinking about very weak hands—right on the border between calling and folding. Stronger hands see a bigger +EV boost from the ante, which is why they’re more willing to 3-bet, but weak hands see almost no EV gain. When we call an opening raise (early in a tournament) with a hand like J6s or 74s, we’re hoping either to strike gold or sometimes take the pot away from our opponent if he missed the flop. But face it, most of the time we’re going just to check and fold. We both know that on average, he’ll have the better hand on the flop. We can’t change that. Trying to bluff the pot away from him too often with weak hands is just spewing chips. All those times we check and fold, it doesn’t matter if there’s an ante or not; we lose the same amount.
What about when we don’t check and fold? Sometimes we’ll win a bigger pot if there’s an ante. Sometimes we’ll lose a bigger pot. The ante creates a bigger starting pot on the flop, so his continuation bets will be bigger. That means worse implied odds on our draws and it’s more expensive to chase them and miss. Sometimes we’ll float, try to check-raise, or do something else to steal the pot and it won’t work. Oops, we just lost more, because those bets and raises are bigger with an ante. But the clincher is this: With more dead money in the pot, he’s less likely to give it up post-flop.
Put it all together and you see that with weak hands, there’s very little EV change with an ante, because the dead money we gain when we win the pot is countered by the fact that we’re going to win less often. The flip side is that our stronger hands get a bigger bonus from the ante, because our opponent is less likely to give it up and is more likely to pay us off.
Now let’s turn the tables and see how the opener responds to a 3-bet. Because we have to consider all the different 3-bet sizes, I only compare our original example (10,000 stacks, 100/200/25 blinds, opening raise to 500) with the same situation, but 6,000-chip stacks.
The first table are the 4-bets we saw in the previous section where we responded to 3-bets of various sizes. The second table is new, where we reduce the effective stacks to 6,000.
One thing to notice is that with 6,000 stacks, 4-bet bluffing is a lot less common. The reason? Lack of fold equity. Even when he puts in a half- pot 3-bet, that’s a bet of 1,150 or almost 20% of his stack. When he makes a larger 3-bet, he’s even more committed. Surprisingly, the equilibrium solution occasionally flat-calls a big 3-bet, even though it’s 40% of our stack. It does this occasionally with good aces, plus some medium pocket pairs and KK. The solution expects that everything goes in the middle post-flop most of the time, but figures it gets the best of it when someone gives up, given that the opener has position.