As I said in the beginning of this chapter, the purpose of looking at all these equilibrium solutions is not so we can imitate them. What I’m hoping to do is illuminate some trends, so you can see which factors are important in making your 3- and 4-betting decisions. These trends will be important when we talk about exploitive play in the next section.
One of the trends I’d like to highlight is how often you can defend your blind. Next up are a series of graphs that show how often the big blind defends his blind (by either calling or raising) in the equilibrium solutions. When considering blind defense, the four most important factors are:
• His opening range. • Your direct pot odds.
• Your implied odds, which I express as a stack-to-pot ratio (SPR). This is your stack size (if you just call) divided by the size of the pot (if you just call).
• The tournament pressure.
For this first graph, I assume you’re getting 3-to-1 pot odds and your SPR is six.
You can see how often the equilibrium solution defends his blinds (either calls or folds) depending on the opener’s range. For example, if there’s low tournament pressure (a bubble factor of 1.25) and the opener is raising 20%, the solution defends his blind about 30% of the time. This can rise to essentially 100% if the opener has a very wide opening range and there’s no pressure. Three-to-1 pot odds is huge when there’s no pressure.
Next we look at the effect of how deep the stacks are. We keep the pot odds at 3-to-1 and fix his opening range at 10%, then see how the blind defense changes as a function of SPR.
Without pressure, we can often get paid off when we hit, but people are much more selective post-flop when the pressure is high. Our implied odds shrink with pressure. Even a little bit of pressure makes a big difference. At an SPR of 10, the solution defends with 53% of hands under no pressure, but only 34% with low pressure.
Now let’s see how sensitive we are to his raise size.
Again, we see a similar pattern where we don’t care as much about pot odds when there’s pressure. This chart is good support for the strategy of having a very large (3- to 4-BB) standard raise size at the beginning of a tournament, then decreasing that raise size as the tournament progresses and the pressure increases. In the beginning with no pressure, our large raise size gives poor direct odds (and cuts down SPR), keeping blind defense reasonable. Later on when the pressure increases, we have a smaller standard raise size, since it’s incorrect for players to defend too often, no matter what the size.
Another thing we could look at is the implied odds that suited connectors need to call in a variety of situations. Let’s first take a look at a typical suited connector, 76s, with only light tournament pressure. This graph shows the implied odds you need to call depending on how good your direct pot odds are.
Several lines here correspond to how strong the opening raiser is. The SPR given is simply the minimum you need to make calling better than folding. For example, against a moderate mid-position raise, the equilibrium solution recommends a call if it’s getting 2.5-to-1 pot odds, as long as its SPR is at least about 3.5. The dotted lines show you the range suggested by the Rule of 5 and 10, which means if it costs us 5% of our stack or less, we call; if it costs 10% of our stack or more, we fold; in between, we have a judgment call (first suggested in Pot-Limit & No-Limit Poker, by Stewart Ruben and Bob Ciaffone, pg. 65).
This doesn’t always mean that calling is the best option; raising could be better either above or below these lines, but calling is better than folding.
As you can see, the equilibrium solution thinks that the Rule of 5 and 10 is pretty accurate facing an average raise with typical pot odds. The solution is more liberal with calling when the opener’s range is very wide; it’s now kind of “pre-flop floating” here and willing to bluff or semi-bluff on a lot of flops. Note that the solution won’t call against a very tight player without a lot of odds. It thinks that the opener’s post-flop play will be so good that it won’t get paid off enough. In an equilibrium, this is probably right. However, real opponents who are this tight are usually willing to go down fighting with top pair. It will be much easier to stack them, meaning that it’s probably right to call with about 5% of your stack.
What about other types of suited connectors or pocket pairs against which we typically apply the Rule of 5 and 10 (or 3 and 6)? This graph now shows the implied odds needed to call for various hand types.
Take a look for the hand you’d like to call with on the left and find the line that corresponds to its description. For example, to call with 85s, you need an SPR of about eight. Calling with 53s needs at least 10 SPR. Again, the dotted lines show the calling range suggested by the Rule of 5 and 10.
So it looks like the equilibrium solution thinks that the Rule of 5 and 10 is pretty good for a lot of suited 0-gaps and 1-gaps; they all need about the same SPR to be profitable. The Rule was never intended for tiny connectors less than 54s; you can see how their profitability drops off dramatically. Off-suit connectors fare a lot worse than I would have expected. I probably play off-suit connectors more than most people (usually calling up to 4% to 5% of my stack), but I’ve been noticing lately that my results with them are less than stellar. I’ll be changing my game starting today. You should be aware that the equilibrium does call or bluff 3-bet with off-suit connectors when the opener has a wider range. Against a wide range, a hand like 87o can hit top or second pair with a weak kicker and be a favorite.