Suppose, further, that both players know that the BB has Ax here always (this isn’t the best hand to demonstrate this, as BB has plenty of busted draws, but whatever, pretend it’s a standard ace high board with no draws and it’s gone c/c c/c c/b if you’re hung up on that), and that BTN has 50% Qx+, and 50% air. This is, naturally, an idealized situation, but we’ll get to the practical applications at the end. For now, you should all know what the Nash Equilibrium of this situation is (Equilibrium isn’t just an endgame thing). Do you? If not, let’s calculate it.
For demonstration purposes: Let’s assume as the button, we have three choices on some site with crummy software: Betting t500, betting t1000, or betting t2000 all-in to the pot of t1000. Let’s say we decide to bet t2000, or not at all.
In Nash Equilibrium, after you make your river bet, your opponent should be indifferent between calling and folding. Hence, as the button, along with betting t2000 for all of our value hands, we want to bet t2000 with the percentage of our air that makes it so folding and calling have the same EV for our opponent. The math on that:
-500 = (2500)x – (2500)(1-x) -500 = 5000x – 2500
2000 = 5000x
x = 2/5
Thus, at Nash Equilibrium, BTN shoves what makes a BB call correct 2/5 of the time. So, BTN shoves 100% of her value and shoves 2/3 of her air. This makes BB indifferent between calling and folding, meaning that 5/6 of the time, BTN has an expectation of +t500, and 1/6 of the time, BTN concedes the pot for an expectation of -t500. Hence, EV playing this strategy is +t333.
Let’s note what happens if we try to make our bets only in 1000, or only in 500:
If we’re betting 1000, again, we should be doing so with 100% of our Qx+, and some % of our air.
BB should be indifferent between calling and folding in equilibrium.
-500 = (1500)x – (1500)(1-x) -500 = 3000x – 1500
1000 = 3000x
x = 1/3
Thus, we want to bet what makes BB call correctly 1/3 of the time. So, we shove 100% of our value hands and 50% of our air. This makes BB indifferent between calling and folding, meaning that 3/4 of the time, we have an expectation of +t500, and 1/4 of the time, we concede the pot for an expectation -t500. This means our expectation playing this strategy is +t250.
At equilibrium, betting pot is inferior to overbetting 2x pot.
If we’re betting 500, you know the drill…
-500 = (1000)x – (1000)(1-x) -500 = 2000x – 1000
500 = 2000x
x = 1/4
So we bet what makes BB call correctly 1/4 of the time, which means betting 100% of value and 33.3% of air. 2/3 of the time we have an expectation of +t500, 1/3 of the time it’s -t500. Expectation playing this strategy is +t167. The bigger the bet, the bigger the EV.
Let’s take it to the extreme, though. Say we’re really deep-stacked, perhaps at a cash game with 10k behind and this river decision. This is also applicable in smaller HUSNG pots. If we make a big overbet shove (crazy, reckless stuff, right?), what’s our EV?
-500 =(10500)x – (10500)(1-x) -500 = 21000x – 10500
10000 = 21000x
x = 47.6%
Which means we bet 100% of our value and 90.8% of our air, giving us an expectation of +453, the best yet.
The general theme: When it’s clear your opponent is highly unlikely to have anything other than bluffcatcher, your chipstack is a weapon. You want to use your chips to allow you to remain unexploitable as you bluff with a bigger percentage of your air hands.
Two quick points to make, which I won’t bother to show the math on:
1. At equilibrium, overbetting is better than any combination of small/large bets with different types of hands.
2. Shoving is still a Nash Equilibrium regardless of what percentages BTN has air and value; it doesn’t have to be 50/50.
OK, you get it. But you read 2+2 strat so you can take more maney. When does this help me take money?
The downside: I am a strong advocate that in the vast majority of games, Nash Equilibria are going to be useless. Nash relies on both players having perfect information about each other’s strategies. This is ridiculous. If a Nash Equilibrium calls for you to do something 90% of the time, you might as well do it 100% of the time – nobody’s going to know. Mediocre players who have no history, aren’t attempting to balance ranges, and do not respond appropriately when you unbalance yours, make it a critical error to not go for MORE than the +t300 expectation that t2000 overbet shoving guarantees you in this hand. There also will be concerns about variance and putting your stack at risk for too slight of edges. All these are legitimate concerns: There’s simply no reason to play any Nash, be it end-game or early- game, if deviating from it affords you a better winrate because of the incompetence of your opponent.
However, as you move up in stakes and get more and more history between common opponents, who tend to understand and react appropriately to your frequencies, you need to start
playing more equilibrium strategies. The Nash Equilibrium here is overbetting, and that’s a significant part of why you see it at higher stakes, taking advantage of players who would never, ever check the river with a big hand in the example above. Stop thinking about Nash in poker as only relevant to the endgame and start thinking more about it in early-game decisions when you’re up against a tough opponent, especially one you have history with. I would argue that the “you only have a bluffcatcher” situation is pretty commonplace, but regardless, there are numerous other spots that make thinking about equilibrium and how its occasionally counterintuitive conclusions more than mental masturbation, but hookers and blow from all of the bing blang blaow you’re going to be singing.