I am a strong advocate of exploitive poker strategy. That is, I believe that you ought to tailor your play to exploit specific mistakes that you believe your opponents will make, even if this means that you are yourself open to exploitation. By definition, the more skillful player will do a better job of exploiting his opponents than they will do of exploiting him.
However, should you find yourself in a sticky spot against a player whom you believe to be more skillful than you are, you may occasionally prefer to employ game theory to help you avoid getting exploited rather than focus on exploiting that opponent.
Full Tilt No-Limit Hold’em, $20.00 BB (5 handed) – Full-Tilt Converter Tool from FlopTurnRiver.com
MP ($2087)
Button ($4197)
SB ($4659)
Hero (BB) ($8452)
UTG ($6115)
Preflop: Hero is BB with 7
, A
2 folds, Button raises to $70, 1 fold, Hero calls $50
Flop: ($150) A
, 3
, 4
(2 players)
Hero checks, Button bets $120, Hero calls $120
Turn: ($390) K (2 players)
Hero checks, Button bets $372, Hero calls $372
River: ($1134) 4 (2 players)
Hero checks, Button bets $2200
The way I’ve played my hand so far makes it pretty clear that I have a weak Ace. The question is what my opponent will do about that. I generally don’t give my opponents credit for being able to overbet bluff the river, at least not with a good frequency. They might do it occasionally, but the vast majority of the time that they overbet the pot, they will have the nuts or very close to it. So in this spot, I would expect them to be overbetting for value, daring me to call with my weak top pair, but not to try to bluff me off of top pair, at least not with an overbet.
This is an easy strategy to exploit. With top pair weak kicker facing a bet like this from a standard opponent, I fold.
Notice how exploiting this tendency of my opponents leaves me open to exploitation as well. If my opponents were to stop overbetting their strong hands and overbet the river any time they don’t have a strong hand, I would be in trouble. But for the most part I trust in my ability to adapt to my opponents more quickly than they adapt to me.
Against this particular opponent, however, I didn’t feel so confident. I knew him to be smart, creative, and daring. I was sure he put me on exactly the kind of hand that I had, but I didn’t know what he would try to do about it. Thus, I was lost as to what I ought to do on the river.
Because he’s capable of betting for value or bluffing like this, my decision to call has to be based on what percentage of the time he will be bluffing. His bet lays me odds. If he will bluff here more than 33% of the time, I should call. Otherwise, I should fold.
The problem is that I don’t know how often he is bluffing. The best that I can do, then, is choose a calling frequency that cannot be exploited. Since the pot is laying me 2:3, I should call 1/3 of the time.
No matter how often my opponent bluffs in this spot, then, he cannot exploit me. If he bluffs more than 1/3 of the time, he’ll steal a few more pots but he’ll lose too much on his bluffs. If he bluffs less than 1/3 of the time, he’ll win more on his value bets but won’t steal as many pots as he could.
If I knew that he was making one or the other of these mistakes, I could adapt my calling frequency to take advantage of them more fully. But then if he figured out what I was doing he could adapt his bluffing frequency to exploit me. Since I was not confident in my ability to win this battle of wits, I decided to default to an unexploitable calling strategy. I would look at the clock, and if the last digit was 7-9, I would call. If it was 1-6, I would fold. If it was 0, I would look at the next digit.
The time was 9:47. I called. He showed K8o, and I won the pot. But that’s not the important part.
What about a blocking bet on the river instead of check-calling?
With villain showing strength on the flop and turn, a blocking bet would only ensue a shove imo. Andrew, I’m just wondering how the rest of the session proceeded. From the psychological standpoint, did you opponent shut down on the rest of the bluffs and pretty much stay away from you? My real question is how did this call affect the rest of your session at this table? Just trying to figure out how certain dynamics that occur in a hand can affect your perception as a player for the rest of the session.
Interesting, well-written entry.
“Because he’s capable of betting for value or bluffing like this, my decision to call has to be based on what percentage of the time he will be bluffing. His bet lays me odds. If he will bluff here more than 33% of the time, I should call. Otherwise, I should fold.”
I’m confused. Shouldn’t he have to bluff 40% or greater to call given that you’re receiving 3-to-2?
“Since the pot is laying me 2:3, I should call 1/3 of the time.”
I don’t get this part. Shouldn’t it be 2/5ths of the time?
This is equilibrium only for some of his ranges. His distribution of hands that gets to the river with this sequence of hands DOES matter. To take an extreme example, say he gets there (i.e. checked to on the river with this board and these prior betting actions) with a range that has 90% equity vs. you. In equilibrium, he should shove every time, including the times that he's bluffing, and you would have to shove every time.
The non-trivial part is when his hand scoops the pot some strictly positive % of the time that's LESS than 60% (I think it's safe to say this is the case you're dealing with). 60% is that point where, even if he bluffs with all of his losing hands, a call is only break even (you win 3 when you're good, lose 2 when he's got it):
3*.4 – 2*.6 = 0
Then if he has a winning hand less than 60% of the time, you have to make his losing hands be indifferent between bluffing and checking back. He's risking 2 to win 1, which must be successful 2/3 of the time to show a profit, so you should call 1/3 of the time.
The interesting part about this hand is that a significant % of the time he has a hand that's chopping with you. This allows him to shove and basically get a freeroll for 1/2 of the pot if he's sure you have a bare ace. Let s be the % of the time he has a split, and x be the % of the time he has a scoop. Then if s + 2x > 1.2, you have to fold to the bet. Because even if he's bluffing with all of his losing hands, calling loses 2 x% of the time, wins .5 s% of the time, and wins 3 1 – s – x % of the time. To call you need
3 (1 – s – x) – 2x + .5s > 0
3 – 3s – 3x – 2x + .5s > 0
3 > 5x + 2.5s
1.2 > 2x + s
As a check, note that when s = 0, you need x < .6, as before.
bruechips
Edit: the first paragraph should obviously end “you would have to fold every time” instead of “you would have to shove every time.”
spritpot nails it, but Andrew does deal with those issues (not the split) in the post. Andrew’s thrust is sound. If:
1) I know I can only beat a bluff; and
2) My opponent knows I can only beat a bluff; and
3) I know my opponent’s range is sufficiently wide; then
game theory says my calling frequency should match the pot odds because that makes my opponent indifferent about betting/checking/amount of bet no matter what he has.
Sorry, I confused my calling odds with his bluffing odds, but that wasn’t really the point of the post anyway. Darren and Sprit explained the game theory elements better than I could have- thanks guys!
Geoff,
Eugene took the words out of my mouth regarding a blocking bet. It would be pretty transparent, and I’d probably find myself facing the exact same situation, using game theory to determine how often I must call an even larger bet.
In fact, it’s an even worse spot for me because the bet is larger. When my hand is relatively face-up, the best I can do is minimize my losses with the right calling frequency. But it’s a +EV spot for Villain, and moreso when the pot and bet size are larger. As it happens, this is addressed in my most recent 2+2 article.
Eugene,
Great question. You reminded me that I meant to post another hand alongside this one. I’ll make a new post now that addresses exactly this.