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A Level 3 Call
This hand occurred during an online heads up match with $3 and $6 blinds and $600 effective stacks. My opponent, a good but not great player, raised to $18 on the button, and I called with Kc 6c from my BB. The flop came 9h 6d 4c, giving me middle pair with a good kicker and a back door flush draw. I checked and called a bet of $28.
A 7c on the turn improved my flush draw, and I checked again. My opponent also checked, and the river was the 8s. This was a bad card, not so much because was an overcard to my pair, but because it put four to a straight on the board. I checked again, and my opponent bet $70 into a pot of about $74.
Because I expected this player to be thinking on Level 2, as even most good players do when it comes to value betting the river, I figured he would not make a thin value bet here. My passive play had represented a weakish one-pair hand that would be pretty scared by the river. Thus, I expected his range to consist of exactly straights and bluffs. I further determined that this was such a good card for him to bluff that he would bet hands like Ace-high that had some but very little showdown value, since my hand looked a lot like a small pair.
This was Level 3 thinking on my part: I considered the likely strength of his hand and also the kind of hand I had represented. A Level 4 thinker could exploit me here by value betting something like two pair. I did not think this player would be capable of that, however. I called the bet, and he showed me 2h 4h. As I suspected, he had put enough thought into my possible holdings to realize that even though he had a pair he could probably only win the pot with a bluff. He was using Level 2 logic with his river bet, and by thinking one level ahead of him, I made the correct decision.