What I am asking is actually how should one bet if he knows the cards of the opponents. Since the question is too broad I would to know what is the best action in the following scenario, but if there are resources for a more general approach I would like to know them.

Let's say that after the flop it's only you and another player, in a tournament hand, which talks first and checks. Flop is Jh, 10s, 6d.

You don't know the exact cards of this player but is predictable enough you can say that has 2 high cards like AK or AQ based on pre flop betting and the fact that he didn't bet post flop. However, you managed to hit a pair because let's say you have 6,7 clubs.

The exact cards are not important but I am trying to give a realistic hand, where you have close to 60% chance of winning and your opponent close to 40%. Let's further assume that your stack at this point is half of the opponent and about as big as the pot so if my calculations are correct it's profitable for both to call an all in. My question is, is it your best option at this point to go all in? Is it to check and bet only in the final round? Does it depend on how late this hand is in the tournament?

  • The question is indeed broad. Actually a hand like AK will probably bet the flop, given his usual preflop action which should be raise. This is called Cbet. But given your low stack (as its equal to current pot), that gives you a SPR of 1, means you should get all-in or call an all-in, mostly because you hit something. Practically, your stack and your pair hit governs here.
    – user1165
    May 27 '15 at 4:24
  • @vlzvl I expect that most people would play tight in this situation to avoid getting out of the game. Assuming that you read your opponent correctly and you know if you are winning or not at any point it is really a matter of how much you should risk, which you say should be everything. Your link is useful though.
    – kon psych
    May 27 '15 at 5:05
  • about the risk, that depends. Not risking to just enter ITM? no, it's not worth waiting for better hands than the one you already have. Risking to move up the money ladder, provided there are other smaller stacks. Perhaps, that a job for ICM. Personally, when my stack gets such low and i'm close to bubble, i definitely call this hand since i'm not gonna live enough anyway in the ITM if im that tight. If im late in tourney, ICM and chip-to-value probably governs there, but generally i try to be more loose before ITM and tighter late in tourney
    – user1165
    May 27 '15 at 6:05

This question is incomplete. Your information is defined not just as your belief distribution over your opponent's hands (1st order knowledge), but also your belief about his belief distribution about your hand (2nd order knowledge), and so on. The extreme case where you have perfect infinite-order knowledge is called "common knowledge", and is impossible to achieve in hold'em poker.

To take the example you gave, assume that you know your opponent's hand but he doesn't know yours. If you've been playing tight in his recall of the previous hands, you may have reason to believe that he will put you on, say, an over-pair if you go all in (in this case, AA/KK/QQ). This makes it irrational for him to call given that belief. Then this makes it rational for you to go all-in, because you will win the pot 100% of the time if you perfectly predicted his behavior. That's a case of exploiting asymmetric information (where you have better information than the opponent), which is the fundamental play of poker.

You might be interested in what optimal play looks like as information approximates common knowledge. To do that, just assume you both turn your cards up. In this case, your expected value is monotonically increasing with the amount of money you bet (which is always true in perfect information heads-up play if your odds are >0.5, regardless of whether opponent will fold or not), so you should go all in. It also turns out that he'll call (his expected value from calling is monotonically increasing with amount already in pot; in this example, the pot size is large enough relative to your maximum bet). Given 0.6*(1+1+1)<2, your expected value is actually worse off than the above case where only you had information about his hand.

In fact, take your example again, imagine if we are to change the settings and decrease the proportion of money x in the pot relative to your stack size, so that you wouldn't be worse off compared to the case in the second paragraph. Set 0.6*(x+2)=1+x, you get x=0.5. In other words, you're worse off under full information whenever x>0.5, i.e. amount of money already in pot is more than half your stacks. Let's also find the point where it no longer makes sense for him to call all-in: 0.4(x+2)=1, x=0.5, in other words he won't call all-in if x<0.5. These ranges, as you can see, have no overlap. Depending on pot size, he'll either do the same (by folding when x<0.5) or be better off (calling when x>0.5) when he has better information about your hand. This statement is in fact true generally in heads up, regardless of how you tweak the odds and cards. The fundamental theorem of poker states that better information on the part of opponents cannot hurt them (even if it's just a belief about beliefs).

The underlying reason is because heads-up poker is a two-player zero-sum game. Things are far more complicated in multi-way pots, and I won't try to expend on that here, but generally you can find cases where having better information makes you worse off.

  • Given the opponent's hand I assume he will call any bet expecting to get lucky, I had to clarify that. I need to read the 4th paragraph more carefully but I get the main point. My question is about a tournament setting so it doesn't seem right at first to go all in if let's say you are in the bubble and there are a few players with much lower stacks than you do.
    – kon psych
    May 28 '15 at 7:47

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