[DJ Sensei]

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For example, if you have AK on a QTx flop and bet half the pot into 2 opponents, they will both have to fold roughly 33% of the time [adjusted for the times JT just calls/would have bet if you checked, and the turn card is an ace, so it's actually around 30%] for the bet to make any sense. If you bet the pot, they must both fold slightly under 50% of the time.

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I'm not following you.

You bet so as to offer opponents 3:1 and they both have to fold 1/3 of the time to make the bet profitable.

You bet so as to offer opponents 2:1 and they both have to fold 1/2 of the time to make the bet profitable.

Yet not accounting for other factors, this event will only happen roughly 1/6 of the time (2 players each with 3 possible decisions {call, raise, fold}, 6 possible outcomes one of which is both fold).

On this basis alone, are either of the bets rational?

I have to believe that players are more inclined to fold if one bets the pot (or better still overbets the pot) rather than half the pot. But facing two opponents one would have to have some combination of factors working in his or her favor (e.g., much bigger stack, proven ability to knock people out with semi-bluffs, or simply being in the outright lead at that point).

A lot would seem to depend on the preflop action and subjective probabilities of their likelihood of folding versus caling or raising.

In other words, using your example with AK, presumably you raised and were called by two players preflop. The flop more or less completely missed you.

So what is the probability that flop hit one of the others? Or what is the probability that you were behind from the start (eg., one or both opponents is on a medium pair) and are now down to perhaps 10 outs at best with the QT flop.

Under these circumstances, wouldn't you need a higher probability of both opponents folding in order to make your bet profitable?

I guess what I'm getting at is that this is a much more complicated situation facing two opponents, one of whom presumably cold called and the other overcalled preflop. Head-up is a whole different and probably simpler story.

I tend to be thick in the head on theory, so maybe I'm not looking at this in the right way. Hence my response in an effort to exchange dialog and learn.

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The fact that they both have to fold 33% of the time when given 3:1 pot odds (with a half pot bet) is based on expected value. Lets assume that if they call or raise, you lose the hand, and if they fold, you win the hand. (A very simple assumption, but surprisingly good in most cases)

So lets say the pot is 1000. You bet 500. Now, lets say they both fold 1/3 of the time and the other 2/3 of the time, one or the other calls or raises.

EV = 1/3 * (+1000) + 2/3 * (-500) = 0

thus, its an neutral EV play if they fold exactly 1/3 of the time. If they fold less than this, you're losing chips in the long run by making the play. If they fold more, you're winning chips.


Now, onto probability of them both folding. It gets more complicated with both people involved, but lets continue to assume that the only way you win the pot is if they both fold. If the two opponents act independently (another very false assumption, but one that is good enough for this simulation). Lets say they both fold 2/3 of the time independently. Then the probability that they both fold is 2/3 * 2/3 = 4/9. This is greater than 1/3, so the play would be profitable. If each of them independently folds 1/2 the time, they both fold 1/2 * 1/2 = 1/4 of the time, thus its a bad play. (Note that this alone is a very good example of why continuation bets work much better against a single opponent)


Now you mentioned that betting the pot is more likely to chase off opponents. This is true, since theyre getting worse pot odds. However, you need to win the pot outright 1/2 the time with a full pot bet to make it profitable to bet that much. (By the same EV calculations as above, betting 1000 into a 1000 pot, your EV = 1/2 * (+1000) + 1/2 * (-1000) = 0 if they both fold half the time, its positive if they fold more than half the time, and negative if they fold less than half the time)
Now, if they're acting independently, you need them each to fold about 3/4 of the time to make a full pot bet profitable: (3/4 * 3/4 = 9/16 chance that both fold, 9/16 is greater than 1/2, so the play is +EV)

As you noted, the probability that any particular play will work is very subjective. As such, these calculations are a good exercise in expected value, but are no substitute for a good read on your opponents.