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In their conventional meaning, two-way bets are an irrational fiction.  Let me define that meaning for those who have not touched the topic:

Two-way bet: a bet that in one instance will occasionally induce calls from hands worse than others that your opponent will fold.

That is, for no apparent reason, we expect that a bet will induce our opponents to fold some hands better than our holding and call with some worse than our holding.  We cannot bet with say, a pair of aces on an ace-high, four diamond board and expect to get called by an ace with a worse kicker but fold out pockets fours with a diamond.  This is nonsensical - our opponents will never rationally elect to fold hands stronger than other hands with which they would call.  They will tend to draw some hazy line and fold hands below the line to our bet and call with hands above.  If they are not doing this with any kind of patterning, then we will not be able to predict either that they will magically call with worse than our ace ten and fold some better but non-nut hands.

These kinds of two-way bets do not exist.

Situations do exist where we can make one bet that accomplishes multiple objectives, but only when we are targeting multiple ranges.  These could be the multiple ranges that exist in a multi-way pot or they could be the multiple levels of one person's range prior to the river.  The complexity of no limit hold'em causes this, but the original envisioning of two-way bets studies the point in the game at which all things converge - the river.  On earlier streets, complexities that result from facing multiple opponents, from facing multiple parts of multiple ranges and from the complication of playing future streets allow us to achieve a result that is on its face irrational - to fold out "better" and get calls by "worse."

How we define "better" and "worse" will help explain how this is possible.

Targeting multiple ranges in multi-way pots:
This concept should be intuitive and familiar to most players.  Even in one session, many times we have opportunities to invite bad players to play against us while shutting out good players.  Talented players will be folding some hands in their range that we like to eliminate while bad players will call with hands that we like to face.  For preflop play, we eliminate some hands that dominate us while inviting in others that we dominate.  There are perhaps countless reasons that will drive the better players to fold some of the hands we like to fold out (positional considerations, difficulty playing postflop, etc.) and other reasons that the bad player will cite to justify a call (desire to gamble, fondness for a particular hand, etc.).  We exploit our opponents' different thought processes to fold out better hands held by one and get calls by worse hands by the other.  This creates profitable postflop opportunities where they would not exist prior, since we now face a range against which we are profitable rather than face two against which we are not.  This would not make sense versus one range, but facing two or more, we can achieve this.

Keep in mind that "better" and "worse" are a bit relative preflop and do not only apply to hot and cold equity, but to the versatility and ease of play of certain holdings postflop.  We can eliminate hands that play well versus us and invite in hands that play worse.

Postflop opportunities arise also.  The conditions we take advantage of often look similar to the preflop situations in which we find these moves to be profitable. We exploit the differences in thought process employed by two players to isolate ourselves against the weaker range and eliminate the stronger one.

Ex.: assume that a tight player will give a great deal of credit to a check raise by us and will fold all but very good overpairs, nut draws, and sets.  A third player, a fish, will not give any consideration to anything but his own cards.  The first player opens in the cutoff of a six-handed game, the button folds, the fish in the small blind calls and we overcall in the big blind with seven six of clubs.  The flop is seven-three-two with two diamonds.  The fish checks, we check, the raiser continuation bets three-fourths pot.  The fish calls and we check-raise four and a half times his bet.  He now folds pocket eight, nines, tens and jacks and all pairs of sevens that have us outkicked.  He will continue with pocket queens or better and nut draws.  In addition to folding out much of his range that has us beaten, we will protect our hand versus some of his semibluffs with air, such as overcards, that would complete on the turn or river with some frequency and we protect ourselves from having to attempt sometimes difficult bluffcatching with a very weak holding.  We accomplish all this and we get to play our top pair with position on a bad player who will continue with a very wide range against which we have great equity and will play well on future streets.

The next section extends this discussion by noting how we can exploit differences in how players will play different segments of their overall range.

Targeting multiple levels of one person's range:
On the flop and turn, many situations exist that allow us to fold out hands that have more equity against our particular holding than hands with which our opponent will call.  How is this possible?  It works because our opponent's range has several layers, and within each layer a call/fold threshold still exists, but the net effect is that it will fall at unique points within each layer such that we will be able to induce calls by hands that fare worse against our hand and folds by hands that fare better against it.

Prior to the river, a range can be divided into several segments depending upon what hole card combinations someone will arrive at the flop holding.  Our opponents will typically have some assortment of made hands, draws, and complete air.  Sometimes these will overlap a bit when our opponents have pair plus draw hands, but depending on the relative strengths of the made hand (how strong is their pair?) versus their draw (is it to the nuts?) the holding will be better characterized as one or the other.  Often our opponents will have equity-static weak made hands that they will fold to resistance early in a hand or to continued resistance on the turn, since their hand cannot really improve, but they will continue blindly to call down with draws, since their decisions will be fairly easy.  That isn't to say that most people play missed or completed draws well on later streets, but in their mind the decisions will be easy - they have either nothing (a missed draw), or a very good five card hand, which is the effective nuts.  On the turn, ace high will have very decent equity against many draws and very poor equity against many pairs, such that if our opponents call with draws and fold with pairs that a bet will show profit in both instances.  On certain boards a second barrel will indeed elicit this result.

  PICTURE

In the above illustration, I have depicted our equity with something like ace king unimproved on a ten-seven-three-two board with two diamonds versus various draws that our opponents might hold (eight nine, jack queen of diamonds, ace six of diamonds, etc.) and our equity versus some of their made hands (jack ten, pocket sixes, nine sevens).

On the turn, we will be a favorite against drawing hands.  These hands will have sometimes as few as ten outs (jack eight, eight six, etc.), but never more than twenty-one (eight nine of diamonds when we do not hold a diamond).  These hands will never have more than fifty percent equity against us on the turn and will almost always have much less than forty percent - a bet against this range will show an immediate profit.

Contrast our equity versus draws with how we fare versus made hands.  With ace king high, we will have six, three or zero outs to improve to the best hand when we face a made hand that is a pair or better.  We will never have better than 14 percent equity on the turn.

Note that our opponents will often call with nearly all of the draws that they hold and fold many of the made hands that they hold.  Why is this?  Our opponents do not know that we have ace king high and are playing against our entire range of hands, just as we are playing against the entire range of theirs.  Hands like pocket sixes begin to feel extremely marginal on the turn when they face a second bet, whereas a hand like pocket jacks still appears very strong versus our range for betting two streets.  Yet our equity with our particular holding, ace king, is the same against both, and if we can get our opponent to fold enough of his made hands, a bet with ace king will show a profit.

When our opponents hold a draw like ace eight of diamonds, they will often call again, even though this hand has only twelve outs to improve to a hand better than ace king high.  It is worth noting that our opponent will face some reverse implied odds issues when an ace falls on the river, but we will as well sometimes when our opponents have hands like ace ten or ace seven.  Our role as the aggressor will frequently allow us to play better on the subsequent streets than our opponents.  This is a somewhat complicated topic, though, and I will not elaborate further at the moment.

This idea that we will make better decisions than them on subsequent streets is important.  In addition to inducing calls from "worse" and folds from "better," we will also know roughly which hands are calling and folding and be able to play the turn and river better than our opponent.  If our opponent is calling with draws and folding his made hands, we can pick off bluffs when the draws miss and not offer implied odds when they hit.

By the river, all hands are equity-static.  There are no future streets and the equity of their holding versus our holding or range is now fixed.  This is important because the various layers of our opponent's range merge into one complete range comprised of all possible holdings at that point.  There can be no decisions to "bluff call" or continue drawing, for instance.  They may elect to bluff with some hands that they would otherwise fold, but they will not call with hands that have no equity for any rational reason nor will they elect to fold hands with which they should call.  If they are doing the former, they are not doing the latter and vice versa.