The Problems with a Polarized Range by Matthew Janda

http://www.cardrunners.com/article/110/the-problems-with-a-polarized-range/

The Problems with a Polarized Range

                A player polarizes his range when he raises all his very strong hands and some bluffs, and calls with all his medium strength hands. While the former range is very easy to play, the latter is extremely difficult to play against competent opponents who know their opponent has no strong hands in his range. Since good players will attempt to exploit their opponents using polarized ranges by over-betting to maximize their bluff frequency, you should attempt to avoid polarizing your ranges against opponents good enough to punish you for having no very strong hands in their range. 

                In order to best illustrate this concept, it’s best to start off with an extreme example of the difficulties you’ll face when you only have medium strength hands in your range. Suppose you are playing 6-max with 100 big blind stacks and under the gun opens for 3.5 big blinds and you are the only caller from the button. Your preflop range is JJ-22, KQs, JQs, JTs, T9s, 98s, AQ, AJs, and ATs, which you believe to be very close to a theoretically optimal flatting range and your opponent knows this. Suppose the flop comes 2h 2s 2d.

                Unfortunately, through no fault of your own, you received the worst possible flop for your range and your range contains no very strong hands. Raising this flop would make little sense, since your opponent knows he cannot be beat when he has JJ-AA, and has 28% equity against the top hand in your range with AK, AQ, and KQ. Instead you are forced to call or fold on the flop, and will frequently face turn and river bets and will often end up folding the best hand. Suppose you decide to flat JJ-77, AQ, and KQs. Once again, this is the same situation one finds himself in when he polarizes his range by only calling with medium strength hands.

                Your opponent will want to maximize his bluff frequency, and in order to do this he will bet equal fractions of the pot on all three streets. The pot will be 8.5bb on the flop, and in order to make the pot 201.5bb on the river your opponent must make the pot grow at the rate of “r” each street, where 8.5r³ = 201.5bb. Since r = 2.87, your opponent will want to be around 93% of the pot on each street (since his 93% pot sized bet and your call will make the pot grow at the desired rate). Since your opponent cannot recklessly fire all three streets, your opponent will have to give up bluffing some air hands on the turn and some air hands on the river. Let’s take a look at what frequency your opponent can be bluffing on every street before you can start calling with your medium strength hands profitably.   

Your opponent will want to make you indifferent to calling on the river, so 67.5% of his 93% pot sized bets on the river must be for value. An easy was to visualize why your opponent needs to be betting for value 67.5% of the time on the river when he makes a 93% pot sized bet is to imagine the pot is $100 on the river and your opponent makes a $93 bet. When you call your opponents bet and win, you’ll win both the original pot and his river bet for $193. When you call and lose, you’ll lose only $93 (the money you already lost in the pot is dead money). If you win $193 only 32.5% of the time and lose $93 the remaining 67.5% of the time when you call with your medium pocket pairs, you will be indifferent to calling  and folding since ($193 x 32.5%) - ($93 x 67.5%)  = 0.  Since calling is effectively the same thing as folding on the river and have an expected value of 0, in order to not be getting odds to call the turn your opponent must follow through on the river 67.5% of the time (because every time you face a river bet, calling and folding will both have an expected value of 0). The same logic applies on the flop, so in order to make you indifferent to calling the flop with JJ your opponent must follow through on the turn 67.5% of the time. In other words,

1)      On the river, 67.5% of your opponent’s bets must be for value.

2)      On the turn, 45.5% (0.675 x 0.675) of your opponent’s turn betting range must be able to value bet the river.

3)      On the flop, 30.7% of your opponent’s flop betting range (0.675 x 0.675 x 0.675) must be able to value bet the river.

Note how on each street our opponents range consists of more and more value hands and less bluffs, so the more betting rounds which remain the more your opponent can bluff. You can see how problematic this is to play against. While you will surely hit a six outer with AQ or spike a two outer with a pocket pair on some turns or rivers, your opponent will often do the same to you (especially since most of his bluffs have such high equity). It’s not particularly difficult for him to be able to value bet the river with 30.7% of the hands he continuation bets on the flop. Many players at NL$200 and lower think “Well, pocket fives have over 50% equity against an UTG’s continuation bet range on a 222 or 664 flop, and since I’m getting better than 2:1 this is an easy call.” Yet this is not the case. Even if your hand has a lot of equity against your opponents continuation betting range on the flop, too often you’ll be forced to fold it on the turn or river,  or end up making an unprofitable river call. There are some flops from certain positions you just cannot prevent your opponent from firing 100% of his range on. If you are lucky enough to hold 66 or TT on a 6c 6h Td flop in the small blind against a cutoff open, you need to check-call those hands to defend against your opponent who knows your range is weak and will likely be double and triple barreling many turns and rivers.

Now instead imagine another situation where the cutoff opens and you are the only caller from the button. The flop comes Qh 7s 2c, and once against your opponent continuation bets. Since this flop connects with your range around as well as his, your opponent will probably bet less than 90% of the pot and won’t be able to get it in by the river unless he overbets the turn or river.  In order to avoid the situation you were faced with in the previous example, you should refrain from polarizing your range and instead flat most of your sets on the flop. While AQ and KQ might be good hands to raise (along with some air) on the flop, it’s more useful to flat with your sets and raise them on the turn. This prevents your range from consisting only of hands QJ and worse and polarizing your range. In addition, sets don’t need fear falling behind when an overcard falls or your opponent drills two pair.

                Nevertheless, there are still situations when polarizing ones range is correct.  Two of the most common ones are when...

1)      Your opponent isn’t good enough to realize your range is polarized, or won’t punish you if he does.

2)      You are in position and the board has a lot of draws, so the vast majority of turns will put some nut type hands in your range as soon as the turn card hits.

For example, suppose the cutoff opens and you are the only caller from the button. The flop comes Ts 9s 5h, and the cutoff makes his standard sized continuation bet. Polarizing your range by raising all your two pairs, sets, monster draws, and air is perfectly acceptable in this case, because just about every turn card will put some nut type hands in your range. Any spade gives you a possible flush, any ten or nine brings possible trips, and any ace, king, queen, jack, or eight gives possible two pairs or straights. So 32 of the possible 49 cards which can turn, or about 2/3 the deck, will put nut type hands in your range and protect you from an opponent who wants to exploit polarized ranges by over-betting. Likewise, slow playing a set or two pair will often result in you either losing action or getting drawn out on many of the previously mentioned turn cards, making slow playing usually a bad idea.

                Recognizing when your opponents range is polarized provides a great way to increase your win rate.  If you do open from middle position and get a caller from the SB, don’t be afraid to bet large amounts on a very dry 955 flop (whether it’s rainbow or there are two of one suite), especially when your stack is a bit over 100 big blinds. You know your opponent can’t be strong unless he has 99 or 55, which only make up 4 hand combinations. Likewise, don’t be afraid to punish opponents when you know they can’t be strong after they check-call on a draw heavy board. if you know your opponent would never call with a hand better than KQ on a Qc Ts 6c 2s board on the turn (since he would likely check-raise AQ or better on the flop and the 2 on the turn doesn’t ever improve him), don’t be afraid to over-bet jam the river against an opponent as a bluff when the river comes a 3d. While you and your opponents will undoubtedly not play perfectly, realizing the theoretical problems polarized ranges have when facing over-bets will both allow you to better exploit your weaker opponents, and slowly gear your play style towards a more theoretically optimal style to use against opponents equally capable of exploiting you.

                Thanks for reading, and good luck at the tables.

By Matthew Janda