Poker: Ranges Revealed

This blog discusses a fundamental concept in poker, ranges. The term is bandied around a lot at a recreational poker table, occasionally even correctly. Typical adjectives that go along with the word ‘range’ include tight range, wide range, condensed range, polarised range, range advantage, linear range and balanced range. I’ll try to explain these terms, to help my own understanding of them in the game of Texas no limit holdem.

A range of hands is the collection of starting cards a player may hold in any given spot, excluding all the combinations that were folded earlier. Given his or her actions in the hand to date, a good player will take their next action in a way that considers the full range of hands that they’re likely to have at that point, not just the 2 hidden cards they happen to hold. Similarly, the goal of hand reading is to put your opponent on a likely range of hands. Although Phil Helmuth (famous poker player) may talk about his white magic and ability to put his opponent on, say, Ace Jack suited, exactly, this is more of a parlour trick. Poker is a game of very incomplete information — trying to guess exactly what the other player has is a mugs game.

While this may seem an obvious way of analysing the game, it’s probably only in the last 10 to 15 years that it’s become the universal approach. The thinking has progressed roughly as follows. The 1st level of sophistication, as a beginner today might think, is to base your betting decisions on the 2 cards you’re holding. “I am at the river and I have a terrible hand, so I check and give up”. The 2nd level of sophistication is to consider what your opponent is holding / thinking. “I am at the river and I have a terrible hand, but my opponent doesn’t know that and anyway they may have a weak hand too, so I bluff”. The opponent may then take this to the 3rd level and figure that you’re bluffing, so will call.

You can take this to the 4th level, 5th level, and so on ad infinitum, and it essentially comes down your confidence that you can exploit your opponent’s ability to only take it to the nth level by going to the n+1th level yourself. And if you overestimate your opponent’s brilliance and go to the 11th level, while they only go to the 9th level, you still get it wrong!

A computer solution side-steps all this guessing by playing in an unexploitable way; it could even announce its actions for its range, face-up, and would still be unexploitable. In the above river decision, for example, the game theory optimal play would value bet a portion of the range that is very strong, balancing this by bluffing with the perfect percentage of the range that gives the opponent just enough incentive to pay off the value bet. The opponent is financially indifferent to calling or folding to your bet.

So thinking in terms of ranges is clearly useful if the players around the felt are doing sensible things, with ranges that reflect their position, the betting action, chip stacks, etc. There are 13 * 13 = 169 starting hands, for a total of 1326 combinations. Against a good player, you might be pretty confident for example that they’ll call your raise from late position or the blinds with the following range exclusively:

The 2-card combinations that are highlighted are the ones that the player will call with, for example AJs (Ace Jack, both of the same suit). The greyed-out cells are either folded (e.g. 72o, the 7 and 2 of different suits) or raised (e.g. AA, the ‘pocket rockets’) and will not call your raise.

This might be regarded as a tight range, as it only contains 22% of the possible 1326 combinations. It is also a condensed (or capped) range, as it has all the middling cards, excluding the best and worst possible holdings. Note that it has more suited than unsuited combinations, for the extra flush possibilities these offer.

Despite this conservative tight play, the pre-flop raiser will tend to have the range advantage on the flop, meaning that if no more betting took place the raiser would be more likely to win the pot than the caller i.e. the raiser’s range has more equity on average. This is because by calling a bet you have the chance to win that bet plus what was already in the pot. You therefore have pot odds to call despite having less than 50% chance of winning.

This gives the pre-flop raiser the incentive to make a high percentage of continuation bets (“c-bets”) on the flop, though this should by no means be taken as a rule. Note that while it’s important to play your range and not your actual holdings, that doesn’t literally mean that you bet every combination in your range exactly the same. That is not what a computer GTO solution looks like (see appendix). Occasionally in poker we have a showdown and you do unfortunately have to show your cards — your actual holding becomes relevant!

The goal instead in range selection is to construct a balanced range, so that the hands that go into your checking range, calling range and raising range are all considered together. Done right, as you go to the next street your opponent can never be completely sure that you don’t hold a 3, say, or whether you have a strong or weak hand.

The opposite of a tight range, a wide range, is what I tend to see in the game I play! The main criterion for selecting this range is slightly different too, namely the amount of whiskey consumed that night. This begs the question whether thinking in terms of ranges is even useful, if your opponents may still have any 2 cards when the flop comes.

I think that your opponents can still be counted on to act somewhat rationally. If everyone limps in pre-flop they are less likely to hold the AA. You can’t be certain of this, of course, but nothing is certain in poker anyway. And if someone is limping their pocket AAs you have the consolation of knowing that this is a bad play in the long run, even if you lose a lot of money this hand because you didn’t expect it.

Often you can narrow down a loose passive player’s range very accurately. If they 3-bet you pre-flop it probably means their range is AA, KK or QQ, and nothing else. Recreational players may not define their range much pre-flop; you can’t deduce much from their position and they want to see the flop with any 2 cards. However, they may often be less tricky post-flop. By the river you should be able to make some guesses as to the range they have.

The opposite of a condensed range is a polarised range — a key poker concept, especially for river bets. A polarised range is either very strong or very weak. A condensed range (middling cards only) should be played passively by checking. Betting a condensed range doesn’t achieve anything: stronger opponents’ hands call and weaker hands fold. A polarised range on the other hand should be played aggressively: you can value bet the strong part of the range, while you get fold equity by bluffing with the weak part of the range.

Earlier streets, the flop and turn, tend to be less polarised: the bottom of your range still has some equity, with 2 more cards still to come to improve, while the top of your range can still be outdrawn. Your range and betting strategy therefore tends to be more of a linear (or merged) range on the flop than on later streets, containing a mixture of strong, medium and weak hands.

To sum up, this blog aimed to explain the terms tight range, wide range, condensed range, polarised range, range advantage, linear range and balanced range. I hope I sold that idea that ‘range’ is a useful way of thinking about poker strategy — even in low stakes poker!

Appendix — how to read piosolver outputs

If you thought the above calling range example of limited practical use for low stakes poker, you would be absolutely convinced of the uselessness of a computer piosolver output in all its glory — and I would agree. Even a professional player couldn’t memorise and apply a game theory optimal solution perfectly, and needs to apply heuristics (e.g. ‘generally bet big in this spot’) and approximations. As a mathematical object, however, I do find these outputs aesthetically pleasing. Here’s an example of one:

Button raises pre-flop, you call in big blind. Flop comes Th 9c 5h. Pot is 9 chips. You check, button makes a bet of 6 chips. What do you do? [Source upswingpoker.com]

The greyed out cells are combinations that are not in your range (because you folded or raised pre-flop). Unlike the simpler example we now have mixed strategies: A6o is only half greyed out because it was only called half of the time pre-flop. 73s is a green (call) 25% of the time and blue (fold) 75% of the time. Some of the recommended plays are fairly predictable: if you have 55 and the flop is T95 rainbow you have a set and should raise. (Technically you need to have a randomiser at the table with you and raise small 20% of the time and raise big 80% of the time. In practice you would raise big 100% of the time). This strong raise is carefully balanced out by bluff raises from hands that completely missed the board (e.g. A3s) so that the opponent has incentive to call.

Across your entire range you would fold nearly 50% of the time, call with your single pairs like KTo, call or raise sometimes with your straight draws like J8s, and raise a lot with your two pairs and sets. My purpose in showing this was not to discuss why the computer might come to this solution in this spot, simply to show that it is possible to create a perfectly balanced set of actions, across multiple bet sizes. The strategy is unexploitable even if the opponent had a copy of this output to hand and so knew the computer’s strategy (though not the computer’s exact hole cards, obviously).

I do find these outputs very interesting and clever, though I don’t think they’re that useful for low stakes poker and I don’t own a copy of piosolver. It is expensive for a start, $250 for the basic version (unlike chess computers, by contrast, which are essentially free). There’s a fair amount of set up required, to key in all the players’ ranges and allowable bet sizes, and they only really work for heads up pots, which don’t occur that often in my games.