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# Renton Theorem aka ABCD Theorem

In poker, we attempt to make money by playing the best we can, making what we feel is the most profitable play in every scenario, making as few mistakes as possible. Assuming you know the exact hand range of your opponent, there is almost always a clear best play, and if you like money you should make that play every time, right?

Yes, on a basic level this is true. Whenever we have the nuts we should bet/raise/reraise, whenever we have nothing we should fold. If we have nothing and feel like we can make our opponent fold a better hand enough of the time, we should bet as a bluff.

However, theres another level of poker that goes beyond this one hand. Not only does your opponent have a range, but you do as well. Not only do you want to make the best play with THIS hand, THIS time, but you want to make the most money with your range, EVERY time. You want to make the most profitable plays when you are in this spot. You want the average EV of your play with your range in this spot to be as high as possible.

A pretty common example of this was in an infamous Samoleus/EmpireMaker2 thread on 2p2 a while back. Samoleus criticized EM2 for 3-betting QJs in button vs blind scenarios. His reasoning was that "QJs has too much value in calling." He believed that it was more profitable to call QJs than to waste it on a 3-bet.

The other less obvious reason why "QJs has too much value in calling" is that EM2’s range consists of subdivided ranges in which he 3-bets/calls/folds, and putting QJs into the 3-bet range reduces the number of profitable plays he makes as a whole.

Renton Theorem:

In any no limit hold’em scenario where there is money left to be bet, hero’s range is divided into subranges A, B, C, and D, where:

A = hero’s ‘nut’ range consisting of hands to be aggressively bet/raised for value.

B = the range of hands that aren’t as strong as range A and benefit from passive play and/or pot control.

C = the range of hands that have a nominal amount of value, but can stand little or no action.

D = hands with little or no value

The four ranges are determined by the resultant play that is optimal for the range as a whole.

Subranges A, B, C, and D are directly adjacent in terms of playability/strength (i.e. the bottom of ‘A’ borders with the top of ‘B’). All of the hands in a given subrange should be played the same (barring randomizing your play), and this is how the range is defined.

What do I mean by "optimal for the range as a whole?" Is that different from "optimal in a vaccum? Let’s start with an example similar to the QJs above.

Example 1:

Thinking opposition opens a wide range (we’ll say 30% of holdings) and we are on the button. We have 3 ways of exploiting this player:

1. 3-bet for value

2. 3-bet as a bluff

3. call and exploit postflop

We don’t have many reads on this player but we can safely assume that like most players, he is going to fold his open to 3-bets an exploitably large amount.

So as an example within the example, lets say our hand is 98s. What is the optimal play in a vaccum?

Hard to say. Since we think our friend is going to fold a ton to 3-bets, it is highly likely that 3-betting is optimal, due to a high amount of preflop fold equity. However, 98s is a great hand postflop, and can continue on a very high %age of flops, so we are also certain that its profitable to coldcall with. So do we 3-bet or call? Probably 3-bet if we feel that given gameflow he’s gonna fold 85% of time.

However, even if he folds a massive amount of the time, the best play for our entire range is to call.

In this situation our ranges are subdivided as follows:

A = {QQ+ AK}. These are hands we’d be glad to stack off with, and should 3-bet for value and get it in.

B = {55-JJ, AJ-AQ, KQ, suited broadways, suited connectors, some 22-44}. These hands aren’t comfortable stacking off and are certainly profitable to coldcall.

C = {22-44, gappers, offsuit aces/broadways}. These are hands that are slightly too weak/unplayable to call, and we elect to 3-bet these as bluffs/semibluffs.

D = {the rest}. Have almost no value and we fold.

We can play ranges A, B, and C profitably. So our duty, in order to maximize the amount we exploit our opponent, is to make A + B + C add up to the highest possible percentage. ‘A’ is a static value range. ‘B’ consists of all the hands we feel we can profitably call that aren’t in range ‘A’. ‘C’ consists of the widest possible percentage of remaining hands that we can 3-bet and get away with it, and are chosen from hands just below ‘B’ strength due to maximizing value when called.

By ‘get away with it’ I mean that since we’re up against thinking opposition, we want to play as aggressively as we can whilst avoiding exploitation at all costs. If villain wizens up and starts 4-bet bluffing us, thats very bad, and we are no longer exploiting him without readjusting which takes time and causes us to make mistakes in the transition.

Example 2:

Seat 1: AmPHisbaenA (\$414.25 in chips)

Seat 2: gl79 (\$402.60 in chips)

Seat 3: Tnx4urMoney (\$456.10 in chips)

Seat 4: Kodack (\$186 in chips)

Seat 5: 69MadMike69 (\$53.70 in chips)

Seat 6: jhndh541 (\$106.05 in chips)

Seat 7: Renton555 (\$524.60 in chips)

Seat 8: jfager007 (\$394 in chips)

Seat 9: whaaatever (\$253 in chips)

gl79: posts small blind \$2

Tnx4urMoney: posts big blind \$4

*** HOLE CARDS ***

Dealt to Renton555 [9d 9c]

Kodack: folds

jhndh541: folds

Renton555: raises \$8 to \$12

jfager007: folds

whaaatever: calls \$12

AmPHisbaenA: folds

gl79: folds

Tnx4urMoney: folds

*** FLOP *** [Ac 8s Jd]

Renton555: checks

whaaatever: bets \$24

Renton555: folds

In this hand, betting the flop is absolutely certainly profitable, as this flop bitchslaps our range and he has to fold the vast majority of his. Betting may even be best. However, checking is profitable with 99, since he likely checks down worse pairs.

Without going too deep into this, let me create the subranges.

A = {AJ, A8, AA, JJ, 88, AK, AQ, T9} Hands we bet and continue to a raise.

B = {A2-AT, QQ-KK, Jx} Hands we check call for pot control and deception.

C = {77, 99, TT, 87ish} Hands we check fold for showdown value.

D = {air} Hands we bet as a bluff.

Even though betting 99 is profitable and maybe best, we prefer to bet all our air and try to check down 99, and hence have a less exploitable cbet.

Extrapolating further:

Say we bet ranges A and D and get raised. Then we have a whole new set of subranges.

A = {AJ, sets} Our nut range, we 3-bet all in or call and get it in on turns, depending on what we feel is most profitable.

B = {AK, AQ, A8, T9} We call and reevaluate.

C = {air like KcQc} We 3-bet as a bluff and shove turns we improve.

D = {rest of air} We fold.

It goes on and on.

The Renton theorem is something I think good players think about a ton. I’ve never seen anything like this quantified very clearly, and only recently started seeing it this clearly myself. Hopefully this isn’t totally redundant info to you guys.

gl