Matt Bodie, Poker

My writing about poker, game theory, and occasionally sports.

[mattbodie]

Exploitative vs GTO Bet Sizing

I’ve been dabbling with the Solver+ iOS app recently to replay some of my trouble hands and see what the Pio solutions are for these spots.  It’s not a perfect solver, but it allows me to learn the general GTO line in any given situation.  For example, in a standard raised pot, or SRP situation, the app will generally give you 3 choices, check, make a small Cbet, or make a large Cbet or pot-sized bet.  The app does not allow you to put in your exact bet size as when you played the hand, but the fundamental strategy is there to learn from.  Usually, the correct solution is to make the small Cbet in most scenarios I’ve come across so far.   

One concept that I’ve been struggling with internally is changing my overall table approach with GTO bet sizing - ie, smaller bet sizing.  Over the years, through training and practice, I have simply defaulted to betting 3/4 pot on just about any street.  It’s naturally tough for me to change this habit that has been implanted in my general strategy from day 1.  My flawed thought process (or theory) goes something like this: if I bet 3/4 pot or more, then I’ll usually be giving my opponent the roughly the correct odds to call with an OESD or FD on the flop, but I’ll be giving them the incorrect odds to call on the turn.  Regardless of how flawed this may be, I have been highly successful over the years deploying this general strategy both in live and online games.  

Lately, since diving in to Solver+ and changing my betting strategy accordingly, if just feels as if I’m getting looked up more on boards where I’m just wanting my Cbet to take it down.  It feels like I’m bleeding money here where my normal sized, bigger Cbet would have a better chance of doing what I want it to do.  With that said, I ran a couple filters in PT4 just from this past week to see my Cbetting scenario and success rate.  I found the results to be very interesting.  

The first filter I ran is for hands where I 2bet PF and then made a flop Cbet between 51-91% the size of the pot.  Even though the sample size is not much because of my new strategy, here were the results:

In the 2nd filter, I ran the same thing except for 19-49% the pot size for my Cbet’s.  Here were the results:

Obviously, the 2nd filter is a much larger sample size, because I am actively trying to employ this strategy.  However, look at the win rate in BB/100.  I nearly doubled my win rate in similar hands even though it “felt” like I was losing or not playing correctly.  The fact is that yes, these sample sizes are small, but it just felt uncomfortable because it’s not my norm.  

My point with all of this is that it is ok to deviate from a GTO strategy in order to exploit certain players.  If we feel that a certain villain will call small Cbets light but fold out most of his weak range to a bigger Cbet, then it is ok to Cbet bigger.  A great example is looking at some of the other good villains I play against.  One in particular plays very GTO against me - she’s always betting small on the early streets, and her bigger bets are either for value or they get the job done and get me to fold.  However, I have seen her treat fishy players much differently, and that is ok to make exploitative bets when necessary.  

One interesting nugget though is once you start an exploitative line, it’s usually best to continue that line.  In the following hand, I made an exploitative flop raise of what I thought was a weak donk bet.  As the hand played out, I incorrectly fired the turn, and checked back the river, only to be shown AK (which is clearly not a GTO play from the villain).  See below output:

In this case, my “exploitative play” clearly backfired.  However, what is super interesting is that I could have made a GTO play on the river - and that GTO play is an all-in bluff.  Look at the EV comparison below.  Even though I likely still lose the hand, it would have been higher EV to bluff all in as opposed to checking back because villain would likely fold out weaker kings.  See below:

This concept is so basic, but a little mind blowing to me because these are the types of plays that I’m just simply not making enough of the time.  GTO is not all small splashing bets, it takes guts to go with a read and make this type of overbet, knowing full well that we won’t always be right in doing it.  

[mattbodie]

Alpha

The first letter of the alphabet.  Perhaps a good place to start with understanding the basics of GTO.  Alpha, when applied to poker, represents how often a bluff has to work for it to break even, assuming the bluff has 0% equity.  The opposite of alpha is MDF, or Minimum Defense Frequency.  MDF refers to the optimal calling frequency or calling % that a villain must call with to break even when faced with a bet.  Assume that “b” stands for the amount of the bet, and “p” stands for what is in the pot before the bet is made.  Here are the equations for Alpha and MDF:

Alpha = b / (b + p)

MDF = p / (b + p)

That’s great you might say.  A couple of equations for determining how often a bluff has to be successful and how often a villain should defend or call.  Why are these numbers so important and why are they the foundation of GTO play in poker?  

One of my biggest weaknesses or “leaks” in my poker game and study habits is my tendency to assume what I read is correct and not dig into the math of how to play the game.  So instead of just assuming that the formulas for Alpha and MDF in “Modern Poker Theory” (MPT) are correct (they are), I decided to run some numbers and learn the math behind the formulas for myself.  This is already strengthening my understanding of poker theory, and it’s quite simple really.  

I started out not fully understanding the chart on page 79 of MPT.  The chart refers to the size of an all in bet in a simple poker toy game.  The size of the pot is $100, and P1 has a range of AA and QQ, while P2′s hand is KK.  P1 can decide to check or bet, and P2 can decide to call with KK.  Very simple, but I was initially not fully getting why P1′s EV goes up with the size of the bet.  Here is the chart:

Again, my tendency has always been to browse these types of charts and just accept that they are correct without understanding why.  Well, here’s why this chart is correct, and the math starts with Alpha.  

In the toy game used in the text, we assume the P1 bets $100 to keep all the math simple.  In this scenario, villain calls and folds at 50% with KK.  So it’s easy to calculate the EV of a $100 bet for P1:

EV ($100 bet) = [ (EVfold) * (P2fold%) ] + [ (EVcalled) * (P2call%) ]

EV ($100 bet) = (100 * .5) + (200 * .5)

EV ($100 bet) = 50 + 100 = $150

$150 / 2 = $75 (because P1 has AA and QQ 50% of the time)

What stumped me is if you take any bet size on this chart and run the same EV calculation, you don’t get the same answer as the chart.  That is because as the bet size goes up, the villain’s MDF goes down and no longer calls at a 50% clip.  So P1′s Alpha goes up for each separate bet size.  All the numbers are related, but in order to calculate this chart, the starting point is Alpha.  

I ran several iterations of bet sizes of $150, $200, $300, and $500.  I failed trying to do 150 math in my head and decided to take the time to write the rest out.  As you can see from my notes, once I got to $500, and I understood and proved the formulas to myself, I started to rely on the equations for Alpha.  

On a very basic level, as is the case in this toy game, the value of Alpha should be considered before making any zero equity bluff, or any river bluff.  It may not be easy to estimate a player’s MDF in real time, but it’s important to note that the bigger the bet, the more of a chance it needs to force a fold.  What’s not intuitive is that the hero’s EV actually goes up with the size of the overbet.  This is such a simple concept yet commonly misunderstood from casual players like myself.  It’s amazing how poker theory can be better explained by understanding the math of the game.  

[mattbodie]

Textbook

A number of books are quickly becoming more and more available containing formal research, documentation, and teaching about GTO principles.  I’ve passively read a few of them and am beginning to really study and dive into more to develop a more robust understanding of the concept.  But like any academic topic, let’s start with a few textbook definitions of GTO.  

To begin, it’s important to note that GTO does not just apply to poker.  It applies to all sorts of competition and conflict that we as humans have to or love to partake in.  A great summary of this is Michael Acevedo’s definition of Game Theory from his book “Modern Poker Theory,” which states:

“Game Theory: A whole field of math and science that studies mathematical models of conflict or cooperation between intelligent, rational decision makers.  It can be applied to economics, military tactics, politics, psychology, biology, computer science, and card games like poker.”

This comprehensive take on Game Theory is interesting as it can be applied to just about any competition one can think of due to the fact that it is a mathematical based science.  However, what is hidden in the definition is that it makes one enormous assumption - that the competitors are both “intelligent, rational decision makers.”  When applying Game Theory to military conflict, politics, or athletics, this assumption is more often than not true.  However, in a gambling game such as poker, the mental motivations and preconceived strategy notions of all players are absolutely not always intelligent and rational.  It’s important to take this into account when relying on solver outputs during hand analysis.  Sometimes it makes sense not to 3bet a nit with KK even though a solver will almost always recommend doing so at perhaps a 95%+ clip.  Nits are not playing the game on an intelligent (and certainly not rational) level.  

So how can this complicated study of mathematical modeling known as Game Theory be applied to poker, a game where clearly not all the participants are rational thinkers?  Let’s take a step back and look at David Sklansky’s immortal definition of the Fundamental Theorem of Poker: “Everytime you play a hand differently from the way you would have played it if you could see all your opponents’ cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, they lose.  Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.”

That’s very wordy, but very basic in stating that poker is a game of incomplete information.  Because there is incomplete information, we have this study of mathematics or Game Theory to fall back on.  But how do we do that if our opponents are irrational?  

In his brief introduction in “Modern Poker Theory,” Acevedo states that poker for every player is about making a monetary profit, and that “Profit = opponent mistakes - our mistakes.”  Every poker player, even the best, world-class, nosebleed players, make mistakes.  So it’s all about minimizing your mistakes compared to that of your competition.  To minimize one’s mistakes, Acevedo states that the GTO premise is to understand the game at its core to accomplish the following:

- Fix leaks in your own game and find what others are doing wrong.

- Adjust to exploit them. 

This somewhat ties into Sklansky’s Fundamental Theory in that one’s strategy can become so pure that one can force an opponent to play their hands differently from how they would if they could see your cards.  

Since we know that we cannot see our opponents’ cards, nor understand our opponents’ rational (or irrational) motivations, I’ll draw on one more noted expert in poker Game Theory, Andrew Brokos.  In his fantastic book, “Play Optimal Poker,” Brokos does not make the initial assumption of rationality in his initial dictionary definition of Game Theory, quoted from the Oxford Dictionary: “the branch of mathematics concerned with the analysis of strategies for dealing with competitive situations where the outcome of a participant’s choice of action depends critically on the actions of other participants.”  This definition does not assume that one’s opponent is rational, but it does infer that mathematical models exist for dealing with situations that depend on one’s opponent’s moves.  

On the very next page, Brokos differs from the textbook definition of game theory by saying “Game theory offers a completely different framework for thinking about poker.  It assumes that any time there is a mathematically correct play, your opponent will make it.”

On the surface, Brokos is agreeing with Acevedo’s definition of Game Theory in that one’s opponent is a rational-thinking opponent trying to make a profit.  I believe that if we make this assumption during hand analysis, review, and solver output, then it will only make us better at the game of poker.  If we can defeat strong, rational and mathematically sound villains, then we will be better equipped to play the game at higher levels.  We won’t have to worry about the “non-rational” opponents because beating them should already be second nature if you’re reading this.  Cheers.

[mattbodie]

What is GTO?

GTO stands for “Game Theory Optimal.”  It’s the biggest buzzword in poker strategy right now, but what the hell exactly does GTO mean?  Is it a specific strategy to play?  A high-level of thought process?  A robotic method of playing hands?  C-betting 1/3-1/2 pot?

I came up playing online poker pre-Black Friday, and in those days, GTO really wasn’t really a thing at all.  I used to C-bet 3/4 pot on every hand and not think twice about it!  Occasionally, you’d hear the term “game theory” thrown around in a podcast or training video, but in my mind, it was really only applicable to gambling scenarios, specifically, contrarian strategies.  I could never really comprehend how it could be applied to poker because every hand of poker has so many unique variables that there could never be an “optimal” line to take.  

Fast forward a few years and welcome to the world of poker solvers.  While I am a complete rookie when it comes to solvers, I am starting to understand their power in analyzing optimal poker play.  One will never fully be able to solve every poker problem possible, but the more hands that one can analyze, the more prepared for similar situations that player will be in any given hand.  

I realize that I am very late to the poker-solver-GTO trend/party.  Since the time that I had to give up online poker back in 2011, I’ve obviously focused on improving my live game.  With live play, you don’t get in nearly the amount of volume that you do multi-tabling online.  That said, I chose to ignore the GTO noise over the past couple years and simply focus my efforts on beating bad players and making money at the table.  I lost my thirst to understand the game from a theoretical standpoint and simply opted for profitable, bankroll-building live strategy.  

However, this past year PokerStars opened a legal site in my home state of PA, and I’ve put in 52,149 hands with a 9.64 bb/100.  It’s not crazy volume, nothing like I used to do back in the day, but I’d argue that it’s more hands than I’ve played live in the past 9 years since Black Friday.  

So with my new-found access to volume and online profit once again, I now have that thirst and energy to just get better at the game.  Enter GTO.  And let’s loop back to my initial question, what is GTO?  Well, I aim to figure that out exactly in the coming weeks/months/years and write about it here.