• Poker Optimal Bluffing Frequency Definition
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If you've never played poker, you probably think that it's a game for degenerate gamblers and cigar-chomping hustlers in cowboy hats. That's certainly what I used to think. It turns out that poker is actually a very complicated game indeed. Poker originated in Europe in the middle ages. The early forerunners of poker originated in Europe in the middle ages, including brag in England and pochen. And David Sklansky, in his book The Theory of Poker, states 'Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting.' Poker Game Theory Application #1 – Optimal Bluffing Frequency As made famous by David Sklansky the idea is that you should bluff with a busted hand on the river at the same frequency as the odds you are offering your opponent from the pot. The bluffing ratio of one bluff for every two legitimate bets and the calling frequency of 50% are a general results for all situations in which Player B will bet or bluff the size of the pot. The optimal ratios will change depending on the size of the bet in relation to the pot, but are independent of other factors.

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Poker Optimal Bluffing Frequency Definition

An Example: Optimal Bluffing and Calling Frequencies

Much has been written about game theoretic optimal frequencies forbluffing, and calling a possible bluff. This serves as a nice example ofhow the underlying principles of game theory can be used as a startingpoint for a poker algorithm, but then must eventually be transcended toachieve the highest playing levels.

It can be shown that the theoretical maximum guaranteed profit from agiven poker situation can be attained by bluffing, or calling a possiblebluff, with a predetermined probability. The relative frequency of theseactions is based only on the size of the bet in relation to the size ofthe pot. To ensure the best result against perfect play, the action mustbe unpredicatable, and one way to accomplish this is by selecting aparticular range of hands to act upon, which will occur uniformly atrandom.

In the following example, we imagine two players involved in a hand ofpot-limit Draw poker (where either may bet an amount up to the currentsize of the pot). Player B has called with a one-card draw to a flush,against Player A who currently has the best hand. To simplify the math,we will assume that Player B will win the showdown if she makes the flush,but will lose otherwise. We will further assume that the probability ofcompleting the flush is exactly 0.20, or one in five. The question is howthe hand should be played after the draw.

The first principle is that Player A should not bet, because Player B willsimply fold if she missed on the draw, but will call (or raise) if shemade the flush. Since there is no profit in Player A betting, we canassume without loss of generality that Player B is first to act after thedraw. The correct strategy for Player B is to bet (the size of the pot)whenever she makes the flush, and also to bet occasionally when the drawfailed. The optimal frequency of bluffs by Player B and calls by Player Aare computed with a game theoretic analysis. For each pair of frequenciesthe overall expectation (expressed as a fraction of the total pot beforethe draw) can be calculated. Table 2 gives a sampling of these valuesover the full range of bluffing and calling frequencies.

Table 2: Expected Values for a Four-Flush Draw: Bluffing vs CallingFrequencies

Legend:

BR = ratio of bluffs to legitimate bets

ABF = absolute bluff frequency (fraction)

Poker Optimal Bluffing Frequency Analyzer

CFr = absolute calling frequency (fraction)

(expected values are expressed as a fraction of the total pot,

given a 0.2 legitimate betting frequency, and pot-sized bet)

If Player B never bluffs and Player A never calls, it has the same effectas having no betting round after the draw, and the expected value is 0.20of the pot for B and 0.80 for A. We can see from the table that to obtainthe guaranteed maximum, Player B should bet 30% of the time - 20% withthe flush and an additional 10% as bluffs, selected at random. Player Acan always ensure his optimal expectation of 0.70 by calling exactly 50%of the time Player B bets. The bluffing ratio of one bluff for every twolegitimate bets and the calling frequency of 50% are a general resultsfor all situations in which Player B will bet or bluff the size ofthe pot. The optimal ratios will change depending on the size of the betin relation to the pot, but are independent of other factors.

While these are optimal strategies, they are not maximalstrategies. A maximal strategy is directed toward exploiting weaknessesin the opponent, whereas an optimal strategy implicitly assumes perfectplay on the part of the opponent.

The game theoretic approach is valid if the opponent is a very strongplayer, or perhaps an unknown player, but is certainly not the way tomaximize net profit in the long run. In a typical game of poker, gametheory is not an appropriate strategy, because it also guarantees that aplayer makes no more than the expected value from the particulargame situation. This effectively ensures that the opponent also playsoptimally, regardless of her approach to the game.

As an example of maximizing strategies, we observe how a strong pokerplayer handles this type of situation. If faced with a bet from a playerwho never bluffs, a strong player will usually fold a marginal hand,knowing she cannot win. Conversely, she will often call a chronicbluffer, even with only a mediocre holding. In the role of Player B, astrong player will frequently bluff against an overly conservative player,but will seldom try to bluff a player who almost always calls. The netresult is an expectation higher than the optimal 0.3, and the tabledemonstrates just how profitable these strategy adjustments can be inpractice.

An algorithm based on game theoretic principles will provide a solid basisfor betting strategy. Nevertheless, to advance to the highest levels, aprogram must be able to understand each opponent's playing style, and beable to adapt to the specific game conditions.

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Thu Feb 12 14:00:05 MST 1998

There’s a buzz word in the poker world these days that you may have heard, but many new players aren’t yet familiar with it. That buzz is GTO.

GTO stands for Game Theory Optimal. What it means is using an unexploitable strategy, which cannot be countered by your opponent. In this article I’m going to explain what it is, tell you why you should learn about it, and then explain why you shouldn’t be focusing on it in games.

So for starters, what does that mean, an unexploitable strategy? You will also hear the word balanced or balancing used in these discussions. I think the easiest way to explain this is with a basic example:

Let’s say there is $100 in the pot, and you bet $100 on the river. Your opponent, who has a medium strength hand (a bluff catcher, something like JT on a board of QJ963), has to now decide whether or not to call this final river bet. They have to call $100 to win the $200 now in the pot. Therefore the call must be right 1/3rd of the time. If it is right exactly this often, the call breaks even over the long run (losing $100 each of the two times it’s wrong, and winning $200 the one time it’s right). And they have a hand strength that will lose to everything you are betting for value, and beat everything you’re bluffing with. Now, if they had a read that you were a conservative player that never bluffs the river, then they can easily fold to this bet. Conversely, if you were a known wild bluffer, they can easily call knowing they’ll catch you well more than 1/3rd of the time with a bluff. Let’s say, however, that you bet your range of hands on the river such that you are value betting 2/3rds of the time and bluffing 1/3rd. This is the unexploitable strategy… your opponent is now indifferent to calling or folding. If they call all the time, they will break even by catching you bluffing at the precise frequency that the pot odds are offering them, netting them zero won/lost over the long term (and if they fold all the time by definition they win/lose zero on the river over the long term). So you are bluffing at a GTO frequency, making this river bet unexploitable… it doesn’t matter if they call or fold.

Why should you learn about GTO strategies? I think you may start to see from the example, that learning about GTO strategies involves topics like basic math, odds, ranges, and frequencies. As you study these things, you’ll develop a much stronger sense of constructing solid ranges, so when you get to the river in a hand you have a more balanced range… in the case of the example above, picking the appropriate number of value bets vs. bluffs. And learning what types of hands are better to bluff with given the board texture, situation, and bet sizing. A firm grasp of these concepts will help you to make much better decisions, understand situations and ranges better, and help control your opponents through keeping proper frequencies, and exploiting their frequencies (like in the example above, if you were a known bluffer or a known non-bluffer, those are lopsided frequencies that can be exploited by calling down frequently, or not calling down with any marginal made hands at all. These are all concepts that fall under the umbrella of GTO play. Learning this will not only strengthen your understanding of the game a great deal, but also prepare you to play against other very strong players who also understand these concepts. If you are in a tournament heads up against Fedor Holz, playing a GTO strategy will prevent Fedor (or any expert) from being able to exploit you.

Now that we’ve talked about what GTO means, and why it’s a good thing to learn more about, let’s talk about why you should not be focusing on doing this in your games. This comment may surprise you. This GTO business sounds pretty nice. I should work on it away from the tables to strengthen my game. But now in the heat of battle, not use it? That’s right. The reason is, for most readers, you will be playing against opponents who make many frequent mistakes. And thus, although a GTO strategy will be profitable against them, an exploitative strategy will do even better. For instance, in our example above, we know that when we bet $100 into $100, the GTO bluffing frequency is 33% to make our opponent indifferent to calling or folding. If our opponent is a world class player who excels at reading their opponents, this forced indifference is a good thing… by default they cannot get the best of us. Most of your opponents, however, will not be Fedor Holz or Phil Ivey. Especially in the micro and small stakes games, they will be making many frequency mistakes.

Poker Optimal Bluffing Frequency Calculator

So for instance, if your opponent were a tight/conservative player who never calls big river bets without a monster hand, then we will make quite a bit more money over time by bluffing at a higher frequency than 33%. In fact against this player a much better strategy might be to value bet much tighter/stronger hands, and add in more bluffs from our range so that we may be betting with 20% value bets (our strongest hands only) and 80% bluffs. This is a frequency that is well out of balance, and our opponent could easily exploit us by calling down much lighter (a common adjustment to someone who bluffs too much). But unlike the world class player, they won’t recognize this and exploit us, they’ll just keep folding too much. Or on the other side of that coin, if our opponent is a calling station that calls down with any pair or even ace highs, then we can exploit that by betting much wider for value (2nd pair may be an easy value bet vs. this type of calling station), and not bluffing much if at all. Again, a strong player can easily exploit us if our river bets are always value bets and never bluffs by simply folding all medium strength bluff catchers to our bet. But our calling station friend won’t, they’ll just call, providing us much more value long term than the GTO value bet/bluff frequency would.

The study of GTO strategies isn’t for the beginning player. But once you have the basics down and as you advance your game, it becomes an integral part of a more advanced poker strategy.

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