“There is nothing so terrible as activity without insight.”
– Johann Wolfgang von Goethe

There’s no escaping it- from a purely financial standpoint poker is a negative sum game. This simple fact presents a conundrum that every mainstream poker training business tries very hard to ignore: It is impossible to have as a sincere mission the objective to ‘teach the world to play great poker’.

“Teach the world to play great poker” – the reader may laugh at loud at the very idea of it. Yet the absurdity of this phrase belies the seriousness of a structural problem that pervades the poker training industry: There is an inevitable conflict of interest that arises between expanding the customer base of a training site and the benefit derived from that training site by each individual customer. In this way, the very success of a poker training site punishes its customers!

In this article I am going to introduce a metastrategic concept that is essential to the success of your poker career- the value of strategic information. After reading the article you will understand why so many players resent mainstream poker training sites for cannibalizing the games, and how to estimate the value of emerging opportunities in poker. You will also discover why a different approach to strategic information distinguishes Cardquant from mainstream poker training sites.

Of necessity this article will only appeal to a fraction of my readership, but those are the very people that I am most eager to make a connection with. So if you really enjoy this article, and want to see the next article in the series come along shortly, please take the time to introduce yourself in the comments. Let’s get started…

This article is the first in the Cardquant Identity Series– a series in which I will introduce a number of new concepts to my readership, explain how they relate to my work at Cardquant, and how my work at Cardquant relates to my larger vision for my scientific research and my philosophical writing.

Information Distribution Curves

It is well established that poker games are games of incomplete information. What is seldom explicitly stated, much less formalized, is that each variant of poker has its own associated information distribution among the competing population, and that this distribution has a strong deterministic influence on the distribution of profits from that game variant.

I have developed the ‘Information Distribution Curve’ as a visual way to represent the various structural features that determine the value of information in a game variant.

Let’s take a look at an example of an Information distribution curve for a poker variant in order to familiarize the reader with the key features of such curves.

Poker Information Distribution Curve

The curve shown ranks the population in order of the level of strategic information about the game that they retain at a given moment in time. The ‘x’ axis spans the population, terminating at the 100th percentile. The ‘y’ axis indicates the quantity of information that each player retains, as a fraction of the theoretical perfect strategic understanding. Whether this perfect understanding is attainable, or is in fact some form of infinity for any given variant, is a philosophical digression that is not significant for the discussion at hand.

I have marked out three key points on this curve:

  • Imax :- This indicates the participant who has the highest level of strategic information for the game variant.
  • Ihero:- This point represents you, the hero, indicating your current level of strategic information for a game variant in the context of the wider population.
  • Iopp:- This point indicates the level of strategic information for a single opponent on the curve. Any point on the curve which is not you represents a potential opponent.

The double-headed blue arrow indicates ∆I: the difference in information level between you and a given opponent, with a positive value giving you the advantage and a negative value conferring an advantage to your opponent. In a heads-up game, only one ∆I would be relevant but in a 6-handed cash game, or a multiplayer tournament, the level of strategic information for every one of your opponents in the game matters (although each opponent’s level is not equally weighted- another complication).

The information landscape for any given game variant will develop over time. Each single-headed arrow on the curve indicates a feature that will change across time:

  • The black vertical arrow adjacent to Imax indicates the rate of discovery dImax/dt of strategic information at the leading edge of the game variant.
  • The black vertical arrow adjacent to Ihero indicates the rate dIhero/dt at which you, the hero, learns new strategic information. That similar such arrows could be drawn for every one of your opponents follows trivially.
  • The white horizontal arrow on the curve indicates the ‘information drift’, which is a term I have coined to describe the rate at which the high level information accessible to those at the top of the population flows to those lower down in the population. For now a qualitative description will suffice, although there are some natural ways to formalize information drift which I will briefly discuss below.

Before we continue there is one other essential feature common to all game variants that I have not indicated on the diagram above- the information ceiling. The information ceiling is the theoretical limit to which it is possible to formalize any particular game variant. The reason that tic-tac-toe is not played for money, and in fact rarely even played at all other than with children, is because the information ceiling is so low that perfect strategy is accessible to all people with even a basic capacity for reason.

Now that we know how to read an Information Distribution Curve it’s time to explore some ways in which you can use the Information Distibution Curve concept to make better metastrategic decisions in your poker career.

Metastrategic Applications of Information Distribution Curves

I’m going to write a formal definition and more thorough discussion of metastrategy in a later article in this series, but for now think of metastrategy as the analysis of the structural factors that determine what the real (as opposed to the ludic) rules and payoffs are in a game, and consequently whether a game is worth playing.

I can assert unequivocally that your metastrategic decisions are the most important ones that you will make in your poker career because the relationship between metastrategy and strategy is hierarchical, with the former dominating the latter.

For any given poker variant it is desirable to be at as high a level of ‘I’ as is possible.
This fact should be obvious, and is generally well understood by all those players who seek to improve their game by studying training materials, or attempting to research it themselves. From this observation it follows that the value of playing in a particular game (as distinguished from the value of the game variant generally) is strongly determined by the difference in I (∆Iho1,o2…on) between you and your opponents.

In an ‘open-borders’ online poker world the set of opponents that you face will have values of ‘I’ that are representative of the population for that game variant. Factors such as the stakes that you play, the time of day that you play, and whether you play on weekends or in holiday seasons will affect which subset of the population you face for a given session, but the accessibility of such online games ensures that the Information Distribution Curve dominates the value derivable from a given game variant.

A poker game variant is said to be ‘information saturated’ when there is very little difference between the level of strategic information available to the top of the population compared with that available to the bottom. The information distribution curve for such a game is shown below:

The Limit Holdem Information Distribution Curve

Information Saturated Game

This game variant is information saturated because the difference ∆I0100 between the very strongest and the very weakest players participating in the variant is too small for even the best players to maintain a high win-rate. Without a large influx of new fish, or a profound conceptual breakthrough that raises the information ceiling or an acceleration of the rate of development at the leading edge dImax/dt, then there is little profit in the open-source game.

The information distribution curve above I understand to be an accurate representation of the current state of Limit Holdem as we move into the year 2020.

It is clear that there is very little profit to be made playing open-borders games in an information saturated game variant. A clear understanding of the concepts introduced so far enables one to see that the most profitable opportunities in poker will be found in either closed games, in game variants with a high information ceiling, or in game variants with a low information drift. Since these three identifiable game properties are not mutually exclusive it follows that the most lucrative poker games will combine all of them.

I will briefly discuss each property in turn below:

Closed games

Closed games are those where access is restricted based on criteria determined by the game’s gatekeeper. In closed games it is possible for you to have a ∆Iheroo1,o2…on that is far more favourable than those available in open-borders games drawn from the same game variant. Examples of such games are any and all home games, many live games spread in casinos, and the emerging app-via-agent game. It is important to understand that if the main reason that a particular game is highly profitable is because it is closed, then that game will either be difficult to get access to, involve an unusually high level of risk (ranging from illegal underground games to unregulated games to unscrupulous opponents), or both.