Knowing that each phase of a spot makes a certain quantity of information available is not enough: that quantity has to be measured. And to measure it you need a tool able to break information down into its smallest parts.
That tool does not come from mathematics. It comes from semiotics and the philosophy of languages, where a group of researchers studies communication according to the principles of componential semantics.
A tool is needed that breaks meaning down
The task is to define, for every phase of the spot, «the value of probability that is realised within many combinatorial possibilities». It is Anolli’s definition of information, and it calls for a mechanism of decomposition and signification: something that takes a code, splits it into the minimal units that compose it, and measures how much each of them carries.
The linguistic characteristics of Texas Hold’em poker allow those same principles to be used to process and quantify the information that is progressively revealed over the course of the spot, and in relation to it to attribute an index of the potential skills involved. It is the step that leads from the completeness of information to a measurement.
From Leibniz to Hjelmslev: componential analysis
Componential semantics has its origin in Leibniz’s componential analysis, which allows meaning to be broken down into its elementary components. It was applied successfully by the Russian linguist Trubetzkoy to the decomposition of linguistic vocal sounds, and later refined with the model of Necessary and sufficient conditions, the NSC, of the Danish linguist Hjelmslev.
Like the linear mathematical model seen earlier, componential semantics and the NSC too have their limits in explaining the complex field of communication. But on the linguistic system of Texas Hold’em poker and on its communicative dynamics they turn out to be a perfectly applicable model.
The two conditions, and why poker satisfies them
The model demands two necessary conditions: that meaning be decomposable into semantic traits for its formation, and that the number of those traits constitute a limited inventory.
The linguistic code of Texas Hold’em satisfies both perfectly. It develops through the random combination of 1326 possible combinations, and the order of arrangement does not matter.
There is more, and it concerns the NSC model: every combination has a univocal, absolute meaning, determined precisely by its constitutive components. A pair of aces means only and exclusively a pair of aces, with no possibility of ambiguity. It is a property natural languages do not have, and it makes the deck a more docile object of analysis than a language.
The informative unit is the single card
The purpose of these tools is to break a code down into elementary units of meaning that cannot be decomposed further, and which combined among themselves generate new meanings according to the grammar that governs them.
In the case of poker, and in consideration of the Fundamental Theorem, it is a matter of defining the smallest quantity of information expressible by the smallest constitutive unit of the fifty-two card deck. That unit is the single card, and in this context it takes the name of Informative Unit, the Ui.
It is a relatively simple step: it means breaking the deck down into each of its minimal constitutive parts, the single element. It is what Hjelmslev, in the field of linguistics, calls a phoneme.
The deck is an alphabet of fifty-two letters
An element taken on its own offers a rather reduced value of information, limited to its nature. You can consider every elementary component as a letter of an alphabet of fifty-two letters.
Take the letter A of our alphabet, though the concept holds for any letter. On its own it expresses the basic information that concerns it by nature — first letter of the alphabet, a vowel — but it is not able, alone, to express a complete sense: it needs to combine with the others to give rise to meanings.
For the deck of cards it works exactly the same way. Every element, taken singly and not decomposable further, emits a limited quantity of information of its own — its value within the deck, its placement, the symbols depicted on it — but it needs to combine with the other components to generate the combinations on which a spot develops.
The informative value of a card is 1.92 per cent
The next step is to quantify the Informative Value, the Vi, of the informative unit: the quantitative capacity of information potentially expressible by every single element present in the deck.
It is not a low value, and it matters to understand why. It does not refer to how much that card says about itself, but to the many informative opportunities that the combinations of the seven known cards are able to generate against the complete deck.
The calculation relates the complete information contained in the deck, expressed in percentage form and therefore equal to one hundred per cent, to the fifty-two elements that compose it: one hundred divided by fifty-two, 1.92 per cent. It is the portion of information expressible by every single card, and the one it is potentially possible to access.
From here it becomes possible to measure how much information each phase of the spot makes accessible, and therefore which skills can be identified in each phase. It is the degree of access to that information that makes the difference between one player and another, and it is on that difference that the indices are built, as the dissertation demonstrates step by step.