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Models and their
functions
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Games of chance are
developed in their physical consumer-ready form on the basis of
mathematical models, which stand as the premises of their existence
and represent their physical processes. As an example (of
physical-process representation), roulette betting is represented as
follows:
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A roulette bet is
defined as a finite family
 ,
where  is
a subset of the set R of roulette numbers, which allows a
single placement according to the configuration of the roulette
table (such as for straight, split, corner column,
color
bet, etc.),
 is
the payout of ,
and  is
the stake of the placement
.
All bets B so defined form the bet space. Then, for each bet
B, a profit function
 is
defined as follows:  R,
 ,
where R is the set of real numbers and
 is
the characteristic function of set A. Variable
 stands
for the outcome of a game. The value of function
is
called the profit of bet B (Barboianu, 2007). For each
bet B, function may
take negative or non-negative values. The profit function expresses
the net amount the player wins or loses after the spin as a result
of the player’s bet (applying the convention that profit can also be
negative, that is, a loss). This formal system is part of a
mathematical model of the roulette game (there is more than one
model involved in the game of roulette) representing some of the
physical processes of betting, namely chip placements and their
overall financial result after the spin. Using the mathematical
structure of the bet space and the properties of the profit
function, a relation of equivalence between bets can be defined; in
the real world, the relation represents a natural financial
equivalence between bets. The inferred properties of the equivalence
of the bets in the mathematical model yield criteria of optimization
of roulette betting with respect to personal money and time
management. This mathematical model of a roulette bet can be
generalized and adapted further to other types of bets specific to
other games.
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Mathematical models
can additionally represent gambling systems related to several
plays, several games, and generally any quantifiable gambling
activity. For instance, still from roulette, the Martingale system
(keeping the same bet and raising its stake with the same multiplier
successively until the first success) is modeled through a
geometrical progression whose partial sums of terms obey a certain
inequality, which in fact ensure an overall positive profit when a
bet is won. In the Martingale example, not only do we have a
representation of a physical process (the described betting
system) through a mathematical model, but also a mathematical
explanation (through the same model) of the trust this roulette
system is granted by gamblers.
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- Two types of models
- Games of chance and gambling as a
quantifiable activity are represented in applied mathematics through
specific mathematical models which can be distinguished through two
main categories with respect to the purposes they serve, which I
call:
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1.
Probabilistic and
statistical models
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2.
Functional models
- While probabilistic and
statistical models serve for the applications related to the games’
outcomes occurring under conditions of uncertainty, functional
models serve to represent the physical systems and processes that
make the games actually function as well as for applications related
to the functioning.
- As examples, computing the
probability of hitting a specific number or at least a specific
quantity of winning numbers at a specific lottery is workable within
a probabilistic model, which assumes establishing the right
probability field within which to work with the appropriate discrete
probability distribution. Computing statistical means and errors (in
the statistical sense) is workable within a statistical model
by establishing the distribution of the random variable which
describes the outcome and using mathematical means and measures such
as expected value, deviation, dispersion, or variance. Card movement
in poker is described through a functional model dealing with
combinations of symbols and values specific to cards. Roulette
complex bets are represented in a functional model as elements of a
mathematical structure with vectorial and topological features.
Paylines of a multiline slot machine are represented as lines in a
Cartesian grid or paths in a graph, and their mutual independence is
described through topological properties, all these still within a
functional model (Barboianu, 2013a).
- Below are the purposes and
mathematical governing theories involved in each of the two
categories of mathematical models used in gambling.
- 1. Probabilistic and statistical
models. Purposes: quantification of the gambling uncertainty,
measuring risk and possibility, prediction, computation of the
parameters characterizing games, of the means and statistical
errors, providing practical statistics (collections of data),
optimization, providing strategy and optimal play. Governing
theories: Measure Theory, Probability Theory, Mathematical
Statistics, Real Analysis, Decision Theory.
- 2. Functional models. Purposes:
description of the gaming processes and of the functioning of the
games, optimization, providing strategy and optimal play, providing
the necessary theoretical support for the probabilistic and
statistical models. Governing theories: Set Theory,
Combinatorics, Number Theory, Algebra, Topology, Geometry, Graph
Theory, Real Analysis.
- A game and the gambling activity
related to that game are represented by several mathematical models,
each of them serving a purpose for the applied mathematician. This
multimodel feature holds even for the same subcategory of models.
For instance, computing the strength of a poker hand in terms of
probabilities assumes establishing several different probability
fields for the events related to one’s own hand and those events of
type “at least one” related to the opponents’ hands, which means a
different probabilistic model for each application. Every game of
chance is represented by models from both the first and second
categories, even though a model from the latter category may be
trivial. This happens because outcomes and uncertainty are specific
to games of chance by definition (thus explaining the existence of
first-category models) and every probabilistic/statistical model
needs a functional model in order to ensure the grounding
mathematical structures necessary for the governing theories of the
probabilistic/statistical model to be applied. For example, any
probability computation within a probabilistic model needs a
priori a grounding model representing the gaming events to be
measured, which must belong to a Boolean structure, and this latter
model is a functional one.
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- Prevalence of the probabilistic
and statistical models
- There is a prevalence of models of
the first category in the interest of all parties involved in the
study of gambling – researchers, game producers and operators, and
players. This prevalence is explainable first through the fact that
these models provide measures, estimations, and predictions for the
financial results of the gambling activity, which in turn generate
the most important indicators for the commercial aspect of the
phenomenon. These models actually provide game producers and
operators with a mathematical “guarantee” that a certain game can be
run with no risk of ruin for the house over the long run; for the
players, probabilities and statistical indicators are the most
important criteria in making gaming decisions. Second, games of
chance have simple processes of functioning. For commercial reasons,
they are designed to be as undemanding as possible with
straightforward sets of rules and short timeframes of the sessions;
according to this uncomplicated design, the functional models
representing them are usually simple, sometimes trivial (unlike
other types of games – for instance strategy games such as chess –
whose complexity is modeled through richer mathematical models).
There are also exceptions to this functional-model-simplicity rule
– that is, games whose apparently simple functioning hides complex
mathematical models. Such is the case with roulette betting, in
which complex bets are represented as elements of a mathematical
structure generating vectorial and topological spaces and classes of
equivalence within which various further applications can be
developed. This exception applies also to multiline slots, where
probability applications related to groups of paylines rely on a
representation of the display as part of a discrete mathematical
structure (Cartesian grid or graph) generating a metric space (and
implicitly a topology), within which properties such as connection,
neighboring, and independence are defined to serve the probabilistic
models (Barboianu, 2013a, 2013b).
- This prevalence of the models of
the first category in the study of gambling is also reflected in the
content of the existent courses of gambling mathematics – either
experimental or school based – where gambling is presented
exclusively as a plain, direct application of Probability Theory and
Mathematical Statistics.
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- Epistemic view of the act of
mathematical modeling
- Mathematical modeling is the main
mean of inference in science, and modern to contemporary accounts of
the role of mathematics in scientific explanation of physical
phenomena have argued for its indispensability (see Quine, 1981;
Colyvan, 1998; Baker, 2009;
Saatsi, 2011). The inference based on
mathematical models, which is the core method of scientific
reasoning, is possible first because mathematics is a rich source of
structures, and second because of the representation function
of a mathematical model, which allows mathematical structures to be
recognized as embedded in the physical world. Here is how such
inference works in brief, in terms of Bueno and Colyvan’s (2011)
inferential conception of mathematical modeling:
- Step 1. A mapping is
established from the investigated physical system to a convenient
mathematical structure, a step called immersion. The purpose
is to relate the relevant aspects of the physical system (which is
thus idealized) with the appropriate mathematical theory.
- Step 2. Consequences
are drawn from the formal systems within the mathematical theory
that deals with those systems (the governing theory), through
logical inference. This step is called derivation.
- Step 3. A mapping is
established from the mathematical structure to the initial physical
system (not necessarily the inverse of the immersion mapping), and
the mathematical consequences that were obtained in the derivation
step are interpreted in terms of the initial (non-idealized)
physical system. This step is called interpretation.
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Figure. The inferential
conception of applied mathematics
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- The mapping account (also called
structural account) establishes homomorphic or isomorphic
relations between mathematical structures from within a mathematical
theory and mathematical structures recognized in the idealized
physical system (Bueno & French, 2012). This feature of the mapping
account justifies epistemically the inference based on the
mathematical model. The practical execution of the three steps above
is the object of applied mathematics, and the setup of the mappings
from steps 1 and 3 is called mathematical modeling.
- There are also other accounts of
the application of mathematics in physical universe, including
Frege’s (1884, 1889) semantic account and Pincock’s (2004) internal-relation
account. Yet, the structural account is the most representative in
the case of gambling, as well as in the most of the domains and
sciences where the mathematical apparatus is required.
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- References:
Baker, A. (2009). Mathematical
Explanation in Science. British Journal for the Philosophy of Science,
60(3), pp. 611-633.
Barboianu, C. (2007). Complex
bets. In Roulette Odds and Profits: The Mathematics of Complex Bets
(pp. 24–30). Craiova, Romania: Infarom.
- Barboianu C., (2013a).
Configuration of the display. In The mathematics of slots: Configurations,
combinations, probabilities (pp. 18–36). Craiova, Romania:
Infarom.
Barboianu C., (2013b).
The probabilities of the winning events related to several lines. In The mathematics of slots: Configurations,
combinations, probabilities (pp. 107–125). Craiova, Romania:
Infarom.
Bueno, O. &
Colyvan, M., 2011. An Inferential Conception of the Application of
Mathematics. Noûs, 45(2), pp. 345-374.
Bueno, O. & French, S., 2012. Can Mathematics Explain Physical Phenomena?
The British Journal for the Philosophy of Science, 63(1), pp. 85-113.
Colyvan, M. (1998). In Defence
of Indispensability, Philosophia Mathematica, 6(1), pp. 39–62.
Frege, G.,
1884. Die Grundlagen
der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff
der Zahl, Breslau: W. Koebner. Translated as The Foundations of Arithmetic: A logico-mathematical
enquiry into the concept of number, by J.L. Austin, Oxford: Blackwell,
second revised edition, 1974.
Frege, G.,
1889. Begriffsschrift,
eine der arithmetischen nachgebildete Formelsprache des reinen Denkens,
Halle a. S.: Louis Nebert. Translated as Concept Script, a formal language of pure thought modelled
upon that of arithmetic, by S. Bauer-Mengelberg in J. vanHeijenoort
(ed.), From Frege to Gödel: A Source Book in Mathematical Logic,
1879–1931, Cambridge, MA: Harvard University Press, 1967.
Pincock, C., 2004. A New Perspective on
the Problem of Applying Mathematics. Philosophia Mathematica,
12(3), pp. 135-161.
Quine, W.V. (1981). Things and Their Place in Theories. In Theories
and Things, (pp.1-23). Cambridge: Harvard University Press.
Saatsi, J. (2011). The Enhanced Indispensability Argument:
Representational versus Explanatory Role of Mathematics in Science.
British Journal for the Philosophy of Science, 62(1), 143-154
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This entry should be cited as:
- Barboianu, C.
(2014).
Mathematical Models of Games of Chance and Gambling.
Retrieved from
http://probability.infarom.ro/models.html .
back to Problem
Gambling index
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Author |
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The author of this page is Catalin Barboianu
(PhD). Catalin is a games mathematician and problem gambling researcher,
science writer and consultant for the mathematical aspects of gambling
for the gaming industry and problem-gambling institutions.
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