Abstract
This paper introduces Hermite’s polynomials, in the description of quantum games. Hermite’s polynomials are associated with gaussian probability density. The gaussian proba- bility density represents minimum dispersion. I introduce the concept of minimum entropy as a paradigm of both Nash’s equilibrium (maximum utility MU) and Hayek equilibrium (mini- mum entropy ME). The ME concept is related to Quantum Games. Some questions arise after carrying out this exercise: i) What does Heisenberg’s uncertainty principle represent in Game Theory and Time Series?, and ii) What do the postulates of Quantum Mechanics indicate in Game Theory and Economics?.
PACS numbers:
03.67.Lx; 03.65.Ge
AMS numbers:
91A22; 91A23; 91A40
1 Introduction
The quantum games and the quantum computer are closely-related. The science of Quantum Computer is one of the modern paradigms in Computer Science and Game Theory [5,6,7,8,23,41,42,53]. Quantum Com- puter increases processing speed to have more effective data-base search. The quantum games incorporate Quantum Theory to Game Theory algorithms. This simple extrapolation allows the Prisoner’s Dilemma to be solved and demonstrates that the cooperative equilibrium [44,47,51,57] is viable and stable with a probability different from zero.
In Eisert [7,8] the analogy between Quantum Games and Game Theory is expressed as: “At the most abstract level, game theory is about numbers of entities that are effi ciently acting to maximize or minimize. For a quantum physicist, it is then legitimate to ask: what happens if linear superpositions of these actions are allowed for?, that is, if games are generalized into the quantum domain. For a particular case of the Prisoner’s Dilemma, we show that this game ceases to pose a dilemma if quantum strategies are implemented for”. They demonstrate that classical strategies are particular quantum strategy cases.
Eisert, Wilkens, and Lewenstein [8], not only give a physical model of quantum strategies but also express the idea ofidentifying moves using quantum operations and quantum properties. This approach appears to be fruitful in at least two ways. On one hand, several recently proposed quantum information application theories can already be conceived as competitive situations, where several factors which have opposing motives interact. These parts may apply quantum operations using bipartite quantum systems. On the other hand, generalizing decision theory in the domain of quantum probabilities seems interesting, as the roots of game theory are partly rooted in probability theory [43,44]. In this context it is ofinterest to investigate what solutions are attainable if superpositions of strategies are allowed [18,41,42,50,51,57]. A game is also related to the transference ofinformation. It is possible to ask: what happens if these carriers ofinformation are applied to be quantum systems, where Quantum information is a fundamental notion ofinformation? Nash’s equilibria concept as related to quantum games is essentially the same as that of game theory but the most important difference is that the strategies appear as a function of quantum properties of the physical system [41,42].
This paper essentially explains the relationship which exists among Quantum Mechanics, Nash’s equi- libria, Heisenberg’s Uncertainty Principle, Minimum Entropy and Time Series. Heisenberg’s uncertainty principle is one of the angular stones in Quantum Mechanics. One application of the uncertainty principle in Time Series is related to Spectral Analysis “The more precisely the random variable VALUE is deter- mined, the less precisely the frequency VALUE is known at this instant, and conversely”. This principle indicates that the product of the standard deviation of one random variable xt by the standard deviation ofits frequency w is greater or equal to 1/2.
This paper is organized as follows:
2 Quantum Games and Hermite’s Polynomials
Let Γ = (K, S, v) be a game to n−players, with K the set of players k = 1,..., n. Thefinite set Sk of cardinality lk ∈ N is the set of pure strategies of each player where k ∈ K, skjk ∈ Sk, jk = 1, ..., lk and S = ΠKSk represents the set of pure strategy pro fi les with s ∈ S an element of that set, l = l1, l2,..., ln represents the cardinality of S, [12,43,55,56].
The vector function v : S → Rn associates every profile s ∈ S, where the vector of utilities v(s) = (v1(s),..., vn(s))T , and vk(s) designates the utility of the player k facing the profile s. In order to understand calculus easier, we write the function vk(s) in one explicit way vk(s) = vk(j1, j2,..., jn). The matrix vn,l represents all points of the Cartesian product Πk∈KSk. The vector vk(s) is the k– column of v.
If the mixed strategies are allowed then we have:
the unit simplex of the mixed strategies of player k ∈ K, and 
the probability vector. The set of pro fi les in mixed strategies is the polyhedron Δ with 
, where 
and 
. Using the Kronecker product ⊗ it is possible to write1:
where
the probability vector. The set of pro fi les in mixed strategies is the polyhedron Δ with , where and . Using the Kronecker product ⊗ it is possible to write1:
The n− dimensional function 
associates with every pro fi le in mixed strategies the vector of expected utilities
where 
is the expected utility of the player k. Every 
represents the expected utility for each player’s strategy and the vector uk is noted 
.
associates with every pro fi le in mixed strategies the vector of expected utilities
is the expected utility of the player k. Every represents the expected utility for each player’s strategy and the vector uk is noted .
The triplet 
designates the extension of the game Γ with the mixed strategies. We get Nash’s equilibrium (the maximization of utility [3,43,55,56,57]) if and only if, ∀k, p, the inequality 
is respected.
designates the extension of the game Γ with the mixed strategies. We get Nash’s equilibrium (the maximization of utility [3,43,55,56,57]) if and only if, ∀k, p, the inequality is respected.Another way to calculate the Nash’s equilibrium [43,47], is leveling the values of the expected utilities of each strategy, when possible.
If the resulting system of equations doesn’t have solution (p(−k))∗ then we propose the Minimum Entropy Method. This method is expressed as Minp (Σk Hk(p)), where 
standard deviation and Hk(p∗) entropy of each player k.
standard deviation and Hk(p∗) entropy of each player k.
2.1 Minimum Entropy Method
Theorem 1
(Minimum Entropy Theorem). The game entropy is minimum only in mixed strategy Nash’s equilibrium. The entropy minimization program Minp (Σk Hk(p)), is equal to the standard deviation minimization program Minp (Πkσk (p)),when 
has gaussian density function or multinomial logit.
has gaussian density function or multinomial logit.According to Hayek, equilibrium refers to the order state or minimum entropy. The order state is the opposite of entropy (disorder measure). There are some intellectual in fl uences and historical events which inspired Hayek to develop the idea of a spontaneous order. Here we present the technical tools needed in order to study the order state.
Case 1
If the probability density of a variable X is normal: N (µk,σk), then its entropy is minimum for the minimum standard deviation (Hk )min ⇔ (σk)min. ∀k = 1, ..., n.
Proof.
Let the entropy function 
and p(x) the normal density function. Writing this entropy function in terms of minim−u∞m standard deviation we have.
developing the integral we have
and p(x) the normal density function. Writing this entropy function in terms of minim−u∞m standard deviation we have.
For a game to n− players the total entropy can be written as follows:
after making a few more calculations, it is possible to demonstrate that
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The entropy or measure of the disorder is directly proportional to the standard deviation or measure of the uncertainty [1,13,27]. Clausius [48,49], who discovered the entropy idea, presents it as both an evolutionary measure and as the characterization of reversible and irreversible processes [45].
Case 2
If the probability function of 
is a multinomial logit of parameter λ (rationality parameter) [35,36,37,38,39,40], then its entropy is minimum and its standard deviation is minimum for λ → ∞ [21,22].
is a multinomial logit of parameter λ (rationality parameter) [35,36,37,38,39,40], then its entropy is minimum and its standard deviation is minimum for λ → ∞ [21,22].Proof.
Let 
the probability for k ∈ K, jk = 1, ..., lk
where 
represents the partition utility function [1,45].
the probability for k ∈ K, jk = 1, ..., lk
represents the partition utility function [1,45].The entropy Hk(pk), expected utility 
, and variance 
will be different for each player k.
, and variance will be different for each player k.
Using the explicit form of 
, we can get the entropy, the expected utility and the variance:
The equation 
can be obtained using the last seven equations; it explains that, when the entropy diminishes, the parameter (rationality) λ increases.
, we can get the entropy, the expected utility and the variance:
can be obtained using the last seven equations; it explains that, when the entropy diminishes, the parameter (rationality) λ increases.The rationality increases from an initial value of zero when the entropy is maximum, and it drops to its minimum value when the rationality toward the in finite value [38,39,40]: Limλ→∞Hk(pk(λ)) = min(Hk)
The standard deviation is minimum in the Nash’s equilibria [21,22,23].
If the rationality increase, then Nash’s equilibria can be reached when the rationality spreads toward to its in finite: Limλ→∞σk (λ)= 0.
Using the logical chain that has just been demonstrated, we can conclude that the entropy diminishes when the standard deviation diminishes:
after making a few more calculations, it is possible to demonstrate that
■
Remark 1
The entropy Hk for gaussian probability density and multinomial logit are written as 
and 
and Case 3
The special case of Minimum Entropy is when 
and the utility function value of each strategy 
, are the same for all jk, and for all k, see Binmore, Myerson [3,48].
and the utility function value of each strategy , are the same for all jk, and for all k, see Binmore, Myerson [3,48].In the special case of Minimum Entropy when 
, the gaussian density function can be approximated by Dirac’s Delta 
.
, the gaussian density function can be approximated by Dirac’s Delta .The function is called “Dirac’s function”. Dirac’s function is not a function in the usual sense. It represents an infinitely short, infinitely strong unit-area impulse. It satisfies = 
and can be obtained at the limit of the function
is called “Dirac’s function”. Dirac’s function is not a function in the usual sense. It represents an infinitely short, infinitely strong unit-area impulse. It satisfies = and can be obtained at the limit of the function
Case 4
If we can measure the standard deviation 
,then Nash’s equilibrium represents minimum standard deviation (σk)min , ∀k ∈ K.
,then Nash’s equilibrium represents minimum standard deviation (σk)min , ∀k ∈ K.
Theorem 2
Minimum Dispersion. The gaussian density function f (x) = |φ(x)|2 represents the solution of one differential equation given by 
related to minimum dispersion of a lineal combination between the variable x, and a Hermitian operator 
related to minimum dispersion of a lineal combination between the variable x, and a Hermitian operator Proof.
This proofis extended in Pena and Cohen-Tannoudji [4,48]. Let 
be Hermitian operators 2 which do not commute, because we can write,
be Hermitian operators 2 which do not commute, because we can write,
Easily we can verifi ed that the standard deviations satisfy the same rule of commutation:
Let J (α) be the positive function defined as expected value ⟨⟩.
By hypothesis the Hermitian operators can be written as 
. Using this property,
the expression J (α) has the minimum value when J ’(α) = 0, thus the value of α and Jmin are
simplifying Jmin
. Using this property,
We are interested in knowing the explicit form of the function φ(x) when the inequality is transformed into equality and 
, where 
for which we need to solve the following differential equation.
if 
, and 
then 
and
the solution is
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, where for which we need to solve the following differential equation.
, and then and
Remark 2
The gaussian density function f (x) is optimal in the sense of minimum dispersion.
According to the last theorem 
is a gaussian random variable with probability density fk (x), ∀k, and in order to calculate 
. We can write
is a gaussian random variable with probability density fk (x), ∀k, and in order to calculate . We can write
Remark 3
The central limit theorem which establishes that “The mean 
of the random variable 
has normal density function”, is respected in the Minimum Entropy Method.
of the random variable has normal density function”, is respected in the Minimum Entropy Method.2.2 Quantum Games Elements
A complete relationship exists between Quantum Mechanics and Game Theory. Both share an random nature. Due to this nature it is necessary to speak of expected values. Quantum Mechanics defines observed values or expected values in Cohen-Tannoudji [4].
According to Quantum Mechanics an uncertainty exists in the simultaneous measure of two variables such as: position and momentum, energy and time [4]. A quantum element that is in a mixed state is made respecting the overlapping probabilistic principle of pure states. For its side in Game Theory a player that uses a mixed strategy respects the vNM (Von Newmann and Morgenstern) function or overlapping probabilistic of pure strategies according to Binmore [3,43]. The state function in quantum mechanics does not have an equivalent in Game Theory, therefore we will use state function as an analogy with Quantum Mechanics. The comparison between Game Theory and Quantum Mechanics can be shown in an explicit way in Table 1. If the definition of rationality in Game Theory represents an optimization process, we can say that quantum processes are essentially optimal, therefore the nature of both processes is similar
Table 1.
Quantum Mechanic and Game Theory properties
Let k = 1, ..., n be players with lk strategies for every one 
. According to the theorem of Minimum Dispersion, the utility 
converges to Nash’s equilibria and follows a normal probability density 
, where the expected utility is 
. According to the theorem of Minimum Dispersion, the utility converges to Nash’s equilibria and follows a normal probability density , where the expected utility is Definition 1
(Tactics or Sub- Strategies) Let 
be the set of sub-strategies 
with a probabilitygiven by 
where 
and 
when k ≠ l. A strategy jk is constituted of several sub-strategies 
be the set of sub-strategies with a probabilitygiven by where and when k ≠ l. A strategy jk is constituted of several sub-strategies With the introduction of sub-strategies the probability of each strategy is described in the following way 
and we do not need any approximation of the function 
where:
and 
and we do not need any approximation of the function where:
Theorem 3
The normal probability density ρ(u)= |ψ (x, λ)|2 can be obtained using Hermite orthogonal polynomials Hk (x). The parameter 
, indicates the probability value of playing jk strategy. The state function ϕk(x,λ) measuresthe behavior of jk strategy (jk one level in Quantum Mechanics [4,9,31]).The dynamic behavior of jk strategy can be written as 
. One approximation is 
, indicates the probability value of playing jk strategy. The state function ϕk(x,λ) measuresthe behavior of jk strategy (jk one level in Quantum Mechanics [4,9,31]).The dynamic behavior of jk strategy can be written as . One approximation is The function ψ(x, λ) is determined as follows:
where: 
and
and
Proof.
The Hermite polynomials properties will be necessary to express ρ(u) in function of Hk(x).
Using generating function ψ(x, λk) we can write
Let ψ(x, λk) be the generating function
In order to get 
with the equations (QG1, PH1, PH2) and:
with the equations (QG1, PH1, PH2) and:
If 
then wk = 0; we can write:
then wk = 0; we can write:
Comparing (Q2) and (Q3) is easy to conclude
■
The 
parameter represents the probability of 
sub-strategy of each k player. The utility of vNM can be written as:
parameter represents the probability of sub-strategy of each k player. The utility of vNM can be written as:
Using the operators a, a+ let us write the equations:
The equation (Q6.1) indicates that the operator 
decreases the level of k player. The operator a can decrease the sub-strategy description 
decreases the level of k player. The operator a can decrease the sub-strategy description
The equation (Q7.1) indicates that the operator 
increases the level of k player [4,46]. The operator a+ can increase the sub-strategy description 
increases the level of k player [4,46]. The operator a+ can increase the sub-strategy description The Hermite polynomials form an orthogonal base and they possess desirable properties that permit us to express a normal probability density as a linear combination of them.
2.3 Hermite’s Polynomials Properties
The next properties are taken from Cohen-Tannoudji, Pena and Landau [4,31,46]
Hermite polynomials obey the next differential equation [14]:
The principal solution of Hermite’s differential equation3 is
Taking the explicit value of 
−derivative of (PH1):
−derivative of (PH1):
From which we obtain the decreasing operator:
and the increasing operator:
Recurrence formulas:
and normalization condition:
The next condition expresses the orthogonality property of Hermite’s polynomials:
The generating function is obtained using the Taylor development.
Quantum Games generalize the representation of the probability in the utility function of vNM (von Newmann and Morgenstern). To understand Quantum Games it is necessary to use properties of quantum operators in harmonic oscillator and the properties of Hilbert space. The harmonic oscillator is presented as the paradigm in Quantum Games.
2.4 Quantum Games Properties
According to Quantum Mechanics [4,9,31,46], a quantum system follows the next postulates or axioms:
Axiom 1
At a fixed time to, the state of a physical system is defined by specifying a ket |ψ(to)⟩ beloging to the state space (Hilbert space) V.
Axiom 2
Every measurable physical quantity Λ is described by the operator A acting in V ; this operator is an observable parameter.
Axiom 3
The only possible result of the measurement of a physical quantity Λ is one of the eigenvalues of the corresponding observable A.
Axiom 4
Whena physical quantity Λ is measured on a system in the normalized state |ψ(t)⟩ = Σ cj |ϕj⟩, the probability P (bj) of obtaining the non-degenerate eigenvalue cj of the corresponding observable A is: P(cj) = |⟨ϕj,ψ⟩|2 where |ϕj⟩ is the normalized eigen vector of A associated with the eigen value cj.
Let n be players with k = 1, ...,n , lk strategies and 
of each player k.
of each player k.Superposition principle:
Time evolution:
Inner product:
Quantum property
Mean value
Normality condition
3 Time Series and Uncertainty Principle of Heisenberg
Remember some basic definitions about probability and stochastic calculus.
A probability space is a triple (Ω, 
, P) consisting of [10,13,30]:
, P) consisting of [10,13,30]:
- A set Ω that represents the set of all possible outcomes of a certain random experiment.
- A family of subsets of Ω with a structure of σ-algebra:
- –
- –
- –
- A function P : Ω → [0, 1] such that:
- –
- –
- If
form a finite or countably infinite collection of disjointed sets (that is, Aj = Ø if i ≠ j) then
The elements of the σ-algebra are called events and P is called a probability measure. A random variable is a function:
are called events and P is called a probability measure. A random variable is a function:
-measurable, that is, 
for all B in Borel’s σ algebra of R, noted ß(R).If X is a random variable on (Ω, , P) the probability measure induced by X is the probability measure PX (called law or distribution of X) on ß(R) given by
, P) the probability measure induced by X is the probability measure PX (called law or distribution of X) on ß(R) given by
PX(B) = P {ω : X(ω) ∈ B}, B ∈ ß(R)
The numbers PX (B), B ∈ß(R), completely characterize the random variable X in the sense that they provide the probabilities of all events involving X. =ß(R), with P the Lebesgue-Stieltges measure corresponding to F.
Definition 2
since, for a < b, F(b) − F(a) = P {ω : a < X(ω) ≤ b} = PX (a, b], F is a distribution function corre- sponding to the Lebesgue Stieltjes measure PX. Thus among all distribution functions corresponding to the Lebesgue-Stieltges measure PX, we choose the one with F (∞) = 1, F (−∞) = 0. In fact we can always supply the probability space in a canonical way; take Ω = R, The distribution function of a random variable X is the function F = FX from R to [0, 1] given by
F(x) = P {ω : X(ω) ≤ x} x ∈ R
=ß(R), with P the Lebesgue-Stieltges measure corresponding to F.Definition 3
A stochastic process X = {X(t),t ∈ T} is a collection of random variables on a common probability space (Ω, , P) indexedby a parameter t ∈ T ⊂ R, which we usually interpret as time [10,27,28,29,30]. It can thus be formulated as a function X : T × Ω → R.
, P) indexedby a parameter t ∈ T ⊂ R, which we usually interpret as time [10,27,28,29,30]. It can thus be formulated as a function X : T × Ω → R.The variable X(t, •) is −measurable in ω ∈ Ω for each t ∈ T; henceforth we shall often follow established convention and write Xt for X(t). When T is a countable set, the stochastic process is really a sequence of random variables Xt1, Xt2, ..., Xtn, ..
is a density of F, that is, f is nonnegative and 
for all x; furthermore, F′ = f everywhere. Thus in this case, f and h are “Fourier transform pairs”:
−measurable in ω ∈ Ω for each t ∈ T; henceforth we shall often follow established convention and write Xt for X(t). When T is a countable set, the stochastic process is really a sequence of random variables Xt1, Xt2, ..., Xtn, ..Definition 4
If a random variable X = X(t) has the density ρ(x) and f : X → Y is one-to-one, then y = f(x) has the density g(y) given by 
see Fudenberg and Tirole [12], (chapter 6).Thedensitiesρ(x) and g(y) show that “If X is a random variable and f−1(y) exists, then Y is a random variable and conversely.”
see Fudenberg and Tirole [12], (chapter 6).Thedensitiesρ(x) and g(y) show that “If X is a random variable and f−1(y) exists, then Y is a random variable and conversely.”Definition 5
for all points a, b (a < b) at which F is continuous. Ifin addition, h is Lebesgue integrable on (−∞, ∞), then the function f given by
Inversion formula: If h is a characteristic function of the bounded distribution function F, and F (a, b] = F(b) − F(a) then
for all x; furthermore, F′ = f everywhere. Thus in this case, f and h are “Fourier transform pairs”:
Definition 6
If X is an random variable., f a Borel measurable function [on ( 
, )] , then f(X) is an random variable [25]. If W is r.v., h(W) a Borel measurable function [on ( , )] , then h(W) is r.v.
, )] , then f(X) is an random variable [25]. If W is r.v., h(W) a Borel measurable function [on ( , )] , then h(W) is r.v.Theorem 4
The Possibility Theorem: 
. “The more precisely the random variable VALUE x is determined, the less precisely the frequency VALUE w is known at this instant, and vice versa”. Let X = {X(t),t ∈ T} be the stochastic process with a density function f(x) and the random variable w with a characteristic function h(w). If the density functions can be written as f(x) = ψ(x)ψ(x)∗ and h(w) = φ(w)φ(w)∗, then: 
. “The more precisely the random variable VALUE x is determined, the less precisely the frequency VALUE w is known at this instant, and vice versa”. Let X = {X(t),t ∈ T} be the stochastic process with a density function f(x) and the random variable w with a characteristic function h(w). If the density functions can be written as f(x) = ψ(x)ψ(x)∗ and h(w) = φ(w)φ(w)∗, then: If the functions ψ(x = ±∞) = 0 and φ(w = ±∞) = 0 then f(x = ±∞) = 0 and h(w = ±∞) = 0.
Proof.
The next proofis based in Frieden [11].
Let the function ψ(x) defined as,
Here w is the frequency, different to ω ∈ Ω.
Then φ(w) is
According to Fourier’s transform properties, we have:
where
By Parseval Plancherel theorem
Using Schwartz’s inequality
and the probability properties 
, it is evident that:
, it is evident that:
Let 
be the standard deviation of frequency w, and 
the standard deviation of the aleatory variable x
with Parseval Plancherel theorem
be the standard deviation of frequency w, and the standard deviation of the aleatory variable x
Making the product (Q7.1)×(Q7.2) and using Q8, it follows that
Let G = xψ and H = ψ′ in Q5 shows that the right side of (Q9.2) exceeds a certain quantity A
We will note 
. If the variable A is given by A = 
then 
. By definition F′ = f = ψψ′
. If the variable A is given by A = then . By definition F′ = f = ψψ′
Notice that A + A∗ = −1 and using the complex number property 2 Re (A) = −1 or 2|A|cosθ = −1 where θ phase of A, we can write 4|A|2 = 
. Since −1 ≤ cosθ ≤ 1 then |A|2 ≥ 
and therefore:
■
. Since −1 ≤ cosθ ≤ 1 then |A|2 ≥ and therefore:
Remark 4
Let xt be a time series with a spectrum of frequencies wj, where each frequency is an random variable. This spectrum of frequencies can be obtained with a minimum error (standard deviation of frequency). This minimum error σw for a certain frequency w is given by the following equation 
. The expected value E [wj] of each frequency can be determined experimentally.
. The expected value E [wj] of each frequency can be determined experimentally.An occurrence of one particular frequency wj is defined in a confidence interval given by
This last remark agrees and supplements the statement of Hamilton [17].
“We should interpretnot as the portion of the variance of X that is due to cycles with frequency exactly equal to wj, but rather as the portion of the variance of X that is due to cycles with frequency near of wj, ” where:
The only condition that imposes the Possibility Theorem is exhaustive and exclusive 
, because it includes the Minimum Entropy Theorem or equilibrium state theorem 
. It is necessary to analyze the different cases of the time series xt using the Possibility Theorem. A time series xt evolves in the dynamic equilibrium if and only if 
. A time series evolves out of the dynamic equilibrium if and only if 
.
, because it includes the Minimum Entropy Theorem or equilibrium state theorem . It is necessary to analyze the different cases of the time series xt using the Possibility Theorem. A time series xt evolves in the dynamic equilibrium if and only if . A time series evolves out of the dynamic equilibrium if and only if .
Table 2.
Evolution of time series, out of the equilibrium 
. In this table we can see the different changes in σx and σw.
. In this table we can see the different changes in σx and σw.
Table 3.
Evolution of time series, in the equilibrium 
. In this table we can see the different changes in σx and σw.
. In this table we can see the different changes in σx and σw.
Remark 5
Hirschman’sform of the uncertainty principle. The fundamental analytical effect of Heisenberg’s principle is that the probability densities for x and w cannot both be arbitrarily narrow [4], H(W) + 
where: 
and 
. When ψ(x) and φ(w) are gaussian H(W) = B + Log(σW) and H(X) = C + Log(σX), Hirschman’s inequality becomes Heisenberg’s principle, then inequalities are transformed in equalities and the minimum uncertainty is minimum entropy. In Quantum Mechanics the minimum uncertainty product also obeys a minimum entropy sum.
where: and . When ψ(x) and φ(w) are gaussian H(W) = B + Log(σW) and H(X) = C + Log(σX), Hirschman’s inequality becomes Heisenberg’s principle, then inequalities are transformed in equalities and the minimum uncertainty is minimum entropy. In Quantum Mechanics the minimum uncertainty product also obeys a minimum entropy sum.4 Applications of the Models
4.1 Hermites’s Polynomials Application
Let Γ = (K, S, v) be a 3−player game, with K the set of players k = 1, 2, 3. Thefinite set Sk of cardinality lk ∈ N is the set of pure strategies of each player where k ∈ K, skjk ∈ Sk, jk = 1, 2, 3 and S = S1 ×S2 ×S3 represent a set of pure strategy pro fi les with s ∈ S as an element of that set and l = 3 ∗ 3 ∗ 3 = 27 represents the cardinality of S. The vectorial function v : S → R3 associates with every profile s ∈ S the vector of utilities v(s) = (v1(s),..., v3(s)), where vk(s) designates the utility of the player k facing the profi le s. In order to get facility of calculus we write the function vk(s) in an explicit way vk(s)= vk (j1, j2,..., jn).The matrix v3,27 represents all points of the Cartesian product ΠKSk see Table 4. The vector vk (s) is the k- column of v. The graphic representation of the 3-player game is shown in Figure 1.
In these games we obtain Nash’s equilibria in pure strategy (maximum utility MU, Table 4) and mixed strategy (Minimum Entropy Theorem MET, Table 5 and Table 6). After finding the equilibria we carried out a comparison with the results obtained from applying the theory of quantum games developed previously.
Figure 1.
3-player game strategies
Table 4.
Maximum utility (random utilities): maxp (u1 + u2 + u3)
Nash Utilities and Standard Deviations
Nash Equilibria: Probabilities, Utilities
Kroneker Products
Table 5.
Minimum entropy (random utilities): minp(σ1 + σ2 + σ3) ⇒ minp (H1 + H2 + H3)
Nash Utilities and Standard Deviations
Nash Equilibria: Probabilities, Utilities
Kroneker Products
Table 6.
Minimum entropy (stone-paper-scissors): minp(σ1 + σ2 + σ3) ⇒ minp (H1 + H2 + H3)
Nash Utilities and Standard Deviations
Nash Equilibria: Probabilities, Utilities
Kroneker Products
4.2 Time Series Application
The relationship between Time Series and Game Theory appears when we apply the entropy minimization theorem (EMT). This theorem (EMT) is a way to analyze the Nash-Hayek equilibrium in mixed strategies. Introducing the elements rationality and equilibrium in the domain of time series can be a big help, because it allows us to study the human behavior re fl ected and registered in historical data. The main contributions of this complementary focus on time series has a relationship with Econophysics and rationality.
Human behavior evolves and is the result of learning. The objective of learning is stability and optimal equilibrium. Due to the above-mentioned, we can affi rm that if learning is optimal then the convergence to equilibrium is faster than when the learning is sub-optimal. Introducing elements of Nash’s equilibrium in time series will allow us to evaluate learning and the convergence to equilibrium through the study of historical data (time series).
One of the branches of Physics called Quantum Mechanics was pioneered using Heisemberg’s uncer- tainty principle. This paper is simply the application of this principle in Game Theory and Time Series.
Econophysics is a newborn branch of the scientifi c development that attempts to establish the analogies between Economics and Physics, see Mantenga and Stanley [33]. The establishment of analogies is a creative way of applying the idea of cooperative equilibrium. The product of this cooperative equilibrium will produce synergies between these two sciences. From my point of view, the power of physics is the capacity of equilibrium formal treatment in stochastic dynamic systems. On the other hand, the power of Economics is the formal study of rationality, cooperative and non-cooperative equilibrium.
Econophysics is the beginning of a unifi cation stage of the systemic approach of scientifi c thought. I show that it is the beginning of a unifi cation stage but remains to create synergies with the rest of the sciences.
Let 
be a covariance-stationary process with the mean E[xt] = µ and jth covariance γj
be a covariance-stationary process with the mean E[xt] = µ and jth covariance γj
E[(xt − μ)(xt−j − μ)] = γj
If these autocovariances are absolutely summable, the population spectrum of x is given by [21].
If the γj ’s represent autocovariances of a covariance-stationary process using the Possibility Theorem, then
and Sx(w) will be nonnegative for all w. In general for an ARMA(p, q) process: xt = c + ϕ1xt−1 + ϕ2xt−2 + … + ϕpxt−p + εt + θ1εt−2 + … + θqεt−q
Sx(w) ∈ [Sx(w − σw), Sx(w + σw)]
The population spectrum Sx(w) ∈ [Sx(w − σw), Sx(w + σw)] is given by Hamilton [17].
where 
and w is a scalar.
and w is a scalar.Given an observed sample of T observations denoted x1,x2,.., xT , we can calculate up to T − 1 sample autocovariances γj from the formulas: 
and
and
The sample periodogram can be expressed as:
When the sample size T is an odd number, xt will be expressed in terms of periodic functions with M = (T − 1)/2 representing different frequencies 
The coefficients 
can be estimated with OLS regression.
can be estimated with OLS regression.
The sample variance of xt can be expressed as:
The portion of the sample variance of xt that can be attributed to cycles of frequency wj is given by:
with 
as the sample periodogram of frequency wj.
as the sample periodogram of frequency wj.Continuing with the methodology proposed by Hamilton [17]. we develop two examples that will allow us to verify the applicability of the Possibility Theorem.
Example 5
and
In the second step, we find the value of the parameters 
, j = 1,..., (T − 1) for the time series xt according to Q21 (Table 7, Figure 6).
, j = 1,..., (T − 1) for the time series xt according to Q21 (Table 7, Figure 6).
Example 6
Let {vt}t = 1,..,37 be a random variable, and w12 a random variable with gaussian probability density N (0, 1). Both variables are related as continues
By simplicity of computing, we suppose that εv and Et has gaussian probability density N (0, 1). It is evident that σw12 = σε = 1 After a little computing we get the estimated value of 
= 1.1334. The product 
= 1.1334 verifi es the Possibility Theorem and permits us to compute: 
= 1.1334. The product = 1.1334 verifi es the Possibility Theorem and permits us to compute:
w12 ∈ [E[w12] − σw min, E[w12] + σw min]
The spectral analysis can not give the theoretical value of E[w12] = 2.037. The experimental value of E[w12] = 2.8 . You can see the results in Figure 8, Figure 9 and Table 10.
Table 7.
Time series values of xt,yt,zt,ut
Table 8.
Frequencies, variances and sample periodogram of xt
Analysis of xt
Table 9.
Verification of Possibility Theorem for series xt, yt, zt, ut
Application of Possibility Theorem for time series vt
Figure 2.
yt = 2cos(w1(t − 1)) + 4 sin(w12(t − 1))
Figure 3.
zt = 4 sin(w12(t − 1))
Figure 4.
ut = 3
Figure 5.
Constitutive elements of time series xt
Figure 6.
Autocorrelations of time series xt
Figure 7.
Periodogram of time series xt
Figure 8.
Time series vt = sin(E[w12] + εt)(t − 1)) + εt
Figure 9.
Spectral analysis of vt
Conclusion
- Hermite’s polynomials allow us to study the probability density function of the vNM utility inside of a n− player game. Using the approach of Quantum Mechanics we have obtained an equivalence between quantum level and strategy. The function of states of Quantum Mechanics opens a new focus in the theory of games such as sub-strategies.
- An immediate application of quantum games in economics is related to the principal-agent rela- tionship. Specifi cally we can use m types of agents for the case of adverse selection models. In moral risk models quantum games could be used for a discrete or continuous set of efforts.
- In this paper we have demonstrated that the Nash-Hayek equilibrium opens new doors so that entropy in game theory can be used. Remembering that the primary way to prove Nash’s equilibria is through utility maximization, we can affi rm that human behavior arbitrates between these two stochastic- utility (benefits) U (p(x)) and entropy (risk or order) H(p(x)) elements. Accepting that the stochastic-utility/entropy
relationship is equivalent to the well-known bene fits/cost we present a new way to calculate equilibria: 
, where p(x) represents probability function and x = (x1,x2,..., xn) represents endogenous or exogenous variables. - In all time series xt , where cycles are present, it is impossible to know the exact value of the frequency wj of each one of the cycles j. We can know the value of frequency in a certain confidence interval given by wj ∈ [wj − σw min, wj + σw min] where
. “The more precisely the random variable VALUE x is determined, the less precisely the frequency VALUE w is known at this instant, and conversely” - This paper, which uses Kronecker product ⊗, represents an easy, new formalization of game (K, ∆, u(p)), which extends the game Γ to the mixed strategies.
ACKNOWLEDGMENTS
To my father and mother.
This study and analysis was made possible through the generous financial support of TODO1 SERVICES INC (www.todo1.com) and Petroecuador (www.petroecuador.com.ec).
The author is grateful to Gonzalo Pozo (Todo1 Services-Miami), the 8th Annual Conference in Computing in Economics and Finance, July 2002, (the assistants), AIX France, the assistants to 10th Annual Conference of the SouthEast SAS® User Group USA,(the assistants). M2002, the 5th annual data mining technology conference SAS® 2002 USA (the assistants).
The opinions and errors are solely my responsibility.
The author also thanks his PhD advisor, Prof. Jean Louis Rulliere (GATE director, Universite Lumiere Lyon II France).
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- 1We use bolt in order to represent vector or matrix.
- 2The hermitian operator
have the next property: 
, the transpose operator 
is equal to complex conjugate operator 
- 3
- H0(x) = 1 , H1(x) = 2x ,
- H2(x) = 4x2 − 2, H3(x) = 8x3 − 12x
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