Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy

Jiménez, Edward

doi:10.3390/e5040313

Open AccessArticle

Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy

by

Edward Jiménez

^1,2,3

¹

Experimental Economics, Todo1 Services Inc, Miami Fl 33126, USA

²

GATE, UMR 5824 CNRS - France

³

Research and Development Department, Petroecuador, Quito-Ecuador

Entropy 2003, 5(4), 313-347; https://doi.org/10.3390/e5040313

Submission received: 15 November 2002 / Accepted: 5 November 2003 / Published: 15 November 2003

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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

Keywords:

Quantum Games; Minimum Entropy; Time Series; Nash-Hayek equilibrium

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 ex_trapolation 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, Wil_kens, 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 contex_t 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 x_t by the standard deviation ofits frequency w is greater or equal to 1/2.

This paper is organized as follows:

Section 2: Quantum Games and Hermite’s Polynomials, Section 3: Time Series and Heisenberg’s Prin- ciple, Section 4: Applications of the Models, and Section 5: Conclusion.

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 S_k of cardinality l_k ∈ N is the set of pure strategies of each player where k ∈ K, s_{kj_k} ∈ S_k, j_k = 1, ..., l_k and S = Π_KS_k represents the set of pure strategy pro fi les with s ∈ S an element of that set, l = l₁, l₂,..., l_n 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 v^k(s) designates the utility of the player k facing the profile s. In order to understand calculus easier, we write the function v^k(s) in one explicit way v^k(s) = v^k(j₁, j₂,..., j_n). The matrix v_n,l represents all points of the Cartesian product Π_k∈KS_k. The vector v^k(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 Entropy 05 00313 i002

the probability vector. The set of pro fi les in mixed strategies is the polyhedron Δ with Entropy 05 00313 i003

, where

and

. Using the Kronecker product ⊗ it is possible to write1:

where

The n− dimensional function Entropy 05 00313 i008

associates with every pro fi le in mixed strategies the vector of expected utilities

where

is the expected utility of the player k. Every Entropy 05 00313 i011

represents the expected utility for each player’s strategy and the vector u^k is noted Entropy 05 00313 i012

.

The triplet Entropy 05 00313 i014

designates the ex_tension 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 Entropy 05 00313 i015

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 Min_p (Σ_k H_k(p)), where Entropy 05 00313 i018

standard deviation and H_k(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 Min_p (Σ_k H_k(p)), is equal to the standard deviation minimization program Min_p (Π_kσ_k (p)),when Entropy 05 00313 i020

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 (µ_k,σ_k), then its entropy is minimum for the minimum standard deviation (H_k )min ⇔ (σ_k)min. ∀k = 1, ..., n.

Proof.

Let the entropy function Entropy 05 00313 i021

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

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

■

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 Entropy 05 00313 i026

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, j_k = 1, ..., l_k

where

represents the partition utility function [1,45].

The entropy H_k(p^k), expected utility Entropy 05 00313 i030

, and variance Entropy 05 00313 i031

will be different for each player k.

Using the explicit form of Entropy 05 00313 i033

, we can get the entropy, the expected utility and the variance:

The equation Entropy 05 00313 i035

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]: Lim_λ→∞H_k(p^k(λ)) = min(H_k)

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: Lim_λ→∞σ_k (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 H_k for gaussian probability density and multinomial logit are written as Entropy 05 00313 i038

and

Case 3

The special case of Minimum Entropy is when Entropy 05 00313 i041

and the utility function value of each strategy Entropy 05 00313 i042

, are the same for all j_k, and for all k, see Binmore, Myerson [3,48].

In the special case of Minimum Entropy when Entropy 05 00313 i043

, the gaussian density function can be approximated by Dirac’s Delta Entropy 05 00313 i044

.

The function Entropy 05 00313 i044

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 Entropy 05 00313 i044

=

and can be obtained at the limit of the function

Case 4

If we can measure the standard deviation Entropy 05 00313 i047

,then Nash’s equilibrium represents minimum standard deviation (σ_k)_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 Entropy 05 00313 i049

related to minimum dispersion of a lineal combination between the variable x, and a Hermitian operator Entropy 05 00313 i050

Proof.

This proofis ex_tended in Pena and Cohen-Tannoudji [4,48]. Let Entropy 05 00313 i051

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 Entropy 05 00313 i059

. Using this property,

the expression J (α) has the minimum value when J ’(α) = 0, thus the value of α and J_min are

simplifying J_min

We are interested in knowing the explicit form of the function φ(x) when the inequality is transformed into equality and Entropy 05 00313 i063

, where

for which we need to solve the following differential equation.

if

, and

then

and

the solution is

■

Remark 2

The gaussian density function f (x) is optimal in the sense of minimum dispersion.

According to the last theorem Entropy 05 00313 i071

is a gaussian random variable with probability density f_k (x), ∀k, and in order to calculate Entropy 05 00313 i072

. We can write

Remark 3

The central limit theorem which establishes that “The mean Entropy 05 00313 i074

of the random variable Entropy 05 00313 i075

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

**Table 1.** Quantum Mechanic and Game Theory properties
Quantum Mechanics		Game Theory
Particle: k = 1, ..., n		Player : k = 1, ..., n
Quantum element		Player type
Interaction		Interaction
Quantum state: j = 1, ..., l_k		Strategy: j = 1, ..., l_k
Energy e		Utility u
Superposition of states		Superposition of strategies
State function		Utility function
Probabilistic and optimal nature		Probabilistic and optimal nature
Uncertainty Principle		Minimum Entropy
Matrix operators		Matrix operators
Variational Calculus,		Optimal Control,
Information Theory		Information Theory
Complexity		Complexity
variety: number of particles n		variety: number of players n
variability: number ofinteractions n!		variability: number ofinteractions n!
quantitative: Mathematical model		quantitative: Mathematical model
Observable value: E[e]		Observable value: E[u]
The entities communicate efficiently		The players interact efficiently
Entropy		Minimum Entropy

Let k = 1, ..., n be players with l_k strategies for every one Entropy 05 00313 i076

. According to the theorem of Minimum Dispersion, the utility Entropy 05 00313 i077

converges to Nash’s equilibria and follows a normal probability density Entropy 05 00313 i078

, where the expected utility is Entropy 05 00313 i079

Definition 1

(Tactics or Sub- Strategies) Let Entropy 05 00313 i080

be the set of sub-strategies Entropy 05 00313 i081

with a probabilitygiven by Entropy 05 00313 i082

where

and

when k ≠ l. A strategy j_k is constituted of several sub-strategies Entropy 05 00313 i085

With the introduction of sub-strategies the probability of each strategy is described in the following way Entropy 05 00313 i086

and we do not need any approximation of the function Entropy 05 00313 i087

where:

and

Theorem 3

The normal probability density ρ(u)= |ψ (x, λ)|² can be obtained using Hermite orthogonal polynomials H_k (x). The parameter Entropy 05 00313 i090

, indicates the probability value of playing j_k strategy. The state function ϕ_k(x,λ) measuresthe behavior of j_k strategy (j_k one level in Quantum Mechanics [4,9,31]).The dynamic behavior of j_k strategy can be written as Entropy 05 00313 i091

. One approximation is Entropy 05 00313 i092

The function ψ(x, λ) is determined as follows:

where:

and

Proof.

The Hermite polynomials properties will be necessary to express ρ(u) in function of H_k(x).

Using generating function ψ(x, λ_k) we can write

(Q1)

Let ψ(x, λ_k) be the generating function

(Q2)

In order to get Entropy 05 00313 i098

with the equations (QG1, PH1, PH2) and:

If

then w_k = 0; we can write:

(Q3)

Comparing (Q2) and (Q3) is easy to conclude

(Q4)

(Q5)

■

The

parameter represents the probability of Entropy 05 00313 i105

sub-strategy of each k player. The utility of vNM can be written as:

(Q5.1)

(Q5.2)

Using the operators a, a⁺ let us write the equations:

(Q6)

(Q6.1)

The equation (Q6.1) indicates that the operator Entropy 05 00313 i110

decreases the level of k player. The operator a can decrease the sub-strategy description Entropy 05 00313 i111

(Q7)

(Q7.1)

The equation (Q7.1) indicates that the operator Entropy 05 00313 i114

increases the level of k player [4,46]. The operator a⁺ can increase the sub-strategy description Entropy 05 00313 i115

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 nex_t properties are taken from Cohen-Tannoudji, Pena and Landau [4,31,46]

Hermite polynomials obey the nex_t differential equation [14]:

(PH0)

The principal solution of Hermite’s differential equation3 is

(PH1)

Taking the explicit value of Entropy 05 00313 i118

−derivative of (PH1):

(PH2)

(PH2.1)

From which we obtain the decreasing operator:

(PH3)

and the increasing operator:

(PH4)

Recurrence formulas:

(PH5)

and normalization condition:

(PH6)

The next condition expresses the orthogonality property of Hermite’s polynomials:

The generating function is obtained using the Taylor development.

(PH7)

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 nex_t postulates or axioms:

Axiom 1

At a fixed time t_o, the state of a physical system is defined by specifying a ket |ψ(t_o)〉 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 c_j |ϕ_j〉, the probability P (b_j) of obtaining the non-degenerate eigenvalue c_j of the corresponding observable A is: P(c_j) = |〈ϕ_j,ψ〉|² where |ϕ_j〉 is the normalized eigen vector of A associated with the eigen value c_j.

Let n be players with k = 1, ...,n , l_k strategies and Entropy 05 00313 i127

of each player k.

Superposition principle:

(QG1)

Time evolution:

(QG2)

Inner product:

(QG3)

Quantum property

(QG4)

Mean value

(QG5)

Normality condition

(QG6)

3 Time Series and Uncertainty Principle of Heisenberg

Remember some basic definitions about probability and stochastic calculus.

A probability space is a triple (Ω, Entropy 05 00313 i134

, 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, A_j = Ø if i ≠ j) then

The elements of the σ-algebra Entropy 05 00313 i134

are called events and P is called a probability measure. A random variable is a function:

-measurable, that is, Entropy 05 00313 i142

for all B in Borel’s σ algebra of R, noted ß(R).

If X is a random variable on (Ω, Entropy 05 00313 i134

, P) the probability measure induced by X is the probability measure P_X (called law or distribution of X) on ß(R) given by

P_X(B) = P {ω : X(ω) ∈ B}, B ∈ ß(R)

The numbers P_X (B), B ∈ß(R), completely characterize the random variable X in the sense that they provide the probabilities of all events involving X.

Definition 2

The distribution function of a random variable X is the function F = F_X from R to [0, 1] given by

F(x) = P {ω : X(ω) ≤ x} x ∈ R

since, for a < b, F(b) − F(a) = P {ω : a < X(ω) ≤ b} = P_X (a, b], F is a distribution function corre- sponding to the Lebesgue Stieltjes measure P_X. Thus among all distribution functions corresponding to the Lebesgue-Stieltges measure P_X, 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, Entropy 05 00313 i134

=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, Entropy 05 00313 i134

, 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 Entropy 05 00313 i134

−measurable in ω ∈ Ω for each t ∈ T; henceforth we shall often follow established convention and write X_t for X(t). When T is a countable set, the stochastic process is really a sequence of random variables X_t₁, X_t₂, ..., X_tn, ..

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 Entropy 05 00313 i144

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

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 points a, b (a < b) at which F is continuous. Ifin addition, h is Lebesgue integrable on (−∞, ∞), then the function f given by

is a density of F, that is, f is nonnegative and Entropy 05 00313 i147

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 ( Entropy 05 00313 i143

,

)] , then f(X) is an random variable [25]. If W is r.v., h(W) a Borel measurable function [on ( Entropy 05 00313 i143

,

)] , then h(W) is r.v.

Theorem 4

The Possibility Theorem: Entropy 05 00313 i149

. “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: Entropy 05 00313 i150

If the functions ψ(x = ±∞) = 0 and φ(w = ±∞) = 0 then f(x = ±∞) = 0 and h(w = ±∞) = 0.

Proof.

The nex_t proofis based in Frieden [11].

Let the function ψ(x) defined as,

(Q0)

Here w is the frequency, different to ω ∈ Ω.

Then φ(w) is

(Q1)

According to Fourier’s transform properties, we have:

(Q2)

where

(Q3)

By Parseval Plancherel theorem

(Q4)

Using Schwartz’s inequality

(Q5)

and the probability properties Entropy 05 00313 i157

, it is evident that:

(Q6)

Let

be the standard deviation of frequency w, and Entropy 05 00313 i160

the standard deviation of the aleatory variable x

(Q7.1)

(Q7.2)

with Parseval Plancherel theorem

(Q8.1)

(Q8.2)

Making the product (Q7.1)×(Q7.2) and using Q8, it follows that

(Q9.)

(Q9.2)

Let G = xψ and H = ψ′ in Q5 shows that the right side of (Q9.2) exceeds a certain quantity A

(Q10)

We will note Entropy 05 00313 i168

. If the variable A is given by A = Entropy 05 00313 i169

then

. By definition F′ = f = ψψ′

(Q11)

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 = Entropy 05 00313 i172

. Since −1 ≤ cosθ ≤ 1 then |A|² ≥ Entropy 05 00313 i173

and therefore:

(Q12)

■

Remark 4

Let x_t be a time series with a spectrum of frequencies w_j, 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 σ_w for a certain frequency w is given by the following equation Entropy 05 00313 i175

. The expected value E [w_j] of each frequency can be determined experimentally.

An occurrence of one particular frequency w_j is defined in a confidence interval given by

(Q13)

This last remark agrees and supplements the statement of Hamilton [17].

“We should interpret not as the portion of the variance of X that is due to cycles with frequency exactly equal to w_j, but rather as the portion of the variance of X that is due to cycles with frequency near of w_j, ” where:

(Q14)

The only condition that imposes the Possibility Theorem is exhaustive and exclusive Entropy 05 00313 i179

, because it includes the Minimum Entropy Theorem or equilibrium state theorem Entropy 05 00313 i180

. It is necessary to analyze the different cases of the time series x_t using the Possibility Theorem. A time series x_t evolves in the dynamic equilibrium if and only if Entropy 05 00313 i181

. A time series evolves out of the dynamic equilibrium if and only if Entropy 05 00313 i182

.

Table 2. Evolution of time series, out of the equilibrium Entropy 05 00313 i215

. In this table we can see the different changes in σ_x and σ_w.

**Table 2.** Evolution of time series, out of the equilibrium . In this table we can see the different changes in σ_x and σ_w.
Cases	σ_x	σ_w	Entropy
1	↑	↑	↑
2	↑	↓	↑
3	↓	↑	↑
4	↓	↓	↑
5	0	∞	trivial

Table 3. Evolution of time series, in the equilibrium Entropy 05 00313 i218

. In this table we can see the different changes in σ_x and σ_w.

**Table 3.** Evolution of time series, in the equilibrium . In this table we can see the different changes in σ_x and σ_w.
Cases	σ_x	σ_w	Entropy
1	↑	↓	↑
2	↓	↑	↓
3	max	min	min
4	min	min	max
5	0	∞	trivial

Remark 5

Hirschman’sform of the uncertainty principle. The fundamental analy_tical effect of Heisenberg’s principle is that the probability densities for x and w cannot both be arbitrarily narrow [4], H(W) + Entropy 05 00313 i183

where:

and

. When ψ(x) and φ(w) are gaussian H(W) = B + Log(σ_W) and H(X) = C + Log(σ_X), 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 S_k of cardinality l_k ∈ N is the set of pure strategies of each player where k ∈ K, s_{kj_k} ∈ S_k, j_k = 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 v^k(s) designates the utility of the player k facing the profi le s. In order to get facility of calculus we write the function v^k(s) in an explicit way v^k(s)= v^k (j₁, j₂,..., j_n).The matrix v_3,27 represents all points of the Cartesian product Π_KS_k see Table 4. The vector v^k (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): max_p (u¹ + u² + u³)

Nash Utilities and Standard Deviations

Nash Utilities and Standard Deviations
Max_p (u¹ + u² + u³) = 20.4

u¹	u²	u³
8.7405	9.9284	1.6998
σ₁	σ₂	σ₃
6.3509	6.2767	3.8522
H₁	H₂	H₃
4.2021	4.1905	3.7121

Nash Equilibria: Probabilities, Utilities

Nash Equilibria: Probabilities, Utilities
Player 1			Player 2			Player 3
p¹₁	p¹₂	p¹₃	p²₁	p²₂	p²₃	p³₁	p³₂	p³₃
1	0	0	1	0	0	0	00	1
u¹₁	u¹₂	u¹₃	u²₁	u²₂	u²₃	u³₁	u³₂	u³₃
8.7405	1.¹₁20	-3.8688	9.9284	-1.2871	-0.5630	-1.9587	5.7426	1.6998
p¹₁u¹₁	p¹₂u¹₂	p¹₃u¹₃	p²₁u²₁	p²₂u²₂	p²₃u²₃	p³₁u³₁	p³₂u³₂	p³₃u³₃
8.7405	0.0000	0.0000	9.9284	0.0000	0.0000	0.0000	0.0000	1.6998

Kroneker Products

Kroneker Products
j₁	j₂	j₃	v¹(j₁,j₂,j₃)	v²(j₁,j₂,j₃)	v³(j₁,j₂,j₃)	p¹_j1	p²_j2	p³_j3	p¹_j1p²_j2	p¹_j1p³_j3	p²_j2p³_j3	u¹(j₁,j₂,₃)	u²(j₁,j₂,₃)	u³(j₁,j₂,₃)
1	1	1	2.9282	-1.3534	-1.9587	1.0000	1.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	-1.9587
1	1	2	6.2704	3.2518	5.7426	1.0000	1.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	5.7426
1	1	3	8.7405	9.9284	1.6998	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	8.7405	9.9284	1.6998
1	2	1	4.1587	6.9687	4.1021	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1	2	2	3.8214	2.7242	8.6387	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1	2	3	-3.2109	-1.2871	-4.1140	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	0.0000	-1.2871	0.0000
1	3	1	3.0200	2.3275	6.8226	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1	3	2	-2.7397	3.0191	6.6629	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1	3	3	1.1781	-0.5630	5.3378	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	0.0000	-0.5630	0.0000
2	1	1	3.2031	-1.5724	-0.9757	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
2	1	2	1.9478	2.9478	6.7366	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
2	1	3	1.1120	6.4184	5.0734	0.0000	1.0000	1.0000	0.0000	0.0000	1.0000	1.1120	0.0000	0.0000
2	2	1	5.3695	5.7086	-0.7655	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
2	2	2	2.4164	1.6853	7.1051	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
2	2	3	5.2796	2.5158	-4.7264	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
2	3	1	-4.0524	-4.5759	5.8849	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
2	3	2	3.8126	-1.2267	4.8101	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
2	3	3	-1.4681	10.8633	0.2388	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	1	1	-0.4136	-2.6124	4.5470	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	1	2	2.6579	1.7204	0.7272	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	1	3	-3.8688	4.0884	11.2930	0.0000	1.0000	1.0000	0.0000	0.0000	1.0000	-3.8688	0.0000	0.0000
3	2	1	2.1517	4.8284	14.1957	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	2	2	6.8742	-1.8960	7.4744	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	2	3	2.9484	2.1771	0.0130	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	3	1	3.9191	-4.1335	7.4357	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	3	2	-3.8252	3.0861	4.5020	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
3	3	3	3.6409	3.4438	5.4857	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Table 5. Minimum entropy (random utilities): min_p(σ₁ + σ₂ + σ₃) ⇒ min_p (H₁ + H₂ + H₃)

Nash Utilities and Standard Deviations

Nash Utilities and Standard Deviations
Minp (σ1+σ2+σ3) = 1.382

u¹	u²	u³
0.85	2.19	4.54
σ₁	σ₂	σ₃
1.19492	0.187	0.000
H₁	H₂	H₃
2.596	1.090	0.035

Nash Equilibria: Probabilities, Utilities

Nash Equilibria: Probabilities, Utilities
Player 1			Player 2			Player 3
p¹₁	p¹₂	p¹₃	p²₁	p²₂	p²₃	p³₁	p³₂	p³₃
0.0000	0.5188	0.4812	0.4187	0.0849	0.4964	0.2430	0.3779	0.3791
u¹₁	u¹₂	u¹₃	u²₁	u²₂	u²₃	u³₁	u³₂	u³₃
2.8549	1.1189	0.5645	2.3954	2.1618	2.0256	4.5421	4.5421	4.5421
p¹₁ u¹₁	p¹₂ u¹₂	p¹₃ u¹₃	p²₁ u²₁	p²₂ u²₂	p²₃ u²₃	p³₁ u³₁	p³₂ u³₂	p³₃ u³₃
0.0000	0.5805	0.2717	1.0031	0.1835	1.0054	1.1036	1.7163	1.7221

Kroneker Products

Kroneker Products
j₁	j₂	j₃	v₁(j₁,j₂,j₃)	v₂(j₁,j₂,j₃)	v₃(j₁,j₂,j₃)	p¹_j1	p²_j2	p³_j3	p¹_j1p²_j2	p¹_j1p³_j3	p²_j2p³_j3	u₁(j₁,j₂,j₃)	u₂(j₁,j₂,j₃)	u₃(j₁,j₂,j₃)
1	1	1	2.93	-1.35	-1.96	0.0000	0.4187	0.2430	0.0000	0.0000	0.1017	0.2979	0.0000	0.0000
1	1	2	6.27	3.25	5.74	0.0000	0.4187	0.3779	0.0000	0.0000	0.1582	0.9922	0.0000	0.0000
1	1	3	8.74	9.93	1.70	0.0000	0.4187	0.3791	0.0000	0.0000	0.1588	1.3877	0.0000	0.0000
1	2	1	4.16	6.97	4.10	0.0000	0.0849	0.2430	0.0000	0.0000	0.0206	0.0858	0.0000	0.0000
1	2	2	3.82	2.72	8.64	0.0000	0.0849	0.3779	0.0000	0.0000	0.0321	0.1226	0.0000	0.0000
1	2	3	-3.21	-1.29	-4.11	0.0000	0.0849	0.3791	0.0000	0.0000	0.0322	-0.1034	0.0000	0.0000
1	3	1	3.02	2.33	6.82	0.0000	0.4964	0.2430	0.0000	0.0000	0.1206	0.3642	0.0000	0.0000
1	3	2	-2.74	3.02	6.66	0.0000	0.4964	0.3779	0.0000	0.0000	0.1876	-0.5139	0.0000	0.0000
1	3	3	1.18	-0.56	5.34	0.0000	0.4964	0.3791	0.0000	0.0000	0.1882	0.2217	0.0000	0.0000
2	1	1	3.20	-1.57	-0.98	0.5188	0.4187	0.2430	0.2172	0.1260	0.1017	0.3259	-0.1982	-0.2119
2	1	2	1.95	2.95	6.74	0.5188	0.4187	0.3779	0.2172	0.1960	0.1582	0.3082	0.5778	1.4633
2	1	3	1.11	6.42	5.07	0.5188	0.4187	0.3791	0.2172	0.1967	0.1588	0.1765	1.2624	1.1021
2	2	1	5.37	5.71	-0.77	0.5188	0.0849	0.2430	0.0440	0.1260	0.0206	0.1108	0.7196	-0.0337
2	2	2	2.42	1.69	7.11	0.5188	0.0849	0.3779	0.0440	0.1960	0.0321	0.0775	0.3303	0.3129
2	2	3	5.28	2.52	-4.73	0.5188	0.0849	0.3791	0.0440	0.1967	0.0322	0.1699	0.4948	-0.2082
2	3	1	-4.05	-4.58	5.88	0.5188	0.4964	0.2430	0.2575	0.1260	0.1206	-0.4887	-0.5768	1.5153
2	3	2	3.81	-1.23	4.81	0.5188	0.4964	0.3779	0.2575	0.1960	0.1876	0.7151	-0.2405	1.2385
2	3	3	-1.47	10.86	0.24	0.5188	0.4964	0.3791	0.2575	0.1967	0.1882	-0.2763	2.1366	0.0615
3	1	1	-0.41	-2.61	4.55	0.4812	0.4187	0.2430	0.2015	0.1169	0.1017	-0.0421	-0.3055	0.9163
3	1	2	2.66	1.72	0.73	0.4812	0.4187	0.3779	0.2015	0.1819	0.1582	0.4206	0.3129	0.1465
3	1	3	-3.87	4.09	11.29	0.4812	0.4187	0.3791	0.2015	0.1825	0.1588	-0.6142	0.7460	2.2758
3	2	1	2.15	4.83	14.20	0.4812	0.0849	0.2430	0.0409	0.1169	0.0206	0.0444	0.5646	0.5800
3	2	2	6.87	-1.90	7.47	0.4812	0.0849	0.3779	0.0409	0.1819	0.0321	0.2205	-0.3448	0.3054
3	2	3	2.95	2.18	0.01	0.4812	0.0849	0.3791	0.0409	0.1825	0.0322	0.0949	0.3972	0.0005
3	3	1	3.92	-4.13	7.44	0.4812	0.4964	0.2430	0.2389	0.1169	0.1206	0.4727	-0.4833	1.7762
3	3	2	-3.83	3.09	4.50	0.4812	0.4964	0.3779	0.2389	0.1819	0.1876	-0.7175	0.5612	1.0754
3	3	3	3.64	3.44	5.49	0.4812	0.4964	0.3791	0.2389	0.1825	0.1882	0.6852	0.6284	1.3104

Table 6. Minimum entropy (stone-paper-scissors): min_p(σ₁ + σ₂ + σ₃) ⇒ min_p (H₁ + H₂ + H₃)

Nash Utilities and Standard Deviations

Nash Utilities and Standard Deviations
Minp (σ1+σ2+σ3) = 1.382

u¹	u²	u³
0.5500	0.5500	0.5500
σ₁	σ₂	σ₃
0.0000	0.0000	0.0000
H₁	H₂	H₃
0.0353	0.0353	0.0353

Nash Equilibria: Probabilities, Utilities

Nash Equilibria: Probabilities, Utilities
Player 1			Player 2			Player 3
p¹₁	p¹₂	p¹₃	p²₁	p²₂	p²₃	p³₁	p³₂	p³₃
0.3333	0.3333	0.3333	0.3333	0.3333	0.3333	0.3300	0.3300	0.3300
u¹₁	u¹₂	u¹₃	u²₁	u²₂	u²₃	u³₁	u³₂	u³₃
0.5500	0.5500	0.5500	0.5500	0.5500	0.5500	0.5556	0.5556	0.5556
p¹₁ u¹₁	p¹₂ u¹₂	p¹₃ u¹₃	p²₁ u²₁	p²₂ u²₂	p²₃ u²₃	p³₁ u³₁	p³₂u³₂	p³₃u³₃
0.1833	0.1833	0.1833	0.1833	0.1833	0.1833	0.1833	0.1833	0.1833

Kroneker Products

Kroneker Products
j₁	j₂	j₃	v₁(j₁,j₂,j₃)	v₂(j₁,j₂,j₃)	v₃(j₁,j₂,j₃)	p¹_j1	p²_j2	p³_j3	p¹_j1p²_j2	p¹_j1p³_j3	p²_j2p³_j3	u₁(j₁,j₂,j₃)	u₂(j₁,j₂,j₃)	u₃(j₁,j₂,j₃)
stone	stone	stone	0	0	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.0000	0.0000
stone	stone	paper	0	0	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.0000	0.1111
stone	stone	sccisor	1	1	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.0000
stone	paper	stone	0	1	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.1100	0.0000
stone	paper	paper	0	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.1100	0.1111
stone	paper	sccisor	1	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.1111
stone	sccisor	stone	1	0	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.0000	0.1111
stone	sccisor	paper	1	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.1111
stone	sccisor	sccisor	1	0	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.0000	0.0000
paper	stone	stone	1	0	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.0000	0.0000
paper	stone	paper	1	0	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.0000	0.1111
paper	stone	sccisor	1	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.1111
paper	paper	stone	1	1	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.0000
paper	paper	paper	0	0	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.0000	0.0000
paper	paper	sccisor	0	0	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.0000	0.1111
paper	sccisor	stone	1	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.1111
paper	sccisor	paper	0	1	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.1100	0.0000
paper	sccisor	sccisor	0	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.1100	0.1111
sccisor	stone	stone	0	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.1100	0.1111
sccisor	stone	paper	1	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.1111
sccisor	stone	sccisor	0	1	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.1100	0.0000
sccisor	paper	stone	1	1	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.1111
sccisor	paper	paper	1	0	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.0000	0.0000
sccisor	paper	sccisor	1	0	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.0000	0.1111
sccisor	sccisor	stone	0	0	1	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.0000	0.1111
sccisor	sccisor	paper	1	1	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.1100	0.1100	0.0000
sccisor	sccisor	sccisor	0	0	0	0.3333	0.3333	0.3300	0.1111	0.1100	0.1100	0.0000	0.0000	0.0000

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[x_t] = µ and jth covariance γ_j

E[(x_t − μ)(x_t−j − μ)] = γ_j

(Q16)

If these autocovariances are absolutely summable, the population spectrum of x is given by [21].

(Q17)

If the γ_j ’s represent autocovariances of a covariance-stationary process using the Possibility Theorem, then

S_x(w) ∈ [S_x(w − σ_w), S_x(w + σ_w)]

(Q18)

and S_x(w) will be nonnegative for all w. In general for an ARMA(p, q) process: x_t = c + ϕ₁x_t−1 + ϕ₂x_t−2 + … + ϕ_px_t−p + ε_t + θ₁ε_t−2 + … + θ_qε_t−q

(Q19)

The population spectrum S_x(w) ∈ [S_x(w − σ_w), S_x(w + σ_w)] is given by Hamilton [17].

(Q20)

where

and w is a scalar.

Given an observed sample of T observations denoted x₁,x₂,.., x_T , we can calculate up to T − 1 sample autocovariances γ_j from the formulas: Entropy 05 00313 i190

and

(Q21)

The sample periodogram can be expressed as:

(Q22)

When the sample size T is an odd number, x_t will be expressed in terms of periodic functions with M = (T − 1)/2 representing different frequencies Entropy 05 00313 i193

(Q23)

The coefficients Entropy 05 00313 i195

can be estimated with OLS regression.

(Q24)

The sample variance of x_t can be expressed as:

(Q25)

The portion of the sample variance of x_t that can be attributed to cycles of frequency w_j is given by:

(Q26)

with

as the sample periodogram of frequency w_j.

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

As a first step we build four time series knowing all their parameters a_j,b_j and Entropy 05 00313 i200

j = 1, ..., M (Table 7, Figure 2, Figure 3, Figure 4 and Figure 5),where

and

In the second step, we find the value of the parameters Entropy 05 00313 i203

, j = 1,..., (T − 1) for the time series x_t according to Q21 (Table 7, Figure 6).

In the third step, we find Entropy 05 00313 i204

(Table 8, Figure 7), only for the frequencies w₁, w₃, w₅, w₁₂:

In the fourth step, we compute the values σ_x,σ_w and Entropy 05 00313 i205

resp. for y, z, u, (Table 9).

In the fifth step, we verify that the Possibility Theorem is respected, Entropy 05 00313 i207

resp for y,z,u, (Table 9).

Example 6

Let {v_t}_{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 ε_v and E_t has gaussian probability density N (0, 1). It is evident that σ_w₁₂ = σ_ε = 1 After a little computing we get the estimated value of Entropy 05 00313 i209

= 1.1334. The product Entropy 05 00313 i210

= 1.1334 verifi es the Possibility Theorem and permits us to compute: Entropy 05 00313 i211

w₁₂ ∈ [E[w₁₂] − σ_w _min, E[w₁₂] + σ_w _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 x_t,y_t,z_t,u_t

**Table 7.** Time series values of x_t,y_t,z_t,u_t
P a rame te rs o f x_t
j = t	γ_j	ρ_j	E [w_j]	γ_j·cos (E[w₁]*j)	γ_j·cos (E[w₃]*j)	γ_j·cos (E[w₅]*j)	γ_j·cos (E[w₁₂]*j)	x_t	y_t	z_t	u_t	cos(w₁(t-1))	cos(w₃(t-1))	sen(w₅(t-1))	sen(w₁₂(t-1))
1	3,557	0,210	0,170	3,505	3,105	2,350	-1,601	-1,000	2,000	0,000	3,000	1,000	1,000	0,000	0,000
2	-0,030	-0,002	0,340	-0,029	-0,016	0,004	0,018	4,425	5,543	3,572	3,000	0,986	0,873	0,751	0,893
3	8,571	0,505	0,509	7,482	0,364	-7,101	8,447	-0,919	-1,330	-3,216	3,000	0,943	0,524	0,992	-0,804
4	-5,076	-0,299	0,679	-3,950	2,285	4,913	1,485	2,062	1,069	-0,677	3,000	0,873	0,042	0,560	-0,169
5	-9,188	-0,542	0,849	-6,071	7,612	4,138	6,635	6,228	5,381	3,825	3,000	0,778	-0,450	-0,252	0,956
6	4,167	0,246	1,019	2,185	-4,152	1,553	3,928	-0,746	-1,446	-2,767	3,000	0,661	-0,828	-0,893	-0,692
7	-4,030	-0,238	1,189	-1,503	3,673	-3,800	0,510	0,848	-0,285	-1,334	3,000	0,524	-0,996	-0,928	-0,333
8	-8,301	-0,490	1,358	-1,749	4,937	-7,248	6,880	6,781	4,714	3,968	3,000	0,373	-0,911	-0,333	0,992
9	5,607	0,331	1,528	0,238	-0,713	1,183	4,894	0,942	-1,817	-2,238	3,000	0,211	-0,595	0,488	-0,560
10	-1,355	-0,080	1,698	0,172	-0,505	0,805	-0,058	0,469	-1,868	-1,953	3,000	0,042	-0,127	0,978	-0,488
11	-7,601	-0,448	1,868	2,225	-5,913	7,574	6,929	4,232	3,742	3,996	3,000	-0,127	0,373	0,804	0,999
12	7,379	0,435	2,038	-3,322	7,272	-5,329	5,738	-4,394	-2,231	-1,645	3,000	-0,293	0,778	0,085	-0,411
13	3,541	0,209	2,208	-2,106	3,339	0,149	0,749	-7,756	-3,415	-2,515	3,000	-0,450	0,986	-0,692	-0,629
14	-4,731	-0,279	2,377	3,415	-3,126	-3,680	4,579	-2,107	2,720	3,910	3,000	-0,595	0,943	-0,999	0,977
15	7,312	0,431	2,547	-6,058	1,542	7,207	4,826	-5,688	-2,448	-1,005	3,000	-0,722	0,661	-0,628	-0,251
16	0,948	0,056	2,717	-0,864	-0,277	0,497	0,354	-4,957	-4,662	-3,005	3,000	-0,828	0,211	0,169	-0,751
17	-12,981	-0,765	2,887	12,561	9,369	3,796	12,935	4,468	1,888	3,710	3,000	-0,911	-0,293	0,851	0,928
18	-3,060	-0,180	3,057	3,049	2,961	2,788	-1,601	1,807	-2,271	-0,335	3,000	-0,968	-0,722	0,956	-0,084
19	-4,856	-0,286	-	4,839	4,700	4,426	-2,551	-1,674	-5,401	-3,408	3,000	-0,996	-0,968	0,412	-0,852
20	-14,516	-0,856	-	14,048	10,483	4,257	14,462	3,491	1,411	3,404	3,000	-0,996	-0,968	-0,411	0,851
21	0,759	0,045	-	-0,692	-0,223	0,398	0,282	-1,337	-1,592	0,344	3,000	-0,968	-0,722	-0,956	0,086
22	5,394	0,318	-	-4,469	1,135	5,316	3,569	-6,360	-5,536	-3,713	3,000	-0,911	-0,293	-0,852	-0,928
23	-4,600	-0,271	-	3,321	-3,038	-3,580	4,450	0,372	1,342	2,999	3,000	-0,829	0,210	-0,170	0,750
24	6,986	0,412	-	-4,155	6,586	0,301	1,462	-1,157	-0,431	1,013	3,000	-0,722	0,660	0,628	0,253
25	8,725	0,515	-	-3,929	8,600	-6,295	6,797	-5,931	-5,101	-3,911	3,000	-0,595	0,943	0,999	-0,978
26	-5,611	-0,331	-	1,644	-4,367	5,591	5,109	0,035	1,608	2,508	3,000	-0,450	0,986	0,692	0,627
27	3,347	0,197	-	-0,426	1,249	-1,992	0,137	-1,436	1,067	1,653	3,000	-0,293	0,778	-0,084	0,413
28	6,054	0,357	-	0,256	-0,766	1,271	5,290	-6,978	-4,251	-3,997	3,000	-0,127	0,373	-0,804	-0,999
29	-8,491	-0,501	-	-1,788	5,046	-7,410	7,026	0,454	2,030	1,945	3,000	0,042	-0,127	-0,978	0,486
30	0,753	0,044	-	0,281	-0,686	0,710	-0,097	3,473	2,667	2,246	3,000	0,211	-0,594	-0,488	0,561
31	4,354	0,257	-	2,282	-4,338	1,626	4,108	0,177	-3,221	-3,967	3,000	0,373	-0,911	0,332	-0,992
32	-10,875	-0,641	-	-7,183	9,013	4,888	7,837	7,218	2,374	1,326	3,000	0,524	-0,996	0,928	0,331
33	-0,935	-0,055	-	-0,728	0,422	0,905	0,276	8,367	4,094	2,773	3,000	0,661	-0,829	0,893	0,693
34	4,485	0,264	-	3,915	0,188	-3,718	4,422	-0,409	-2,266	-3,822	3,000	0,778	-0,451	0,253	-0,956
35	-9,859	-0,581	-	-9,295	-5,165	1,261	5,846	1,169	2,414	0,668	3,000	0,873	0,042	-0,559	0,167
36	0,000	-	-	-	-	-	-	1,551	5,107	3,221	3,000	0,943	0,524	-0,992	0,805
37	-	-	-	-	-	-	-	-5,717	-1,597	-3,568	3,000	0,986	0,873	-0,751	-0,892

Table 8. Frequencies, variances and sample periodogram of x_t

Analysis of x_t

Analysis of x_t

Variance: s_t²
E[(x_t-E[x_t])²]	16,958
(a₁²+a₃²+b₅²+b₁	16,500
γ⁰	16,958
T	37,000


Frequencies		Wavelength		Coefficients		(a_j²+b_j²)/2		Sample Periodogram		(4*pj/T)s_x(E[w_j])
E[w₁]	0,170	l₁	37,000	a₁	2,000	a₁²	2,000	S_x(E[w₁])	4,961	1,685
E[w₃]	0,509	l₂	12,333	a₃	-3,000	a₃²	4,500	S_x(E[w₃])	21,988	7,468
E[w₅]	0,849	l₃	7,400	b₅	2,000	b₅²	2,000	S_x(E[w₅])	8,350	2,836
E[w₁₂]	2,038	l₄	3,083	b₁₂	4,000	b₁₂²	8,000	S_x(E[w₁₂])	45,375	15,410

Table 9. Verification of Possibility Theorem for series x_t, y_t, z_t, u_t

**Table 9.** Verification of Possibility Theorem for series x_t, y_t, z_t, u_t
Heisenberg's Uncertainty Principle

σ_xσ_w = 3,348 >1/2						σ_yσ_w = 4,234 >1/2

E[x_t]	0,000	σ_x=E[(x_t-E[x_t])²]		4,118		E[y_t]	0,000	E[(y_t-E[y_t])²]		3,206
E[w]	0,892	σw=E[(w-E[w])²]		0,813		E[w]	1,104	E[(w-E[w])²]		1,321

σ_wmin	0,121					σ_wmin	0,156

		Lower[w	Min[j]	Upper[w_j]	Max[j]			Lower[w_j]	Min[j]	Upper[w_j]	Max[j]
E[w₁]	0,170	0,048	0,285	0,291	1,715
E[w₃]	0,509	0,388	2,285	0,631	3,715
E[w₅]	0,849	0,728	4,285	0,970	5,715	E[w₁]	0,170	0,014	0,082	0,326	1,918
E[w₁₂]	2,038	1,916	11,285	2,159	12,715	E[w₁₂]	2,038	1,882	11,082	2,194	12,918

σ_zσ_w = 0,000 <1/2						σ_uσ_w = 0,000 <1/2

E[z_t]	0,000	σz=E[(z_t-E[z_t])²]		2,867		E[u_t]	3,000	σu=E[(u_t-E[u_t])²]		0,000
E[w₁₂]	2,038	σw=E[(w-E[w])²]		0,000		E[w]	∞	σw=E[(w-E[w])²]		−
σwmin	0,174					σwmin	∞

		Lower[wMin[j]		Upper[w_j]	Max[j]			Lower[w_j]	Min[j]	Upper[w_j]	Max[j]
E[w₁₂]	2,038	1,863	10,973	2,038	12,000	E[w_i]	2,038	0,000	0,000	∞	∞

Application of Possibility Theorem for time series v_t

Application of Possibility Theorem for time series v_t
Heisenberg's Uncertainty Principle

σ_vσ_w12= 1,1334 > 1/2


E[v_t]	0,0439	σ_v=E[(v_t-E[v_t])²]		1,1334
E[w₁₂]	2,0377	σ_w=E[(w₁₂-E[w₁₂])²]		1,0000

σ_wmin	0,4412	T	37	γ0	1,1334

		Lower[w₁₂]	Min[j]	Upper[w₁₂]	Max[j]
E[w12]	2,0377	1,5966	9,4021	2,4789	14,5979

Figure 2. y_t = 2cos(w₁(t − 1)) + 4 sin(w₁₂(t − 1))

Figure 3. z_t = 4 sin(w₁₂(t − 1))

Figure 4. u_t = 3

Figure 5. Constitutive elements of time series x_t

Figure 6. Autocorrelations of time series x_t

Figure 7. Periodogram of time series x_t

Figure 8. Time series v_t = sin(E[w₁₂] + ε_t)(t − 1)) + ε_t

Figure 9. Spectral analysis of v_t

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,..., x_n) represents endogenous or exogenous variables.
In all time series x_t , where cycles are present, it is impossible to know the exact value of the frequency w_j of each one of the cycles j. We can know the value of frequency in a certain confidence interval given by w_j ∈ [w_j − σ_w _min, w_j + σ_w _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”
Using Game Theory concepts we can obtain the equilibrium condition in time series. The equilib- rium condition in time series is represented by the Possibility Theorem (Table 7, Table 8, Table 9 and Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7).
This paper, which uses Kronecker product ⊗, represents an easy, new formalization of game (K, ∆, u(p)), which ex_tends 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 nex_t 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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Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy

Abstract

1 Introduction

2 Quantum Games and Hermite’s Polynomials

2.1 Minimum Entropy Method

2.2 Quantum Games Elements

2.3 Hermite’s Polynomials Properties

2.4 Quantum Games Properties

3 Time Series and Uncertainty Principle of Heisenberg

4 Applications of the Models

4.1 Hermites’s Polynomials Application

4.2 Time Series Application

Conclusion

ACKNOWLEDGMENTS

Bibliography

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