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

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

GATE, UMR 5824 CNRS - France

Research and Development Department, Petroecuador, Quito-Ecuador

Received: 15 November 2002
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Accepted: 5 November 2003
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Published: 15 November 2003

This paper introduces Hermite's polynomials, in the description of quantum games. Hermite's polynomials are associated with gaussian probability density. The gaussian probability 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 (minimum 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?.

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_{t}rapolation 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_{k}ens, 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:

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_{kjk} ∈ S_{k}, j_{k} = 1, ..., l_{k} and S = Π_{K}S_{k} represents the set of pure strategy pro fi les with s ∈ S an element of that set, l = l_{1}, l_{2},..., l_{n} represents the cardinality of S, [12,43,55,56].

The vector function v : S → R^{n} associates every profile s ∈ S, where the vector of utilities v(s) = (v^{1}(s),..., v^{n}(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}(j_{1}, j_{2},..., j_{n}). The matrix v_{n,l} represents all points of the Cartesian product Π_{k∈K}S_{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 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 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 u^{k} is noted .

The triplet designates the ex_{t}ension 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 Min_{p} (Σ_{k} H_{k}(p)), where standard deviation and H_{k}(p∗) entropy of each player k.

(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_{k}σ_{k} (p)),when 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.

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.

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

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].

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 , 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.

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})

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
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The entropy H_{k} for gaussian probability density and multinomial logit are written as and

In the special case of Minimum Entropy when , 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

If we can measure the standard deviation ,then Nash’s equilibrium represents minimum standard deviation (σ_{k})_{min} , ∀k ∈ K.

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

This proofis ex_{t}ended in Pena and Cohen-Tannoudji [4,48]. Let 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 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 , where for which we need to solve the following differential equation.
if , and then and
the solution is
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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 f_{k} (x), ∀k, and in order to calculate . We can write

The central limit theorem which establishes that “The mean of the random variable has normal density function”, is respected in the Minimum Entropy Method.

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

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 . 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

(Tactics or Sub- Strategies) Let be the set of sub-strategies with a probabilitygiven by where and when k ≠ l. A strategy j_{k} 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

The normal probability density ρ(u)= |ψ (x, λ)|^{2} can be obtained using Hermite orthogonal polynomials H_{k} (x). The parameter , 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 . One approximation is

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

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

Let ψ(x, λ_{k}) be the generating function

In order to get with the equations (QG1, PH1, PH2) and:

If then w_{k} = 0; we can write:

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

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

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.

The principal solution of Hermite’s differential equation3 is

Taking the explicit value of −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.

According to Quantum Mechanics [4,9,31,46], a quantum system follows the nex_{t} postulates or axioms:

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.

Every measurable physical quantity Λ is described by the operator A acting in V ; this operator is an observable parameter.

The only possible result of the measurement of a physical quantity Λ is one of the eigenvalues of the corresponding observable A.

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 of each player k.

Superposition principle:

Time evolution:

Inner product:

Quantum property

Mean value

Normality condition

Remember some basic definitions about probability and stochastic calculus.

A probability space is a triple (Ω, , 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 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 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**
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, =ß(R), with P the Lebesgue-Stieltges measure corresponding to F.

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

The variable X(t, •) is −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_{t}_{1}, X_{t}_{2}, ..., X_{tn}, ..**Definition 4** **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
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”:

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.”

Inversion formula: If h is a characteristic function of the bounded distribution function F, and F (a, b] = F(b) − F(a) then

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.

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:

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

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:

Let be the standard deviation of frequency w, and the standard deviation of the aleatory variable x
with Parseval Plancherel theorem

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′

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:
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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 . 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

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:

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 x_{t} using the Possibility Theorem. A time series x_{t} evolves in the dynamic equilibrium if and only if . A time series evolves out of the dynamic equilibrium if and only if .

Cases | σ_{x} | σ_{w} | σ_{x}σ_{w} | Entropy |
---|---|---|---|---|

1 | ↑ | ↑ | ↑ | |

2 | ↑ | ↓ | ↑ | |

3 | ↓ | ↑ | ↑ | |

4 | ↓ | ↓ | ↑ | |

5 | 0 | ∞ | trivial |

Cases | σ_{x} | σ_{w} | σ_{x}σ_{w} | Entropy |
---|---|---|---|---|

1 | ↑ | ↓ | ↑ | |

2 | ↓ | ↑ | ↓ | |

3 | max | min | min | |

4 | min | min | max | |

5 | 0 | ∞ | trivial |

Hirschman’sform of the uncertainty principle. The fundamental analy_{t}ical 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(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.

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_{kjk} ∈ S_{k}, j_{k} = 1, 2, 3 and S = S_{1} ×S_{2} ×S_{3} 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 → R^{3} associates with every profile s ∈ S the vector of utilities v(s) = (v^{1}(s),..., v^{3}(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} (j_{1}, j_{2},..., j_{n}).The matrix v_{3,27} represents all points of the Cartesian product Π_{K}S_{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.

Max_{p} (u^{1} + u^{2} + u^{3}) = 20.4 | ||||||
---|---|---|---|---|---|---|

u^{1} | u^{2} | u^{3} | ||||

8.7405 | 9.9284 | 1.6998 | ||||

σ_{1} | σ_{2} | σ_{3} | ||||

6.3509 | 6.2767 | 3.8522 | ||||

H_{1} | H_{2} | H_{3} | ||||

4.2021 | 4.1905 | 3.7121 |

Player 1 | Player 2 | Player 3 | ||||||
---|---|---|---|---|---|---|---|---|

p^{1}_{1} | p^{1}_{2} | p^{1}_{3} | p^{2}_{1} | p^{2}_{2} | p^{2}_{3} | p^{3}_{1} | p^{3}_{2} | p^{3}_{3} |

1 | 0 | 0 | 1 | 0 | 0 | 0 | 00 | 1 |

u^{1}_{1} | u^{1}_{2} | u^{1}_{3} | u^{2}_{1} | u^{2}_{2} | u^{2}_{3} | u^{3}_{1} | u^{3}_{2} | u^{3}_{3} |

8.7405 | 1.^{1}_{1}20 | -3.8688 | 9.9284 | -1.2871 | -0.5630 | -1.9587 | 5.7426 | 1.6998 |

p^{1}_{1}u^{1}_{1} | p^{1}_{2}u^{1}_{2} | p^{1}_{3}u^{1}_{3} | p^{2}_{1}u^{2}_{1} | p^{2}_{2}u^{2}_{2} | p^{2}_{3}u^{2}_{3} | p^{3}_{1}u^{3}_{1} | p^{3}_{2}u^{3}_{2} | p^{3}_{3}u^{3}_{3} |

8.7405 | 0.0000 | 0.0000 | 9.9284 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.6998 |

j_{1} | j_{2} | j_{3} | v^{1}(j_{1},j_{2},j_{3}) | v^{2}(j_{1},j_{2},j_{3}) | v^{3}(j_{1},j_{2},j_{3}) | p^{1}_{j1} | p^{2}_{j2} | p^{3}_{j3} | p^{1}_{j1}p^{2}_{j2} | p^{1}_{j1}p^{3}_{j3} | p^{2}_{j2}p^{3}_{j3} | u^{1}(j_{1},j_{2},_{3}) | u^{2}(j_{1},j_{2},_{3}) | u^{3}(j_{1},j_{2},_{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 |

Minp (σ1+σ2+σ3) = 1.382 | ||||
---|---|---|---|---|

u^{1} | u^{2} | u^{3} | ||

0.85 | 2.19 | 4.54 | ||

σ_{1} | σ_{2} | σ_{3} | ||

1.19492 | 0.187 | 0.000 | ||

H_{1} | H_{2} | H_{3} | ||

2.596 | 1.090 | 0.035 |

Player 1 | Player 2 | Player 3 | ||||||
---|---|---|---|---|---|---|---|---|

p^{1}_{1} | p^{1}_{2} | p^{1}_{3} | p^{2}_{1} | p^{2}_{2} | p^{2}_{3} | p^{3}_{1} | p^{3}_{2} | p^{3}_{3} |

0.0000 | 0.5188 | 0.4812 | 0.4187 | 0.0849 | 0.4964 | 0.2430 | 0.3779 | 0.3791 |

u^{1}_{1} | u^{1}_{2} | u^{1}_{3} | u^{2}_{1} | u^{2}_{2} | u^{2}_{3} | u^{3}_{1} | u^{3}_{2} | u^{3}_{3} |

2.8549 | 1.1189 | 0.5645 | 2.3954 | 2.1618 | 2.0256 | 4.5421 | 4.5421 | 4.5421 |

p^{1}_{1} u^{1}_{1} | p^{1}_{2} u^{1}_{2} | p^{1}_{3} u^{1}_{3} | p^{2}_{1} u^{2}_{1} | p^{2}_{2} u^{2}_{2} | p^{2}_{3} u^{2}_{3} | p^{3}_{1} u^{3}_{1} | p^{3}_{2} u^{3}_{2} | p^{3}_{3} u^{3}_{3} |

0.0000 | 0.5805 | 0.2717 | 1.0031 | 0.1835 | 1.0054 | 1.1036 | 1.7163 | 1.7221 |

j_{1} | j_{2} | j_{3} | v_{1}(j_{1},j_{2},j_{3}) | v_{2}(j_{1},j_{2},j_{3}) | v_{3}(j_{1},j_{2},j_{3}) | p^{1}_{j1} | p^{2}_{j2} | p^{3}_{j3} | p^{1}_{j1}p^{2}_{j2} | p^{1}_{j1}p^{3}_{j3} | p^{2}_{j2}p^{3}_{j3} | u_{1}(j_{1},j_{2},j_{3}) | u_{2}(j_{1},j_{2},j_{3}) | u_{3}(j_{1},j_{2},j_{3}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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 |

Minp (σ1+σ2+σ3) = 1.382 | ||||
---|---|---|---|---|

u^{1} | u^{2} | u^{3} | ||

0.5500 | 0.5500 | 0.5500 | ||

σ_{1} | σ_{2} | σ_{3} | ||

0.0000 | 0.0000 | 0.0000 | ||

H_{1} | H_{2} | H_{3} | ||

0.0353 | 0.0353 | 0.0353 |

Player 1 | Player 2 | Player 3 | ||||||
---|---|---|---|---|---|---|---|---|

p^{1}_{1} | p^{1}_{2} | p^{1}_{3} | p^{2}_{1} | p^{2}_{2} | p^{2}_{3} | p^{3}_{1} | p^{3}_{2} | p^{3}_{3} |

0.3333 | 0.3333 | 0.3333 | 0.3333 | 0.3333 | 0.3333 | 0.3300 | 0.3300 | 0.3300 |

u^{1}_{1} | u^{1}_{2} | u^{1}_{3} | u^{2}_{1} | u^{2}_{2} | u^{2}_{3} | u^{3}_{1} | u^{3}_{2} | u^{3}_{3} |

0.5500 | 0.5500 | 0.5500 | 0.5500 | 0.5500 | 0.5500 | 0.5556 | 0.5556 | 0.5556 |

p^{1}_{1} u^{1}_{1} | p^{1}_{2} u^{1}_{2} | p^{1}_{3} u^{1}_{3} | p^{2}_{1} u^{2}_{1} | p^{2}_{2} u^{2}_{2} | p^{2}_{3} u^{2}_{3} | p^{3}_{1} u^{3}_{1} | p^{3}_{2}u^{3}_{2} | p^{3}_{3}u^{3}_{3} |

0.1833 | 0.1833 | 0.1833 | 0.1833 | 0.1833 | 0.1833 | 0.1833 | 0.1833 | 0.1833 |

j_{1} | j_{2} | j_{3} | v_{1}(j_{1},j_{2},j_{3}) | v_{2}(j_{1},j_{2},j_{3}) | v_{3}(j_{1},j_{2},j_{3}) | p^{1}_{j1} | p^{2}_{j2} | p^{3}_{j3} | p^{1}_{j1}p^{2}_{j2} | p^{1}_{j1}p^{3}_{j3} | p^{2}_{j2}p^{3}_{j3} | u_{1}(j_{1},j_{2},j_{3}) | u_{2}(j_{1},j_{2},j_{3}) | u_{3}(j_{1},j_{2},j_{3}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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 |

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}

If the γ_{j} ’s represent autocovariances of a covariance-stationary process using the Possibility Theorem, then
and S_{x}(w) will be nonnegative for all w. In general for an ARMA(p, q) process: x_{t} = c + ϕ_{1}x_{t−1} + ϕ_{2}x_{t−2} + … + ϕ_{p}x_{t−p} + ε_{t} + θ_{1}ε_{t−2} + … + θ_{q}ε_{t−q}

S_{x}(w) ∈ [S_{x}(w − σ_{w}), S_{x}(w + σ_{w})]

where and w is a scalar.

Given an observed sample of T observations denoted x_{1},x_{2},.., x_{T} , we can calculate up to T − 1 sample autocovariances γ_{j} from the formulas: and

The sample periodogram can be expressed as:

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

The coefficients can be estimated with OLS regression.

The sample variance of x_{t} can be expressed as:

The portion of the sample variance of x_{t} that can be attributed to cycles of frequency w_{j} is given by:

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.

and

In the second step, we find the value of the parameters , j = 1,..., (T − 1) for the time series x_{t} according to Q21 (Table 7, Figure 6).

In the third step, we find (Table 8, Figure 7), only for the frequencies w_{1}, w_{3}, w_{5}, w_{12}:

Let {v_{t}}_{t = 1,..,37} be a random variable, and w_{12} 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_{w}_{12} = σ_{ε} = 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:

w_{12} ∈ [E[w_{12}] − σ_{w} _{min}, E[w_{12}] + σ_{w} _{min}]

The spectral analysis can not give the theoretical value of E[w_{12}] = 2.037. The experimental value of E[w_{12}] = 2.8 . You can see the results in Figure 8, Figure 9 and Table 10.

P a rame te rs o f x_{t} | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

j = t | γ_{j} | ρ_{j} | E [w_{j}] | γ_{j}·cos (E[w_{1}]*j) | γ_{j}·cos (E[w_{3}]*j) | γ_{j}·cos (E[w_{5}]*j) | γ_{j}·cos (E[w_{12}]*j) | x_{t} | y_{t} | z_{t} | u_{t} | cos(w_{1}(t-1)) | cos(w_{3}(t-1)) | sen(w_{5}(t-1)) | sen(w_{12}(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 |

Variance: s_{t}^{2} | |
---|---|

E[(x_{t}-E[x_{t}])^{2}] | 16,958 |

(a_{1}^{2}+a_{3}^{2}+b_{5}^{2}+b_{1} | 16,500 |

γ^{0} | 16,958 |

T | 37,000 |

Frequencies | Wavelength | Coefficients | (a_{j}^{2}+b_{j}^{2})/2 | Sample Periodogram | (4*pj/T)s_{x}(E[w_{j}]) | |||||
---|---|---|---|---|---|---|---|---|---|---|

E[w_{1}] | 0,170 | l_{1} | 37,000 | a_{1} | 2,000 | a_{1}^{2} | 2,000 | S_{x}(E[w_{1}]) | 4,961 | 1,685 |

E[w_{3}] | 0,509 | l_{2} | 12,333 | a_{3} | -3,000 | a_{3}^{2} | 4,500 | S_{x}(E[w_{3}]) | 21,988 | 7,468 |

E[w_{5}] | 0,849 | l_{3} | 7,400 | b_{5} | 2,000 | b_{5}^{2} | 2,000 | S_{x}(E[w_{5}]) | 8,350 | 2,836 |

E[w_{12}] | 2,038 | l_{4} | 3,083 | b_{12} | 4,000 | b_{12}^{2} | 8,000 | S_{x}(E[w_{12}]) | 45,375 | 15,410 |

Heisenberg's Uncertainty Principle | ||||||||||||

σ_{x}σ_{w} = 3,348 >1/2 | σ_{y}σ_{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[w_{1}] | 0,170 | 0,048 | 0,285 | 0,291 | 1,715 | |||||||

E[w_{3}] | 0,509 | 0,388 | 2,285 | 0,631 | 3,715 | |||||||

E[w_{5}] | 0,849 | 0,728 | 4,285 | 0,970 | 5,715 | E[w_{1}] | 0,170 | 0,014 | 0,082 | 0,326 | 1,918 | |

E[w_{12}] | 2,038 | 1,916 | 11,285 | 2,159 | 12,715 | E[w_{12}] | 2,038 | 1,882 | 11,082 | 2,194 | 12,918 | |

σ_{z}σ_{w} = 0,000 <1/2 | σ_{u}σ_{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[w_{12}] | 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[w_{12}] | 2,038 | 1,863 | 10,973 | 2,038 | 12,000 | E[w_{i}] | 2,038 | 0,000 | 0,000 | ∞ | ∞ | |

Heisenberg's Uncertainty Principle | |||||

σ_{v}σ_{w12}= 1,1334 > 1/2 | |||||

E[v_{t}] | 0,0439 | σ_{v}=E[(v_{t}-E[v_{t}])^{2}] | 1,1334 | ||

E[w_{12}] | 2,0377 | σ_{w}=E[(w_{12}-E[w_{12}])^{2}] | 1,0000 | ||

σ_{wmin} | 0,4412 | T | 37 | γ0 | 1,1334 |

Lower[w_{12}] | Min[j] | Upper[w_{12}] | Max[j] | ||

E[w12] | 2,0377 | 1,5966 | 9,4021 | 2,4789 | 14,5979 |

- 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 = (x
_{1},x_{2},..., 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” - This paper, which uses Kronecker product ⊗, represents an easy, new formalization of game (K, ∆, u(p)), which ex
_{t}ends the game Γ to the mixed strategies.

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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^{1}We use bolt in order to represent vector or matrix.^{2}The hermitian operator have the nex_{t}property: , the transpose operator is equal to complex conjugate operator^{3}- H
_{0}(x) = 1 , H_{1}(x) = 2x , - H
_{2}(x) = 4x^{2}− 2, H_{3}(x) = 8x^{3}− 12x

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