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

^{1}

^{2}

^{3}

## Abstract

**:**

## 1 Introduction

_{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.

_{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].

_{t}by the standard deviation ofits frequency w is greater or equal to 1/2.

## 2 Quantum Games and Hermite’s Polynomials

_{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].

^{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.

^{k}is noted .

_{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.

^{(−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.

#### **2.1 Minimum Entropy Method**

**Theorem 1**

_{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.

**Case 1**

_{k},σ

_{k}), then its entropy is minimum for the minimum standard deviation (H

_{k})min ⇔ (σ

_{k})min. ∀k = 1, ..., n.

**Proof.**

**Case 2**

**Proof.**

_{k}= 1, ..., l

_{k}

_{k}(p

^{k}), expected utility , and variance will be different for each player k.

_{λ→∞}H

_{k}(p

^{k}(λ)) = min(H

_{k})

_{λ→∞}σ

_{k}(λ)= 0.

**Remark 1**

_{k}for gaussian probability density and multinomial logit are written as and

**Case 3**

**Case 4**

_{k})

_{min}, ∀k ∈ K.

**Theorem 2**

^{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

**Proof.**

_{t}ended in Pena and Cohen-Tannoudji [4,48]. Let be Hermitian operators 2 which do not commute, because we can write,

_{min}are

_{min}

**Remark 2**

_{k}(x), ∀k, and in order to calculate . We can write

**Remark 3**

#### **2.2 Quantum Games Elements**

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 |

_{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

**Definition 1**

_{k}is constituted of several sub-strategies

**Theorem 3**

^{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

**Proof.**

_{k}(x).

_{k}) we can write

_{k}) be the generating function

_{k}= 0; we can write:

^{+}let us write the equations:

^{+}can increase the sub-strategy description

#### **2.3 Hermite’s Polynomials Properties**

#### **2.4 Quantum Games Properties**

_{t}postulates or axioms:

**Axiom 1**

_{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**

**Axiom 3**

**Axiom 4**

_{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}.

_{k}strategies and of each player k.

## 3 Time Series and Uncertainty Principle of Heisenberg

- 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

_{X}(called law or distribution of X) on ß(R) given by

_{X}(B) = P {ω : X(ω) ∈ B}, B ∈ ß(R)

_{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**

_{X}from R to [0, 1] given by

_{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.

**Definition 3**

_{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**

^{−1}(y) exists, then Y is a random variable and conversely.”

**Definition 5**

**Definition 6**

**Theorem 4**

**Proof.**

^{2}= . Since −1 ≤ cosθ ≤ 1 then |A|

^{2}≥ and therefore:

**Remark 4**

_{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.

_{j}is defined in a confidence interval given by

“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:

_{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 .

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

1 | ↑ | ↑ | ↑ | |

2 | ↑ | ↓ | ↑ | |

3 | ↓ | ↑ | ↑ | |

4 | ↓ | ↓ | ↑ | |

5 | 0 | ∞ | trivial |

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

1 | ↑ | ↓ | ↑ | |

2 | ↓ | ↑ | ↓ | |

3 | max | min | min | |

4 | min | min | max | |

5 | 0 | ∞ | trivial |

**Remark 5**

_{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.

## 4 Applications of the Models

#### **4.1 Hermites’s Polynomials Application**

_{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.

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 |

#### 4.2 Time Series Application

_{t}] = µ and jth covariance γ

_{j}

_{t}− μ)(x

_{t−j}− μ)] = γ

_{j}

_{j}’s represent autocovariances of a covariance-stationary process using the Possibility Theorem, then

_{x}(w) ∈ [S

_{x}(w − σ

_{w}), S

_{x}(w + σ

_{w})]

_{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}

_{1},x

_{2},.., x

_{T}, we can calculate up to T − 1 sample autocovariances γ

_{j}from the formulas: and

_{t}will be expressed in terms of periodic functions with M = (T − 1)/2 representing different frequencies

_{t}can be expressed as:

_{t}that can be attributed to cycles of frequency w

_{j}is given by:

_{j}.

**Example 5**

_{t}according to Q21 (Table 7, Figure 6).

_{1}, w

_{3}, w

_{5}, w

_{12}:

**Example 6**

_{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

_{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:

_{12}∈ [E[w

_{12}] − σ

_{w}

_{min}, E[w

_{12}] + σ

_{w}

_{min}]

_{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 |

## 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 = (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.

## ACKNOWLEDGMENTS

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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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Jiménez, E. Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy. *Entropy* **2003**, *5*, 313-347.
https://doi.org/10.3390/e5040313

**AMA Style**

Jiménez E. Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy. *Entropy*. 2003; 5(4):313-347.
https://doi.org/10.3390/e5040313

**Chicago/Turabian Style**

Jiménez, Edward. 2003. "Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy" *Entropy* 5, no. 4: 313-347.
https://doi.org/10.3390/e5040313