8/12/2019 e 5040313

1/35

E n tr opy 2003, 5, 313-347

EntropyISSN 1099-4300

c 2003 by MDPI

www.mdpi.org/entropy

Quantum Games: Mixed Strategy Nashs Equilibrium RepresentsMinimum Entropy

Edward Jimnez1,2,3

1Experimental Economics, Todo1 Services Inc, Miami Fl 331262GATE, UMR 5824 CNRS - France.3

Research and Development Department, Petroecuador, Quito-EcuadorE-mail: ejimenezecu@yahoo.comPresent address: Paul Rivet y 6 de Diciembre (Edif EL PINAR), Quito-Ecuador.

Received: 15 Nov 2002 / Accepted: 5 Nov 2003 / Published: 15 Nov 2003

Abstract:This paper introduces Hermites polynomials, in the description of quantum games.

Hermites polynomials are associated with gaussian probability density. The gaussian proba-bility density represents minimum dispersion. I introduce the concept of minimum entropy asa paradigm of both Nashs equilibrium (maximum utility MU) and Hayek equilibrium (mini-mum entropy ME). The ME concept is related to Quantum Games. Some questions arise aftercarrying out this exercise: i) What does Heisenbergs uncertainty principle represent in GameTheory and Time Series?, and ii) What do the postulates of Quantum Mechanics indicate inGame Theory and Economics?.

PACS numbers:03.67.Lx, 03.65.GeAMS numbers:91A22, 91A23, 91A40.

Keywords:Quantum Games, Minimum Entropy, Time Series, Nash-Hayek equilibrium.

8/12/2019 e 5040313

2/35

E n tr opy 2003, 5, 313-347 314

1 Introduction

The quantum games and the quantum computer are closely-related. The science of Quantum Computer isone of the modern paradigms in Computer Science and Game Theory [5-8, 23, 41, 42, 53]. Quantum Com-puter increases processing speed to have more effective data-base search. The quantum games incorporate

Quantum Theory to Game Theory algorithms. This simple extrapolation allows the Prisoners Dilemmato be solved and demonstrates that the cooperative equilibrium [44, 47, 51, 57] is viable and stable with aprobability different from zero.

In Eisert[7,8] the analogy between Quantum Games and Game Theory is expressed as: At the mostabstract level, game theory is about numbers of entities that are efficiently acting to maximize or minimize.For a quantum physicist, it is then legitimate to ask: what happens if linear superpositions of these actionsare allowed for?, that is, if games are generalized into the quantum domain. For a particular case of thePrisoners Dilemma, we show that this game ceases to pose a dilemma if quantum strategies are implementedfor. They demonstrate that classical strategies are particular quantum strategy cases.

Eisert, Wilkens, and Lewenstein [8], not only give a physical model of quantum strategies but alsoexpress the idea of identifying moves using quantum operations and quantum properties. This approachappears to be fruitful in at least two ways. On one hand, several recently proposed quantum informationapplication theories can already be conceived as competitive situations, where several factors which haveopposing motives interact. These parts may apply quantum operations using bipartite quantum systems. Onthe other hand, generalizing decision theory in the domain of quantum probabilities seems interesting, asthe roots of game theory are partly rooted in probability theory [43, 44]. In this context it is of interest toinvestigate what solutions are attainable if superpositions of strategies are allowed [18, 41, 42, 50, 51, 57].A game is also related to the transference of information. It is possible to ask: what happens if these carriersof information are applied to be quantum systems, where Quantum information is a fundamental notion ofinformation? Nashs equilibria concept as related to quantum games is essentially the same as that of gametheory but the most important difference is that the strategies appear as a function of quantum properties ofthe physical system [41, 42].

This paper essentially explains the relationship which exists among Quantum Mechanics, Nashs equi-libria, Heisenbergs Uncertainty Principle, Minimum Entropy and Time Series. Heisenbergs uncertaintyprinciple is one of the angular stones in Quantum Mechanics. One application of the uncertainty principlein 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 principleindicates that the product of the standard deviation of one random variable xtby the standard deviation ofits frequencywis greater or equal to 1/2.

This paper is organized as follows:Section 2: Quantum Games and Hermites Polynomials, Section 3: Time Series and Heisenbergs Prin-

ciple, Section 4: Applications of the Models, and Section 5: Conclusion.

2 Quantum Games and Hermites Polynomials

Let ! = (K,S,v)be a game to n!players, withKthe set of players k = ",...,n. The finite set Skofcardinalitylk" Nis the set of pure strategies of each player where k" K,skjk" Sk, jk = ",...,lkandS = #KSkrepresents the set of pure strategy profiles withs" S an element of that set,l = l1, l2,...,lnrepresents the cardinality ofS,[12, 43, 55, 56].

8/12/2019 e 5040313

3/35

E n tr opy 2003, 5, 313-347 315

The vector function v : S# Rn associates every profile s" S,where the vector of utilities v(s)= (v1(s),...,vn(s))T, and vk(s)designates the utility of the player k facing the profile s. In order tounderstand calculus easier, we write the function vk(s)in one explicit way vk(s) = vk(j1, j2,...,jn).Thematrixvn,lrepresents all points of the Cartesian product #k!KSk.The vectorvk(s)is thek! column ofv.

If the mixed strategies are allowed then we have:

$(Sk) =

(pk " Rlk :

lkXjk=1

pkjk = "

)the unit simplex of the mixed strategies of playerk" K,andpk = (pkjk)the probability vector. The set

of profiles in mixed strategies is the polyhedron $ with $= #k!K$(Sk), wherep = (p1j1, p2j2

..., pnjn),andpk = (pk1, p

k2,...,p

kln

)T. Using the Kronecker product $ it is possible to write1:

p = p1$p2$...$ pk"1$pk$pk+1$ ...$ pnp("k) = p1

$p2

$...

$pk"1

$lk

$pk+1

$...

$pn

lk = (", ", ...,")T, lklk,1

ok = (0, 0, ..., 0)T,oklk,1

where

lk = (", ",...,")T,lklk,1

ok = (0, 0,..., 0)T,oklk,1

Then! dimensional functionu: $# Rn associates with every profile in mixed strategies the vector ofexpected utilities

u(p) =

u1 (p,v(s)) ,...,un (p,v(s))T

where uk(p,v(s))is the expected utility of the playerk.Every ukjk = ukjk

(p("k),v(s))represents the

expected utility for each players strategy and the vectoruk is noteduk = (uk1, uk2,...,u

kn)

T.

uk =

lkXjk=1

ukjk(p("k), v(s))pkjk

u = v0p

uk = lk $ vkp("k)The triplet(K,$,u(p))designates the extension of the game !with the mixed strategies. We get Nashs

equilibrium (the maximization of utility [3, 43, 55, 56, 57]) if and only if, %k, p, the inequality uk(p#)&uk(pk#

,p("k))is respected.

1 We use bolt in order to represent vector or matrix.

8/12/2019 e 5040313

4/35

E n tr opy 2003, 5, 313-347 316

Another way to calculate the Nashs equilibrium [43, 47], is leveling the values of the expected utilitiesof each strategy, when possible.

uk1 p("k), v(s) = uk2 p("k), v(s)= ... = ukjk p("k), v(s)lkXjk=1

pkjk = " %k= ",...,n

2k =

lkXjk=1

ukjk

p("k), v(s)

! uk2pkjk = 0If the resulting system of equations doesnt have solution

p("k)

#then we propose theMinimum Entropy

Method. This method is expressed as M inp(P

k Hk(p)) ,where 2k(p

#)standard deviation and Hk(p#)entropy of each player k.

2k(p#) ' 2k

pk#

,p("k)

or

Hk(p#) ' Hk

pk#

,p("k)

2.1 Minimum Entropy Method

Theorem 1 (Minimum Entropy Theorem). The game entropy is minimum only in mixed strategy Nashsequilibrium. The entropy minimization program Minp(

PkHk(p)), is equal to the standard deviation

minimization programM inp(#kk(p)), when ukjkhas 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 influences and historical events whichinspired Hayek to develop the idea of a spontaneous order. Here we present the technical tools needed inorder to study the order state.

Case 1 If the probability density of a variable Xis normal: N(k, k), then its entropy is minimum for theminimum standard deviation(Hk)min ( (k)min. %k= ",...,n.

Proof. Let the entropy functionHk =! R+$"$ p(x)ln(p(x))dxand p(x)the normal density function.Writing this entropy function in terms of minimum standard deviation we have.

Hk= !Z +$"$

p(x)

!"ln

"p

22k

!!

x! uk)2k

!2#$ dx

developing the integral we have

8/12/2019 e 5040313

5/35

E n tr opy 2003, 5, 313-347 317

Hk = ln)

2k

Z +$"$

p(x)dx

+ "

22kZ +$"$ (x! uk)2p(x)dx

= ln)

2k

+ 2k22k

=

"

2+ ln

)2

+ ln k

For a game ton!players the total entropy can be written as follows:

nXk=1

Hk = n

"

2+ ln

)2

+ ln(#nk=1 k)

MinpXk

Hk(p)! ( M inp(#kk(p))after making a few more calculations, it is possible to demonstrate that

M inp

nXk=1

k(p)

!* M inp(#kk(p))

The entropy or measure of the disorder is directly proportional to the standard deviation or measureof the uncertainty [1, 13, 27]. Clausius [48, 49], who discovered the entropy idea, presents it as both anevolutionary measure and as the characterization of reversible and irreversible processes [45].

Case 2 If the probability function ofukjk(p("k), v(s)) is a multinomial logit of parameter (rationality

parameter) [35-40], then its entropy is minimum and its standard deviation is minimum for #+ [21,22].

Proof. Letpkjk the probability fork " K, jk= ",...,lk

pk

jk =

eukjk

(p!k)

Plkjk=1

eukjk(p!k) =

eukjk

(p!k)

Zk

whereZk

,uk(p"k)

=Plk

jk=1eukjk

(p!k) represents the partition utility function [1, 45].

The entropy Hk(pk),expected utility E{ukjk(p"k)} = uk(p) , and variance V ar{ukjk(p

"k)}will bedifferent for each playerk.

8/12/2019 e 5040313

6/35

E n tr opy 2003, 5, 313-347 318

Hk(pk) = !

lkXjk=1

pkjk ln(pkjk

)

uk(p) =lkX

jk=1

pkjkukjk

V ar

ukjk(p"k)

=

lkXjk=1

pkjk(ukjk! uk)2

Using the explicit form of pkjk , we can get the entropy, the expected utility and the variance:

Hk(pk) = ln(Zk)! uk(p)

uk(p) =

ln(Zk)

uk(p)

= V ar{ukjk(p

"k)}= 2k

The equation Hk(pk)

=!2k can be obtained using the last seven equations; it explains that, when the

entropy diminishes, the parameter (rationality)increases.

The rationality increases from an initial value of zero when the entropy is maximum, and it drops to itsminimum value when the rationality toward the infinite value [38, 39, 40]: Lim%$Hk(pk()) = min(Hk)

The standard deviation is minimum in the Nashs equilibria [21, 22, 23].

If the rationality increase, then Nashs equilibria can be reached when the rationality spreads toward to

its infinite:Lim%$k() = 0.Using the logical chain that has just been demonstrated, we can conclude that the entropy diminishes

when the standard deviation diminishes:

(Hk(p) = (Hk(p))min) ( (k() = (k)min)

M inp

Xk

Hk(p)

! ( Minp(#kk(p))

after making a few more calculations, it is possible to demonstrate that

M inp nXk=1

k(p)!* M inp(#kk(p))Remark 1 The entropy Hk for gaussian probability density and multinomial logit are written as

12

+

ln)

2+ ln k andln(Plk

jk=1eukjk )! uk

8/12/2019 e 5040313

7/35

E n tr opy 2003, 5, 313-347 319

Case 3 The special case of Minimum Entropy is when 2k = 0 and the utility function value of each strategy

ukjk(p("k), v(s)) =uk,are the same for alljk,and for allk, see Binmore, Myerson [3, 48].

In the special case of Minimum Entropy when 2k# 0, the gaussian density function can be approximatedby Diracs Deltax! uk.

The function

x! uk

is called Diracs function. Diracs function is not a function in the usual

sense. It represents an infinitely short, infinitely strong unit-area impulse. It satisfies

x! uk

=+ ifx = uk0ifx6=uk

and can be obtained at the limit of the function

x! uk

=Limk"0

"

k)

2e"x!uk#2k

2

Case 4 If we can measure the standard deviation2k 6= 0, then Nashs equilibrium represents minimumstandard deviation(k)min , %k " K.

uk1(p("k), v(s)) 6= uk2(p

("k), v(s))6=... 6=ukn(p("k), v(s))

lkXjk=1

pkjk = "

2

k =

lk

Xjk=1

(uk

jk(p(

"k)

, v(s))! uk

)2

p

k

jk >0

Theorem 2 Minimum Dispersion. The gaussian density functionf(x) =|(x)|2 represents the solution

of one differential equation given byx"hxi22x

+ x

(x) = 0 related to minimum dispersion of a lineal

combination between the variablex, and a Hermitian operator!i

x

.

Proof. This proof is extended inPena and Cohen-Tannoudji [4, 48]. Let

bA,

bBbe Hermitian operators2

which do not commute, because we can write,

hbA,bBi= bAbB !bBbA= iCEasily we can verified that the standard deviations satisfy the same rule of commutation:

2 The hermitian operatorbA have the next property:ebA =bA! , the transpose operatorebA is equal to complex conjugate operatorbA!.

8/12/2019 e 5040313

8/35

8/12/2019 e 5040313

9/35

E n tr opy 2003, 5, 313-347 321

ifbA= x,bB = !i x ,and DbBE= 0then DbCE= " andx! hxi

22x+

x

(x) = 0

the solution is

(x) =

"

x)

2

1/2e"

(x!hxi)2

42

f(x) = |(x)|2 = "

x)

2e" (x!hxi)

2

22x

Remark 2 The gaussian density functionf(x)is optimal in the sense of minimum dispersion.

According to the last theoremukjk =x is a gaussian random variable with probability densityfk(x), %k,and in order to calculateE

hukjk

i= uk =k. We can write

fk(x) = "

k)

2e"(

x!uk)2

22k

Remark 3 The central limit theorem which establishes that The mean k of the random variable ukjk hasnormal 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 randomnature. Due to this nature it is necessary to speak of expected values. Quantum Mechanics defines observedvalues or expected values in Cohen-Tannoudji[4].

According to Quantum Mechanics an uncertainty exists in the simultaneous measure of two variablessuch 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 playerthat uses a mixed strategy respects the vNM (Von Newmann and Morgenstern) function or overlappingprobabilistic of pure strategies according toBinmore[3, 43]. The state function in quantum mechanics doesnot have an equivalent in Game Theory, therefore we will use state function as an analogy with QuantumMechanics. The comparison between Game Theory and Quantum Mechanics can be shown in an explicitway in Table 1. If the definition of rationality in Game Theory represents an optimization process, we cansay that quantum processes are essentially optimal, therefore the nature of both processes is similar

8/12/2019 e 5040313

10/35

E n tr opy 2003, 5, 313-347 322

Table 1. Quantum Mechanic and Game Theory properties

Quantum Mechanics

Particle:k = ",...,nQuantum elementInteractionQuantum state: j = ",...,lkEnergy eSuperposition of statesState functionProbabilistic and optimal natureUncertainty PrincipleMatrix operators

Variational Calculus,Information TheoryComplexityvariety: number of particles nvariability: number of interactions n!quantitative: Mathematical model

Observable value: E[e]The entities communicate efficientlyEntropy

Game Theory

Player : k = ",...,nPlayer typeInteractionStrategy:j = ",...,lkUtilityuSuperposition of strategiesUtility functionProbabilistic and optimal natureMinimum EntropyMatrix operators

Optimal Control,Information Theory

Complexityvariety: number of players nvariability: number of interactionsn!quantitative: Mathematical model

Observable value: E [u]The players interact efficientlyMinimum Entropy

Letk = ",...,nbe players withlkstrategies for every onejk=mk1,...,mklk

. According to the theorem of

Minimum Dispersion,the utility ukjk , converges to Nashs equilibria and follows a normal probability density

k(u) = e!1

2( uk

!kk)2

k&2 ,where the expected utility is Ehukjki= k.Definition 1 (Tactics or Sub- Strategies) LetMkjk be the set of sub-strategies m

kjk" Mkjk , %k, jk with a

probability given by

bk

mkjk

2where mkjk" Mk ; ,lkjk=1Mkjk =Mk, Mk ={0, ", ...,+} and Mkjk -Mkjl = .

whenk 6=l. A strategyjk is constituted of several sub-strategiesmkjk

.

With the introduction of sub-strategies the probability of each strategy is described in the following way

bkjk

2

=

Pmkjk!Mkjk bk

mkjk 2

and we do not need any approximation of the function$

Pmkjk=0ck

mkjk(t)mkjk

(x, ),

where:

(x, ) =P$

mkjk=0c

kmkjk

(t)mkjk(x, ), ck

mkjk(t) = bk

mkjkeiwkt,

ck

mkjk

2=

bk

mkjk

2=pk

mkjk

andP$

mkjk=0

bk

mkjk

2= ",%k.

8/12/2019 e 5040313

11/35

8/12/2019 e 5040313

12/35

E n tr opy 2003, 5, 313-347 324

Ifckmkjk

=bkmkjk

thenwk = 0;we can write:

"

pk)

2e"

(x!k)2

2 (Q3)

=$X

mkjk=0

!"(k)

mkjk q)2mkjk"1qmkjk !

)n

#$!" e

"x2

2 Hmkjk x&

2q

mkjk !)

2mkjk)

#$

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

bkmkjk=

(k)mkjk

q)2

mkjk"1q

mkjk !)

k

(Q4)

mkjk (x, k) =

e"x2

2 Hmkjk x&2q

mkjk !)

2mkjk)

(Q5)

The

bk

mkjk

2parameter represents the probability of mkjk sub-strategy of each k player. The utility of

vNM can be written as:

Eh

ukjk

i = uk(p"k) =

lkXjk=1

bkjk

2ukjk (Q5.1)

bkjk 2 = Xmkjk

!Mkjkbkmkjk 2 (Q5.2)

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

amkjk(x, k) =

(x + x

))23/2 mkjk (x, k) (Q6)

amkjk(x, k) =

)nmkjk"1

(x, k) (Q6.1)

The equation (Q6.1) indicates that the operatora= (x+

x)

(&2)

3/2decreases the level of kplayer. The operator

acan decrease the sub-strategy descriptionmkjk# mkjk ! ".

a+mkjk(x, ) =

x!

x

)23/2mkjk (x, ) (Q7)

a+mkjk(x, ) =

)n + "mkjk+1

(x, ) (Q7.1)

8/12/2019 e 5040313

13/35

E n tr opy 2003, 5, 313-347 325

The equation (Q7.1) indicates that the operatora+ = (x" x)

(&2)

3/2increases the level of kplayer [4, 46]. The

operatora+ can increase the sub-strategy descriptionmkjk# mkjk + ".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 Hermites Polynomials Properties

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

Hermite polynomials obey the next differential equation [14]:

H00

mkjk(x)! 2xH0mkjk (x) + 2m

kjk

Hmkjk(x) = 0, mkjk = 0, ...,+ (PH0)

The principal solution of Hermites differential equation3 is

Hmkjk (x) = (!")ex2

d

mkjk

e"x2

dxmkjk

(PH1)

Hmkjk(x) = (2x)m

kjk! m

kjk

(mkjk ! ")"!

(2x)mkjk"2

+mkjk(m

kjk! ")(mkjk ! 2)(mkjk! 3)

2! (2x)m

kjk"4

+

Taking the explicit value ofmkjk!derivative of (PH1):

dmkjk Hmkjk

(x)

dxmkjk = m

kjk !2

mkjk (PH2)

dmkjk Hmkjk

( x&2

)

dxmkjk

= mkjk !)

2mkjk (PH2.1)

From which we obtain the decreasing operator:

dHmkjk(x)

dx = 2nHmkjk"1

(x) (PH3)

and the increasing operator:

2x(x! ddx

)Hmkjk(x) =Hmkjk+1

(x) (PH4)

Recurrence formulas:

3H0(x) = 1 , H1(x) = 2x,H2(x) = 4x

2! 2 , H3(x) = 8x3! 12x

8/12/2019 e 5040313

14/35

E n tr opy 2003, 5, 313-347 326

Hmkjk+1(x) = 2xHmkjk

(x)! 2mkjkHmkjk"1(x) (PH5)and normalization condition:

Z +$"$

e"x2

H2mkjk(x)dx (PH6)

=

Z +$"$

e"x2

dmkjk Hmkjk

(x)

dxmkjk

dx= mkjk !2mkjk)

The next condition expresses the orthogonality property of Hermites polynomials:

De"

x2

2 Hmkjk(x), e"

x2

2 Hmkjk(x)E

= Z +$

"$ e"x2

Hmkjk (x)Hm

kjl(x)dx= 0 %m

k

jk 6=m

k

jl

The generating function is obtained using the Taylor development.

F(x, k) = ex2e"(x"k)

2

=Xmkjk

(k)mkjk

mkjk ! Hmkjk

(x) (PH7)

Quantum Games generalize the representation of the probability in the utility function of vNM (vonNewmann and Morgenstern). To understand Quantum Games it is necessary to use properties of quantumoperators in harmonic oscillator and the properties of Hilbert space. The harmonic oscillator is presentedas the paradigm in Quantum Games.

2.4 Quantum Games Properties

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

Axiom 1 At a fixed timeto ,the state of a physical system is defined by specifying a ket|(to)i beloging tothe state space (Hilbert space) V.

Axiom 2 Every measurable physical quantity % is described by the operatorA acting inV; this operatoris an observable parameter.

Axiom 3 The only possible result of the measurement of a physical quantity% is one of the eigenvalues of

the corresponding observableA.

Axiom 4 When a physical quantity % is measured on a system in the normalized state |(t)i=P

cjj

,the probabilityP(bj) of obtaining the non-degenerate eigenvaluecj of the corresponding observableA is:

P(cj) ='

j, 2

wherej

is the normalized eigen vector ofA associated with the eigen value cj.

Letnbe players withk = ",...,n,lkstrategies andmkjk" Mkjkof each playerk.

8/12/2019 e 5040313

15/35

E n tr opy 2003, 5, 313-347 327

Superposition principle:

k =$X

mkjk=0

ckmkjkmkjk

,D

mkjk, mkjl

E= mkjkm

kjl

(QG1)

Time evolution:

ckmkjk=bkmkjk

eiwkt, i=)!" (QG2)

Inner product:

ckmkjk=D

k, mkjk

E (QG3)

Quantum property

ckmkjk 2 = bkmkjk 2 =pkmkjk (QG4)$X

mkjk=0

pkmkjk=

lkXjk=1

pkjk

= "

pkjk

=X

mkjk!Mkjk

pkmkjk

Mean value

E[xk] =xk ='

k |x|k (QG5)Normality conditionk =

'k, k

=k2

(QG6)

3 Time Series and Uncertainty Principle of Heisenberg

Remember some basic definitions about probability and stochastic calculus.

A probability space is a triple(&,=, P)consisting of [10, 13, 30]:

A set & that represents the set of all possible outcomes of a certain random experiment.

A family = of subsets of& with a structure of-algebra:

! " = A " ==* Ac " = A1, A2,... " ==* ,$i=1Ai" =

A functionP : [0, "]such that:

8/12/2019 e 5040313

16/35

E n tr opy 2003, 5, 313-347 328

P(!) = 0, P(&) = "

If A1, A2,... " = form a finite or countably infinite collection of disjointed sets (that is, Ai-Aj =! ifi 6=j) then

P(,$i=1Ai) =$X

i=1

P(Ai)

The elements of the-algebra = are called events andPis called a probability measure.

A random variable is a function:

X : R # X()

=-measurable, that is,X"1(B) " = for allBin Borelsalgebra ofR, noted (R).If Xis a random variable on(&,=, P)the probability measure induced byXis the probability measure

PX(called law or distribution ofX) on (R)given by

PX(B) =P{: X() " B} , B" (R)The numbers PX(B), B"(R),completely characterize the random variable Xin the sense that they

provide the probabilities of all events involvingX.

Definition 2 The distribution function of a random variableX is the function F = FX from R to[0, "]given by

F(x) = P{: X() ' x} x " Rsince, fora < b, F(b) ! F(a) = P{: a < X() ' b}= PX(a, b], Fis a distribution function corre-

sponding to the Lebesgue Stieltjes measurePX .Thus among all distribution functions corresponding to theLebesgue-Stieltges measurePX,we choose the one withF(+) = ", F(!+) = 0.In fact we can alwayssupply the probability space in a canonical way; take & = R,= =(R),with Pthe Lebesgue-Stieltgesmeasure corresponding toF.

Definition 3 A stochastic process X = {X(t), t " T} is a collection of random variables on a commonprobability space (&,=, P) indexed by a parametert" T/ R, which we usually interpret as time [10,27-30]. It can thus be formulated as a functionX:T R.

The variable X(t, )is =!measurable in " &for each t" T;henceforth we shall often followestablished convention and writeXtforX(t).When Tis a countable set, the stochastic process is really asequence of random variablesXt1, Xt2, ..., Xtn, ..

8/12/2019 e 5040313

17/35

E n tr opy 2003, 5, 313-347 329

Definition 4 If a random variableX = X(t) has the density (x) andf : X# Y is one-to-one, theny = f(x) has the density g(y) given by g(y) =

(f!1(y))f0(f!1(y)) = (f

"1(y)) (f"1)0(y), see Fudenberg and

Tirole [12], (chapter 6). The densities(x) andg(y) show that IfX is arandom variable andf"1(y)exists, thenYis a random variable and conversely.

Definition 5 Inversion formula: Ifh is a characteristic function of the bounded distribution functionF,andF(a, b] =F(b)! F(a)then

F(a, b] = limc%$

"

2

Z +c"c

e"iwa ! e"iwbiu

h(w)dw

for all points a, b (a < b)at which Fis continuous. If in addition, h is Lebesgue integrable on(!+,+) ,then the functionfgiven by

f(x) = "2Z +$"$

e"iwxh(w)dw

is a density ofF, that is, f is nonnegative and F(x) =R+x"$ f(t)dt for all x;furthermore, F

0 = feverywhere. Thus in this case,fand hare Fourier transform pairs:

h(w) = z(f(x)) = "

2

Z +$"$

eiwxf(x)dx

f(x) = z"1(h(w)) = "

2

Z +$"$

e"iwxh(w)dw

Definition 6 IfX is an random variable., fa Borel measurable function [on (

8/12/2019 e 5040313

18/35

E n tr opy 2003, 5, 313-347 330

Herewis the frequency, different to " & .Then(w)is

(w) = z [(x)] = "

)2 Z +$

"$(x)eiwxdx

%x, (w=

+) = 0 (Q1)

According to Fouriers transform properties, we have:

n(w)

wn =

in)2

Z +$"$

xn(x)eiwxdx= inz [xn(x)] (Q2)

n(x)

xn =

(!i)n)2

Z +$"$

wn(w)e"iwxdw= (!i)n "1z [wn(w)]

where:

zn(x)

xn

= ")

2

Z +$"$

n(x)xn

eiwxdx= (iw)n (w) (Q3)

z"1

n(w)

xn

=

")2

Z +$"$

n(w)

xn e"iwxdw= (!ix)n (x)

By Parseval Plancherel theoremZ +$"$

(w)(w)#dw=

Z +$"$

(x)(x)#dw (Q4)

Using Schwartzs inequality

Z +$"$

G#Hdx2'Z +$"$

|G|2dx 0Z +$"$

|H|2dx (Q5)

and the probability propertiesR+$"$ f(t)dt= ",

R+$"$ h(w)dw= ", it is evident that:Z +$

"$(w)(w)#dw= ",

Z +$"$

(x)(x)#dw= " (Q6)

Let2wbe the standard deviation of frequencyw,and 2xthe standard deviation of the aleatory variable

x.

2w =Z +$"$

w2(w)(w)#dw (Q7.1)

2x =

Z +$"$

x2(x)(x)#dx (Q7.2)

with Parseval Plancherel theorem

8/12/2019 e 5040313

19/35

E n tr opy 2003, 5, 313-347 331

Z +$"$

(x)

x

(x)#

x dx =

Z +$"$

(w(w)) (w(w))# dw (Q8.1)

Z +$"$ 00#dx = Z +$

"$ w2

(w)(w)#dw (Q8.2)

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

2w2x =

Z +$"$

w2(w)(w)#dw

Z +$"$

x2(x)(x)#dx (Q9.)

=

Z +$"$

00#dx

Z +$"$

x2(x)(x)#dx (Q9.2)

LetG= xandH= 0 in Q5 shows that the right side of (Q9.2) exceeds a certain quantityA

Z +$"$

|x|2dx 0Z +$"$

|0|2

dx &Z +$"$

x#0dx2

(Q10)

We will note 2w2x =

R+$"$

00#dxR+$"$ x

2(x)(x)#dx& |A|2 .If the variable Ais given by A =R+$"$ x

#0dxthen |A|2 =R+$"$ x

2dx |#| |00#| .By definitionF0(x) =f=#

f0 = 0#+ 0#

A + A# =

Z +$"$

xdx (0#+ 0#) =

Z +$"$

xf0(x)dx

A + A# = !Z

+$

"$f(x)dx=

!" (Q11)

Notice that A+A# =!" and using the complex number property 2 R e (A) =!" or2 |A| cos =!"wherephase ofA, we can write4 |A|2 = 1

cos2 . Since !" ' cos ' " then |A|2 & 1

4and therefore:

2x2w&

"

4 (Q12)

Remark 4 Letxt be a time series with a spectrum of frequencieswj,where each frequency is an randomvariable. This spectrum of frequencies can be obtained with a minimum error (standard deviation of fre-

quency). This minimum errorw for a certain frequency wis given by the following equation wmin= 1

2x.

The expected valueE[wj]of each frequency can be determined experimentally.

An occurrence of one particular frequencywjis defined in a confidence interval given by

wj"

E[wj]! "2x

, E[wj] + "

2x

(Q13)

8/12/2019 e 5040313

20/35

8/12/2019 e 5040313

21/35

E n tr opy 2003, 5, 313-347 333

4 Applications of the Models

4.1 Hermitess Polynomials Application

Let ! = (K,S,v)be a3!player game, withKthe set of playersk = ", 2, 3. Thefinite setSkof cardinalitylk" Nis the set of pure strategies of each player where k " K, skjk" Sk, jk = ", 2, 3and S= S1S2S3represent a set of pure strategy profiles with s " S as an element of that set and l= 30303 = 27representsthe cardinality ofS. The vectorial functionv : S# R3 associates with every profiles"Sthe vector ofutilitiesv(s) = (v1(s),...,v3(s)), wherevk(s)designates the utility of the playerkfacing the profiles. Inorder to get facility of calculus we write the function vk(s)in an explicit wayvk(s) =vk(j1, j2,...,jn).Thematrixv3,27represents all points of the Cartesian product #KSksee Table 4.The vectorvk(s)is the k-column ofv. The graphic representation of the 3-player game is shown in Figure 1.

In these games we obtain Nashs equilibria in pure strategy (maximum utility MU, Table 4) and mixedstrategy (Minimum Entropy Theorem MET, Tables 5, 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

8/12/2019 e 5040313

22/35

E n tr opy 2003, 5, 313-347 334

Table 4. Maximum utility (random utilities):maxp(u1 + u2 + u3)

Nash Utili ties and Standard Deviations

Maxp (u1+u

2+u

3) = 20.4

u1

u2

u3

8.7405 9.9284 1.6998

1111 2222 3333

6.3509 6.2767 3.8522

H1 H2 H3

4.2021 4.1905 3.7121

Nash Equilibria: Probabilities, Utilities

Player 1 Player 2 Player 3

p11 p12 p

13 p

21 p

22 p

23 p

31 p

32 p

33

1 0 0 1 0 0 0 0 1

u11 u12 u

13 u

21 u

22 u

23 u

31 u

32 u

33

8.7405 1.1120 -3.8688 9.9284 -1.2871 -0.5630 -1.9587 5.7426 1.6998

p11u11 p

12u

12 p

13u

13 p

21u

21 p

22u

22 p

23u

23 p

31u

31 p

32u

32 p

33 u

33

8.7405 0.0000 0.0000 9.9284 0.0000 0.0000 0.0000 0.0000 1.6998

Kroneker Products

j1 j2 j3 v1(j1,j2,j3

)

v2(j1,j2,j3

)

v3(j1,j2,j3)

p1j1 p

2j2 p

3j3 p

1j1p

2j2 p

1j1p

3j3 p

2j2p

3j3 u1

(j1,j2,j3

)

u2(j1,j2,j3

)

u3(j1,j2,j3

)

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 0.00003 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

8/12/2019 e 5040313

23/35

E n tr opy 2003, 5, 313-347 335

Table 5. Minimum entropy (random utilities):minp(1 + 2 + 3) * minp(H1 + H2 + H3)

Nash Ut i l i t ies and Standard Deviat ion s

Minp (((( 1111 + ++ + 2222 + + + + 3333 )))) = 1.382

u1

u2

u3

0.85 2.19 4.54

1 2 31.19492 0.187 0.000

H1 H2 H3

2.596 1.090 0.035

Nash Equi l ib r ia: Probabi l i t ies, Uti l i t ies

Play er 1 Play er 2 Play er 3

p 11 p1

2 p1

3 p2

1 p2

2 p2

3 p3

1 p3

2 p3

3

0 .00 00 0 .5 18 8 0 .4 81 2 0 .41 87 0 .0 84 9 0 .4 96 4 0 .2 43 0 0 .37 79 0 .3 79 1

u1

1 u1

2 u1

3 u2

1 u2

2 u2

3 u3

1 u3

2 u3

3

2 .85 49 1 .1 18 9 0 .5 64 5 2 .39 54 2 .1 61 8 2 .0 25 6 4 .5 42 1 4 .54 21 4 .5 42 1

p 11u1

1 p1

2u1

2 p1

3u1

3 p2

1u2

1 p2

2u2

2 p2

3u2

3 p3

1u3

1 p3

2u3

2 p3

3 u3

3

0 .00 00 0 .5 80 5 0 .2 71 7 1 .00 31 0 .1 83 5 1 .0 05 4 1 .1 03 6 1 .71 63 1 .7 22 1

Kroneker Produc ts

j1 j2 j3 v1(j1,

j2,

j3)

v2(j1,

j2,

j3)

v3(j1,

j2,

j3)

p1j1 p2j2 p

3j3 p

1j1p

2j2 p

1j1p

3j3 p

2j2p

3j3 u

1(j1,

j2,

j3)

u2(j1,

j2,

j3)

u3(j1,

j2,

j3)

1 1 1 2.93 -1 .35 -1 .9 6 0 .0 00 0 0 .4 18 7 0 .2 43 0 0 .0 00 0 0 .0 00 0 0 .1 01 7 0 .2 97 9 0 .0 00 0 0 .0 00 0

1 1 2 6.27 3.25 5.74 0 .0 00 0 0 .4 18 7 0 .3 77 9 0 .0 00 0 0 .0 00 0 0 .1 58 2 0 .9 92 2 0 .0 00 0 0 .0 00 0

1 1 3 8.74 9.93 1.70 0 .0 00 0 0 .4 18 7 0 .3 79 1 0 .0 00 0 0 .0 00 0 0 .1 58 8 1 .3 87 7 0 .0 00 0 0 .0 00 0

1 2 1 4.16 6.97 4.10 0 .0 00 0 0 .0 84 9 0 .2 43 0 0 .0 00 0 0 .0 00 0 0 .0 20 6 0 .0 85 8 0 .0 00 0 0 .0 00 0

1 2 2 3.82 2.72 8.64 0 .0 00 0 0 .0 84 9 0 .3 77 9 0 .0 00 0 0 .0 00 0 0 .0 32 1 0 .1 22 6 0 .0 00 0 0 .0 00 0

1 2 3 -3 .2 1 -1 .2 9 -4 .1 1 0 .0 00 0 0 .0 84 9 0 .3 79 1 0 .0 00 0 0 .0 00 0 0 .0 32 2 - 0. 10 34 0 .0 00 0 0 .0 00 0

1 3 1 3.02 2.33 6.82 0 .0 00 0 0 .4 96 4 0 .2 43 0 0 .0 00 0 0 .0 00 0 0 .1 20 6 0 .3 64 2 0 .0 00 0 0 .0 00 0

1 3 2 -2.74 3.02 6.66 0 .0 00 0 0 .4 96 4 0 .3 77 9 0 .0 00 0 0 .0 00 0 0 .1 87 6 - 0. 51 39 0 .0 00 0 0 .0 00 0

1 3 3 1.18 -0.56 5.34 0 .0 00 0 0 .4 96 4 0 .3 79 1 0 .0 00 0 0 .0 00 0 0 .1 88 2 0 .2 21 7 0 .0 00 0 0 .0 00 0

2 1 1 3.20 -1 .57 -0 .9 8 0 . 51 88 0 .4 18 7 0 .2 43 0 0 .2 17 2 0 .1 26 0 0 .1 01 7 0 .3 25 9 -0 .1 98 2 -0 .2 11 9

2 1 2 1.95 2.95 6.74 0 .5 18 8 0 .4 18 7 0 .3 77 9 0 .2 17 2 0 .1 96 0 0 .1 58 2 0 .3 08 2 0 .5 77 8 1 .4 63 3

2 1 3 1.11 6.42 5.07 0 .5 18 8 0 .4 18 7 0 .3 79 1 0 .2 17 2 0 .1 96 7 0 .1 58 8 0 .1 76 5 1 .2 62 4 1 .1 02 1

2 2 1 5.37 5.71 -0.77 0 .5 18 8 0 .0 84 9 0 .2 43 0 0 .0 44 0 0 .1 26 0 0 .0 20 6 0 .1 10 8 0 .7 19 6 -0 .0 33 7

2 2 2 2.42 1.69 7.11 0 .5 18 8 0 .0 84 9 0 .3 77 9 0 .0 44 0 0 .1 96 0 0 .0 32 1 0 .0 77 5 0 .3 30 3 0 .3 12 9

2 2 3 5.28 2.52 -4.73 0 .5 18 8 0 .0 84 9 0 .3 79 1 0 .0 44 0 0 .1 96 7 0 .0 32 2 0 .1 69 9 0 .4 94 8 -0 .2 08 2

2 3 1 -4.05 -4.58 5.88 0 . 51 88 0 .4 96 4 0 .2 43 0 0 .2 57 5 0 .1 26 0 0 .1 20 6 - 0. 48 87 -0 .5 76 8 1 .5 15 3

2 3 2 3.81 -1.23 4.81 0 .5 18 8 0 .4 96 4 0 .3 77 9 0 .2 57 5 0 .1 96 0 0 .1 87 6 0 .7 15 1 -0 .2 40 5 1 .2 38 5

2 3 3 -1 .47 1 0.86 0 .2 4 0 .5 18 8 0 .4 96 4 0 .3 79 1 0 .2 57 5 0 .1 96 7 0 .1 88 2 - 0. 27 63 2 .1 36 6 0 .0 61 5

3 1 1 -0.41 -2.61 4.55 0 . 48 12 0 .4 18 7 0 .2 43 0 0 .2 01 5 0 .1 16 9 0 .1 01 7 - 0. 04 21 -0 .3 05 5 0 .9 16 3

3 1 2 2.66 1.72 0.73 0 .4 81 2 0 .4 18 7 0 .3 77 9 0 .2 01 5 0 .1 81 9 0 .1 58 2 0 .4 20 6 0 .3 12 9 0 .1 46 5

3 1 3 -3 .87 4 .09 11 .2 9 0 .4 81 2 0 .4 18 7 0 .3 79 1 0 .2 01 5 0 .1 82 5 0 .1 58 8 - 0. 61 42 0 .7 46 0 2 .2 75 8

3 2 1 2.15 4.83 14.20 0 .4 81 2 0 .0 84 9 0 .2 43 0 0 .0 40 9 0 .1 16 9 0 .0 20 6 0 .0 44 4 0 .5 64 6 0 .5 80 0

3 2 2 6.87 -1.90 7.47 0 .4 81 2 0 .0 84 9 0 .3 77 9 0 .0 40 9 0 .1 81 9 0 .0 32 1 0 .2 20 5 -0 .3 44 8 0 .3 05 4

3 2 3 2.95 2.18 0.01 0 .4 81 2 0 .0 84 9 0 .3 79 1 0 .0 40 9 0 .1 82 5 0 .0 32 2 0 .0 94 9 0 .3 97 2 0 .0 00 5

3 3 1 3.92 -4.13 7.44 0 .4 81 2 0 .4 96 4 0 .2 43 0 0 .2 38 9 0 .1 16 9 0 .1 20 6 0 .4 72 7 -0 .4 83 3 1 .7 76 2

3 3 2 -3.83 3.09 4.50 0 .4 81 2 0 .4 96 4 0 .3 77 9 0 .2 38 9 0 .1 81 9 0 .1 87 6 - 0. 71 75 0 .5 61 2 1 .0 75 4

3 3 3 3.64 3.44 5.49 0 .4 81 2 0 .4 96 4 0 .3 79 1 0 .2 38 9 0 .1 82 5 0 .1 88 2 0 .6 85 2 0 .6 28 4 1 .3 10 4

8/12/2019 e 5040313

24/35

E n tr opy 2003, 5, 313-347 336

Table 6. Minimum entropy (stone-paper-scissors):minp(1 + 2 + 3) * minp(H1 + H2 + H3)

Nash Utilities and Standard Deviations

Minp ((((1111++++2222++++3333)))) = 0

u1

u2

u3

0.5500 0.5500 0.5500

1 2 3

0.0000 0.0000 0.0000

H1 H2 H3

0.0353 0.0353 0.0353

Nash Equilibria: Probabilities, Utilities

Player 1 Player 2 Player 3

p1

1 p1

2 p1

3 p2

1 p2

2 p2

3 p3

1 p3

2 p3

3

0.3333 0.3333 0.3333 0.3333 0.3333 0.3333 0.3300 0.3300 0.3300

u1

1 u1

2 u1

3 u2

1 u2

2 u2

3 u3

1 u3

2 u3

3

0.5500 0.5500 0.5500 0.5500 0.5500 0.5500 0.5556 0.5556 0.5556

p1

1u1

1 p1

2u1

2 p1

3u1

3 p2

1u2

1 p2

2u2

2 p2

3u2

3 p3

1u3

1 p3

2u3

2 p3

3 u3

3

0.1833 0.1833 0.1833 0.1833 0.1833 0.1833 0.1833 0.1833 0.1833

Kroneker Products

j1 j2 j3 v1(j

1,j2,j3

)

v2(j

1,j2,j3

)

v3(j

1,j2,j3

)

p1j1 p2j2 p

3j3 p

1j1p

2j2 p

1j1p

3j3 p

2j2p

3j3 u

1(j1,j2,j3

)

u2(j

1,j2,j3

)

u3(j

1,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 scci sor 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 scci sor 1 1 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.1100 0.1100 0.1111

stone scc isor s tone 1 0 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.1100 0.0000 0.1111

stone scc isor paper 1 1 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.1100 0.1100 0.1111

stone scc isor scc isor 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 scci sor 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 scci sor 0 0 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.0000 0.0000 0.1111

paper scc isor s tone 1 1 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.1100 0.1100 0.1111

paper scc isor paper 0 1 0 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.0000 0.1100 0.0000

paper scc isor scc isor 0 1 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.0000 0.1100 0.1111

scc isor s tone s tone 0 1 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.0000 0.1100 0.1111

scc isor s tone paper 1 1 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.1100 0.1100 0.1111

scc isor s tone scc isor 0 1 0 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.0000 0.1100 0.0000scc isor paper s tone 1 1 1 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.1100 0.1100 0.1111

scc isor paper paper 1 0 0 0.3333 0.3333 0.3300 0.1111 0.1100 0.1100 0.1100 0.0000 0.0000

scc isor paper scc isor 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

8/12/2019 e 5040313

25/35

E n tr opy 2003, 5, 313-347 337

4.2 Time Series Applicati on

The relationship between Time Series and Game Theory appears when we apply the entropy minimizationtheorem (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 reflected and registered in historical data. The main contributionsof 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 optimalequilibrium. Due to the above-mentioned, we can affirm that if learning is optimal then the convergenceto equilibrium is faster than when the learning is sub-optimal. Introducing elements of Nashs equilibriumin time series will allow us to evaluate learning and the convergence to equilibrium through the study ofhistorical data (time series).

One of the branches of Physics called Quantum Mechanics was pioneered using Heisembergs 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 scientific development that attempts to establish the analogies

between Economics and Physics, see Mantenga and Stanley [33]. The establishment of analogies is acreative way of applying the idea of cooperative equilibrium. The product of this cooperative equilibriumwill produce synergies between these two sciences. From my point of view, the power of physics is thecapacity of equilibrium formal treatment in stochastic dynamic systems. On the other hand, the power ofEconomics is the formal study of rationality, cooperative and non-cooperative equilibrium.

Econophysics is the beginning of a unification stage of the systemic approach of scientific thought. I showthat it is the beginning of a unification stage but remains to create synergies with the rest of the sciences.

Let {xt}t=$t="$be a covariance-stationary process with the meanE[xt] =andjth covariancej

E[(xt ! )(xt"j ! )] = j (Q16)If these autocovariances are absolutely summable, the population spectrum ofx is given by [21].

Sx(w) = "

2

(0 + 2

$Xj=1

jcos(jw)

) (Q17)

If thejs represent autocovariances of a covariance-stationary process using the Possibility Theorem,then

Sx(w) " [Sx(w ! w), Sx(w+w)] (Q18)

and Sx(w)will be nonnegative for all w. In general for an ARMA(p, q) process:xt= c+1xt"1 +2xt"2+...+pxt"p+t+1t"1+2t"2+...+qt"q

E[t] =

2 for t=

0 otherwise

(Q19)

The population spectrumSx(w) " [Sx(w ! w), Sx(w+w)]is given byHamilton[17].

8/12/2019 e 5040313

26/35

E n tr opy 2003, 5, 313-347 338

Sx(w) = 2

2

(" +1e"iw +2e

"i2w +...+qe"iqw)

("! 1e"iw ! 2e"i2w ! ...! qe"iqw)

0(" +1e

iw +2ei2w +...+qe

iqw)

("! 1eiw ! 2ei2w ! ...! qeiqw) (Q20)wherei=

)!" andwis a scalar.Given an observed sample ofTobservations denotedx1, x2,.., xT,we can calculate up to T! " sample

autocovariancesjfrom the formulas: x= 1T

PTt=1xtand

bj = "TTX

t=j+1

(xt ! x) (xt"j ! x) for j = 0, ",..T! " (Q21)

The sample periodogram can be expressed as:

bSx(w) = "2

(b0 + 2 T"1Xj=1

bjcos(jw)) (Q22)When the sample size Tis an odd number, xt will be expressed in terms of periodic functions with

M= (T! ")/2representing different frequencieswj = (2T)j < .

xt=b+ MXj=1

nbajcos(wj(t! ")) +bbjsin(wj(t! "))o+ut (Q23)The coefficients

baj,

bbjcan be estimated with OLS regression.

baj = 2T

TXt=1

xtcos(wj(t! ")) j = ", 2,...,M

bbj = 2T

TXt=1

xtsin(wj(t! ")) j = ", 2,...,M (Q24)

The sample variance ofxtcan be expressed as:

"

T

T

Xt=1(xt ! x)2 = "

2

M

Xj=1(ba2j+bb2j) (Q25)The portion of the sample variance ofxtthat can be attributed to cycles of frequencywjis given by:

"

2(ba2j +bb2j) =4TbSx(wj) (Q26)

withbSx(wj)as the sample periodogram of frequencywj.

8/12/2019 e 5040313

27/35

E n tr opy 2003, 5, 313-347 339

Continuing with the methodology proposed by Hamilton[17]. we develop two examples that will allowus to verify the applicability of the Possibility Theorem.

Example 5 As a first step we build four time series knowing all their parameters aj, bj andwj = (2T

)j,j = ",...,M(Table 7, Figures 2,3,4,5), where

xt = 2 cos(w1(t ! "))! 3cos(w3(t! "))+2 sin(w5(t! ")) + 4 sin w1(t! "))

and

yt = 2 cos(w1(t! ")) + 4 sin(w12(t! "))zt = 4 sin(w12(t

!"))

ut = 3

In the second step, we find the value of the parametersbj, j = ", ..., (T! ")for the time series xtaccording to Q21 (Table 7, Figure 6).

In the third step, we findbSx(wj)(Table 8, Figure 7), only for the frequencies w1, w3, w5, w12:In the fourth step, we compute the valuesx, wand wmin=

12x

.resp. fory,z, u,(Table 9).

x= "

T

TXt=1

(xt ! x)2 , w = "4

4Xj=1

(wj ! w)2

In the fifth step, we verify that the Possibility Theorem is respected, xw

& 12resp fory,z, u,(Table

9).

Example 6 Let{vt }t=1,..,37be a random variable, andw12a random variable with gaussian probabilitydensityN(0, ").Both variables are related as continues

vt = sin ((E[w12] +t)(t! ")) +tw12 = E[w12] +t, E[w12] =

2

37"2

By simplicity of computing, we suppose that tand thas gaussian probability density N(0, ").It isevident thatw12= = " After a little computing we get the estimated value ofbv = "."334. The productbvw12= "."334verifies the Possibility Theorem and permits us to compute: wmin= 12bv = 0.44"2

w12 " [E[w12]! wmin, E[w12] +wmin]The spectral analysis can not give the theoretical value ofE[w12] = 2.037.The experimental value of

E[w12] = 2.8. You can see the results in Figures 8,9 and Table 10.

8/12/2019 e 5040313

28/35

E n tr opy 2003, 5, 313-347 340

Table 7. Time series values ofxt, yt, zt, ut

P a ra m e t e r s o f x t

j = t j j E [w j] j*co s ( E [ w1 ] * j ) j*co s ( E [ w 3 ] * j ) j*co s ( E [ w5 ]*j) j*co s ( E [ w12 ] * j ) x t y t z t u t co s ( w 1 ( t -1 ) ) c o s ( w3 ( t -1 ) ) s e n ( w5 ( t - 1 )) se n ( w12 ( t -1) )

1 3 ,5 5 7 0 ,2 1 0 0 ,1 70 3 ,50 5 3 ,1 05 2 ,3 50 -1 ,60 1 -1 ,0 0 0 2 ,0 0 0 0 ,0 0 0 3 ,0 0 0 1 ,0 0 0 1 ,00 0 0 ,0 00 0 ,0 00

2 -0 ,03 0 -0 ,0 0 2 0 ,3 40 -0 ,02 9 -0 ,0 16 0 ,0 04 0 ,01 8 4 ,4 2 5 5 ,5 4 3 3 ,5 7 2 3 ,0 00 0 ,9 8 6 0 ,87 3 0 ,7 51 0 ,8 933 8 ,5 7 1 0 ,5 0 5 0 ,5 09 7 ,48 2 0 ,3 64 -7 ,10 1 8 ,44 7 -0 ,9 1 9 -1 ,3 3 0 -3 ,2 1 6 3 ,0 00 0 ,9 4 3 0 ,52 4 0 ,9 92 -0 ,8 04

4 -5 ,07 6 -0 ,2 9 9 0 ,6 79 -3 ,95 0 2 ,2 85 4 ,9 13 1 ,48 5 2 ,0 6 2 1 ,0 6 9 -0 ,6 7 7 3 ,0 00 0 ,8 7 3 0 ,04 2 0 ,5 60 -0 ,1 69

5 -9 ,18 8 -0 ,5 4 2 0 ,8 49 -6 ,07 1 7 ,6 12 4 ,1 38 6 ,63 5 6 ,2 2 8 5 ,3 8 1 3 ,8 2 5 3 ,0 00 0 ,7 7 8 -0 ,45 0 -0 ,2 52 0 ,9 56

6 4,1 6 7 0 ,2 4 6 1 ,0 19 2 ,18 5 -4 ,1 52 1 ,5 53 3 ,92 8 -0 ,7 4 6 -1 ,4 4 6 -2 ,7 6 7 3,0 00 0 ,6 6 1 -0 ,82 8 -0 ,8 93 -0 ,6 92

7 -4 ,03 0 - 0,2 38 1 ,1 89 -1 ,50 3 3 ,6 73 -3 ,80 0 0 ,51 0 0 ,8 48 -0 ,2 85 -1 ,3 34 3 ,0 00 0 ,5 24 -0 ,99 6 -0 ,9 28 -0 ,3 33

8 -8 ,30 1 - 0,4 9 0 1 ,3 58 -1 ,74 9 4 ,9 37 -7 ,24 8 6 ,88 0 6 ,7 8 1 4 ,7 1 4 3 ,9 6 8 3 ,0 00 0 ,3 7 3 -0 ,91 1 -0 ,3 33 0 ,9 92

9 5 ,6 0 7 0 ,3 3 1 1 ,5 28 0 ,23 8 -0 ,7 13 1 ,1 83 4 ,89 4 0 ,9 4 2 -1 ,8 1 7 -2 ,2 3 8 3 ,0 00 0 ,2 1 1 -0 ,59 5 0 ,4 88 -0 ,5 60

1 0 -1 ,35 5 - 0,0 80 1 ,6 98 0 ,17 2 -0 ,5 05 0 ,8 05 -0 ,05 8 0 ,4 69 -1 ,8 68 -1 ,9 53 3,0 00 0 ,0 42 -0 ,12 7 0 ,9 78 -0 ,4 88

1 1 -7 ,60 1 -0 ,4 4 8 1 ,8 68 2 ,22 5 -5 ,9 13 7 ,5 74 6 ,92 9 4 ,2 3 2 3 ,7 4 2 3 ,9 9 6 3 ,0 00 -0 ,12 7 0 ,37 3 0 ,8 04 0 ,9 99

1 2 7,3 7 9 0 ,4 3 5 2 ,0 38 -3 ,32 2 7 ,2 72 -5 ,32 9 5 ,73 8 -4 ,3 9 4 -2 ,2 3 1 -1 ,6 4 5 3,0 00 -0 ,29 3 0 ,77 8 0 ,0 85 -0 ,4 11

1 3 3,5 4 1 0 ,2 0 9 2 ,2 08 -2 ,10 6 3 ,3 39 0 ,1 49 0 ,74 9 -7 ,7 5 6 -3 ,4 1 5 -2 ,5 1 5 3,0 00 -0 ,45 0 0 ,98 6 -0 ,6 92 -0 ,6 29

1 4 -4 ,73 1 - 0,2 7 9 2 ,3 77 3 ,41 5 -3 ,1 26 -3 ,68 0 4 ,57 9 -2 ,1 0 7 2 ,7 2 0 3 ,9 1 0 3 ,0 00 -0 ,59 5 0 ,94 3 -0 ,9 99 0 ,9 77

1 5 7,3 1 2 0 ,4 3 1 2 ,5 47 -6 ,05 8 1 ,5 42 7 ,2 07 4 ,82 6 -5 ,6 8 8 -2 ,4 4 8 -1 ,0 0 5 3,0 00 -0 ,72 2 0 ,66 1 -0 ,6 28 -0 ,2 51

1 6 0,9 4 8 0 ,0 5 6 2 ,7 17 -0 ,86 4 -0 ,2 77 0 ,4 97 0 ,35 4 -4 ,9 5 7 -4 ,6 6 2 -3 ,0 0 5 3,0 00 -0 ,82 8 0 ,21 1 0 ,1 69 -0 ,7 51

1 7 -12 ,98 1 -0 ,7 65 2 ,8 87 1 2,56 1 9 ,3 69 3 ,7 96 1 2,93 5 4 ,4 68 1 ,8 88 3 ,7 10 3 ,0 00 -0 ,91 1 -0 ,29 3 0 ,8 51 0 ,9 28

1 8 -3 ,06 0 - 0,1 80 3 ,0 57 3 ,04 9 2 ,9 61 2 ,7 88 -1 ,60 1 1 ,8 07 -2 ,2 71 -0 ,3 35 3,0 00 -0 ,96 8 -0 ,72 2 0 ,9 56 -0 ,0 84

1 9 -4 ,85 6 -0 ,2 8 6 - 4 ,83 9 4 ,7 00 4 ,4 26 -2 ,5 5 1 -1 ,6 7 4 -5 ,4 0 1 -3 ,4 0 8 3 ,0 00 -0 ,99 6 -0 ,96 8 0 ,4 12 -0 ,8 52

2 0 -14 ,51 6 -0 ,8 5 6 - 1 4 ,04 8 1 0 ,4 83 4 ,2 57 1 4 ,46 2 3 ,4 9 1 1 ,4 1 1 3 ,4 0 4 3 ,0 00 -0 ,99 6 -0 ,96 8 -0 ,4 11 0 ,8 51

2 1 0 ,7 5 9 0 ,0 4 5 - -0 ,69 2 -0 ,2 23 0 ,3 98 0 ,2 8 2 -1 ,3 3 7 -1 ,5 9 2 0 ,3 4 4 3 ,0 0 0 -0 ,96 8 -0 ,72 2 -0 ,9 56 0 ,0 86

2 2 5 ,3 9 4 0 ,3 1 8 - -4 ,46 9 1 ,1 35 5 ,3 16 3 ,5 6 9 -6 ,3 6 0 -5 ,5 3 6 -3 ,7 1 3 3 ,0 00 -0 ,91 1 -0 ,29 3 -0 ,8 52 -0 ,9 28

2 3 -4 ,60 0 - 0 ,2 7 1 - 3 ,32 1 -3 ,0 38 -3 ,58 0 4 ,4 5 0 0 ,3 7 2 1 ,3 4 2 2 ,9 9 9 3 ,0 0 0 -0 ,82 9 0 ,21 0 -0 ,1 70 0 ,7 502 4 6 ,9 8 6 0 ,4 1 2 - -4 ,15 5 6 ,5 86 0 ,3 01 1 ,4 6 2 -1 ,1 5 7 -0 ,4 3 1 1 ,0 1 3 3 ,0 0 0 -0 ,72 2 0 ,66 0 0 ,6 28 0 ,2 53

2 5 8 ,7 2 5 0 ,5 1 5 - -3 ,92 9 8 ,6 00 -6 ,29 5 6 ,7 9 7 -5 ,9 3 1 -5 ,1 0 1 -3 ,9 1 1 3,0 0 0 -0 ,59 5 0 ,94 3 0 ,9 99 -0 ,9 78

2 6 -5 ,61 1 -0 ,3 3 1 - 1 ,64 4 -4 ,3 67 5 ,5 91 5 ,1 0 9 0 ,0 3 5 1 ,6 0 8 2 ,5 0 8 3 ,0 0 0 -0 ,45 0 0 ,98 6 0 ,6 92 0 ,6 27

2 7 3 ,3 4 7 0 ,1 9 7 - -0 ,42 6 1 ,2 49 -1 ,99 2 0 ,1 3 7 -1 ,4 3 6 1 ,0 6 7 1 ,6 5 3 3 ,0 0 0 -0 ,29 3 0 ,77 8 -0 ,0 84 0 ,4 13

2 8 6 ,0 5 4 0 ,3 5 7 - 0 ,25 6 -0 ,7 66 1 ,2 71 5 ,2 9 0 -6 ,9 7 8 -4 ,2 5 1 -3 ,9 9 7 3,0 0 0 -0 ,12 7 0 ,37 3 -0 ,8 04 -0 ,9 99

2 9 -8 ,49 1 - 0 ,5 0 1 - -1 ,78 8 5 ,0 46 -7 ,41 0 7 ,0 2 6 0 ,4 5 4 2 ,0 3 0 1 ,9 4 5 3 ,0 0 0 0 ,0 4 2 -0 ,12 7 -0 ,9 78 0 ,4 86

3 0 0 ,7 5 3 0 ,0 4 4 - 0 ,28 1 -0 ,6 86 0 ,7 10 -0 ,09 7 3 ,4 7 3 2 ,6 6 7 2 ,2 4 6 3 ,0 0 0 0 ,2 1 1 -0 ,59 4 -0 ,4 88 0 ,5 61

3 1 4 ,3 5 4 0 ,2 5 7 - 2 ,28 2 -4 ,3 38 1 ,6 26 4 ,1 0 8 0 ,1 7 7 -3 ,2 2 1 -3 ,9 6 7 3 ,0 0 0 0 ,3 7 3 -0 ,91 1 0 ,3 32 -0 ,9 92

3 2 -10 ,87 5 -0 ,6 4 1 - -7 ,18 3 9 ,0 13 4 ,8 88 7 ,83 7 7 ,2 1 8 2 ,3 7 4 1 ,3 2 6 3 ,0 0 0 0 ,5 2 4 -0 ,99 6 0 ,9 28 0 ,3 31

3 3 -0 ,93 5 -0 ,0 5 5 - -0 ,72 8 0 ,4 22 0 ,9 05 0 ,2 7 6 8 ,3 6 7 4 ,0 9 4 2 ,7 7 3 3 ,0 0 0 0 ,6 6 1 -0 ,82 9 0 ,8 93 0 ,6 93

3 4 4 ,4 8 5 0 ,2 6 4 - 3 ,91 5 0 ,1 88 -3 ,71 8 4 ,4 2 2 -0 ,4 0 9 -2 ,2 6 6 -3 ,8 2 2 3,0 0 0 0 ,7 7 8 -0 ,45 1 0 ,2 53 -0 ,9 56

3 5 -9 ,85 9 - 0 ,5 8 1 - -9 ,29 5 -5 ,1 65 1 ,2 61 5 ,8 4 6 1 ,1 6 9 2 ,4 1 4 0 ,6 6 8 3 ,0 0 0 0 ,8 7 3 0 ,04 2 -0 ,5 59 0 ,1 67

3 6 0 ,0 0 0 - - - - - - 1 ,5 5 1 5 ,1 0 7 3 ,2 2 1 3 ,0 0 0 0 ,9 43 0 ,52 4 -0 ,9 9 2 0 ,8 05

3 7 - - - - - - - -5 ,7 1 7 -1 ,5 9 7 -3 ,5 6 8 3 ,0 0 0 0 ,9 86 0 ,87 3 -0 ,7 5 1 -0 ,8 92

Table 8. Frequencies, variances and sample periodogram ofxt

Analysis of xt

Variance: sx

E[(x t-E[x t])2] 16,958

(a12+a3

2+b 5

2+b1 16,500

0 16,958

T 37,000

F re qu en cie s W a v ele ng th Coe ff ic ie nts (aj + bj) /2 Sample Periodogram (4*pj/T)sx(E[wj])

E[w1] 0,170 l1 37,000a1 2,000a12

2,000 Sx(E[w1]) 4,961 1,685

E[w3] 0,509 l2 12,333a3 -3,000a32

4,500 Sx(E[w3]) 21,988 7,468

E[w5] 0,849 l3 7,400b5 2,000b52

2,000 Sx(E[w5]) 8,350 2,836

E[w12 ] 2,038 l4 3,083b12 4,000b12 8,000 Sx(E[w12 ]) 45,375 15,410

8/12/2019 e 5040313

29/35

E n tr opy 2003, 5, 313-347 341

Table 9. Verification of Possibility Theorem for seriesxt, yt, zt, ut

Heisenberg's Uncertainty Principle

xw= 3,348 >1/2 yw= 4,234 >1/2

E[xt] 0,000 x=E[(xt-E[xt])2] 4,118 E[yt] 0,000E[(yt-E[yt])

2] 3,206

E[w] 0,892 w=E[(w-E[w])2] 0,813 E[w] 1,104E[(w-E[w])

2] 1,321

wmin 0,121 wmin 0,156

Lower[w Min[j] Upper[wj] Max[j] Lower[wj Min[j] Upper[wj] 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

zw= 0,000

8/12/2019 e 5040313

30/35

E n tr opy 2003, 5, 313-347 342

y t V s t

-8 , 0 0 0 0

-6 , 0 0 0 0

-4 , 0 0 0 0

-2 , 0 0 0 0

0 ,0 0 0 0

2 ,0 0 0 0

4 ,0 0 0 0

6 ,0 0 0 0

8 ,0 0 0 0

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0

T im e

yt

Figure 2. yt= 2 cos(w1(t! ")) + 4 sin(w12(t! "))

z t V s t

- 5 ,0 0 0 0

- 4 ,0 0 0 0

- 3 ,0 0 0 0

- 2 ,0 0 0 0

- 1 ,0 0 0 0

0 ,0 0 0 0

1 ,0 0 0 0

2 ,0 0 0 0

3 ,0 0 0 0

4 ,0 0 0 0

5 ,0 0 0 0

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0

Tim e

zt

Figure 3. zt= 4 sin(w12(t! "))

u t V s t

0

0 , 5

1

1 , 5

2

2 , 5

3

3 , 5

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0T i m e

ut d d

Figure 4. ut= 3

T im e S e ri e x t

-1 0 , 0 0 0 0

-8 , 0 0 0 0

-6 , 0 0 0 0

-4 , 0 0 0 0

-2 , 0 0 0 0

0 ,0 0 0 0

2 ,0 0 0 0

4 ,0 0 0 0

6 ,0 0 0 0

8 ,0 0 0 0

1 0 ,0 0 0 0

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0

t

xt

x t c o s ( w 1 ( t- 1 ) ) c o s ( w 3 ( t- 1 ) ) s e n ( w 5 ( t -1 ) ) s e n ( w 1 2 ( t - 1 ) )

Figure 5. Constitutive elements of time seriesxt

8/12/2019 e 5040313

31/35

E n tr opy 2003, 5, 313-347 343

A u t o c o r r e l a t i o n s o f x t V s L a g ( j )

- 1 ,0 0 0 0

- 0 ,8 0 0 0

- 0 ,6 0 0 0

- 0 ,4 0 0 0

- 0 ,2 0 0 0

0 ,0 0 0 0

0 ,2 0 0 0

0 ,4 0 0 0

0 ,6 0 0 0

1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7

L a g ( j )

Autocorrelations

Figure 6. Autocorrelations of time seriesxt

Figure 7. Periodogram of time seriesxt

- 3

- 2

- 1

0

1

2

3

4

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0

T i m e v t

Figure 8. Time seriesvt = sin ((E[w12] +t)(t! ")) +t

Figure 9. Spectral analysis ofvt

8/12/2019 e 5040313

32/35

E n tr opy 2003, 5, 313-347 344

5 Conclusion

1 Hermites polynomials allow us to study the probability density function of the vNM utility inside ofan!player game. Using the approach of Quantum Mechanics we have obtained an equivalence betweenquantum level and strategy. The function of states of Quantum Mechanics opens a new focus in the theoryof games such as sub-strategies.

2 An immediate application of quantum games in economics is related to the principal-agent rela-tionship. Specifically we can usemtypes of agents for the case of adverse selection models. In moral riskmodels quantum games could be used for a discrete or continuous set of efforts.

3 In this paper we have demonstrated that the Nash-Hayek equilibrium opens new doors so thatentropy in game theory can be used. Remembering that the primary way to prove Nashs equilibria isthrough utility maximization, we can affirm 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

U(p(x))

H(p(x))relationship is equivalent to the well-known benefits/costwe present a new way to

calculate equilibria: Maxx U(p(x))H(p(x)) ,wherep(x)represents probability function and x = (x1, x2,...,xn)represents endogenous or exogenous variables.4 In all time series xt, where cycles are present, it is impossible to know the exact value of the

frequencywj of each one of the cycles j. We can know the value of frequency in a certain confidenceinterval given by wj" [wj ! wmin, wj+wmin]wherewmin = 12x . The more precisely the randomvariable VALUE xis determined, the less precisely the frequency VALUE wis known at this instant, andconversely

5 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 (Tables 7,8,9 and Figures 3,4,5,6,7).

6 This paper, which uses Kronecker product

$, represents an easy, new formalization of game

(K,$,u(p)),which extends the game ! to the mixed strategies.

ACKNOWLEDGMENTS

To my father and mother.

This study and analysis was made possible through the generousfinancial support of TODO1 SERVICESINC (www.todo1.com) and Petroecuador (www.petroecuador.com.ec).

The author is grateful to Gonzalo Pozo (Todo1 Services-Miami), the 8th Annual Conference in Com-

puting in Economics and Finance, July 2002, (the assistants), AIX France, the assistants to 10th AnnualConference of the SouthEast SAS R User Group USA,(the assistants). M2002, the 5th annual data miningtechnology conference SAS R 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 LumiereLyon II France).

8/12/2019 e 5040313

33/35

E n tr opy 2003, 5, 313-347 345

6 Bibliography

[1] Bacry, H. (1991).Introduction aux concepts de la Mecanique Statistique. Ellipsis, Pars,1991.

[2] Bertalanffy, L., (1976).Teora General de los Sistemas.Fondo de Cultura Econmico. USA, 1976.[3] Binmore, K. (1992).Fun and Games. D.C.Health and Company,1992

[4] Cohen-Tannoudji C., Diu B., Laloe F. (1977).Quantum Mechanics. Hermann, Pars, 1977.

[5] Eisert, J. Wilkens, M. Lewenstein, M. (1999) .Quantum Games and Quantum Strategies, Phys. Rev.Lett 83, 3077, 1999.

[6] Eisert J. and Wilkens M. (2000)Quantum Games. J. Mod. Opt. 47, 137903, 2000.

[7] Eisert, J., Scheel S., Plenio M.B. (2002). Distilling Gaussian states with Gaussian operations is im-possible. Phys. Rev. Lett 89, 137903, 2002.

[8] Eisert, J. Wilkens, M. Lewenstein, M. (2001) .Quantum Games and Quantum Strategies, Working

paper, University of Postdam, Germany, 2001.[9] Feynman, R. (1998).Le cours de Physique de Feynman. DUNOT, 1998.

[10] Fleming, W. Rishel, R. (1975). Deterministic and Stochastic Optimal Control. 1975.

[11] Frieden, B. (2001). Information Sciences, Probability, Statistical, Optics and Data Testing. Springer-Verlag.

[12] Fudenberg, D., Tirole, J. (1991).Game Theory, Massachusetts, The Mit Press, 1991.

[13] Gihman, I. Skorohod, A. (1979). Controlled Stochastic Processes. Springer Verlag, New York, 1979.

[14] Griffel, D.H. (1985). Applied Functional Analysis.John Wiley & Sons, New York, 1985.

[15] Griliches Z, Intriligator M. (1996).Handbook of Econometrics, Vol 1, Elsevier, 1996.[16] Hahn, F. (1982).Stability, Handbook of Mathematical Economics.Vol II, Chap 16, 1982.

[17] Hamilton J.D. (1994).Time Series Analysis, Princeton University Press, New Jersey, 1994, Chap(6).

[18] Jiangfeng, Du and all (2002).Experimental Realizations of Quantum Games on a Quantum Computer,Mimeo, University of Science and Technology of China, 2002.

[19] Jimnez, E. (1996). Aplicacin de Redes Neuronales Artificiales en Ssmica del Pozo,Petroecuador,Quito - Ecuador, 1996.

[20] Jimnez, E., Rullire, J.L (2002a). Unified Game Theory, Paper presented in 8th CONFERENCE INCOMPUTING IN ECONOMICS AND FINANCE AIX 2002.

[21] Jimnez, E., Rullire J.L (2002b).Unified Game Theory, I+D Innovacion, UIDT ISSN 1390-2202,Vol 5, 10:39-75. Petroecuador, Quito 2002.

[22] Jimnez, E. (2003a). Preemption and Attrition in Credit/Debit Cards Incentives: Models and Experi-ments, proceedings, 2003 IEEE International Conference on Computational Intelligence for FinantialEngineering, Hong Kong, 2003.

[23] Jimnez, E (2003b). Quantum Games and Minimum Entropy, Lecture Notes in Computer Science

8/12/2019 e 5040313

34/35

E n tr opy 2003, 5, 313-347 346

2669, p 216-225. Springer, Canada 2003.

[24] Jimnez, E (2003d). Factor-K: La Economia Experimental como Paradigma de Transparencia, I+DInnovacion, UIDT ISSN 1390-2202, Vol 6, 12:1-23. Petroecuador, Quito 2003.

[25] Kay, Lai Chung. (1978)A Course in Probability Theory, Harcourt, Brace & World, INC.

[26] Kamien, M., Schwartz, N. (1998). Dynamic Optimization., Elsevier, Amsterdam, 1998.

[27] Karatzas, I., Shreve, S. (1991).Brownian Motion and Stochastic Calculus., Second Edition, Springer-Verlag, 1991.

[28] Karlin S., Taylor H. (1980).A First Course in Stochastic Processes., Academic Press Inc, Usa, 1980.

[29] Karlin S., Taylor H. (1981).A Second Course in Stochastic Processes., Academic Press Inc, Usa, 1981.

[30] Kloeden, P. Platen, E. (1995). Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1995.

[31] Landau. L (1978).Mecanica Cuntica, Editorial Revert, 1978.

[32] Lvy, E. (1998). Dictionnaire de Physique., PUF, 1998.[33] Mantenga, R. Stanley, E (2001).An Introduction to Econophysics,Cambridge University Press, 2001.

[34] Malliaris, A. Brock, W. (1982). Stochastic Methods in Economics in Finance., Elsevier, Netherlands,1982.

[35] McFadden, D. (1976). Quantal Choice Analysis : A survey , Annals of Economicals and Social Mea-surement 5/4., p.363-389, 1976.

[36] McFadden, D. (1984).Econometric Analysis of Qualitative Response Models., Handbook of Econo-metrics Vol II, Chap 24, 1984.

[37] McFadden, D. (2001).Economic Choices., AER, Vol 91 NO.3, 351-378, 2001.

[38] McKelvey, R., Palfrey, T. (1995).Quantal Response Equilibria for Normal Form Games, Games andeconomics Behavior., 10:6-38, 1995.

[39] McKelvey, R., Palfrey, T. (1996). Quantal Response Equilibria., California Institute of Technology,current version 1996.

[40] McKelvey, R., Palfrey, T. (1998). Quantal Response Equilibria for Extensive Form Games. Experi-mental Economics, 1,9-41, 1998.

[41] Meyer, D. (1999). Quantum Strategies. Phys. Rev. Lett. 82 (1999) 1052-1055, 1999.

[42] Meyer, D. (2000). Quantum Games and Quantum Algorithms, To appear in the AMS ContemporanyMathematics volume: Quantum Computation and Quantum Information Science, 2000.

[43] Myerson, R. (1991). Game Theory Analysis of Conflict., Massachusetts, Harvard University Press,1991.

[44] Neyman, A. (2000). Repeated Games with Bounded Entropy., Games and Economic Behavior, Ideal,30, 228-247, 2000.

[45] NG, C. NG, H. (1995). Introduction la Mecanique Statistique., Masson, 2me dition, 1995.

8/12/2019 e 5040313

35/35

E n tr opy 2003, 5, 313-347 347

[46] Pena, L (1979).Introduccin a la Mecnica Cuntica, Compania Editorial Continental, Mxico, 1979.

[47] Rasmusen, E. (1989). Games and Economic Information: An Introduction to Game Theory. , Oxfort:Basil Blackwell, 1989.

[48] Rnyi, A. (1966).Calcul des Probabilits., Paris, Dunod, 1966.

[49] Robert, Ch. (1992).LAnalyse statistique Bayesienne., Economica, 1992.

[50] Rubinstein, A. (1989).The Electronic Mail Game: Strategic Behavior Under, Almost Common Knowl-edge., American Economic Review, 79:385-391, 1989.

[51] Rullire, J. L., Walliser, B. (1995). De la spcularit la temporalit en thorie des jeux., RevuedEconomie Politique, 105, pp 601-615, 1995.

[52] Rust, J. (1994). Structural Estimation of Markov Decision Processes., Handbook of Econometrics,Volume IV, Chapter 51, 1994.

[53] SENKO Corporation (1999).Quantum Computer Simulation, SENKO Corporation, Yokohama Japan,1999.

[54] Suijs, J., Borm.P. (1999). Stochastic Cooperative Games: Superadditivity Convexity and CertaintyEquivalents, Games and Economic Behavior., Ideal, 27, 331-345, 1999.

[55] Tirole, J. (1988).Thorie de Lorganisation Industrielle., Tome II, Economica, 1988.

[56] Varian, H. (1995). Analyse Microconomique., Troisime dition, De Boeck Universit, Bruxelles,1995.

[57] Weibul, J.W. (1995).Evolutionary Game Theory., The MIT press, 1995.

c 2003 by MDPI (http://www.mdpi.org). Reproduction for noncommercial purposes permitted.