Quantum Econophysics

Esteban Guevara Hidalgo Departamento de Física, Escuela Politécnica Nacional, Quito, Ecuador

Abstract The von Neumann Equation & the Statistical The relationships between game theory and quantum me- Mixture of States chanics let us propose certain quantization relationships An ensemble is a collection of identically prepared physi- through which we could describe and understand not only cal systems. When each member of the ensemble is char- quantum but also classical, evolutionary and the biological acterized by the same state vector |Ψ(t) it is called pure systems that were described before through the replicator dy- ensemble. If each member has a probability pi of being in namics. Quantum mechanics could be used to explain more the state |Ψi(t) we have a mixed ensemble. Each member correctly biological and economical processes and even it could encloses theories like games and evolutionary dynam- of a mixed ensemble is a pure state and its evolution is given ics. This could make quantum mechanics a more general the- by Schrödinger equation. To describe correctly a statistical ory that we had thought. mixture of states it is necessary the introduction of the den- Although both systems analyzed are described through two sity operator apparently different theories (quantum mechanics and game n theory) it is shown that both systems are analogous and thus ρ(t)= pi |Ψi(t)Ψi(t)| (1) exactly equivalents. So, we can take some concepts and defi- i=1 nitions from quantum mechanics and physics for the best un- derstanding of the behavior of economics and biology. Also, which contains all the physically significant information we we could maybe understand nature like a game in where its can obtain about the ensemble in question. Any two ensem- players compete for a common welfare and the equilibrium bles that produce the same density operator are physically of the system that they are members. indistinguishable. The density operator can be represented in matrix form. p | t ,i Introduction A pure state is specified by i =1for some Ψi( ) = 1, ..., n and the matrix which represents it has all its ele- Could it have an relationship between quantum mechan- ments equal to zero except one 1 on the diagonal. The di- ics and game theories? An actual relationship between agonal elements ρnn of the density operator ρ(t) represents these theories that describe two apparently different systems the average probability of finding the system in the state |n would let us explain biological and economical processes and its sum is equal to 1. The non-diagonal elements ρnp through quantum mechanics, quantum information theory expresses the interference effects between the states |n and and statistical physics. |p which can appear when the state |Ψi is a coherent linear We also could try to find a method which let us make superposition of these states. quantum a classical system in order to analyze it from a Suppose we make a measurement on a mixed ensemble absolutely different perspective and under a physical equi- of some observable A. The ensemble average of A is de- librium principle which would have to be exactly equivalent fined by the average of the expected values measured in to the defined classically in economics or biology. each member of the ensemble described by |Ψi(t) and with Physics tries to describe approximately nature which is probability pi,itmeansAρ = p1 A1 + p2 A2 + ... + the most perfect system. The equilibrium notion in a physi- p A cal system is the central cause for this perfection. We could n n and can be calculated by using make use of this physical equilibrium to its application in A = Tr{ρ(t)A} .(2) conflictive systems like economics. The present work analyze the relationships between quan- The time evolution of the density operator is given by the von Neumann equation tum mechanics and game theory and proposes through cer- tain quantization relationships a quantum understanding of dρ i = H,ρˆ classical systems. dt (3) Copyright c 2007, American Association for Artificial Intelli- which is only a generalization of the Schrödinger equation gence (www.aaai.org). All rights reserved. and the quantum analogue of Liouville’s theorem. The Replicator Dynamics & EGT The bonestone of EGT is the concept of evolutionary sta- Game theory (von Neumann & Morgenstern 1947; Myer- ble strategy (ESS) (Smith 1982; Smith & Price 1973) that son 1991; Nowak & Sigmund 1999) is the study of decision is a strengthened notion of Nash equilibrium. It satisfies the making of competing agents in some conflict situation. It following conditions has been applied to solve many problems in economics, so- E(p, p) >E(r, p), cial sciences, biology and engineering. The central equilib- E p, p E r, p E p, r >E r, r rium concept in game theory is the Nash Equilibrium which If ( )= ( ) then ( ) ( ),(7) is expressed through the following condition where p is the strategy played by the vast majority of the E(p, p) ≥ E(r, p).(4)population, and r is the strategy of a mutant present in small p r Players are in equilibrium if a change in strategies by any frequency. Both and can be pure or mixed. An ESS one of them (p → r) would lead that player to earn less than is described as a strategy which has the property that if all if he remained with his current strategy (p). the members of a population adopt it, no mutant strategy Evolutionary game theory (Smith 1982; Hofbauer & Sig- could invade the population under the influence of natural mund 1998; Weibul 1995) has been applied to the solution selection. If a few individuals which play a different strategy of games from a different perspective. Through the replica- are introduced into a population in an ESS, the evolutionary tor dynamics it is possible to solve not only evolutionary but selection process would eventually eliminate the invaders. also classical games. That is why EGT has been considered like a generalization of classical game theory. Evolutionary Relationships between Quantum Mechanics & game theory does not rely on rational assumptions but on the Game Theory idea that the Darwinian process of natural selection (Fisher A physical or a socioeconomical system (described through 1930) drives organisms towards the optimization of repro- quantum mechanics or game theory) is composed by n ductive success (Hammerstein & Selten 1994). Instead of members (particles, subsystems, players, states, etc.). Each working out the optimal strategy, the different phenotypes member is described by a state or a strategy which has as- in a population are associated with the basic strategies that signed a determined probability (xi or ρij ). The quantum are shaped by trial and error by a process of natural selection mechanical system is described by the density operator ρ or learning. whose elements represent the system average probability of The model used in EGT is the following: Each agent in th being in a determined state. In evolutionary game theory a n-player game where the i player has as strategy space the system is defined through a relative frequencies vector x Si is modelled by a population of players which have to whose elements can represent the frequency of players play- be partitioned into groups. Individuals in the same group ing a determined strategy. The evolution of the density op- would all play the same strategy. Randomly we make play erator is described by the von Neumann equation which is a the members of the subpopulations against each other. The generalization of the Schrödinger equation. While the evo- subpopulations that perform the best will grow and those lution of the relative frequencies is described through the that do not will shrink and eventually will vanish. The replicator dynamics (5). process of natural selection assures survival of the best It is important to note that the replicator dynamics is a players at the expense of the others. The natural selec- vectorial differential equation while von Neumann equation tion process that determines how populations playing spe- can be represented in matrix form. If we would like to com- cific strategies evolve is known as the replicator dynamics pare both systems the first we would have to do is to try to (Hofbauer & Sigmund 1998; Weibul 1995; Cressman 1992; compare their evolution equations by trying to find a matrix Taylor & Jonker 1978) representation of the replicator dynamics (Guevara 2006c) dxi =[fi(x) −f(x)] xi dX T dt ,(5) = G + G ⎡ ⎤ dt ,(8) n n dxi ⎣ ⎦ where the matrix X has as elements = aij xj − aklxkxl xi.(6) dt 1/2 j=1 k,l=1 xij =(xixj) (9) The element xi of the vector x is the probability of play- and ing certain strategy or the relative frequency of individuals n n using that strategy. The fitness function fi = aij xj T 1 j=1 G + G = aikxkxij f x ij specifies how successful each subpopulation is, ( ) = 2 k=1 n a x x k,l=1 kl k l is the average fitness of the population, and n a A 1 ij are the elements of the payoff matrix . The replicator + ajkxkxji dynamics rewards strategies that outperform the average by 2 k=1 increasing their frequency, and penalizes poorly performing n strategies by decreasing their frequency. The stable fixed − aklxkxlxij (10) points of the replicator dynamics are Nash equilibria (Myer- k,l=1 son 1991). If a population reaches a state which is a Nash equilibrium, it will remain there. are the elements of the matrix G + GT . Although equation (8) is the matrix representation of the Density Operator Relative freq. Matrix replicator dynamics from which we could compare and find ρ is Hermitian X is Hermitian a relationship with the von Neumann equation, we can more- Trρ(t)=1 TrX =1 2 2 over find a Lax representation of the replicator dynamics by ρ (t)  ρ(t) X = X calling Trρ2(t)  1 TrX2(t)=1 n 1 (G1)ij = aikxkxij , (11) Although both systems are different, both are analogous 2 k=1 and thus exactly equivalents. n G 1 a x x ( 2)ij = jk k ji, (12) Quantum Replicator Dynamics & the 2 k=1 n Quantization Relationships (G3)ij = aklxkxlxij (13) The resemblances between both systems and the similarity k,l=1 in the properties of their corresponding elements let us to define and propose the next quantization relationships the elements of the matrixes G1, G2 and G3 that compose T n by adding the matrix G + G . The matrixes G1,G2 and xi → i |Ψk pk Ψk |i = ρ , G3 can be also factorized in function of the matrixes Q and ii X k=1 n 1/2 G1 = QX, (14) (xixj ) → i |Ψk pk Ψk |j = ρij . (19) k=1 G2 = XQ, (15) G3 =2XQX, (16) A population will be represented by a quantum system in which each subpopulation playing strategy si will be rep- Q q where is a diagonal matrix and has as elements ii = resented by a pure ensemble in the state |Ψk(t) and with 1 n a x X2 X 2 k=1 ik k. By using the fact that = we can write probability pk. The probability xi of playing strategy si or the equation (8) like the relative frequency of the individuals using strategy si in dX that population will be represented as the probability ρii of = QXX + XXQ− 2XQX |i dt (17) finding each pure ensemble in the state (Guevara 2006c). Through these quantization relationships the replicator and finally, by grouping into commutators and defining Λ= dynamics (in matrix commutative form) (18) takes the form [Q, X] of the equation of evolution of mixed states (3). And also dX =[Λ,X] X −→ ρ dt . (18) , (20) i ˆ The matrix Λ hasaselements (Λ)ij = Λ −→ − H, (21) 1 n n  2 [( k=1 aikxk) xij − xji ( k=1 ajkxk)].Thisma- trix commutative form of the replicator dynamics (18) where Hˆ is the Hamiltonian of the physical system. follows the same dynamic as the von Neumann equation The equation of evolution of mixed states from quantum (3) and the properties of their correspondent elements statistical mechanics (3) is the quantum analogue of the (matrixes) are similar, being the properties corresponding to replicator dynamics in matrix commutative form (18). our quantum system more general than the properties of the classical system. Games through Statistical Mechanics & QIT The next table shows some specific resemblances between quantum statistical mechanics and evolutionary game theory There exists a strong relationship between game theories, (Guevara 2006d). statistical mechanics and information theory. The bonds be- Table 1 tween these theories are the density operator and entropy (Guevara 2006a; 2006b). From the density operator we can Quantum Statistical Mechanics Evolutionary Game Theory construct and understand the statistical behavior about our n system members n population members system by using the statistical mechanics. Also we can de- | s Each member in the state Ψk Each member plays strategy i velop the system in function of its accessible information | p → ρ s → x Ψk with k ij i i and analyze it through information theories under a criterion ρ, ρ X, x i ii =1 i i =1 of maximum or minimum entropy. i dρ H,ρˆ dX ,X Entropy is the central concept of information theories dt = dt =[Λ ] S = −Tr{ρ ln ρ} H = − xi ln xi (Guevara 2006a). The Shannon entropy expresses the av- i erage information we expect to gain on performing a prob- abilistic experiment of a random variable A which takes the In table 2 we show the properties of the matrixes ρ and X. values ai with the respective probabilities pi.Italsocanbe Table 2 seen as a measure of uncertainty before we learn the value P P p p A i ij j ij of . We define the Shannon entropy of a random variable where pi:j = p is the mutual probability. The A by ij n mutual or correlation entropy also can be expressed through the conditional entropy via H(A) ≡ H(p1, ..., pn) ≡− pi log2 pi. (22) i=1 H(A : B)=H(A) − H(A | B), (28) The entropy of a random variable is completely determined H(A : B)=H(B) − H(B | A). (29) by the probabilities of the different possible values that the A random variable takes. Due to the fact thatp =(p1, ..., pn) The joint entropy would equal the sum of each of ’s and n B is a probability distribution, it must satisfy pi =1and ’s entropies only in the case that there are no correlations i=1 A B 0 ≤ p1, ..., pn ≤ 1. The Shannon entropy of the probabil- between ’s and ’s states. In that case, the mutual entropy ity distribution associated with the source gives the minimal or information vanishes and we could not make any predic- number of bits that are needed in order to store the infor- tions about A just from knowing something about B. mation produced by a source, in the sense that the produced The relative entropy H(p  q) measures the closeness of string can later be recovered. two probability distributions, p and q, defined over the same The von Neumann entropy (von Neumann 1927; 1932) random variable A. We define the relative entropy of p with is the quantum analogue of Shannon’s entropy but it ap- respect to q by peared 21 years before and generalizes Boltzmann’s expres- H p  q ≡ p p − p q sion. Entropy in quantum information theory plays promi- ( ) i log2 i i log2 i, nent roles in many contexts, e.g., in studies of the classi- i i cal capacity of a quantum channel (Schumacher & West- H(p  q) ≡−H(A) − pi log2 qi. (30) moreland 1997; Holevo 1998) and the compressibility of a i quantum source (Schumacher 1995; Jozsa & Schumacher The relative entropy is non-negative, H(p  q) ≥ 0, with 1994). Quantum information theory appears to be the basis p q for a proper understanding of the emerging fields of quan- equality if and only if = . The classical relative entropy tum computation (Bennett & DiVincenzo 1995; DiVincenzo of two probability distributions is related to the probability 1995), quantum communication (Bennett & Wiesner 1992; of distinguishing the two distributions after a large but finite C. Bennett 1993), and quantum cryptography (Ekert 1992; number of independent samples (Sanov’s theorem) (Cover C. Bennett & Mermin 1992). & Thomas 1991). Suppose A and B are two random variables. The joint By analogy with the Shannon entropies it is possible to entropy H(A, B) measures our total uncertainty about the define conditional, mutual and relative quantum entropies. pair (A, B) and it is defined by Quantum entropies also satisfies many other interesting properties that do not satisfy their classical analogues. For H(A, B) ≡− pij log2 pij (23) example, the conditional entropy can be negative and its neg- i,j ativity always indicates that two systems are entangled and while indeed, how negative the conditional entropy is provides a H(A)=− pij log2 pij , (24) lower bound on how entangled the two systems are (Nielsen i,j j & Chuang 2000). By other hand, in statistical mechanics entropy can be H(B)=− pij log2 pij , (25) regarded as a quantitative measure of disorder. It takes its i,j i maximum possible value in a completely random ensemble where pij is the joint probability to find A in state ai and B in which all quantum mechanical states are equally likely in state bj. and is equal to zero in the case of a pure ensemble which The conditional entropy H(A | B) is a measure of how has a maximum amount of order because all members are uncertain we are about the value of A, given that we know characterized by the same quantum mechanical state ket. the value of B. The entropy of A conditional on knowing From both possible points of view and analysis (statisti- that B takes the value bj is defined by cal mechanics or information theories) of the same system H(A | B) ≡ H(A, B) − H(B), its entropy is exactly the same. Lets consider a system com- posed by N members, players, strategies, states, etc. This H(A | B) ≡− pij log2 pi|j, (26) system is described completely through certain density op- i,j erator ρ, its evolution equation (the von Neumann equation) Ppij where pi|j = p is the conditional probability that A is and its entropy. Classically, the system is described through i ij the matrix of relative frequencies X, the replicator dynamics in state ai given that B is in state bj. The mutual or correlation entropy H(A : B) measures and the Shannon entropy. For the quantum case we define how much information A and B have in common. The mu- the von Neumann entropy as tual or correlation entropy H(A : B) is defined by S = −Tr{ρ ln ρ} (31) H(A : B) ≡ H(A)+H(B) − H(A, B), and for the classical case H(A : B) ≡− pij log2 pi:j, (27) H = − xii ln xii i,j (32) i=1 which is the Shannon entropy over the relative frequencies and with respect to the parameter β x X vector (the diagonal elements of ). ∂S In general entropy can be maximized subject to different = −β ΔE2 , (43) constrains. In each case the result is the condition the system ∂β must follow to maximize its entropy. Generally, this condi- ∂2S ∂ E ∂2 E = + β . (44) tion is a probability distribution function. We can obtain the ∂β2 ∂β ∂β2 density operator from the study of an ensemble in thermal equilibrium. Nature tends to maximize entropy subject to From Classical to Quantum the constraint that the ensemble average of the Hamiltonian has a certain prescribed value. We will maximize S by re- The resemblances between both systems (described through quiring that quantum mechanics and EGT) apparently different but anal- ogous and thus exactly equivalents and the similarity in the δS − δρ ρ = ii(ln ii +1)=0 (33) properties of their corresponding elements let us to define i and propose the quantization relationships like in section 5. subject to the constrains δTr (ρ)=0and δ E =0.By It is important to note that equation (18) is nonlinear while using Lagrange multipliers its quantum analogue is linear. This means that the quantiza- tion eliminates the nonlinearities. Also through this quanti- δρ ρ βE γ ii(ln ii + i + +1)=0 (34) zation the classical system that were described through a di- i agonal matrix X can be now described through a density op- and the normalization condition Tr(ρ)=1we find that erator which not neccesarily must describe a pure state, i.e. e−βEi its non diagonal elements can be different from zero repre- ρii = (35) senting a mixed state due to the coherence between quantum e−βEk k states that were not present through a classical analysis. which is the condition that the density operator and its ele- Through the relationships between both systems we could ments must satisfy to our system tends to maximize its en- describe classical, evolutionary, quantum and also the bio- tropy S. If we maximize S without the internal energy con- logical systems that were described before through evolu- strain δ E =0we obtain tionary dynamics with the replicator dynamics. We could 1 explain through quantum mechanics biological and econom- ρ ii = N (36) ical processes being a much more general theory that we had thought. It could even encloses theories like games and evo- β → which is the 0 limit (“high - temperature limit”) in lutionary dynamics. equation (35) in where a canonical ensemble becomes a Problems in economy and finance have attracted the in- completely random ensemble in which all energy eigenstates terest of statistical physicists. Kobelev et al (L. Kobelev are equally populated. In the opposite low - temperature β →∞ 2006) used methods of statistical physics of open systems limit tell us that a canonical ensemble becomes for describing the time dependence of economic character- a pure ensemble where only the ground state is populated. β τ istics (income, profit, cost, supply, currency, etc.) and their The parameter is related to the “temperature” as follows correlations with each other. 1 Antoniou et al (I. Antoniou 2002) introduced a new ap- β = . (37) τ proach for the presentation of economic systems with a By replacing ρii obtained in the equation (35) in the von small number of components as a statistical system de- Neumann entropy we can rewrite it in function of the parti- scribed by density functions and entropy. This analysis is −βEk tion function Z = k e , β and E through the next based on a Lorenz diagram and its interpolation by a con- equation tinuos function. Conservation of entropy in time may in- S =lnZ + β E . (38) dicate the absence of macroscopic changes in redistribution From the partition function we can know some parameters of resources. Assuming the absence of macro-changes in that define the system like economic systems and in related additional expenses of re- sources, we may consider the entropy as an indicator of ef- 1 ∂Z ∂ ln Z E = − = − , (39) ficiency of the resources distribution. Z ∂β ∂β Statistical physicists are also extremely interested in eco- ∂ E 1 ∂S nomic fluctuations (Stanley 2000) in order to help our world ΔE2 = − = − ∂β β ∂β . (40) financial system avoid “economic earthquakes”. Also it is suggested that in the field of turbulence, we may find some We can also analyze the variation of entropy with respect to crossover with certain aspects of financial markets. the average energy of the system Statistical mechanics and economics study big ensem- ∂S 1 bles: collections of atoms or economic agents, respectively. = , (41) ∂ E τ The fundamental law of equilibrium statistical mechanics ∂2S 1 ∂τ is the Boltzmann-Gibbs law, which states that the probabil- = − E P (E)=Ce−E/T T 2 τ 2 ∂ E (42) ity distribution of energy is ,where ∂ E is the temperature, and C is a normalizing constant. The main ingredient that is essential for the derivation of the tions. An abstract DNA-type system is defined by a set of Boltzmann-Gibbs law is the conservation of energy. Thus, nonlinear kinetic equations with polynomial nonlinearities one may generalize that any conserved quantity in a big sta- that admit soliton solutions associated with helical geome- tistical system should have an exponential probability dis- try. Aerts and Czachor (Aerts & Czachor ) shown that the tribution in equilibrium (Dragulescu˘ & Yakovenko 2000). set of these equations allows for two different Lax represen- In a closed economic system, money is conserved. Thus, by tations: They can be written as von Neumann type nonlinear analogy with energy, the equilibrium probability distribution systems and they can be regarded as a compatibility con- of money must follow the exponential Boltzmann-Gibbs law dition for a Darboux-covariant Lax pair. Organisms whose characterized by an effective temperature equal to the aver- DNA evolves in a chaotic way would be eliminated by natu- age amount of money per economic agent. Dragulescu˘ and ral selection. They also explained why non-Kolmogorovian Yakovenko demonstrated how the Boltzmann-Gibbs distri- probability models occurring in soliton kinetics are natu- bution emerges in computer simulations of economic mod- rally associated with chemical reactions. Patel (Patel 2001a; els. They considered a thermal machine, in which the differ- 2001b) suggested quantum dynamics played a role in the ence of temperature allows one to extract a monetary profit. DNA replication and the optimization criteria involved in They also discussed the role of debt, and models with bro- genetic information processing. He considers the criteria ken time-reversal symmetry for which the Boltzmann-Gibbs involved as a task similar to an unsorted assembly oper- law does not hold. ation where the Grover’s database search algorithm fruit- Recently the insurance market, which is one of the im- fully applies; given the different optimal solutions for clas- portant branches of economy, have attracted the attention sical and quantum dynamics. Turner and Chao (Turner & of physicists (Darooneh 2006). The maximum entropy Chao 1999) studied the evolution of competitive interactions principle is used for pricing the insurance. Darooneh ob- among viruses in an RNA phage, and found that the fitness tained the price density based on this principle, applied it of the phage generates a payoff matrix conforming to the to multi agents model of insurance market and derived the two-person prisoner’s dilemma game. Bacterial infections utility function. The main assumption in his work is the by viruses have been presented as classical game-like situa- correspondence between the concept of the equilibrium in tions where nature prefers the dominant strategies. Azhar physics and economics. He proved that economic equi- Iqbal (Iqbal 2004) showed results in which quantum me- librium can be viewed as an asymptotic approximation to chanics has strong and important roles in selection of stable physical equilibrium and some difficulties with mechanical solutions in a system of interacting entities. These entities picture of the equilibrium may be improved by consider- can do quantum actions on quantum states. It may simply ing the statistical description of it. TopsØe (TopsØe 1979; consists of a collection of molecules and the stability of so- 1993) also has suggested that thermodynamical equilibrium lutions or equilibria can be affected by quantum interactions equals game theoretical equilibrium. which provides a new approach towards theories of rise of Quantum games have proposed a new point of view for complexity in groups of quantum interacting entities. the solution of the classical problems and dilemmas in game Neuroeconomics (Glimcher 2002; 2003) may provide an theory. Quantum games are more efficient than classical alternative to the classical Cartesian model of the brain and games and provide a saturated upper bound for this effi- behavior (Piotrowski & Sladkowski a) through a rich dia- ciency (Meyer 1999; J. Eisert & Lewenstein 1999; Marinatto logue between theoretical neurobiology and quantum logic & Weber 2000; Flitney & Abbott 2003; Piotrowski & Slad- (Piotrowski & Sladkowski b; Collum 2002). kowski 2003; Iqbal 2004). The results shown in this study on the relationships be- Nature may be playing quantum survival games at the tween quantum mechanics and game theories are a reason of molecular level (Dawkins 1976; Axelrod 1984). It could the applicability of physics in economics and biology. Both lead us to describe many of the life processes through quan- systems described through two apparently different theories tum mechanics like Gogonea and Merz (Gogonea & Merz are analogous and thus exactly equivalents. So, we can take 1999) who indicated that games are being played at the some concepts and definitions from quantum mechanics and quantum mechanical level in protein folding. Gafiychuk and physics for the best understanding of the behavior of eco- Prykarpatsky (Gafiychuk & Prykarpatsky 2004) applied the nomics and biology. Also, we could maybe understand na- replicator equations written in the form of nonlinear von ture like a game in where its players compete for a common Neumann equations to the study of the general properties welfare and the equilibrium of the system that they are mem- of the quasispecies dynamical system from the standpoint of bers. its evolution and stability. They developed a mathematical model of a naturally fitted coevolving ecosystem and a the- On a Quantum Understanding of Classical oretical study a self-organization problem of an ensemble of interacting species. Systems The genetic code is the relationship between the sequence If our systems are analogous and thus exactly equivalents, of the bases in the DNA and the sequence of amino acids our physical equilibrium (maximum entropy) should be also in proteins. Recent work (R. Knight & Landweber 2001) exactly equivalent to our socieconomical equilibrium. If in about evolvability of the genetic code suggests that the code an isolated system each of its accessible states do not have is shaped by natural selection. DNA is a nonlinear dynami- the same probability, the system is not in equilibrium. The cal system and its evolution is a sequence of chemical reac- system will vary and will evolution in time until it reaches the equilibrium state in where the probability of finding the The quantum analogues of the relative frequencies ma- system in each of the accessible states is the same. The sys- trix, the replicator dynamics and the Shannon entropy are tem will find its more probable configuration in which the the density operator, the von Neumann equation and the von number of accessible states is maximum and equally proba- Neumann entropy. Every game (classical, evolutionary or ble. The whole system will vary and rearrange its state and quantum) can be described quantically through these three the states of its ensembles with the purpose of maximize its elements. entropy and reach its maximum entropy state. We could say The bonds between game theories, statistical mechanics that the purpose and maximum payoff of a physical system and information theory are the density operator and entropy. is its maximum entropy state. The system and its members From the density operator we can construct and obtain all the will vary and rearrange themselves to reach the best possible mechanical statistical information about our system. Also state for each of them which is also the best possible state for we can develop the system in function of its information and the whole system. analyze it through information theories under a criterion of This can be seen like a microscopical cooperation be- maximum or minimum entropy. tween quantum objects to improve their states with the pur- Although both systems analyzed are described through pose of reaching or maintaining the equilibrium of the sys- two apparently different theories (quantum mechanics and tem. All the members of our quantum system will play a game theory) both are analogous and thus exactly equiva- game in which its maximum payoff is the equilibrium of lents. So, we can take some concepts and definitions from the system. The members of the system act as a whole be- quantum mechanics and physics for the best understanding sides individuals like they obey a rule in where they prefer of the behavior of economics and biology. Also, we could the welfare of the collective over the welfare of the individ- maybe understand nature like a game in where its players ual. This equilibrium is represented in the maximum system compete for a common welfare and the equilibrium of the entropy in where the system resources are fairly distributed system that they are members. over its members. A system is stable only if it maximizes the We could say that the purpose and maximum payoff of welfare of the collective above the welfare of the individ- a system is its maximum entropy state. The system and its ual. If it is maximized the welfare of the individual above members will vary and rearrange themselves to reach the the welfare of the collective the system gets unstable and best possible state for each of them which is also the best eventually it collapses (Collective Welfare Principle (Gue- possible state for the whole system. This can be seen like a vara 2006c; 2006d; 2006b)). microscopical cooperation between quantum objects to im- Fundamentally, we could distinguish three states in every prove their states with the purpose of reaching or maintain- system: minimum entropy, maximum entropy, and when the ing the equilibrium of the system. All the members of our system is tending to whatever of these two states. The nat- system will play a game in which its maximum payoff is the ural trend of a physical system is to the maximum entropy equilibrium of the system. The members of the system act state. The minimum entropy state is a characteristic of a as a whole besides individuals like they obey a rule in where manipulated system i.e. externally controlled or imposed. 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