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.