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Common Welfare, Strong Currencies and the Globalization Process Esteban Guevara Hidalgo Center for Nonlinear and Complex Systems (Como, Italy) ICTP-TRIL (Trieste, Italy) SION (Quito, Ecuador)

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OVERVIEW The von Neumann Equation & the Statistical Mixture of States The Replicator Dynamics & EGT Relationships between Quantum Mechanics & Game Theory Quantum Replicator Dynamics & the Quantization Relationships Games through Statistical Mechanics & QIT From Classical to Quantum On a Quantum Understanding of Classical Systems Some Crazy ideas & Conclusions Esteban Guevara Hidalgo

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Physical System Hilbert Space Quantum Mechanics Foundations E H:

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Density Operator and the Statistical Mixture of States Pure Ensemble Mixed Ensemble

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Populations Coherences Density Operator Von Neumann Equation:

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Simetric Game G = (S, E) Game Theory

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Strategies & Equilibrium Points Mejor Respuesta a es tq. sea máxima. son mejores respuestas mutuas. Punto de Equilibrio es si y Estricta Mejor Respuesta a es si es su única mejor respuesta. Punto de Equilibrio Estricto es si y son estrictas mejores respuestas mutuas. Mejor Respuesta a la cual es diferente de es llamada Mejor Respuesta Alternativa.

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Nash Equilibrium

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The Replicator Dynamics & EGT

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Evolutionary Stable Strategies (ESS) G = (S, E) If then

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Relative frequencies vector Payoff Function Average Payoff Function The Replicator Dynamics

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Relationships between Quantum Mechanics & Game Theory

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Matrix Representation of the Replicator Dynamics

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X

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X

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X

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Replicator Dynamics Lax Form

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Relationships between Quantum Mechanics & Game Theory

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Specific Resemblances

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Quantization Relationships

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Quantum Replicator Dynamics

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Games through Statistical Mechanics & QIT JOINT ENTROPY Measures our total uncertainty about the pair (A,B) CORRELATION ENTROPY A measure of how much information A and B have in common and have an idea of how its strategies or states are correlated. CONDITIONAL ENTROPY How uncertain we are about the value of A, given that we know the value of B CONDITIONAL ENTROPY MARKOV CHAIN

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Shannon & von Neumann Entropies Entropy can be regarded as a quantitative measure of disorder. MAXIMUM: lnN Completely random ensemble MINIMUM: 0 Pure ensemble Entropy can be MAXIMIZED subject to different constrains. Generally, the result is a probability distribution function. For example: If we maximize subject to the constrains The result is: which is the CONDITION that the density operator must satisfy to our system tends to maximize its entropy. (“high – temperature limit”) completely random ensemble (“low – temperature limit”) pure ensemble Z partition function Games through Statistical Mechanics & QIT

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If our systems are analogous and thus exactly equivalents, our physical equilibrium (maximum entropy) should be also exactly equivalent to our socieconomical equilibrium (NE or ESS). From an absolutely different point of view: Also suggested From an absolutely different point of view: by: (Dragulescu & Yakovenko 2000). In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. (Darooneh 2006). The maximum entropy principle is used for pricing the insurance. Darooneh obtained the price density based on this principle, applied it to multi agents model of insurance market and derived the utility function. The main assumption in his work is the correspondence between the concept of the equilibrium in physics and economics. (TopsØe 1979;1993) also has suggested that thermo dynamical equilibrium equals game theoretical equilibrium. On the Quantum Understanding of Classical Systems

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Physical equilibrium (maximum entropy) The purpose and maximum payoff of a physical system is its maximum entropy state i.e. its equilibrium. The system and its members will vary and rearrange themselves to reach the best possible state for each of them which is also the best possible state for the whole system.

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Microscopical cooperation between quantum objects to improve their states with the purpose of reaching or maintaining the equilibrium of the system. All the members of our quantum system will play a game in which its maximum payoff is the EQUILIBRIUM of the system. They act as a whole besides individuals like they obey a rule in where they prefer the welfare of the collective over the welfare of the individual. On the Quantum Understanding of Classical Systems

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This equilibrium is represented in the maximum system entropy in where the system “resources” are fairly distributed over its members. A system is stable only if it maximizes the welfare of the collective above the welfare of the individual. A system where its members are in NE (or ESS) is exactly equivalent to a system in a maximum entropy state. On the Quantum Understanding of Classical Systems

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Some Crazy Ideas & Conclusions 1. The quantum analogue of the replicator dynamics is the von Neumann equation. 2. A population is represented by a quantum system in which each subpopulation playing strategy … 3. Quantum mechanics could be used to explain more correctly biological and economical processes and even encloses theories like games and evolutionary dynamics. 4. Although both systems analyzed are described through two apparently different theories (quantum mechanics and game theory) both are analogous and thus exactly equivalents. 5. So, we can take some concepts and definitions from quantum mechanics and physics for the best understanding of the behavior of economics and biology.

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6. We could maybe understand nature like a game in where its players compete for a common welfare and the equilibrium of the system that they are members as a whole 7.All the members of our system will play a game in which its maximum payoff is the equilibrium of the system. They act as a whole besides individuals like they obey a rule in where they prefer to work for the welfare of the collective besides the individual welfare. 8. A system is stable only it maximizes the welfare of the collective above the welfare of the individual. If it is maximized the welfare of the individual above the welfare of the collective the system gets unstable an eventually collapses. 9. A system where its members are in NE (or ESS) is exactly equivalent to a system in a maximum entropy state (its stability should be given by the welfare of the collective). Some Crazy Ideas & Conclusions

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10. The GLOBALIZATION process has a behavior exactly equivalent to a system that is tending to an MAXIMUM ENTROPY state. Big common markets (EU, NA, SA,ASIA,O), And strong common currencies ($, €, ₤, ٭,…), With the time The number of markets and currencies ($, €, ₤, ٭,…), will decrease until the system reaches its equilibrium i.e. only one currency in the world. Some Crazy Ideas & Conclusions

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