Quantum Methods in Social Science

The following sections are included:

• Dedication
• Contents
• Preface
• About the Authors

The following sections are included:

• What’s it all about?
  • Three versions of physics
  • Quantizing classical physics
  • Quantum numbers
  • Quantum field theory
• Counting in culture, society, and science
  • As simple as 1, 2, 3
  • And then there were none
• Classical or quantum representation?
  • Data and information
  • Representation I: The classical way
  • Representation II: The quantum way
  • Second quantization and how to count things

Before we get on to exploiting the tools of quantum physics in modeling the way populations of idealized identical items interact, in this chapter, we will look briefly at a very well-known classical model of interacting populations, which is based on the Lotka–Volterra equations (see [17]). We will do this in order to have something with which to compare the outcomes of the quantum models that we will introduce later. It will also help to see the limitations of classical models and help justify our consideration of alternative quantum models later. We will consider populations of idealized countable individuals that can change with time due to various processes such as birth, death, and inter-species competition. We wish to model these populations so that we can predict how may individuals there will be at any given time. To do this, we need a mathematical specification of the population and a mathematical rule that tells us how the population will change. In a classical formulation of the problem, we use ordinary algebra in which the variable of the system, in this case the number of individuals in the population simply represent numbers whose values can, in principle, be known exactly at any instant. We begin with a single species of (effectively) identical individuals. We next outline a simple mathematical model that specifies how this population changes with time due to birth and death. It is a very well-known example from ecology.

In this chapter, we give a short overview of the basic mathematical elements of the quantum mechanical representation of a system. By system, we do not necessarily mean anything physical and it certainly does not have to be thought of as a particle located in space and time. Rather it is simply something that can be characterized in a mathematical way. The basic mathematical elements that we need require no more than what is to be found in almost any modern textbook on quantum mechanics (QM) (see [4]) in which the development of the basic equations is based on symmetry arguments rather than on any analogies with classical mechanics).

The following sections are included:

• The mechanics of counting
  • Repetition, cyclic processes, and clocks
  • Phasors: The complex plane as a clock counter
• Generating a spectrum of integers with the integer operator
• Generating a spectrum of natural numbers with the number operator
• Occupation number notation and properties of â and â⁺
  • Occupation number notation
  • Everything stops at zero
  • Building the state │n〉 from scratch
  • Commutation brackets with functions of â and â⁺

The following sections are included:

• The quantum trader
• Several non-trading traders
• Allowing the traders to trade
• A simple transaction model: Two traders
  • Setting up the Hamiltonian
  • Applying the HEM to the population numbers
  • A direct method of solution
• Key characteristics of the simple quantum two-trader model
• Eigenfrequency approach
• Stable trading partnerships
• Several trading traders
  • A three-trader example
• Independent subsystems: Worlds within worlds
• Asymmetrical trading
  • Different exchange rates
  • A new integrals of the motion
  • A worked example, P=1, Q=2
  • An approximate solution
• Exercises

So far, we have examined systems that, although they have a dynamical aspect, in that the number of countable items involved, e.g., the assets of traders, has varied in time, there has been no indication of spatial movement. In this chapter, we will briefly examine how the quantum Hamiltonian methods we have developed so far can be applied to situations in which populations, they could be people, or non-human animals for example, move about within a spatial region. The good news is that we do not really need to learn any new mathematics. With a little adjustment to the conceptual elements involved, we can apply what we have learned directly to the population migration situation. To do this, we can think of an array of identical cells that are in contact, into which, or out of which, members of a population can move. It makes sense to allow the population to move only between neighboring cells. This process is often called nearest neighbor hopping. The simplest system in which we can apply our methods consists of a one-dimensional line of identical adjacent cells.

The following sections are included:

• Buying and selling models
• Integrals of the motion
• Solving the equations of motion
• Further refinements: Supply and price

In this part of the book, we have explored in a very elementary way the use of the number operator and its companions, the number amplitude operators, that correspond to creation and annihilation operators in QFT, to model some simple systems in which populations of people, goods, cash etc. vary in time, through being exchanged between different elements of the system as a whole. We have picked extremely simple systems with only a few agents to illustrate the principles involved. We have stressed a key property of this approach to be the existence of certain conservation rules. This is a really important aspect of the modeling and something that physicists always look for in dealing with physical systems, whether they be treated classically of by quantum methods. Whether the system involves galaxies or a few atoms, a physicist will insist that fundamental quantities like energy and momentum are conserved as the system evolves. This constraint is vital in deciding on the validity of the model. It is very easy to write lines of computer code with a perfectly correct set of equations, but which overall do not keep some conserved quantity constant. Models of isolated systems that do not constrain totals of energy and momentum to be fixed should certainly be deemed invalid. The reason is that it is possible to demonstrate mathematically that the accepted theories of physics conserve these quantities. Moreover, the conserved quantities tell physicists precisely what parameters they need to model. For example, physics tells us to measure and keep track of the product of mass times velocity (momentum) and one half times mass, times velocity squared (kinetic energy). Using mass times velocity cubed, for example, would be a waste of time, since it is not part of the great accounting process that controls motion in the universe. It is precisely these features of a physical system that tell the physicist that it is a waste of time thinking about a perpetual motion machine, or something that gives energy for nothing. In contrast, many non-physically based systems models have no all-enveloping theory that tells the modeler what parameters to choose for modeling. With the Lokta–Volterra model (LVM) for example, it seems intuitively obvious to pick the population values of rabbits and foxes and write an albeit ad hoc equation for their variation in time. But why not pick the population number squared? There is nothing to tell us not to. The same appears true of financial models. One picks income, or price or tax. Why not use some combination or some function of these quantities? There are no rules, apart from convention or intuition that constrains the choice of such variables. Why should this matter, well, it could lead to a prediction of something for nothing. The equivalent of a perpetual motion machine might then seem possible in finance or economics. Is continuous economic growth really possible? This may look as feasible as a perpetual motion machine did before the rules of physics were properly understood.

The following sections are included:

• Introduction
• Some basic ideas from classical mechanics
  • Energy conservation
  • First analogs with social science
  • The Lagrangian and the action functional
  • The birth of the Hamiltonian
  • Momentum conservation in finance?
  • Conservation: How important is it?
• Applying basic classical mechanics to finance and economics: Some examples
  • Example 1: A synergetics approach in behavioral economics
  • Example 2: An action functional and arbitrage
  • Example 3: Re-modeling equilibrium pricing with a Lagrangian framework
• Option pricing and physics based partial differential equations
  • What are options?
  • Brownian motion and Einstein
  • Exercises
  • Some toolkit concepts which relate to stochastics
  • Taylor series adapted for stochastic processes
  • Exercises
  • Option pricing: The stochastic approach
  • Exercises

In the former chapter, we took a step back, by considering the classical mechanics framework and potential uses of that framework in a social science environment. Note the “step back” relative to the first part of the book, where we actually used second quantization, but in a way which is almost divorced from physics.

The following sections are included:

• Ellsberg paradox and elementary quantum mechanics
  • Exercises
• References — Part II

The following sections are included:

• Introduction
• Classical probability
  • Bayes formula, conditional probability, and formula of total probability
  • Interpretation of probability: Statistical, subjective and their mixing
  • Contextuality of Kolmogorov theory
  • Exercises
• Classical decision making through the Bayesian probability update
  • Subjective and frequentist interpretations of classical Bayesian inference
  • Cromwell rule

The following sections are included:

• Quantum probability: Pure states
  • Exercises
• Quantum decision making through update of the belief state
  • Exercises
• Quantum probability: Mixed states
  • Exercises
• Events in quantum logic

The notion of common knowledge plays a crucial role in various problems of coordination of actions in philosophy, economics (including accounting and capital market research), game theory, statistics, computer science, and artificial intelligence. Surprisingly, this notion was elaborated not so long ago, in the 1950s, starting with the works of Littlewood [79], Schelling [91], Harsanyi [51–53], Nozick [84], and culminating in a detailed account of common knowledge by Lewis [77]; see also [45] for the first mathematical account of the common knowledge problem.

The following sections are included:

• Quantum representation of the states of the world
• Knowing events, quantum representation
  • Exercises
• Common knowledge, quantum representation
• Quantum state update, projection postulate
• Quantum(-like) viewpoint on the Aumann’s theorem
  • Common prior assumption
  • Disagree from quantum(-like) interference
• Agent with information representation based on incompatible question-observables

The following sections are included:

• Examples illustrating agreement on disagree
  • Exercises
• Example: Agreement on disagree from two opinion polls
  • Still possible to agree on the posterior probabilities
  • Is entanglement important?
  • Classical model
• A concluding discussion on approaching agreement between quantum agents

The following sections are included:

• Operator approach to common knowledge formalization
  • What does it mean to know? Quantum probabilistic formulation
• References — Part III

The following section is included:

• Index