## Bajric Sanel

#### Ph.D. Economics & Computer Science

## Quantum Economics Models: A Comprehensive Guide to a New Era in Economic Analysis

The world of economics has always been driven by models – frameworks that allow us to understand complex systems and predict future trends. As we venture further into the 21st century, a new type of economic model is emerging, one that incorporates principles from quantum physics: Quantum Economics Models. These models are fundamentally changing the way we approach economic analysis, promising to deliver more accurate forecasts and deeper insights into economic behavior.

**1. Quantum Game Theory**

One of the most influential applications of quantum mechanics in economics is Quantum Game Theory. This takes the principles of game theory, a mathematical model of strategic interaction, and introduces elements of quantum mechanics.

The Quantum Game Theory model assumes that the players’ strategies are in a state of superposition, akin to how particles exist in multiple states at once in quantum physics. This allows for richer strategy sets and often results in different equilibrium outcomes compared to classical game theory. The mathematical representation of a quantum game can be described by the density matrix ρ which evolves according to the Liouville-Von Neumann equation.

*Theory example:*

Let’s consider a simple game known as the Prisoner’s Dilemma. In the classical version, two prisoners, A and B, can either confess (C) or not confess (NC).

Here are the payoffs:

B confesses | B does not confess | |
---|---|---|

A confesses | -5, -5 | 0, -10 |

A does not confess | -10, 0 | -1, -1 |

In a one-shot game, the Nash equilibrium is that both confess, even though both would be better off not confessing.

Now, let’s bring in quantum mechanics. Instead of the strategies being either confess or not confess, they can be a superposition of both. This is represented by a quantum bit or qubit. We represent the state of a qubit using a vector:

|ψ> = α|0> + β|1>

In this representation, |0> stands for not confessing, |1> for confessing. The complex numbers α and β define the state of the qubit, with |α|^2 giving the probability of not confessing, and |β|^2 the probability of confessing.

The quantum game matrix would now use these quantum strategies, and it’s possible to show that a new equilibrium can be found where neither player has a strict incentive to deviate.

**2. Quantum Finance**

Quantum Finance is another intriguing field where quantum principles are applied to financial models. The most notable example is the Quantum Black-Scholes model which enhances the traditional Black-Scholes model for pricing options.

The Quantum Black-Scholes model is derived from the Schrödinger equation, a key equation in quantum mechanics. Instead of assuming that stock prices follow a geometric Brownian motion as in the classical model, the quantum version accounts for the quantum nature of financial markets, providing a more accurate option pricing model.

**Theory example:**

Consider the Black-Scholes model for pricing options. In the classical Black-Scholes model, the price of a European call option is given by:

C(S, t) = SN(d1) – Xe^(-rt)N(d2)

where: S = spot price of the underlying asset, X = strike price of the option, r = risk-free interest rate, t = time to expiration, N = cumulative distribution function of the standard normal distribution, d1 = (ln(S/X) + (r + σ^2 /2)t) / σ√t, d2 = d1 – σ√t, and σ = volatility of returns of the underlying asset.

In the quantum version, we use the Schrödinger equation to describe the evolution of the stock price. This leads to a modified Black-Scholes equation that includes a quantum potential term. The quantum potential term can capture aspects of market behavior that are not present in the classical model, providing potentially more accurate option prices.

**3. Quantum Bayesianism (QBism)**

Quantum Bayesianism or QBism is an interpretation of quantum mechanics that offers a framework for understanding the subjective experiences of an agent interacting with a dynamic world. It links the realms of quantum mechanics and Bayesian probability theory, offering a potential bridge to economic models that focus on individual behavior and subjective experience.

QBism suggests that probabilities in quantum mechanics are not elements of physical reality but are personal judgements made by the observer. This approach is captured mathematically by the use of Bayesian updating and the use of the quantum state as a state of knowledge.

**Theory example:**

QBism primarily affects how we interpret probabilities in quantum systems. Let’s consider a simple quantum system, like the spin of an electron. The spin can be either up or down. In standard quantum mechanics, we would describe this as a superposition of states:

|ψ> = α|up> + β|down>

Where |α|^2 is the probability of measuring the spin as up and |β|^2 as down.

In QBism, these probabilities do not represent physical reality but are rather the observer’s subjective probabilities. For instance, if an observer believes the spin is more likely to be up, they assign a higher probability to the |up> state. If new information becomes available (say, through measurement or interaction with the system), the observer updates their probabilities using Bayes’ rule, just as in classical Bayesian inference.

**Conclusion**

While these three models illustrate some applications of quantum principles in economics, the field of Quantum Economics is still young and evolving. The intersection of quantum mechanics and economics promises a richer understanding of economic systems, markets, and human behavior. As this field continues to mature, it’s expected that new quantum economics models will emerge, each with its unique mathematical representation and predictive power.