Quantum Many-Worlds Interpretation and Its Extensions to Decision Theory, Game Theory, and Economics

 Quantum Many-Worlds Interpretation and Its Extensions to Decision Theory, Game Theory, and Economics

Abstract

This paper explores the quantum many-worlds interpretation (MWI) and its implications beyond fundamental physics, extending to decision theory, game theory, and economics. Starting from the foundational principles of MWI, we examine how observers interact with branching universes, analyze Newcomb’s paradox in both classical and quantum forms, and discuss applications in quantum game theory and quantum economics. By integrating quantum concepts like superposition, entanglement, and interference, we demonstrate how these ideas resolve classical paradoxes and offer novel frameworks for modeling complex systems. Mathematical formulations are presented using Unicode symbols for clarity.

1. Introduction

The many-worlds interpretation of quantum mechanics, proposed by Hugh Everett III in 1957, posits that the universe branches into multiple parallel realities during quantum measurements, avoiding the wave function collapse postulate. This framework has profound philosophical and practical implications, particularly in understanding decision-making under uncertainty.

In this paper, we synthesize discussions on MWI, observer roles in branching, Newcomb’s paradox, its quantum variant, quantum game theory, and quantum economics. These topics form a coherent progression from quantum foundations to applied interdisciplinary fields, highlighting how quantum principles can reshape classical theories.

2. Quantum Many-Worlds Interpretation

2.1 Core Principles

The state of a quantum system is described by the wave function Ψ, which evolves linearly according to the Schrödinger equation:

iℏ ∂/∂t Ψ = Ĥ Ψ

where Ĥ is the Hamiltonian operator, i is the imaginary unit, and ℏ is the reduced Planck’s constant.

In MWI, measurements do not collapse Ψ; instead, the universe splits into branches corresponding to each possible outcome. For a superposition state:

Ψ = α |up⟩ + β |down⟩

with |α|² + |β|² = 1, the measurement entangles the observer, resulting in:

Ψ = α |up, observer sees up⟩ + β |down, observer sees down⟩

Probabilities arise from the squared amplitudes |α|² and |β|², aligning with the Born rule.

2.2 Advantages and Challenges

MWI maintains mathematical consistency without ad hoc collapse. It resolves the measurement problem via decoherence, where environmental interactions make branches effectively independent. However, it faces criticisms for ontological extravagance (infinite universes) and unverifiability.

Decoherence is modeled by tracing out environmental degrees of freedom, leading to a reduced density matrix ρ that appears classical.

3. Observer’s Role in Branch Optimization

3.1 Limitations on Branch Selection

Observers cannot directly select branches, as branch weights are fixed by wave function amplitudes:

Ψ = α |A⟩ + β |B⟩, with weights |α|² and |β|².

Decisions themselves are quantum processes, branching accordingly.

3.2 Decision Theory in MWI

In the Deutsch-Wallace framework, rational agents maximize weighted utility across branches:

U_total = Σ_i |α_i|² U_i

This resolves paradoxes like Newcomb’s by equating strategies under certain conditions.

For quantum decision trees, strategies influence subsequent branch structures via measurements.

3.3 Applications

In quantum amplification, random number generators use superpositions like (1/√2)|0⟩ + (1/√2)|1⟩ to create balanced branches. Ethical implications redefine free will as shaping branch structures rather than outcomes.

4. Newcomb’s Paradox

4.1 Problem Setup

A predictor places contents in boxes A (fixed $1,000) and B (either $1,000,000 or $0 based on prediction). Choosing only B (one-boxing) yields ~$1,000,000 if predicted correctly; both (two-boxing) yields ~$1,000.

4.2 Conflicting Principles

Expected utility favors one-boxing:

For accuracy p ≈ 1, EU_one = p × 1,000,000 ≈ 1,000,000

EU_two = p × 1,000 + (1-p) × 1,001,000 ≈ 1,000

Dominance favors two-boxing, as it adds $1,000 regardless of B’s content.

4.3 Resolutions

Causal links or pre-commitment support one-boxing; coincidences support two-boxing. Philosophical ties to free will and determinism.

5. Quantum Newcomb’s Paradox

5.1 Quantum Setup

Boxes are entangled, e.g., |Ψ⟩ = (1/√2) (|0_A 1_B⟩ + |1_A 0_B⟩).

Player choice is a quantum operation; measurement collapses the state, ensuring correlation without retrocausality.

5.2 Mathematical Model

Using qubits, predictor applies Hadamard (H) and CNOT gates. Nash equilibria favor one-boxing analogs due to entanglement.

In extended models, quantum strategies achieve deterministic predictions.

5.3 Differences from Classical

Entanglement provides built-in correlation, invalidating dominance in some states.

6. Applications in Quantum Game Theory

6.1 Quantum Strategies

Strategies are unitary operators in SU(2). Initial entangled state:

|Ψ⟩ = cos(γ/2)|00⟩ + i sin(γ/2)|11⟩

Payoffs from Tr(ρ M), where ρ is the final density matrix.

6.2 Quantum Prisoner’s Dilemma

Classical Nash: (2,2); quantum: (3,3) via “magic” strategy U(θ=π/2, φ=0).

6.3 Broader Applications

In auctions, entanglement prevents collusion. In finance, quantum arbitrage uses entangled portfolios. Social applications include quantum voting and climate games.

7. Applications in Quantum Economics

7.1 Quantum Market Models

Prices as expectations: P(t) = ⟨ψ(t)| ˆP |ψ(t)⟩

Entangled assets: |Ψ_AB⟩ models non-local correlations.

7.2 Financial Tools

Quantum Monte Carlo for pricing: Z = ∫ D[path] exp(-S[path]/ℏ)

Risk: Quantum VaR via Tr[ρ ˆVaR(α)]

7.3 Macro and Micro Economics

Quantum multipliers: 1/(1 - ⟨MPC| ˆM |MPC⟩)

Consumer behavior via non-commuting utilities.

7.4 Empirical and Policy Implications

Quantum models explain market anomalies; applications in central banking and carbon markets.

8. Conclusion

MWI provides a unified framework for quantum phenomena, extending to resolve decision paradoxes and enhance game-theoretic and economic models. Quantum entanglement and superposition offer tools for modeling complexity, promising revolutions in interdisciplinary applications. Future work should focus on empirical validation and computational implementations.

References

This synthesis draws from foundational works by Everett (1957), Deutsch & Wallace (on decision theory), Nozick (1969) on Newcomb’s, Piotrowski & Sładkowski (2003) on quantum games, and Orrell & Haven on quantum economics.