Multiscaling Edge Effects in Agent-Based Money Emergence Model

Table of Contents

1. Introduction

This paper investigates an agent-based computational model for money emergence from initial barter trading, inspired by Menger's postulate that money can spontaneously emerge in a commodity exchange economy. The model reveals phenomena interpretable as emergence and collapse of money, along with related competition effects. A key finding is the development of multiscaling in money lifetimes near critical threshold values, drawing parallels to critical phenomena in real financial markets.

2. Model

The agent-based model consists of N agents, each producing one type of good (k=1,...,N). Agent k produces good type k. The elementary interaction involves multiple steps including co-trader search, goods exchange, preference updates, and production/consumption phases.

2.1 Agent Interactions

Each agent maintains buying preferences and engages in transactions that follow a structured sequence. A turn comprises N consecutive transactions, ensuring each agent has opportunity to participate.

2.2 Transaction Mechanism

The transaction process involves: (1) searching for trading partners, (2) exchanging goods based on mutual needs, (3) updating buying preferences, and (4) production and consumption phases.

3. Technical Framework

3.1 Mathematical Formulation

The model's dynamics can be described using preference matrices and utility functions. For agent i with preference vector $P_i = [p_{i1}, p_{i2}, ..., p_{iN}]$ where $p_{ij}$ represents the preference for good j, the transaction utility is given by:

$U_{ij} = \sum_{k=1}^{N} p_{ik} \cdot q_{jk} - \sum_{k=1}^{N} p_{jk} \cdot q_{ik}$

where $q_{jk}$ represents the quantity of good k held by agent j.

3.2 Multiscaling Analysis

The multiscaling behavior near critical thresholds is analyzed using multifractal formalism. The partition function is defined as:

$Z(q,s) = \sum_{\mu} p_{\mu}^q(s) \sim s^{\tau(q)}$

where $\tau(q)$ is the mass exponent and the multifractal spectrum $f(\alpha)$ is obtained through Legendre transformation.

4. Experimental Results

4.1 Money Emergence Patterns

Simulations demonstrate spontaneous elevation of one commodity to money status through a process analogous to physical spontaneous symmetry breaking. The bootstrap mechanism ensures accepted status across all transactions.

4.2 Critical Threshold Behavior

Near critical parameter values, money lifetimes exhibit multiscaling characteristics. This behavior mirrors critical phenomena observed in financial markets, particularly in Forex dynamics where similar complex scaling patterns emerge.

5. Code Implementation

Below is a simplified Python implementation of the agent transaction mechanism:

class Agent:
    def __init__(self, agent_id, goods_preference):
        self.id = agent_id
        self.preferences = goods_preference
        self.inventory = {i: 1 for i in range(len(goods_preference))}
    
    def calculate_utility(self, other_agent):
        utility = 0
        for good_id, pref in enumerate(self.preferences):
            utility += pref * other_agent.inventory.get(good_id, 0)
        return utility
    
    def engage_transaction(self, other_agent):
        if self.calculate_utility(other_agent) > threshold:
            # Execute goods exchange
            self.update_preferences()
            other_agent.update_preferences()
            return True
        return False

def simulate_turn(agents):
    for i in range(len(agents)):
        for j in range(i+1, len(agents)):
            agents[i].engage_transaction(agents[j])

6. Applications and Future Directions

This model has significant implications for understanding financial market dynamics, particularly in decentralized systems like cryptocurrency markets. Future research directions include:

• Extension to multiple currency systems
• Integration with real market data
• Application to blockchain-based economic systems
• Study of regulatory impacts on money emergence

7. Original Analysis

The agent-based money emergence model presented in this study represents a significant contribution to computational economics, particularly in understanding how monetary systems can spontaneously organize from simple barter economies. The model's demonstration of multiscaling effects near critical thresholds provides a mathematical bridge between economic phenomena and physical critical systems, reminiscent of the interdisciplinary approaches seen in works like CycleGAN (Zhu et al., 2017) that connect disparate domains through fundamental mathematical principles.

What makes this research particularly compelling is its validation of Menger's century-old hypothesis using modern computational methods. The bootstrap mechanism identified in the model—where money becomes accepted because it is in a position of money—parallels network effects observed in contemporary digital currencies. This aligns with research from the Santa Fe Institute on complex adaptive systems, which emphasizes how simple local interactions can generate complex global phenomena.

The multiscaling analysis reveals that money lifetimes near critical transitions exhibit fractal characteristics similar to those observed in financial market volatility clustering. This connection to real market behavior, as documented in the European Physical Journal B and Journal of Economic Dynamics and Control, suggests that the model captures essential features of monetary dynamics. The mathematical framework employing partition functions and multifractal spectra provides tools for quantifying economic complexity that could be applied to analyze systemic risk in financial networks.

Compared to traditional economic models that often rely on equilibrium assumptions, this agent-based approach embraces the inherent disequilibrium and path-dependency of economic systems. The model's ability to simulate both money emergence and collapse makes it particularly relevant for understanding cryptocurrency dynamics, where new monetary forms regularly appear and disappear. Future work connecting these findings to empirical data from platforms like Ethereum could yield valuable insights for both economists and policymakers.

8. References

1. Menger, C. (1871). Principles of Economics
2. Yasutomi, A. (1995). Physica D: Nonlinear Phenomena
3. Górski, A.Z. et al. (2007). Acta Physica Polonica B
4. Zhu, J.Y. et al. (2017). CycleGAN: Unpaired Image-to-Image Translation
5. Arthur, W.B. (1999). Science
6. Lux, T. & Marchesi, M. (1999). Nature
7. Mantegna, R.N. & Stanley, H.E. (2000). Introduction to Econophysics