Doxa: A Formal Multi-Agent Simulation Framework for Economic and Strategic Interaction¶

Abstract¶

We introduce Doxa, a formal computational framework for simulating decentralized economic systems populated by heterogeneous, AI-driven agents. The platform integrates agent-based modeling, market microstructure, and dynamic game theory, enabling the study of strategic interaction under bounded rationality.

Doxa is defined by a declarative configuration language, a deterministic simulation engine, and a modular architecture supporting endogenous communication, trust dynamics, and exogenous shocks. Agents are powered by large language models, enabling adaptive reasoning and negotiation in natural language.

We formalize the system as a discrete-time stochastic game with endogenous network formation, and demonstrate how equilibrium-like structures emerge from decentralized interactions rather than being imposed ex ante. The framework enables controlled experimentation in computational economics, mechanism design, and AI-driven markets.

1. Introduction¶

The study of decentralized economic systems requires tools capable of modeling heterogeneous agents, strategic interaction, and non-equilibrium dynamics. Traditional frameworks in computational economics lack:

flexible scenario specification
endogenous communication and trust dynamics
integration of modern AI reasoning capabilities

Doxa addresses these limitations by introducing a unified architecture where:

agents are strategic decision-makers
markets implement explicit clearing mechanisms
interactions generate emergent macro-level phenomena

The system can be interpreted as a computable economy where agents solve constrained optimization problems under uncertainty, with incomplete information and evolving beliefs.

Classical approaches impose equilibrium assumptions. However, real systems operate out of equilibrium, with continuous adjustment driven by local decisions.

Doxa takes a different stance:

Economic structure is not assumed. It is generated.

The framework enables simulation of:

endogenous price formation
trust-based coordination
strategic negotiation under uncertainty

The key innovation is combining:

formal economic structure
programmable environments
LLM-based agent cognition

2. Formal Model ¶

2.1 System Definition¶

A Doxa simulation is defined as:

\[ \mathcal{D} = (\mathcal{A}, \mathcal{R}, \mathcal{M}, \mathcal{E}, \mathcal{T}) \]

where:

( $\mathcal{A}$ ): set of agents
( $\mathcal{R}$ ): resource space
( $\mathcal{M}$ ): markets
( $\mathcal{E}$ ): event processes
( $\mathcal{T}$ ): transition operator

Time is discrete: ( $t = 0,1,\dots,T$ )

2.2 Agent Structure¶

Each agent ( $a \in \mathcal{A}$ ) is defined by:

\[ a = (P_a^t, U_a, \Sigma_a, \mathcal{B}_a^t, \Theta_a) \]

where:

( $P_a^t \in \mathbb{R}_+^{|\mathcal{R}|}$ ): portfolio
( $U_a$ ): utility function
( $\Sigma_a$ ): strategy space
( $\mathcal{B}_a^t$ ): belief state
( $\Theta_a$ ): behavioral parameters

Agents, emergently, try to solve:

\[ \max_{\sigma_a \in \Sigma_a} \mathbb{E}\left[\sum_{t=0}^T \beta_a^t U_a(P_a^t)\right] \]

subject to feasibility constraints. Agents are not guaranteed to solve this problem optimally, but it serves as a normative benchmark for their behavior. Their actions and thoughts are indeed emergent from their interactions with the environment and other agents, but constraints and incentives are designed to encourage them to approximate this optimization process.

The agent acts based on its current portfolio, beliefs, and the context of the environment, which includes market conditions and the actions of other agents. The strategy space ( $\Sigma_a$ ) encompasses all possible actions, including trading, communication, and resource transformation.

It is important to note that while agents are designed to be utility-maximizing, their actual behavior may deviate from this ideal due to bounded rationality, limited information, and the complexity of the environment or, more technically, based on inhereted stochastic elements of LLMs reasoning. This is where the integration of LLMs comes into play, allowing agents to use grasp the environment and make decisions in a more human-like, sometimes suboptimal way or apparently irrational, but still strategically coherent manner.

2.3 Bounded Rationality via LLMs¶

Agents do not solve the optimization problem exactly.

Instead, decisions are approximated by:

\[ \sigma_a^t \sim \pi_{\theta}(P_a^t, \mathcal{B}_a^t, \text{context}) \]

where ( $\pi_{\theta}$ ) is an LLM-induced policy.

Implications:

non-stationary strategies
heuristic reasoning
path dependence

This places Doxa in the class of behavioral computational economies.

2.4 Market Model¶

Markets ( $m \in \mathcal{M}$ ) are defined as tuples:

\[ m = (r, c, \pi, \Pi, \kappa) \]

where:

( $r$ ): traded resource
( $c$ ): numeraire (currency)
( $\pi$ ): price
( $\Pi$ ): price bounds
( $\kappa$ ): clearing mechanism

Supported mechanisms:

continuous double auction (LOB)
periodic clearing
call auctions

Market makers implement:

\[ q^{bid/ask} = \pi \pm s + \kappa \cdot inventory \]

introducing liquidity and stabilizing price dynamics.

Clearing Mechanisms¶

Continuous double auction
Batch clearing
Call auction

Matching function:

\[ \mathcal{M}: \mathcal{O}_t \rightarrow \text{Trades}_t \]

where ( $\mathcal{O}_t$ ) is the set of orders at time ( $t$ ).

Production and Transformation**¶

Agents have production functions:

\[ f_a: \mathbb{R}_{*+}^{|\mathcal{R}|} \rightarrow \mathbb{R}_{*+}^{|\mathcal{R}|} \]

Example:

\[ f_{farmer}(gold) = 4 \cdot corn \]

This introduces:

endogenous supply
interdependence across agents

2.5 Game-Theoretic Interpretation¶

The system defines a dynamic stochastic game:

players: agents
actions: trade, communicate, transform resources
payoffs: utility over portfolios

At each timestep:

\[ G_t = (\mathcal{A}, \Sigma, U, T) \]

where ( $T$ ) is the transition function induced by:

market clearing
agent actions
world events

Equilibria are not imposed but emerge dynamically, allowing study of:

bounded rational equilibria
coordination failures
market inefficiencies

Event Process¶

Events are stochastic processes:

\[ E_t \sim \mathcal{P}(E \mid S_t) \]

State transition:

\[ S_{t+1} = \mathcal{T}(S_t, A_t, E_t) \]

Events introduce:

non-stationarity
regime shifts
exogenous shocks

Equilibrium Concept¶

Classical Nash equilibrium is intractable.

Instead, Doxa approximates:

behavioral equilibrium
empirical stationary distributions
policy convergence states

Relations form a weighted directed graph:

\[ G = (V, E), \quad E = \{(i,j,t_{ij})\} \]

Trust evolves as:

\[ t_{ij}^{t+1} = (1 - \delta)t_{ij}^t + \Delta_{ij} \]

where:

( $\delta$ ): decay rate
( $\Delta_{ij}$ ): event-driven update

Trust affects:

probability of trade acceptance
negotiation outcomes
coalition formation

Trust Network¶

Define a directed weighted graph:

\[ G_t = (\mathcal{A}, E_t) \]

with weights:

\[ t_{ij}^t \in [0,1] \]

Update rule:

\[ t_{ij}^{t+1} = (1 - \delta)t_{ij}^t + \phi(\text{interaction}) \]

where ( $\delta$ ) is a decay factor and ( $\phi$ ) captures the impact of interactions (e.g., successful trade increases trust, failed negotiation decreases it).

Trust affects:

probability of trade
negotiation outcomes
coalition formation

3. System Architecture¶

Doxa implements a modular architecture:

Core Components¶

Engine: deterministic orchestrator of simulation dynamics
Environment: state container (agents, portfolios, markets)
Agents: decision units with LLM-based reasoning
Market Engine: order matching and price formation
Event Scheduler: exogenous shocks and trends
Relation Graph: social structure

Execution Loop¶

At each timestep:

Maintenance and constraint enforcement
Agent decision phase
Market clearing
Event execution
State update and logging

This defines a discrete-time dynamical system.

3.1 Engine Implementation¶

The engine implements:

\[ S_{t+1} = \mathcal{T}(S_t) \]

via the loop:

maintenance constraints
agent decisions
market clearing
event execution
belief update

Determinism¶

Given:

fixed YAML
fixed seeds

the system is reproducible.

3.1 Declarative Scenario Specification¶

Doxa uses a YAML-based domain-specific language enabling full reproducibility.

Formal Structure¶

A scenario is defined as:

\[ \mathcal{S} = (\mathcal{R}, \mathcal{A}, \mathcal{M}, \mathcal{E}) \]

where:

( $\mathcal{R}$ ): resources
( $\mathcal{A}$ ): agents
( $\mathcal{M}$ ): markets
( $\mathcal{E}$ ): events

This allows:

deterministic replay
parameter sweeps
comparative statics

3.2 Event Dynamics and Exogenous Shocks¶

Events introduce non-stationarity:

Types:¶

Shock: instantaneous perturbation
Trend: persistent drift
Conditional: state-triggered

Formally:

\[ P_{t+1} = f(P_t, A_t, E_t) \]

where ( $E_t$ ) modifies transition dynamics.

This enables modeling of:

supply shocks
crises and panic
policy interventions

3.3 Communication and Strategic Interaction¶

Agents exchange structured messages:

\[ m = (i, j, \tau, payload) \]

Protocols include:

propose
accept
reject

This introduces:

incomplete information
signaling games
negotiation equilibria

LLMs approximate reasoning under bounded rationality, enabling natural-language strategy formation.

4. Analytical Properties¶

4.1 Existence of Feasible Paths¶

Given constraint validation, there exists at least one feasible trajectory:

\[ \exists {S_t}_{t=0}^T \]

4.2 Non-Equilibrium Dynamics¶

The system does not guarantee convergence:

cycles may emerge
chaotic dynamics possible
path dependence dominates

4.3 Emergent Price Formation¶

Prices evolve as:

\[ \pi_{t+1} = \pi_t + \Delta(\text{order flow}) \]

No Walrasian auctioneer is assumed.

5. Experimental Design Framework¶

Doxa enables controlled experiments:

Independent Variables¶

agent preferences
market structure
event processes

Dependent Variables¶

price volatility
allocation efficiency
network structure

Metrics¶

allocative efficiency
Gini coefficient
trade volume
convergence speed

6. Case Study: Bilateral Production Economy¶

Two-agent system:

Agent A: transforms gold → corn
Agent B: transforms corn → gold

Structure¶

\[ A \leftrightarrow B \]

This creates:

mutual dependence
endogenous trade necessity

Dynamics¶

shocks → price spikes
scarcity → bargaining power shifts
trust → trade friction

Equivalent to:

a minimal general equilibrium model
with production and strategic interaction

7. Applications¶

Economic Research¶

market efficiency under bounded rationality
liquidity crises
price discovery

Game Theory¶

repeated games with communication
endogenous coalition formation

AI Systems¶

emergent coordination
language-mediated strategy

Policy¶

intervention testing
systemic risk

8. Validation and Consistency Guarantees¶

The system enforces:

feasibility of constraints
resource conservation
market consistency
reachability of objectives

This ensures simulations are:

internally coherent
economically meaningful
reproducible

9. Limitations¶

no formal equilibrium guarantees
scalability constraints
dependence on LLM quality

Agents are:

not fully rational
not stationary
sensitive to prompting

10. Conclusion¶

Doxa reframes economic modeling as:

a programmable, dynamic system of interacting agents

It eliminates:

static equilibrium assumptions
rigid analytical constraints

and replaces them with:

simulation-based inference
emergent structure
adaptive behavior

This positions Doxa as a foundational tool for:

computational economics
AI-driven market design
next-generation economic experimentation

Doxa represents a shift toward programmable economic systems, where:

agents are autonomous and adaptive
markets are explicitly modeled
dynamics emerge from interaction

It bridges:

agent-based modeling
computational economics
modern AI

The framework provides a foundation for studying complex adaptive systems where equilibrium is not assumed but constructed through interaction.

11. Future Works¶

Critical directions:

1. Learning Agents¶

Hybrid LLM + reinforcement learning:

\[ \pi_a = \lambda \pi_{LLM} + (1-\lambda)\pi_{RL} \]

2. Mechanism Design Layer¶

Design optimal markets:

\[ \max_{\kappa} ; \text{Efficiency}(\kappa) \]

3. Distributed Simulation¶

Scaling to ( $ 10^5+ $ ) agents

4. Formal Analysis¶

convergence proofs
stability conditions
equilibrium approximation bounds
distributed simulation architecture
endogenous learning (RL + LLM hybrid)
mechanism design modules
formal equilibrium analysis tools

Doxa: A Formal Multi-Agent Simulation Framework for Economic and Strategic Interaction¶

Abstract¶

1. Introduction¶

**2. Formal Model **¶

2.1 System Definition¶

2.2 Agent Structure¶

2.3 Bounded Rationality via LLMs¶

2.4 Market Model¶

Clearing Mechanisms¶

Production and Transformation**¶

2.5 Game-Theoretic Interpretation¶

Event Process¶

Equilibrium Concept¶

2.6 Social and Trust Dynamics¶

Trust Network¶

3. System Architecture¶

Core Components¶

Execution Loop¶

3.1 Engine Implementation¶

Determinism¶

3.1 Declarative Scenario Specification¶

Formal Structure¶

3.2 Event Dynamics and Exogenous Shocks¶

Types:¶

3.3 Communication and Strategic Interaction¶

4. Analytical Properties¶

4.1 Existence of Feasible Paths¶

4.2 Non-Equilibrium Dynamics¶

4.3 Emergent Price Formation¶

5. Experimental Design Framework¶

Independent Variables¶

Dependent Variables¶

Metrics¶

6. Case Study: Bilateral Production Economy¶

Structure¶

Dynamics¶

7. Applications¶

Economic Research¶

Game Theory¶

AI Systems¶

Policy¶

8. Validation and Consistency Guarantees¶

9. Limitations¶

10. Conclusion¶

11. Future Works¶

1. Learning Agents¶

2. Mechanism Design Layer¶

3. Distributed Simulation¶

4. Formal Analysis¶

2. Formal Model ¶