Title: Dimensional Game Theory: A Framework for Strategic Algorithm Development Author: Nnamdi Michael Okpala, OBINexus Computing Date: May 22, 2025

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Abstract

This document introduces "Dimensional Game Theory"-a novel extension of classical game theory designed for real-world algorithmic applications. By redefining strategies as multi-dimensional constructs, this framework enables dynamic detection of strategic imbalances and real-time adaptive decision-making. We present formal definitions, theorems, and algorithms that substantiate the model and discuss its application across finance, cybersecurity, autonomous systems, and business strategy.

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1. Introduction

Traditional game theory relies on equilibrium concepts and payoff matrices. Dimensional Game Theory builds on this by mapping strategies to defined dimensions, allowing for richer analysis and adaptive behavior in complex environments.

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2. Formal Definitions

• Game: G = (N, A, u) where N = set of players, A = action profiles, u = payoff functions.
• Strategy: Pure (si ? Ai) or Mixed (si over Ai).
• Nash Equilibrium: No player can benefit by changing their strategy unilaterally.

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3. Dimensional Game Theory

• Strategic Dimension (D): An attribute category (e.g., offensive, defensive).
• Dimensional Strategy (sD): A strategy optimized for a specific dimension.

Theorem 1: Perfect Game Outcome

In a zero-sum game, perfect play across all dimensions yields deterministic tie.

Corollary: Strategic Imbalance

A win implies imbalance in at least one dimension.

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4. Algorithmic Implementation

4.1 Dimension Detection

Input: Historical data H = {(si1, si2, oi)}
Output: Dimension Set D
Apply PCA to strategy features.
Extract significant components into D.

4.2 Strategic Adaptation

Input: Game state g, estimated opponent strategy so
Output: Counter-strategy composite
Identify dominant dimensions.
Generate counter-strategies.
Weight by dominance.
Return blended strategy.

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5. Applications

• Finance: Trading based on dimension dominance (momentum, liquidity, etc).
• Cybersecurity: Defense allocation by attack type (social, brute-force).
• Autonomous Vehicles: Navigating based on risk/aggression dimensions.
• Business: Competitive positioning using dimensions like price vs. innovation.

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6. Conclusion

Dimensional Game Theory enables the development of adaptive systems that detect and exploit strategic imbalances. This realigns game theory with computational practicality and opens new frontiers in algorithmic design.

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Future Work

• Advanced dimension detection methods
• Multi-agent reinforcement integration
• Policy-aligned strategic simulations