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Foundational Mathematical Framework

Based on Rasmusen’s “Games and Information” and advanced game theory literature, THE STRATEGIST implements rigorous mathematical models for personal optimization.

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Core Nash Equilibrium Definition for 4-Agent Architecture

For Strategist’s 4-agent system: ∀i ∈ , ∀s’ᵢ ∈ Sᵢ: uᵢ(sᵢ, s₋ᵢ) ≥ uᵢ(s’ᵢ, s*₋ᵢ)** Where no internal agent can improve its utility by unilaterally changing strategy, given other agents’ optimal strategies. Mathematical Interpretation: 
Players: i = {1, 2, 3, 4} = {Optimizer, Protector, Explorer, Connector}
Strategies: sᵢ ∈ Sᵢ (agent i's strategy set)
Utility: uᵢ: S₁ × S₂ × S₃ × S₄ → ℝ (agent i's payoff function)
Equilibrium: s* = (s₁*, s₂*, s₃*, s₄*) where no unilateral deviation improves payoff

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Advanced Equilibrium Concepts for Personal Optimization

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Subgame Perfect Equilibrium

Critical for temporal decision-making. From Rasmusen’s analysis, ensures credible long-term strategies by eliminating non-credible threats. Mathematical Definition: A strategy profile is subgame perfect if it induces Nash equilibrium behavior in every subgame. Personal Application: Eliminates strategies like “I’ll exercise tomorrow if I don’t today” because the “tomorrow self” has no incentive to follow through. Backward Induction Algorithm: 
For time T, T-1, T-2, ..., 1:
  At time t, choose action that maximizes:
  uᵢ(aₜ) + δ · Vᵢ(t+1 | aₜ)
  
Where δ = discount factor, Vᵢ = continuation value

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Correlated Equilibrium

From Aumann’s theory - enables coordination through shared information sources. Mathematical Framework: 
Correlation Device: μ ∈ Δ(A₁ × ... × A₄)
Conditional Strategies: σᵢ(aᵢ | recommendation)
Incentive Compatibility: No agent wants to deviate from recommendation
Personal Application: External data sources (weather, stress levels, social commitments) coordinate agent strategies more effectively than independent decision-making. Example: 
{
  "correlation_device": "weather_forecast",
  "agent_strategies": {
    "rain_day": {
      "optimizer": "indoor_productivity_focus",
      "protector": "cozy_safety_prioritization",
      "explorer": "creative_indoor_projects",
      "connector": "intimate_social_connections"
    },
    "sunny_day": {
      "optimizer": "outdoor_efficiency_tasks", 
      "protector": "vitamin_d_health_optimization",
      "explorer": "adventure_seeking_behaviors",
      "connector": "large_group_social_activities"
    }
  }
}

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Evolutionary Stable Strategies (ESS)

From Maynard Smith’s biological framework, adapted for personal habit formation. Strategy s is ESS if:*
  • π(s, s) > π(s’, s*)** OR
  • π(s, s) = π(s’, s*) AND π(s*, s’) > π(s’, s’)**
Personal Interpretation: A habit configuration is evolutionarily stable if:
  1. It performs better against itself than any alternative habit
  2. If it performs equally well, it performs better against the alternative than the alternative performs against itself
Replicator Dynamics: 
ẋᵢ = xᵢ[f(eᵢ,x) - f(x,x)]

Where:
xᵢ = frequency of behavior i in your behavioral repertoire
f(eᵢ,x) = fitness of behavior i against current behavioral mix
f(x,x) = average fitness across all behaviors

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Advanced Mathematical Structures

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Utility Function Decomposition

Agent-Specific Utility Functions: Optimizer: u₁(s) = Σ(output_value) / Σ(resource_cost) Protector: u₂(s) = baseline_security - Σ(risk_exposure × impact) Explorer: u₃(s) = Σ(novelty_value × growth_potential) - stagnation_penalty Connector: u₄(s) = Σ(relationship_quality × interaction_frequency) - social_isolation_cost

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Strategic Complementarities and Substitutabilities

Supermodular Games: When agents’ strategies are strategic complements: 
∂²uᵢ(sᵢ, s₋ᵢ) / ∂sᵢ∂sⱼ > 0 for j ≠ i
Example: Higher Optimizer effort makes Explorer effort more valuable (learning accelerates with better systems) Submodular Games: When agents’ strategies are strategic substitutes: 
∂²uᵢ(sᵢ, s₋ᵢ) / ∂sᵢ∂sⱼ < 0 for j ≠ i
Example: Higher Protector effort makes Explorer effort less necessary (safety reduces need for exploration)

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Existence and Uniqueness Theorems

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Nash’s Existence Theorem

For finite games: Every finite game has at least one Nash equilibrium (possibly in mixed strategies). For continuous games (Strategist application):
  • Strategy sets Sᵢ are non-empty, convex, compact subsets of Euclidean space
  • Utility functions uᵢ are continuous and quasi-concave in sᵢ
  • Therefore, equilibrium exists

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Uniqueness Conditions

Contraction Mapping: If the best response function is a contraction, equilibrium is unique: 
||BR(s) - BR(s')|| ≤ α||s - s'|| for α < 1
Personal Application: Stable personalities tend toward unique equilibrium configurations.

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Bayesian Games and Incomplete Information

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Type Space Construction (Harsanyi Transformation)

When agents don’t know each other’s true preferences: 
Type Space: Θᵢ = possible preference types for agent i
Belief Function: μᵢ(θ₋ᵢ | θᵢ) = agent i's beliefs about others' types
Strategy: sᵢ(θᵢ) = action as function of type
Personal Application: Early in personal development, agents don’t know true preferences. Bayesian updating reveals personality through observed choices.

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Perfect Bayesian Equilibrium

Requirements:
  1. Sequential Rationality: Agents optimize at every decision point
  2. Belief Consistency: Beliefs derived from strategies using Bayes’ rule where possible
Personal Application: As you learn about yourself, agent coordination improves through better information about internal preferences.

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Mixed Strategies and Randomization

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When to Use Mixed Strategies

Indifference Condition: In equilibrium, agents must be indifferent between strategies in their support. Example: Exercise Timing 
Morning Gym Utility = Evening Gym Utility
p · (high_energy_workout) + (1-p) · (disrupted_schedule) = 
q · (lower_energy_workout) + (1-q) · (consistent_routine)
Solve for optimal mixing probabilities p and q

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Mixed Strategy Equilibrium Computation

For 2×2 subgames between agents: 
Agent 1 mixing: u₁(Up, Left) = u₁(Down, Left) and u₁(Up, Right) = u₁(Down, Right)
Agent 2 mixing: u₂(Up, Left) = u₂(Up, Right) and u₂(Down, Left) = u₂(Down, Right)

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Mechanism Design for Self-Control

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VCG Mechanisms for Internal Coordination

Truthful Reporting: Agents honestly report internal states Efficient Allocation: Resources distributed optimally across life domains Budget Balance: Total energy/attention allocation equals available resources VCG Payment Rule: 
Payment by agent i = Σⱼ≠ᵢ vⱼ(outcome_without_i) - Σⱼ≠ᵢ vⱼ(actual_outcome)
Personal Application: Agents “pay” attention costs based on social welfare they create/destroy.

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Myerson’s Optimal Mechanism

Virtual Valuations: Account for agent type distributions 
ψᵢ(vᵢ) = vᵢ - (1-Fᵢ(vᵢ))/fᵢ(vᵢ)
Revenue Maximization: Allocate resources to agents with highest virtual valuations, not just highest reported values.

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Repeated Games and Temporal Consistency

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Folk Theorem Application

Any individually rational payoff can be sustained if agents are sufficiently patient (δ → 1). Trigger Strategies for Habits:
  • Grim Trigger: Permanent punishment after deviation (too harsh)
  • Tit-for-Tat: Copy last period’s action
  • Generous Tit-for-Tat: Occasional forgiveness (optimal for personal development)

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Dynamic Programming for Life Optimization

Bellman Equation: 
V(state) = max{immediate_utility + δ · E[V(next_state)]}
Personal Application: 
V(health_state, energy_level, relationships) = 
max{u(daily_actions) + δ · E[V(tomorrow_state | daily_actions)]}
Mathematical rigor ensures optimal strategic configurations rather than intuitive guesswork. Next: Computational Methods →