Projection / capability. Model another decision-maker as an endogenous center of choice rather than a fixed disturbance.
Question this mathematics answers. What happens when another agent changes strategy in response to mine?
Equation family
Multi-agent dynamics: ẋ=Ax+B₁u₁+B₂u₂Best response: uᵢ∈argmax_{uᵢ} Uᵢ(uᵢ,u_{−i})Nash equilibrium: each strategy is a best response to the others
Symbols. u₁,u₂ = independent agent actions; Uᵢ = agent i payoff/objective; u_{−i} = other agents’ strategies.
Assumptions. The game/model must specify agents, strategy sets, information, and utilities/objectives. Equilibrium claims are relative to that model.
Where the inputs come from. Payoffs/constraints come from the designed interaction or estimated preferences; explicit consent/refusal is an observed action/authority state, not a model probability.
Short derivation / justification
If agent 2 minimizes J₂(u₂)=(ẋ−r₂)²+ρu₂² and ẋ contains u₁+u₂, differentiating J₂ with respect to u₂ yields an endogenous best response that changes when u₁ changes. That is the strategic turn: u₂ is not fixed noise.
Worked example
In a Prisoner’s-Dilemma payoff table, (D,D) can be Nash because neither player can improve unilaterally even though both would receive more at (C,C). Equilibrium therefore has no moral halo.
Cadet or real-world application. Human-machine teaming, decentralized systems, negotiation, mechanism design, shared-resource policies.
What to look for. Ask whether another participant can choose back, refuse, or respond strategically.
Common misinterpretation. “Predicted agreement is consent” or “equilibrium is fair/optimal.”
Break / domain boundary. Game theory predicts/characterizes strategic consistency under modeled preferences; it does not create authority or legitimacy.