The Interaction layer specifies the structural constraints that arise when multiple decision-makers act within a shared environment. Unlike Choice, which concerns internal coherence of individual decisions, Interaction governs joint admissibility: which combinations of actions, strategies, and outcomes are mutually compatible given the strategic interdependence of agents.
The laws at this level do not aggregate behavior and do not rely on psychological assumptions. Instead, they restrict the logical and incentive structure of multi-agent systems. They determine when a profile of actions can persist without profitable deviation, when strategies are internally coherent across contingencies, and when institutional or mechanism rules successfully align incentives with prescribed behavior.
Interaction laws operate by ruling out incompatible configurations of actions or paths of play. They define feasibility, equilibrium, incentive alignment, and impossibility at the level of strategic coordination. Where Choice ensures that individual decisions are well-defined, Interaction ensures that joint outcomes are stable, admissible, or provably unattainable.
SAT – Structure – Laws / Relations – Interaction (Markets, Strategy & Mechanisms)
| Law | Description | Syntax | Constrained Object | Invariant | Condition |
|---|---|---|---|---|---|
| Nash Equilibrium (Mutual Best Response) | No agent can improve payoff by unilateral deviation given the actions of others. | Equilibrium | State space | Stability – fixed point of best responses | Structural assumptions – defined strategies and payoffs |
| Joint Strategy Feasibility | Only strategy profiles admissible for all agents simultaneously are allowable. | Constraint | State space | Balance – joint admissibility | Institutional / boundary rules – game rules |
| Equilibrium Existence Theorem | At least one equilibrium exists under specified structural conditions. | Identity | State space | Stability – existence of equilibrium | Structural assumptions – compactness, continuity |
| Dominance Principle | Strictly dominated strategies cannot be part of any admissible outcome. | Identity | State space | Optimality – non-dominated strategies | Structural assumptions – payoff comparability |
| Incentive Compatibility | Prescribed actions are aligned with agents’ incentives under the governing rules. | Identity | State space | Optimality – incentive alignment | Institutional / boundary rules – mechanism structure |
| Strategic Consistency | Strategies form coherent contingent plans across all decision nodes. | Identity | Trajectory space | Stability – plan coherence | Structural assumptions – complete contingent strategies |
| Subgame-Perfect Equilibrium | No profitable deviation exists along any feasible continuation path. | Dynamic | Trajectory space | Optimality – pathwise non-improvement | Structural assumptions – sequential rationality |
| Equilibrium Selection Robustness | Small perturbations select equilibria in a systematic, non-arbitrary way. | Dynamic | Trajectory space | Stability – robustness to perturbation | Scale / regime – small perturbations |
| Mixed Strategy Validity | Randomized strategies must satisfy probability coherence constraints. | Probabilistic | Trajectory space | Distribution – valid probability measures | Statistical conditions – independence |
| Impossibility Theorems | Certain sets of desirable interaction properties cannot be jointly satisfied. | Identity | State space | Balance – incompatible requirements | Structural assumptions – conflicting criteria |
Taken together, the laws of Interaction delineate the boundary between coordination and contradiction in multi-agent systems. They specify which strategy profiles can coexist, which equilibria can exist, which deviations are forbidden, and which combinations of desirable properties cannot be simultaneously achieved. Violations at this level do not reflect inefficiency or misbehavior; they indicate strategic incompatibility within the defined rules of interaction.
These laws are inherently conditional. Their force depends on clearly stated assumptions about strategy spaces, information, institutional rules, and admissible deviations. When those conditions hold, the constraints are absolute; when they fail, the law does not weaken—it ceases to apply. This conditionality is not a flaw but a defining feature of strategic structure.
Interaction therefore provides the bridge between individual choice and system-level outcomes. It does not explain aggregate behavior, nor does it prescribe optimal institutions. It establishes the logical limits of coordination, within which markets, mechanisms, and strategic systems must operate before any aggregation or policy analysis can meaningfully occur.