Zij(t){W,G,Y,R,B,C}Z_{ij}(t) \in \{W, G, Y, R, B, C\}: The current relational zone (White, Green, Yellow, Red, Black, Clear),
\gamma: Rationality/temperature parameter (similar to logit choice models),
U~i(Zij)\tilde{U}_i(\cdot | Z_{ij}): Augmented utility conditioned on current zone,
P(siZij(t))P(s_i | Z_{ij}(t)): Likelihood of strategy sis_i given the zone agent ii assigns to agent jj.
Thus, strategy selection is not purely payoff-maximizing, but zone-sensitive. For example:
In Green Zone, cooperation strategies (e.g., Tit-for-Tat) become highly probable.
In Yellow Zone, agents may select ambiguous or mixed strategies (e.g., Grim Trigger, Suspicious Tit-for-Tat).
In Black Zone, retaliatory or destructive strategies dominate.
3. Transition Coupling: Payoff-to-Zone Feedback
The framework also accommodates bidirectional feedback between strategies/payoffs and relational zones. Specifically, we define:
dRij(t)dt=f(i(t),j(t),si(t),sj(t))+Iij(t)Lij(t)\frac{dR_{ij}(t)}{dt} = \alpha \cdot f\left( \pi_i(t), \pi_j(t), s_i(t), s_j(t) \right) + \lambda \cdot I_{ij}(t) - \delta \cdot L_{ij}(t)
Where:
\alpha: Responsiveness of relationship to payoff and strategy dynamics.
f()f(\cdot): A function mapping payoffs and strategic behavior to relational impact (e.g., defecting in cooperative expectation sharp decrease in RR).
Lij(t)\delta \cdot L_{ij}(t): Decay due to latent distrust, memory of past betrayal, or observed inconsistency.
This equation describes relational inertia and momentum: the result of strategies not only impacts current outcomes but reconfigures the relational space of future interactions.
Theoretical Implications:
Challenging the Stationarity Assumption: Nash Equilibrium assumes strategic consistency under static payoffs. Our model reveals how zone drift and relational volatility disrupt equilibrium persistence.
Incorporating Ambiguity and Memory: Repeated games often presume perfect recall or discounting. Our model introduces nonlinear memory, zone stickiness, and relational hysteresis.
From Equilibrium to Trajectory: Rather than solving for a fixed point (Nash, Subgame Perfect Equilibrium), RZE proposes that economic behavior unfolds as trajectories through a zone landscape --- with path-dependence, emergent trust, betrayal cascades, or institutional lock-in.
CHAPTER 5. Simulation and Analysis
A. Multi-Zone Agent Simulation in Long-Term Investment
1. Rationale for Simulation Approach
