We define a formal function to represent the relational value between two economic agents i and j at time t, denoted by:
Rij(t)=k=1nwkVk,ij(t)+Iij(t)+CR_{ij}(t) = \sum_{k=1}^n w_k \cdot V_{k,ij}(t) + \lambda \cdot I_{ij}(t) + C
Variable Definitions:
Rij(t)R_{ij}(t): The relational zone value or cumulative strength of the relationship between agent i and j at time t.
Vk,ij(t)V_{k,ij}(t): The k-th vector component of interaction between i and j (e.g., trust, reciprocity, perceived fairness, strategic ambiguity, etc.) at time t.
wkw_k: The weight assigned to each component VkV_{k}, reflecting its relative importance in the relational context (bounded: k=1nwk=1\sum_{k=1}^n w_k = 1, with wkRw_k \in \mathbb{R}).
Iij(t)I_{ij}(t): A function denoting long-term intent alignment between agents i and j, incorporating forward-looking, ambiguous, or symbolic strategies that may not yield immediate payoffs.
\lambda: A scalar parameter representing the sensitivity of the system to future-oriented alignment; it modulates the influence of Iij(t)I_{ij}(t).
CC: A constant capturing baseline relational proximity, institutional norms, or inherited trust/distrust (initial conditions).
Interpretation and Theoretical Justification:
The Relational Value Function (RVF) provides a formal mechanism for calculating an agent's strategic zone relative to another in dynamic economic environments. Unlike classical utility functions or payoff matrices, which assume immediate and discrete responses, the RVF:
1. Aggregates multiple dimensions of interaction --- trust, commitment, ambiguity tolerance, memory of past behavior, and strategic posture --- into a unified relational construct.
2. Integrates long-term alignment Iij(t)I_{ij}(t) to capture the intentional stance, where economic actors are not only driven by current incentives, but also by strategic signaling, moral posturing, and reputational investments.
3. Allows for zone-based classification:
Rij(t)0R_{ij}(t) \approx 0 White (neutral)
Rij(t)>GR_{ij}(t) > \theta_G Green (cooperative)
Y<Rij(t)<Y+\theta_Y^- < R_{ij}(t) < \theta_Y^+ Yellow (ambiguous)
Rij(t)<RR_{ij}(t) < \theta_R Red (conflictual)
Rij(t)BR_{ij}(t) \ll \theta_B Black (destructive betrayal)
Rij(t)>JR_{ij}(t) > \theta_J & sustained Clear (vision-aligned)
Where \theta_* denotes calibrated threshold values obtained either empirically or via simulation depending on context.
Operational Implication:
By modeling the dynamic value of a relationship using the RVF, we shift the focus of economic interaction from transactional payoff to relational trajectory. This allows:
Analysis of nonlinear trust decay or reinforcement (e.g., betrayal may cause abrupt drop in Rij(t)R_{ij}(t), while trust grows slowly).
Simulation of path-dependent interactions (e.g., once in Red or Black Zone, very high future alignment IijI_{ij} may still fail to repair damage due to memory encoded in past Vk,ij(t)V_{k,ij}(t)).
Evaluation of policy or institutional reforms not only on economic output, but on relational health of economic agents or sectors.
B. Integration with Payoffs and Strategy in Game Theory
While classical game theory models interactions in terms of payoff matrices and strategic equilibrium points, our Relational Zone Economic (RZE) framework extends these formulations by embedding them within a relational zone dynamic. This offers a richer modeling of strategic interaction that accounts for historical memory, future-oriented intent, and relational ambiguity.
