1. Modified Utility Function with Relational Zone Overlay
Let the classical utility or payoff from strategy profile s=(si,sj)s = (s_i, s_j) be denoted:
Ui(s)=i(si,sj)U_i(s) = \pi_i(s_i, s_j)
We extend this by defining an augmented utility function that includes the relational zone value:
U~i(s,t)=i(si,sj)+Rij(t)\tilde{U}_i(s, t) = \pi_i(s_i, s_j) + \beta \cdot R_{ij}(t)
Where:
U~i(s,t)\tilde{U}_i(s, t): Augmented utility for agent ii considering both material payoff and relational context.
\beta: Sensitivity coefficient representing the agent's valuation of relational dynamics in utility terms.
Rij(t)R_{ij}(t): The relational value function from subsection A.
This formulation reflects real-world behaviors such as:
Firms accepting suboptimal short-term profits for strategic trust-building,
Governments tolerating inefficient outcomes to preserve diplomatic alliances,
Consumers choosing ethical products despite price disadvantages.
2. Relational-Zone-Weighted Strategy Selection
Let the probability of choosing a strategy siSis_i \in S_i be influenced by the zone in which agent ii perceives agent jj. Define a zone-weighted strategy function:
P(siZij(t))=eU~i(si,sjZij(t))siSieU~i(si,sjZij(t))P(s_i | Z_{ij}(t)) = \frac{e^{\gamma \cdot \tilde{U}_i(s_i, s_j | Z_{ij}(t))}}{\sum_{s_i' \in S_i} e^{\gamma \cdot \tilde{U}_i(s_i', s_j | Z_{ij}(t))}}
Where:
