Estimating the Long-term Behavior of Biologically Inspired Agent-based Models

The paper defines the transition and production regions B_{V,U}^t and C_{V,U}^t, establishes the key identifications P(s(f(α,X_t))=U | s(α)=V)=P(p(α)∈B_{V,U}^t) and P(s(β)=U | s(α)=V, β∈g(α,X_t))=P(p(α)∈C_{V,U}^t), and then derives the GRR (its Eq. (3)) by splitting the update into transition and production contributions and summing/conditioning over current states V, exactly as in the candidate’s argument . The candidate solution mirrors this derivation via indicator/count decompositions and linearity of expectation, reaching the same expression. A minor notational hiccup in the model solution is the momentary summation over a random index set when taking expectations, which is easily fixed by writing the production count as a sum of indicators before taking expectations. Aside from this small point, both are consistent and essentially identical in structure. The paper’s framework and the update rule Xt+1=f(Xt,Xt)∪g(Xt,Xt) are used as intended .