Supplement to “ Strategic Learning and the Topology of Social Networks ” : Examples

IN THIS SUPPLEMENT, we give two examples showing that the assumptions of bounded out-degree and L-connectedness are crucial. Our approach in constructing equilibria will be to prescribe the initial moves of the agents and then extend this to an equilibrium strategy profile. Define the set of times and histories agents have to respond to as H = {(i t a) : i ∈ V t ∈ N0 a ∈ [0 1] × {0 1}|N(i)|·t}. The set [0 1] × {0 1}|N(i)|·t is interpreted as the pair of the private belief of i and the history of actions observed by agent i up to time t. If a ∈ [0 1] × {0 1}|N(i)|·t , then for 0 ≤ t ′ ≤ t, we let at′ ∈ [0 1] × {0 1}|N(i)|·t denote the history restricted to times up to t ′. We say that a subset H⊆H is history-closed if, for every (i t a) ∈H, we have that for all 0 ≤ t ′ ≤ t that (i t ′ at′) ∈H. For a strategy profile Q̄, denote the optimal expected utility for i under any response as u i (Q̄)= supR̄ ui(R̄), where the supremum is over strategy profiles R̄ such that R =Qj for all j = i in V .