The Adaptive Dynamics of Turnout: Supplementary Material

In this document, we describe in greater detail how we arrived at the continuous-time model. We accomplish several tasks here. First we establish that the turnout functions exist, and we present the continuous-time turnout functions in the case that one party wins all the elections on some interval of time. Second, we derive the α(t) formalism we employed in Section 1.2; this is our most important result, because the differential equations of the continuous-time model follow immediately from this formalism. Third, we establish conditions on the existence of ties in the continuous-time model. Because we will consider turnout in discrete as well as continuous time, it is necessary to distinguish between functions defined on finite sets and infinite sets. We will consider turnout over some finite interval [0, T ], and we suppose that on this interval there are exactly M time steps. Thus, u(t) is defined for t = mδ, where m ∈ 0, 1, 2, . . ., δ = T/M and v(t) is defined for t ≥ 0. We define vδ(t) as a continuous-time interpolation of u(t). For all t ∈ [0, T ],