Order-of-Magnitude Estimates for v and σV

We obtain order-of-magnitude estimates for both v and σV from the typical margin of victory and the typical accuracy of national polls in American and French presidential elections. Let g be the probability density of the voter distribution. Initially, we focus on the expected value of v + X, which is v. Elementary algebra shows that this term, v, shifts the cut-point between supporters of D and of R by the quantity v/[2(r − d)] and thus adds to D's vote share a proportion given by the integral of g over an interval of length v/[2(r − d)].

We use the traditional 1–7 scales of the American National Election Study and French Presidential Election Study. Because the interval just described is typically near the middle of the scale where, empirically, values of g are on the order of 0.20, that integral is of the order of 0.20v/[2(r − d)]. Typically, in American and French elections, d is near 3.0 and r is near 5.0 (or 6.0), and the vote share of the winner ranges from 50 percent to about 62 percent – that is, a range of about twelve percentage points, or 0.12 when expressed as a proportion. Because variation in the winner's margin from election to election is due only in part to valence effects, this represents only an upper bound for the effect of valence characteristics on vote share. For this upper bound for v, we have, very roughly, 0.12 = 0.20v/[2(5.0 – 3.0)], or v ≅ 2.4. Thus values of v in the range from 0 to 2 or 3 appear to be plausible.