Equilibrium of two policy-motivated candidates under electoral uncertainty

Candidates k = 1, 2 have quadratic losses \(u_k(y) = -\|y - a_k\|^2\) and care only about the policy that wins (Wittman 1983). The decisive voter sits at \(m \sim N(c, s^2 I)\) and elects the nearer platform, so candidate 2 wins with probability \(\Phi(((c - (x_1 + x_2)/2)^\top (x_2 - x_1)) / (s \|x_2 - x_1\|))\) (Calvert 1985). Candidate k maximises \(P_k u_k(x_k) + (1 - P_k) u_k(x_{other})\). The equilibrium is found by alternating best responses (BFGS on the analytic gradients) from start (default: each ideal point pulled towards the centre, \(c + t (a_k - c)\) with \(t = s / (s + \|a_k - c\|)\)), then polished by Newton steps on the joint first-order conditions. In one dimension with ideal points \(\pm a\) and \(c = 0\) the platforms are \(\pm a / (1 + 2 a \phi(0) / s)\): divergent, and converging to the centre as \(s\) shrinks.

Usage

PolicyMotivatedEquilibrium(
  ideal_1,
  ideal_2,
  center,
  sd,
  start = NULL,
  tol = 1e-10,
  max_iter = 500L
)

Arguments

ideal_1, ideal_2

Candidate ideal points (vectors of length d).

center

Mean \(c\) of the decisive voter's position.

sd

Standard deviation \(s\) of each coordinate of that position.

start

Optional list of starting platforms.

tol

Stop when no platform moves more than tol.

max_iter

Maximum best-response rounds.

Value

List with x1, x2, win_prob_2, utilities, gradients, rounds, converged.

References

Wittman, D. (1983). Candidate motivation: a synthesis of alternative theories. American Political Science Review 77, 142-157.

Calvert, R. L. (1985). Robustness of the multidimensional voting model: candidate motivations, uncertainty, and convergence. American Journal of Political Science 29, 69-95.

Examples

PolicyMotivatedEquilibrium(-1, 1, 0, 0.5)[c("x1", "x2")]
#> $x1
#> [1] -0.3852423
#> 
#> $x2
#> [1] 0.3852423
#>