Stochasticity and quadratic bias results for HODs obtained by Monte Carlo sampling from a wide volume of HOD parameter space. Separate searches are performed with assembly bias parameters set to zero (red points, 500 HODs) and with the four assembly bias parameters free (blue points, 200 HODs). Left panel: Stochasticity parameters α₁ versus α₀. Middle panel: α₀ versus quadratic bias b₂. Right panel: α₁ versus b₂, showing a tight degeneracy for the HODs without assembly bias (red points). The relation between b₂ and α₁ for these points is fit by the cubic polynomial ${\alpha }_1(b_2)=-1.1b_2^3+0.59b_2^2-0.56b_2+0.01$ (dashed black line). Dashed gray lines at α₀ = 1, α₁ = 0, and b₂ = 0 indicate the values corresponding to Poisson shot noise and no quadratic bias. The HODs shown are selected based on the constraints of Eq. (14), with a 2% tolerance on the values of linear bias and density. We note that the red HODs (no assembly bias) are a sample from a lower-dimensional subspace of the parameter volume from which the blue HODs with assembly bias are drawn. Each point represents the mean parameters of 20 galaxy samples generated from a single HOD.