The criminal-citizen-inspector game cycles — morie_inspection

Replicator dynamics for a rock-paper-scissors payoff with stake a (each strategy beats the next) has \(x^T A x = 0\) on the simplex (Research.P8.Inspection.payoff_zero_sum), the centroid as its only interior rest point (interior_fixed_point, only_interior_rest), and \(V = x_1 x_2 x_3\) as a conserved quantity (conserved): orbits are closed curves around the centroid, so more inspection does not monotonically reduce crime.

Usage

morie_inspection_game(a, x, steps = 1000L, dt = 0.01)

Arguments

a: Stake (non-zero).
x: Starting point on the simplex (three non-negative shares summing to one).
steps, dt: Number and size of fourth-order Runge-Kutta steps.

Value

A list: trajectory (a matrix with steps + 1 rows), invariant (\(V\) along the orbit), invariant_drift (its largest deviation from the start), dV_dt (the exact derivative at the start, zero), field (the vector field at the start) and fixed_point.

References

D'Orsogna MR, Perc M (2015). Statistical physics of crime: a review. Physics of Life Reviews 12, section IV, eq. 6.

Examples

g <- morie_inspection_game(a = 1, x = c(0.5, 0.3, 0.2), steps = 200)
g$invariant_drift
#> [1] 3.334832e-14