chse_fliptime — Leadership flip time and oscillation condition

Syntax

chse_fliptime, mu(varname|#) eta(varname|#) kappa(varname|#)
[rbar(#) h0(varname|#) epsilon(#) generate(stub) replace]

Description

chse_fliptime computes the expected first-passage time t* for the
hierarchy belief h(t) to cross the 0.5 threshold, using the two-player
benchmark model (Section 3, Definition 3.2):

t* = (1/mu_tilde) * ln( (h0 - 0.5) / epsilon )

where mu_tilde = |Re(lambda)| is the effective decay rate from the
linearised Jacobian eigenvalues:

lambda = ( -mu +/- sqrt(mu^2 - 4*eta_bar*kappa_bar) ) / 2

The oscillation condition (Definition 3.1) holds when the discriminant
mu^2 - 4*eta_bar*kappa_bar < 0, giving complex eigenvalues and oscillatory
dynamics. In this case the period is:

T = 2*pi / Im(lambda)

When the hierarchy is stable (h* > 0.5 and no oscillation), t* = inf:
leadership never flips deterministically.

Options

mu(varname|#) mean-reversion strength mu > 0.

eta(varname|#) equilibrium reframing rate eta_bar.

kappa(varname|#) equilibrium resistance rate kappa_bar.

rbar(#) reframing attack rate r_bar. Default 1.0.

h0(varname|#) initial hierarchy belief. Default 0.75.
Must be > 0.5 for player 1 to be the initial leader.

epsilon(#) flip precision threshold. Default 0.01.

generate(stub) creates variables: stub_tstar, stub_disc,
stub_oscillates.

Saved results (scalar call)

r(t_star) Flip time (. = stable/never flips)
r(discriminant) mu^2 - 4*eta*kappa
r(mu_tilde) Effective decay rate
r(h_star) Fixed point h*
r(oscillates) 1 if oscillation condition holds
r(period) Period (if oscillatory)
r(regime) stable / oscillatory / stable_node

Examples

Oscillatory regime (HSI=1.0, disc < 0):

. chse_fliptime, mu(0.6) eta(0.4) kappa(0.4)
// t* = 10.73, regime = oscillatory, period = 23.75

Stable regime (HSI=2.1, disc > 0):

. chse_fliptime, mu(2.0) eta(0.8) kappa(0.2)
// t* = ., regime = stable (h* > 0.5)

Panel of parameter settings:

. chse_fliptime, mu(mu_var) eta(eta_var) kappa(kappa_var) gen(ft) replace
. summarize ft_tstar ft_disc ft_oscillates

References

Nityahapani (2025). Contested Hierarchy with Social Embedding.
Section 3, Definitions 3.1 and 3.2.

Author

Nityahapani

chse package v1.0.0