Stochastic Dynamic Models for Political Regime Change and ...

R n (0) = 1 and R n (t + 1) = R n (t) + λ(1 − f).
O n = 1, constant over time.
The effect of interaction among the individuals in the population is given by the following
parameters depicting the tendency to switch:
Y → X β 1
X → Y β 1
Y → Z β −
X → Z β −
Z → Y β +
Z → X β +
The β parameters must be o( 1 N
) to keep the interaction effect in control. We choose the
following values:
β + = 0.3
β − = 0.2
β 1 = 0.1
With the above, the transition probabilities in the simplified 3-dimensional model can be
evaluated as follows (omitting time subscripts):
P (Y → X)
P (Z → X)
P (X → Z)
P (Y → Z)
P (X → Y )
P (Z → Y )
P (φ)
= γ 1 (R n )(1 − q) Y N + β 1 XY
N
= γ 0 (R n ) Z N + β + XZ
N
X(Y +Z)
= β − N
+ λfp X N
= β −
Y (X+Z)
N
= γ 1 (O n )(1 − q) X N + β 1 XY
N
= γ 0 (O n ) Z N + β + Y Z
N
= 1 - all of the above
+ λ(1 − f)(1 − p) Y N
+ λ(1 − f)p Z
N
+ λf(1 − p) X N
Hence, the change expectations are as follows:
E(∆X) = −(λf + γ 1 (O n )(1 − q)) X N + (γ 1(R n )(1 − q) + λ(1 − f)(1 − p)) Y N
+(γ 0 (R n ) + λ(1 − f)p) Z N − β XY
−
N
+ (β + − β − ) XZ
N
E(∆Y ) = (λf(1 − p) + γ 1 (O n )(1 − q)) X N − (γ 1(R n )(1 − q) + λ(1 − f)(1 − p)) Y N
+γ 0 (O n ) Z N − β XY
−
N
+ (β + − β − ) Y Z
N
E(∆Z) = λfp X N − (γ 0(R n ) + γ 0 (O n ) + λ(1 − f)p) Z N
XY
+2β −
N
− (β + − β − ) XZ
N
− (β + − β − ) Y Z
N
This is by construction closed, E(∆X) + E(∆Y ) + E(∆Z) = 0.
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