Appendix

Proof of Proposition 1

Consider the first case and suppose that \({\hat{e}}\left( s\right)

e s\), and all unpopular leaders are removed from office after one period. The value to the selectorate along the equilibrium path is \(\frac{\rho A+\left( 1-\rho \right) {\bar{A}}\left( \rho \right) +\Delta }{1-\beta }\). Let

$$\begin{aligned} W\left( x\right) =x+\beta \left[ \frac{\rho A+\left( 1-\rho \right) {\bar{A}} \left( \rho \right) +\Delta }{1-\beta }\right] . \end{aligned}$$

Retaining popular incumbent yields \(W\left( A\right) .\) Deviating by removing such an incumbent makes the selectorate worse off since \(W\left( A\right) >W\left( {\bar{A}}\left( \rho \right) \right) \). Now consider whether there could be a worthwhile deviation by retaining an unpopular incumbent rather than picking a new incumbent at random. This will not be the case either since \(W\left( -A\right) <W\left( {\bar{A}}\left( \rho \right) \right) \). Hence there is no worthwhile one-shot deviation for the selectorate. Since the probability that an incumbent is retained is independent of \(\delta \), it is optimal for all incumbents to set \(e

e s\) for all \(c>0\). \(\square \)

Proof of Proposition 2

We first show that it is optimal for the selectorate in such cases to retain the offspring of leaders in this case if they produce \(\Delta \) when the out-of-equilibrium beliefs are that if the leader choose \(e

e s\), then there is an infinite reversion to playing the benchmark equilibrium where \( e

e s\) for all leaders and only popular leaders are retained. In the benchmark equilibrium, the payoff along the equilibrium is

$$\begin{aligned} \frac{\rho A+\left( 1-\rho \right) {\bar{A}}\left( \rho \right) }{1-\beta }. \end{aligned}$$

In the proposed hereditary equilibrium, the payoff is:

$$\begin{aligned} \frac{{\bar{A}}\left( \rho \right) +\Delta }{1-\beta }. \end{aligned}$$

Suppose now that the incumbent leader has an unpopular offspring then retaining that individual is optimal if

$$\begin{aligned} -A+\Delta +\beta \left[ \frac{{\bar{A}}\left( \rho \right) +\Delta }{1-\beta } \right] \ge {\bar{A}}\left( \rho \right) +\beta \left[ \frac{\rho A+\left( 1-\rho \right) {\bar{A}}\left( \rho \right) }{1-\beta }\right] \end{aligned}$$

which reduces to the condition above. Clearly, if this condition holds, it will hold a fortiori if the incumbent’s offspring is popular. This equilibrium exists as long as \(\left( 1-\rho \right) B\ge c\) . This is because if the incumbent deviates to \(e

e s\), then his incumbent will be retained in office with probability \(\rho \). However, if he chooses \(e=s\), then his offspring will hold office for sure. \(\square \)