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\begin{document}

\title{The problem of succession\\
Preliminary draft\\
For discussion only}
\author{Leon Taylor\thanks{%
Send correspondence to Leon Taylor, 7883 Tall Pines Court, Apartment I, Glen
Burnie, Md. 21061; or e-mail to taylorleon@aol.com}}
\maketitle
\tableofcontents

\begin{abstract}
The self-interested ruler will not name a successor. Instead, he will prefer
to rely on a tacit rule of succession, in which longevity confers
legitimacy. [JEL\ D72, D81]
\end{abstract}

\section{Introduction}

For most organizations, steady growth depends in part on the passing of the
crown. \ For instance, Russia's disruptive transitions of power may have
robbed it of stability and wealth \cite{hingley}. The problem of succession
extends beyond governments: Many disputes in admissions, employment and
promotion concern how best to allocate vacant positions.

Where no rule of succession is accepted by all, the organization may fall
back upon an implicit rule:\ If the current leaders have long held power,
then they must deserve to stay in power. For example, after the political
turmoil of the Time of Troubles in Russia, early in the 17th century, the
acceptance of the Romanovs as rulers (if not of a particular Romanov) seems
to have increased with their duration in power.

This note proposes a model of the likelihood that a cadre of leaders will
last under the implicit rule of longevity. With refinement, such models may
eventually help analysts anticipate turning points in countries that lack a
credible constitution.

\section{Analysis}

\subsection{The longevity of the regime}

A \textit{regime} is the period in power of a group that identifies with a
family, area or policy. Let $i$ index the number of years that a given
regime has continued, $i=1,2,...$. Let $X_{n}=i_{n}$ denote that the regime,
in year $n$, has been in power for $i$ years. \ The probability of extending
a regime depends only on the variable $X_{n}$. The longer that a regime has
remained in power, the more likely that it will remain in power for one year
more. \ Denote the probability that the regime will extend from $i$ years to 
$i+1$ as $P_{i,i+1}$. We can draw upon the analysis of Markov processes \cite%
{ross} to state the probability more precisely:

\[
P_{i,i+1}=P\{X_{n+1}=i_{n}+1|X_{n}=i_{n}\}=f(i_{n}),\ f^{\prime }>0. 
\]

Either the regime continues for one more year, or it stops:

\[
P_{i,i+1}+P_{i,i}=1. 
\]

As a special case, regimes that rarely regain power after losing it may be
stylized with

\[
P_{i,i}=1\ when\ X_{n-1}=i. 
\]

The probability of continuing the regime depends simply on longevity. So,
specify the model in a simple way:

\begin{equation}
P_{i,i+1}=1-p^{i_{n}}  \label{model1}
\end{equation}
and

\begin{equation}
P_{i,i}=p^{i_{n}}  \label{model2}
\end{equation}%
where $0<p<1$. \ 

To estimate the model, equate $p$ to the value that just renders $%
P_{i,i+1}<.5$ when a given regime actually ends. The model may then be used
to crudely predict the duration in power of a similar cadre where there is
no widely accepted rule of succession.

An example comes from the tumultuous politics of Mexico between the 1910
revolution and World War II. During that period, several coups deposed
elected leaders (see table) \cite{columbia}. The mean length of term was 3
years ($\sigma =2$). Thus $p^{3}=.5$, or $p=.8$. Under these conditions, a
regime that remained in power for 10 years would have had a 90 percent
chance of continuing for at least one more year.

\begin{tabular}{|l|l|}
\hline
\textbf{President of Mexico} & \textbf{Start date} \\ \hline
Francisco Madero & 1911 \\ \hline
Victoriano Huerta & 1913 \\ \hline
Venustiano Carranza & 1914 \\ \hline
\`{A}lvaro Obreg\`{o}n & 1920 \\ \hline
Plutarco El\`{\i}as Calles & 1924 \\ \hline
\`{A}lvaro Obreg\`{o}n & 1928 \\ \hline
Portes Gil & 1928 \\ \hline
Ortiz Rubio & 1930 \\ \hline
Abelardo Rodr\`{\i}guez & 1932 \\ \hline
L\`{a}zaro C\`{a}rdenas & 1934 \\ \hline
Manuel \`{A}vila Camacho & 1940 \\ \hline
\end{tabular}

\subsection{The ruler's choice of a law of succession}

Why doesn't the ruler adopt a clear law of succession? Because such a law
will remove all uncertainty only if it covers all contingencies; and it is
in the immediate interest of the ruler not to do so. Instead, he can
increase his power by playing off potential successors against one another
for as long as he can.

The ruler seeks a degree of eligibility for successors that will maximize
over time his welfare function, which may (or may not) reflect the welfare
of his nation. This problem, for the ruler, is deterministic:\ Given any
positive probability $P_{i,i+1}$ of extending the regime for a year, the
ruler will plan now to maximize his welfare for his remaining rule,
regardless of what will actually happen.

Let $s$ be the share of all feasible candidates who are declared eligible
for succession, $0\leq s\leq 1$. In short, $s$ is the law of succession. The
ruler receives utility from power, which increases in $s$, a function of
time $t$: $U[s(t)]$, $\partial U/\partial s>0,\ \partial ^{2}U/\partial
s^{2}<0$. But the welfare of the nation, $W$, decreases as $s$ increases,
since an unclear law of succession creates uncertainty. For $W[s(t)]$, $%
\partial W/\partial s<0,\ \partial ^{2}W/\partial s^{2}<0$. When picking the
law of succession at time $0$, the ruler expects to remain in power until
time $T$. The welfare of the nation from time $T$ on may be summarized in
the function $Z[s(T)]$, $\partial Z/\partial s<0,\ \partial ^{2}Z/ds^{2}<0$.

The ruler chooses the function $s(t),\ 0<t\leq T$, given $s(0)=s_{0},$ to
maximize

\begin{equation}
V=\int_{0}^{T}\delta _{1}U[s(t)]+\delta _{2}W[s(t)]\ dt\ +(1-\delta
_{1}-\delta _{2})Z[s(T)],  \label{opt}
\end{equation}
where $\delta _{i}$ are welfare weights.

Any solution $s^{*}(t)<1$ to the static problem in (\ref{opt}) must satisfy

\[
\delta _{1}\frac{\partial U}{\partial s}\leq -\delta _{2}\frac{\partial W}{%
\partial s} 
\]
for $0<t<T$, and

\[
\delta _{1}\frac{\partial U}{\partial s}\leq -\delta _{2}\frac{\partial W}{%
\partial s}-(1-\delta _{1}-\delta _{2})\frac{\partial Z}{\partial s} 
\]%
at $t=T$. The purely self-interested ruler ($\delta =1$) will never narrow
the field of successors ($s^{\ast }(t)=1,\ 0<t\leq T$). If he has some
interest in current national welfare but none in the future, then he will
again pick a constant rule of succession. If he has some interest in the
future, then he will pick a rule of succession that narrows the field
discontinuously at time $T$. \ That is, $s^{\ast }(T)<<s^{\ast }(T-\epsilon
) $. He will name just one successor only if he is purely altruistic ($%
\delta _{1}=0$).

If a permanent rule of succession $L$ has been imposed on the ruler, then
his choice of $s(t)$ is constrained to $0<s(t)\leq L\leq 1$. If the rule is
to be imposed at time $T_{1}$, then $L=1$ for $t<T_{1}$. Neither case
changes the economic substance of the solution:\ The ruler will not name the
successor unless compelled to do so.

\subsubsection{The dynamic problem}

Suppose that narrowing the field of potential successors increases the
probability of an illegal removal, such as an assassination, of share $A$ of
the successors$,\ 0\leq A\leq 1$, as now-disappointed pretenders seek the
throne through violence. Then model $A=A[s^{\prime }(t),t]$ where $\partial
A/\partial s^{\prime }\leq 0$. Also, $\partial A/\partial t\geq 0$: The
increasing longevity of the regime may increase the probability of an
assassination, particularly when the regime permits pretenders few
nonviolent means of challenge.

Broaden the ruler's own welfare function to $U_{1}[s(t),A(t)]=U_{2}[s(t),s^{%
\prime }(t),t].$ Again, $\partial U_{2}/\partial s>0$. But also, $\partial
U_{2}/\partial s^{\prime }=(\partial U_{1}/\partial A)$ $\partial A/\partial
s^{\prime }\geq 0$:\ Adding candidates to the list may make assassination
less likely for the leader as well as for the successors.\footnote{%
If the ruler's life is not at risk, then he may gain bargaining power from
threats to remove some of his potential successors. In that case, $\partial
U_{1}/\partial A>0$. That renders $\partial U_{2}/\partial s^{\prime }\leq 0$%
.}

\ Also assume that $\partial U_{1}/\partial A=0$ when $A=0$:\ The ruler
risks nothing from a mild threat arising from a risk-free environment.
Finally, assume that $\partial U_{2}/\partial t\leq 0$: As time passes, the
ruler loses room for bargaining, until he can no longer delay naming a
successor.

Likewise, rewrite society's welfare function as $%
W_{1}[s(t),A(t)]=W_{2}[s(t),s^{\prime }(t),t]$. Here, $\partial
W_{2}/\partial s<0$:\ Given that the list of potential successors includes
the best candidate, extending the list can only create uncertainty.\footnote{%
At the end of his regime, the ruler has gained all that he can from
negotiations with potential successors. \ As a citizen, it is then in his
interest to select the strongest candidate for the nation.} \ Also,$\
\partial W_{2}/\partial s^{\prime }\geq 0$ in a society where the risk of
assassination is large enough: Widening the list reduces the risk of
assassination and of subsequent uncertainty.\footnote{%
In a society where assassination is unlikely, extending the list of
successors more rapidly may accelerate uncertainty. In that case, $\partial
W_{2}/\partial s^{\prime }\leq 0$.
\par
{}}

Generalize the objective functional to:

\[
V_{1}=\int_{0}^{T}\delta _{1}U_{2}[s(t),s^{\prime }(t),t]+\delta
_{2}W_{2}[s(t),s^{\prime }(t),t]\ dt\ +(1-\delta _{1}-\delta _{2})Z[s(T)]. 
\]%
$A(T)$ is not an argument in $Z$: The perceived probability of an
assassination affects current welfare, but only the actual deed would have
lasting effects.

Now $s^{*}(t)$ must satisfy the Euler equation \cite{chiang}

\begin{equation}
\delta _{1}(\frac{\partial U}{\partial s}-\frac{\partial U_{s^{\prime }}}{%
\partial t})=-\delta _{2}(\frac{\partial W}{\partial s}-\frac{\partial
W_{s^{\prime }}}{\partial t})  \label{euler}
\end{equation}%
for all $t$.

In a society where assassination is common, the ruler may increase his
bargaining power by delaying a sudden expansion of the candidates' list
until near the end of the regime. Thus $\partial U_{s^{\prime }}/\partial
t>0 $. At the same time, as pressure for assassination rises toward the end
of the regime, society will gain more from a sudden release of that
pressure. Thus $\partial W_{s^{\prime }}/\partial t>0$. These conditions
imply that $\partial U/\partial s$ is a larger number than when
assassination is rare -- and thus that $s$ is smaller. \ The intuition is
that the ruler can gain from maintaining a short list of successors until
late in his regime, when a sudden expansion may most benefit him.

\subsubsection{Endtime conditions}

Suppose that the ruler can also choose both the expected length of his
regime $\overline{T}$ and the succession rule at the end $s(\overline{T})$.
Then, at the optimal endpoint, $t=\overline{T}^{\ast }$, $s(t)$ must satisfy
--- in addition to (\ref{euler}) --- the transversality conditions

\begin{equation}
\delta _{1}(U-\partial U/\partial s^{\prime }\ s^{\prime })+\delta
_{2}(W-\partial W/\partial s^{\prime }\ s^{\prime })=0  \label{trans1}
\end{equation}
and

\begin{equation}
\delta _{1}\ \partial U/\partial s^{\prime }+\delta _{2}\ \partial
W/\partial s^{\prime }=0.  \label{trans2}
\end{equation}%
The transversality conditions are satisfied if: The risk of assassination is
reduced to zero ($A=0\rightarrow \partial U/\partial s^{\prime }=\partial
W/\partial s^{\prime }=0$); and the regime continues until the moment in
which it benefits neither the ruler nor the ruled ($U(\overline{T}^{\ast
})=W(\overline{T}^{\ast })=0$).

Suppose that the constitution constrains the length of the regime to $T$ but
that the ruler may choose $s(T)$. For example, negotiations may limit the
duration of dictatorial or colonial rule but let the ruler specify the set
from which his immediate successor will be chosen. That choice must satisfy (%
\ref{euler}) and (\ref{trans2}) but need not satisfy (\ref{trans1}). The
ruler may satisfy the transversality condition by expanding the pool of
eligible pretenders until the risk of assassination is zero, but the
constitution may prevent him from extending the regime until exhausting its
benefits of the moment.

Suppose that the ruler may choose $\overline{T}$ but an imposed rule of
succession constrains $s(\overline{T})$. His choice must satisfy (\ref{euler}%
) and (\ref{trans1}) but need not satisfy (\ref{trans2}). To satisfy the
transversality condition, the ruler must choose to end his regime at a
moment when the pool of eligible pretenders is expanding ($s^{\prime }>0);$
or continue his regime until exhausting all immediate benefits and then fix
the size of the pool of pretenders during that endtime ($U(\overline{T}%
^{\ast })=W(\overline{T}^{\ast })=s^{\prime }(\overline{T}^{\ast })=0$).

The last necessary conditions at $T$ are variants on Kuhn-Tucker conditions.
If the ruler can choose the rule of succession at $T$,\footnote{%
If $0<s^{*}<1$, then $\partial Z/\partial s=0$. If $s^{*}=0$, then $\partial
Z/\partial s\leq 0$. If $s^{*}=1$, then $\partial Z/\partial s\geq 0$. The
condition $s^{*}\partial Z/\partial s\geq 0$ summarizes these cases.}

\begin{equation}
(1-\delta _{1}-\delta _{2})s^{*}(T)\frac{\partial Z}{\partial s}\geq 0.
\label{KTcond}
\end{equation}
If he can choose $T$ ,

\[
(1-\delta _{1}-\delta _{2})\frac{dZ}{d\overline{T}}=(1-\delta _{1}-\delta
_{2})\frac{\partial Z}{\partial s}\frac{\partial s}{\partial \overline{T}}%
\leq 0 
\]
where, if $\partial s/\partial \overline{T}<0$, then $s^{*}(\overline{T})=0$%
. If the ruler puts any weight on the future, if he can choose the length of
his regime, and if the rule of succession narrows the eligible pool toward
the end of his regime, then he will choose exactly one successor. The
intuition is that he continues to rule, and thus to narrow the pool, until
only one successor remains.

\section{Conclusions and reflections}

Only the altruistic ruler will rationally choose to lay down a rule of
succession that covers all contingencies, since he will otherwise gain by
dividing and conquering his potential successors. Rather than plan his
succession, it is in his political interest to cling to power as long as
possible, since his perceived legitimacy increases with longevity.

An implicit rule of succession is inefficient for society in part because it
generates uncertainty. Just as the lack of a monetary rule may encourage
unanticipated inflation and thus discourage saving, the lack of an explicit
rule of succession may discourage people from investing in political
knowledge over the long term. That may hamper the cultivation of a stock of
leaders. Instead, there is an incentive for political speculation. An
implicit rule may have encouraged many of the palatial intrigues for power
that ``had conferred on the Russian 18th century the spirit of a serialized
light opera'' (\cite{hingley}, page 96), until Paul I specified male
primogeniture as the rule of succession for the Romanovs at the turn of the
19th century.

A theoretical question of interest is how to resolve the paradox of rulers
who will not choose their successors and who will resist the imposition of a
successor since that would reduce their power.

\begin{thebibliography}{9}
\bibitem{columbia} Chernow, Barbara A. and George A. Vallasi. \textit{The
Columbia encyclopedia}. Fifth edition. New York:\ Houghton Mifflin, 1993.

\bibitem{chiang} Chiang, Alpha C. \textit{Elements of dynamic optimization. }%
New York:\ McGraw-Hill, 1992.

\bibitem{hingley} Hingley, Ronald. \textit{Russia:\ A concise history}.
London:\ Thames and Hudson, 1991.

\bibitem{ross} Ross, Sheldon M. \textit{Introduction to probability models}.
Fifth edition. Boston:\ Academic Press, 1993.
\end{thebibliography}

\end{document}