The Prince—or Better No Prince? The Strategic Value of Appointing a Successor

Abstract

Appointing a successor (the “prince”) allows the ruler (the “king”) to alter the structures of conflicts that take place between him and his potential challengers, as well as the structures of conflicts that take place among his potential challengers. Motivated by historical examples and using an infinite horizon rulership competition game, we show that while an appointed prince constitutes a powerful and dangerous threat to the incumbent ruler (the elevated threat effect), the appointed prince can also offer the incumbent ruler increased protection against other potential challengers (the barrier effect). We determine conditions when the overall effect of appointing a successor benefits the incumbent ruler and enables him to acquire a larger share of the governance rent in equilibrium.

Keywords

coup ruler governance rent successorship regimes

Introduction

“The king is dead, long live the king!” In a number of countries this proclamation was used to announce the death of the incumbent king and the immediate successorship by a previously designated prince. It hints at an important dimension of a wide range of governance structures, whether they are kingdoms, dictatorships or other autocratic regimes, democratic regimes, or even companies in the business world. The immediate succession of rulership in many of these contexts is assigned to a designated successor: a “prince.” This article studies when it is beneficial for a self-interested incumbent to have a prince than not having a prince. We also study how the appointment of a prince affects the rent captured by the incumbent, the appointed prince, and the potential challengers to the rulership.

Appointing a prince allows the incumbent (the “king”) to alter not only the structures of conflicts that take place between him and his potential challengers but also the structures of conflicts that take place among his potential challengers.¹ The later include those conflicts that only take place after the incumbent’s “death.” The incumbent cares about the structures of those conflicts that take place after his death, and this is true even for an incumbent who has no concern about his legacy but is only motivated to maximize the rent he can capture during his rulership. Even for a self-interested incumbent, the structures of future conflicts for succession among the elite after his death matter because such structures affect the potential challengers’ incentives to revolt against him. Appointing a prince changes the potential challengers’ strategic calculus regarding revolting. This affects the rent that the incumbent has to give them to discourage them from revolting.

More specifically, appointing a prince has two main effects. First, the prince constitutes an additional hurdle for contenders who aim at taking over the power from the incumbent. Appointing a prince may stabilize the regime of the incumbent, as the existence of a prince makes it more difficult for a member of the elite to replace the incumbent. Second, the appointment of an heir apparent places the prince in an elevated position: a position which makes a coup easier for the prince than for one of the many contenders. In case the incumbent leader dies or is removed, the prince is the natural successor. Hence, a prince offers increased protection against the rest of the elite (the barrier effect), but the prince himself becomes a powerful and potentially dangerous contender (the elevated threat effect).

The difficulties for the incumbent leaders to defend their elevated position to avoid “political death”—or even physical death—are well illustrated in autocracies. Eisner (2011) analyzes a large database on major monarchies in Europe ranging from AD 600 to 1800. He finds 219 sure or likely regicides, which was about 15 percent of all deaths. In about 40 percent of these regicides, one of the members of the group associated with the assassination became the king. Eisner points out that this regicide rate is higher than any historical estimates of homicide rates in Europe’s general population during this time period. This regicide rate is also higher than the homicide rate in Ciudad Juarez in Mexico, which is one of the most murderous cities in the contemporary world. Eisner concludes that European kingship prior to the Industrial Revolution was “amongst the most dangerous occupations found anywhere in the world” (p. 564), and also speculates about a possible relationship between violent conflict and succession rules (p. 572). Similarly, Svolik (2012, 4) considers a database of 316 authoritarian leaders governing some time during 1946 and 2008 and finds that 205 of these were removed by insiders by coups and other nonconstitutional means.

Aspects of the incumbents’ violent struggle for power have been described in many important articles in economics (e.g., see Stephan and Ursprung 1998; Azam 2002; Azam and Mesnard 2003; Mehlum and Moene 2004; Acemoglu, Egorov, and Sonin 2009; Virág 2009). The “king” and his “dukes” are not just rival enemies, however. Some collaboration between them is necessary for the provision of elementary governance tasks. An incumbent needs the support of his powerful subordinates against external threats, as well as in the protection of property rights, the provision of other essential public goods, and the smooth operation of the governance system that yields tax revenue and keeps the ruler in power. This suggests that support and collaboration between the incumbent and the elite yields a positive governance rent and also creates a distributional conflict between the incumbent and his elite.² In compensation for their continued support, the incumbent may share the spoils of rulership. In their pioneering work on political survival and the mutual dependency between a ruler and the “selectorate”—the group of players who have the power to support or replace the incumbent—political scientists Bueno de Mesquita et al. (2002, 2003) argue that it is these returns on their support that makes the elite willing to lend their support, rather than trying to replace the incumbent.

The simultaneous existence of collaboration and competition between the incumbent and the elite applies to both autocracy and democracy, to party leaders and their high-ranking party officers in democracy. Many aspects of this interaction and the role of institutional differences have been studied by now. Flores (2009) applies the selectorate theory to study empirically the political survival of foreign ministers. Egorov and Sonin (2011) study the trade-off between loyalty and competence of persons in key positions. Competence helps stabilizing the regime and defending it against external threats. But a more competent deputy is also a threat to the incumbent. We abstract from external threats in this article, but focus on the competition between rival members of the elite who may wage a coup. Flores and Smith (2011) focus on how regime differences affect the hiring and firing of ministers by the leader. Bueno de Mesquita and Smith (2009) apply the selectorate theory to study the implications of revolutionary threats for institutional change. Bueno de Mesquita and Smith (2015) discuss and compare various types of threats to the incumbent’s power. Sekeris (2011) considers a ruler’s choice of supporters from a heterogeneous set of individuals, how the ruler can use this choice to enhance his own rent, and how this choice depends on the ruler’s strength. Francois, Rainer, and Trebbi (2014) pursue related questions focusing on the role of the experience of subordinates, where higher experience may increase their ability to stage a coup. Che, Chung, and Qiao (2014) study a model of leader accountability in which the leader’s own decision regarding whether to perform or to embezzle affects his ability to use the prize of eventual promotion to leadership to discourage his subordinates from embezzling. These considerations may also apply to private governance structures, such as the relationship between CEOs and their immediate subordinates.³

In both economics and political science, scholars have recently emphasized that while the literature on autocratic performance and autocratic stability traditionally focuses on the threat of popular revolts, more attention needs to be devoted to the importance of violent conflicts between the incumbent leader and the elite in determining regime stability and performance.⁴ Despite the fact that succession arrangements are critical in shaping the interaction between the incumbent leader and the elite as well as among the elite, to our knowledge, this article is the first contribution that formally analyzes how the incumbent can strategically choose succession arrangements to maximize his lifetime rents in a dynamic model of mutual collaboration and competition between the incumbent and the elite.

Our analysis will be cast in terms of interactions between the “king”, his “dukes”, and the “prince”. The insights obtained from our analysis regarding how the incumbent balances the trade-off between the elevated threat effect and the barrier effect when making succession decisions, however, are also relevant for other settings characterized by simultaneous collaboration and competition between the incumbent and his powerful subordinates. These include, for example, interactions between party leaders and their high-ranking party officers in democracy, between CEOs and their immediate subordinates in firms, and between leaders and their subordinates in other organizations such as unions and possibly even criminal and terrorist organizations.⁵

The fact that strategic considerations in conflicts surrounding succession are important in shaping societal and organizational governance has been recognized by scholars in many fields. The political scientist Herz (1952, 30) refers to the “crown-prince problem”: to install a crown prince may come along with a transfer of authority, and a strong crown prince may challenge the king even prior to natural death or voluntary demise. Reflecting on succession arrangements in different societies, the anthropologist Goody ([1966] 2010, 2) argues that succession conflicts “may loom so large that some societies, both monarchies and party systems, refuse to name a successor lest he should be tempted to succeed too soon.” The economist Tullock (1987, 151) argued that if a dictator formally appoints a successor, “this gives the successor both strong motives for assassinating him and reasonable security that he will get away with it.” In her study of successions in the Song dynasty in China, the historian Ebrey (2006, 54) observes that both emperors Taizong and Zhenzong were reluctant to name an heir, and argues that this reluctance reflected a deep understanding by these rulers that “the appointment of an heir apparent would change the political dynamics of their court.” Students of corporate governance have also emphasized that the threat of challenges by insider senior executives against the CEOs—especially, challenges by potential successors—is important in affecting the functioning of firms (Ocasio 1994; Shen and Cannella, Jr. 2002). The economist Congleton (2011, 62) suggests that founders of business organizations and political regimes often adopt organizational practices that make themselves irreplaceable, so as to avoid their removal. Such tactics imply that the exit or death of the founder often leads to crisis for the organization or the regime.

Using an infinite horizon rulership competition game, we analyze how appointing a prince changes the structure of rulership contests among possible contestants, and how this feeds back and determines the division of rents for the incumbent. In particular, we ask if the appointment of a prince increases the incumbent’s payoff. A prince is defined here as a designated successor in case of the incumbent’s death or removal from power. The existence of a designated successor changes the elite members’ incentives for a coup. The coup may become more attractive for the designated prince and less attractive for other members of the elite. In turn, this changes the equilibrium distribution of rents. Our key question is if and when the appointment of a prince is in the interest of the incumbent and stabilizes his regime. We show that under both the princeless regime and the regime with a prince, the total probabilities of success in a coup are proportional to the payments that are required to make such a coup unattractive. This implies that the incumbent’s rent in the regime with a prince is higher than in a comparable regime without a prince if the sum of the probabilities of success in a coup is smaller in the regime with a prince. Whether this condition holds depends on the institutional environment and technological conditions.

The literature on succession discussed above clearly recognizes that a successor can pose a threat to the incumbent, and there is also some recognition that appointing a successor can “change the dynamics” among the elite. To our knowledge, however, this article is the first that articulates the barrier effect as an illustration of how appointing a successor can change the dynamics among the potential challengers to the incumbent. Our analysis also highlights the hitherto neglected insight that even a self-interested leader who does not care about his legacy per se has the incentive to use succession decisions to manipulate the structures of succession conflicts among the contenders for the rulership, because the structures of conflicts after his death can still affect the rent that the incumbent can capture during his lifetime.

Our formal analysis focuses on the trade-off between the barrier effect and the elevated threat effect and how it affects the king’s rent in equilibrium. The rationale for appointing a prince may, of course, have many further dimensions. After presenting our analysis, we discuss how our model can be extended to deal with some other dimensions of the succession problem.

The Formal Structure

We consider an infinite horizon rulership game. There is a large set K of players. At the beginning of each period t, there are n + 1 players who are potentially active in this period: one king and n dukes or one king, n − 1 dukes, and one prince. Whether or not there is a prince distinguishes two regimes which we analyze first separately and compare the outcomes subsequently. Players are infinitely lived. All players are identical in their preferences along all dimensions, but in each period only these n + 1 players are assigned to roles in which they can potentially make active choices. All other players are inactive. Some of these inactive players have been dukes, kings, or princes, or a combination of these roles in previous periods, and will never be active in the future. As will become clear, one or more players in a given period may suffer a permanent loss of their status as king, prince, or duke and exit for the rest of the time horizon. All players who have never been in the role of a king, a prince, or a duke previously are in the “replacement set.” Players who are not currently king, prince, or duke in a given period cannot make any decision and receive a period payoff of 0 in this period. If, in the ongoing period, a duke or the king or a prince is forced to exit, the replacement set is used to make random draws from this set to replace these players for the next period.

The Princeless Structure

Consider first a governance structure without a prince. One of the players is the incumbent ruler at the beginning of any given period t = 1, 2, …. We refer to this player as the incumbent in t and denote this player as R_t . Further, there are n dukes at the beginning of each current period, and the set of these is S_t ≡ {S _1t, S _2t, …, S_nt }. Possible actions and their sequencing in each period t are as follows. First, the king firmly commits to pay identical and nonnegative shares, s_t , in the total governance rent to each of the n dukes in the current period.^6,7 We limit the per duke share to $s_{t} \in [0, \frac{1}{n}]$ , as the king cannot pay more than the governance rent. Next, we assume that one of the ex ante identical dukes is determined at random to have an opportunity to revolt. We denote S_it as the duke who is selected in period t.⁸ Duke S_it chooses η_it ∈ {0, 1}, where η _it = 1 denotes the decision to revolt and η _it = 0 denotes the decision not to revolt.

If duke S_it chooses η _it = 0, the economy stays peaceful in the respective period, a governance rent of size 1 accrues and is distributed between the king and his dukes according to the announced shares. The king receives 1 − ns_t , and each duke receives s_t . This then ends period t and period t + 1 starts, without any change in the identity of the king, or in the identity of the dukes.

If duke S_it chooses η _it = 1, the revolt destroys the governance rent in the respective period. Nothing can be distributed between the king and the dukes in the respective period. The revolt is successful with probability f^N. This and all similar revolt success probabilities are exogenous and intertemporally invariant.⁹ In case of success, the king is removed and receives a zero payoff in all future periods. The revolting duke becomes the king at the beginning of the next period. And a new duke is drawn from the replacement set. If the revolt is unsuccessful, the revolting duke is dismissed and receives zero payoff from then on. The non-revolting dukes remain in place independent of the outcome of the revolt.¹⁰ With probability 1 − f ^N , the coup fails. The revolting duke is dismissed and replaced.

Players who are active in a given period may condition their actions on different own or others’ actions in the past, and also on the identity of players in the different roles at different times, and this may lead to a whole wealth of possible equilibrium interaction outcomes. For each period t, we could outline the full history of the game and define the strategies that could be dependent on aspects of this history. We limit the analysis to stationary Markov perfect strategies: that is, we analyze a game in which we constrain the local strategies of players to be functions of players’ own current status (as an incumbent ruler, as a duke who has an opportunity to revolt, or one of the other dukes) and of actions taken in the current period by other players. That is, a player’s local strategy at a period t depends on the player’s role in this period and of other players’ actions taken in this period. If the player i is inactive in that period (member of the replacement set, or a replaced king or duke, or a duke without the opportunity to revolt), the choice set is empty. If the player is the incumbent king, his local strategy is a choice of $s_{t} \in [0, \frac{1}{n}]$ . If the player is the “active” duke with an opportunity to revolt, his local strategy at period t is a function $η_{i t} (s_{t}) : [0, \frac{1}{n}] \to {0, 1}$ . Note that this setup also includes a stationarity property: for instance, the set of local strategies of an incumbent at period t is the same as the set of local strategies of the player who is the incumbent king at period t + k.

Players’ expected payoff in the current and all future periods is the sum of all discounted expected period payoffs and depends on their and the other players’ actions and on the possible random outcomes in the coup-periods. We denote the discount factor δ ∈ (0, 1) and assume that all players use the same discount rate.

We show the following proposition:

Proposition 1: The princeless regime has a peaceful Markov perfect equilibrium (MPE) in stationary strategies with payments to each duke in each period of:

s_{t} = s^{N} \equiv \frac{δ f^{N}}{1 + n δ f^{N}} .

The king’s rent per period in this equilibrium is given by:

r^{N} = \frac{1}{1 + n δ f^{N}} .

Proof: Consider the following candidate for a stationary MPE: the king offers s_t = s^N as given in (1) for all t. The active duke chooses not to revolt if s_t ≥ s^N and chooses to revolt if s_t < s^N .

To confirm optimality of these strategies, we make use of the one-stage deviation principle.¹¹ Assuming that all players choose actions according to the candidate equilibrium in all future periods t + 1, …, we consider payoffs that emerge for possible local deviations from equilibrium play in period t.

First, consider the active duke S_it . He compares his payoff:

s_{t} + δ \frac{s^{N}}{1 - δ},

from η_it = 0 with the expected payoff from η_it = 1, which is given by:

f^{N} δ \frac{(1 - n s^{N})}{1 - δ} .

We find that η_it = 0 is optimal if and only if s_t ≥ s^N.

Second, consider R_t . Again, we use the one-stage deviation principle. Taking candidate equilibrium play of dukes in the period subgame as just described, and equilibrium play of all active players in future periods into account, any s_t > s^N , even though it will prevent revolt, is dominated by s_t = s^N . The alternative to s_t = s^N which has the highest period payoff among all alternatives other than s_t = s^N is s_t = 0. This choice will trigger η_it = 1. The present value of expected payoff from s_t = 0 as a possible deviation from the candidate equilibrium is given by:

δ (1 - f^{N}) \frac{1 - n s^{N}}{1 - δ} .

For any s^N < (1/n), these values are smaller than (1 − ns^N)/(1 − δ)—the present value of rulership in the candidate equilibrium. Hence, there is no optimal individual deviation from the local strategy in the candidate equilibrium for both the king and for the dukes if ns^N < 1. As

n s^{N} = \frac{n δ f^{N}}{1 + n δ f^{N}} < 1,

this condition holds.

The king’s rent is r^N = 1 − ns^N in each period. Inserting the equilibrium value (6) and simplifying yields (2). This completes the proof. ▪

Proposition 1 describes a peaceful MPE in stationary strategies for a regime without a prince. It can be shown that there does not exist any peaceful MPE in stationary strategies in which the king offers a share to each duke that is lower than s^N .¹² Furthermore, note that r^N > s^N : the king is always better-off than each duke in the equilibrium.

The MPE in Proposition 1 has intuitively plausible properties. The rent allocation is determined by the dukes’ threats, that is, by the success probabilities of revolting. Discounting matters. The king’s period rent is positive and is smaller if the future is more important, that is, for a higher δ. The extreme case reveals the intuition. For δ = 0 it is not worthwhile to revolt, even if s_t is negligible, because this reduces payment in the current period to 0 and, for δ = 0, whatever could be gained in the future, is discounted to 0.

The Regime with a Prince

We now consider a governance structure that is identical to the baseline structure, except for the modification which is at the focus of our research: starting with an incumbent king and n dukes, the king can give one of the dukes a special and recurrent role and appoint this duke as his designated successor, or, borrowing from the terminology used in monarchies, a “crown prince” (hereafter simply prince). The existence of the prince changes the game in each period, as the prince is the natural successor for a king who is replaced, and the prince is also in an elevated position that may allow him to organize a coup in any period. This framework gives the prince a pole position, should some misfortune happen to the king.

Compared to the case when he was a duke, the prince’s elevated status can make it easier for him to succeed in launching a coup against the king. Furthermore, when he succeeds he is in the position to get control of the king’s resources which may give him a strategic advantage when facing a possible challenge by a duke. These effects make the prince a more dangerous adversary for the incumbent than a duke. We shall refer to this as the elevated threat effect when the king appoints a prince. At the same time, because the prince possesses a pole position, even if a duke who is not elevated as the prince succeeds in eliminating the king, the prince can become the king and will be armed with the king’s resources when engaging in conflict with a revolting duke. This implies that a revolting duke now needs to eliminate the king and the prince, and the prince now constitutes an additional barrier. This barrier effect makes it more difficult for a duke to become a king, and reduces the duke’s threat. We are interested in comparing these two effects and in determining conditions under which one effect dominates the other.

More precisely, the differences in the setup compared to the no-prince section are as follows. The king chooses (p_t , s_t ). The component p_t is the share in period t’s governance rent offered to the prince. Each of the n − 1 dukes is offered a share s_t . These shares are, again, firmly committed and paid at the end of the concurrent period. The prince chooses not to revolt (η_pt = 0) or to revolt (η_pt = 1) If the prince revolts, this destroys the governance rent in period t completely. Then one of the dukes is chosen at random to have the opportunity to revolt. This “active” duke S_it chooses between η_it = 0 and η_it = 1. If η_pt = 1 and/or if η_it = 1 this again destroys the governance rent in period t. The governance rent is positive and equal to 1 if η_pt = η_it = 0.

If the prince revolts (η_pt = 1), this revolt is successful with probability π _p and unsuccessful with probability 1 − π _p . If the prince is not successful, the prince will be removed and replaced by a new prince at the beginning of the next period; if the prince is successful, the prince replaces the king and the former king is replaced for the current and all future periods. Hence, an attack by the prince leads to a situation in which there is a king or a designated king but no prince for the remainder of the period. This may seem an opportunity for the active duke, as he has to overthrow only one player, rather than two, in order to get a chance to succeed as the incumbent king in the next period. Should a prince who overcame the incumbent king be attacked by the duke, in analogy with the princeless regime we denote the probability that the duke overcomes the prince in this attack as f^N , assuming the same technology as between a duke and a king in the princeless framework. We define f_p = π _p (1 – f^N ). This is the probability by which a prince’s revolt is eventually successful and by which the prince becomes the new incumbent ruler in period t + 1. This reflects the fact that if the prince challenges the incumbent in period t, and, if this challenge is successful, the prince will be attacked by the duke in the same period t.

If the prince does not revolt, the active duke decides whether to revolt. A successful coup by the active duke requires a removal of both the king and the prince. We assume that the duke’s revolt either removes both the king and the prince or none of them. If the coup fails, the king and the prince stay in office, but remove the duke and replace him by a new one. If the coup succeeds, then the duke becomes the new king, and this king draws a new prince and a new duke from the replacement set. We denote the probability of the revolting duke to succeed in this coup as f_s . All inactive dukes keep their position for the next period.

Again, we could describe histories for each player and behavioral strategies more formally, but for brevity we abstain, as we impose Markov restrictions on local strategies. We can state the following proposition:

Proposition 2: A regime with a prince has a peaceful MPE in stationary strategies with payments to the prince of:

p_{t} = p = \frac{δ f_{p}}{1 + (n - 1) δ f_{s} + δ f_{p}},

and to each duke of:

s_{t} = s = \frac{δ f_{s}}{1 + (n - 1) δ f_{s} + δ f_{p}},

in each period if f ^N > f_s . Moreover, the king’s per period rent in this equilibrium is given by:

r = \frac{1}{1 + (n - 1) δ f_{s} + δ f_{p}} .

Proof: Consider the following candidate for a stationary MPE: the king offers (p_t , s_t ) = (p, s) as in (7) and (8) in all periods. When p_t ≥ p and s_t ≥ s, both the prince and the active duke will not revolt. When p_t < p or s_t _< s, both the prince and the duke will revolt.

We make use of the one-stage deviation principle again. Consider first the active duke S_it ’s decision for given offers s_t and p_t if η_pt = 0. The payoff from η_it = 1 is given by:

f_{s} δ \frac{1 - p - (n - 1) s}{1 - δ},

and the payoff from η_it = 0 is given by:

s_{t} + δ \frac{s}{1 - δ} .

The s_t = s that equalizes these present values is given by:

s = \frac{(1 - p) δ f_{s}}{1 + (n - 1) δ f_{s}} .

Given the perspective of playing the peaceful candidate equilibrium in all future periods, if η_pt = 0 the choice η_it = 0 is optimal only if s_t ≥ s.

Turn next to the optimal choice η_it if η_pt = 1. Either the prince was successful and replaced the king, or the prince was unsuccessful and was removed, but not replaced in the ongoing period. In either case, the duke revolts against one player only: the incumbent king or the former prince who just became the king. The probability that the duke is successful in replacing this player is f^N. Let

f^{N} > f_{s},

which says that it is easier to overthrow a single incumbent king than to overthrow an executive government consisting of an incumbent and a prince.

When η_pt = 1, if the active duke revolts when facing an executive government that consists of a single incumbent king now, his payoff will be $f^{N} δ \frac{1 - p - (n - 1) s}{1 - δ} > f_{s} δ \frac{1 - p - (n - 1) s}{1 - δ}$ . We know, however, from the above discussion regarding the active duke’s decision for given offers s_t and p_t when η_pt = 0, that $f_{s} δ \frac{1 - p - (n - 1) s}{1 - δ} = s + δ \frac{s}{1 - δ}$ (which reflects the fact that s is such that it makes the active duke indifferent between revolting or not revolting when the prince does not revolt). Note that when η_pt = 1, the governance rent in period t is 0. Thus, if the duke revolts when facing a single incumbent king, regardless of the value of the offered share s_t in this period, the duke will get a zero payoff in the current period and will therefore get a payoff of $δ \frac{s}{1 - δ}$ by not revolting.

Combining these observations, the active duke chooses η_it = 1, if η_pt = 1 as:

f^{N} δ \frac{1 - p - (n - 1) s}{1 - δ} > f_{s} δ \frac{1 - p - (n - 1) s}{1 - δ} = s + δ \frac{s}{1 - δ} > δ \frac{s}{1 - δ} .

We take this into consideration when we now return to the choice of the prince, as the active duke’s revolt is what a prince expects to happen if the prince challenges the incumbent. The prince observes (p_t, s_t) and chooses η_pt. The size of p_t obviously matters, but s_t is important for the prince, too. This s_t guides the behavior of the active duke in case the prince did not revolt.

First, consider the case s_t ≥ s, for which the active duke does not revolt if the prince did not revolt. If the prince revolts, his payoff is:

δ f_{p} \frac{1 - p - (n - 1) s}{1 - δ} .

This anticipates that the revolt destroys the governance rent in period t and makes the duke revolt, for whatever s_t was promised. If the prince accepts p_t , the duke will not revolt given that s_t ≥ s, and the prince has a payoff from accepting of:

p_{t} + δ \frac{p}{1 - δ} .

Accordingly, the prince will not revolt if:

p_{t} \geq p = δ f_{p} \frac{1 - (n - 1) s}{1 + δ f_{p}},

and revolts otherwise. Inserting this p into (10) and solving for s yields:

s = \frac{δ f_{s}}{1 + (n - 1) δ f_{s} + δ f_{p}} and p = \frac{δ f_{p}}{1 + (n - 1) δ f_{s} + δ f_{p}} .

We have shown that if $p_{t} \geq p = \frac{δ f_{p}}{1 + (n - 1) δ f_{s} + δ f_{p}}$ and $s_{t} \geq s = \frac{δ f_{s}}{1 + (n - 1) δ f_{s} + δ f_{p}}$ , both the prince and the active duke will not revolt.

Now consider s_t < s, the prince’s payoff from revolting is again (11). If the prince does not revolt, then the active duke revolts. The payoff of the prince becomes:

δ (1 - f_{s}) \frac{p}{1 - δ} .

Because $s = \frac{δ f_{s}}{1 + (n - 1) δ f_{s} + δ f_{p}}$ and $p = \frac{δ f_{p}}{1 + (n - 1) δ f_{s} + δ f_{p}}$ , and 1 > δ(1 − f_s),

δ f_{p} \frac{1 - p - (n - 1) s}{1 - δ} > δ (1 - f_{s}) \frac{p}{1 - δ} .

For any offer with s_t < s, the duke is paid too little and will revolt even if the prince does not. This in turn implies that the prince will prefer to revolt if s_t < s, even if p_t ≥ p. Summing up, the best responses of the prince and the active duke are as follows: when p_t ≥ p and s_t ≥ s, both the prince and the duke will not revolt. When p_t < p or s_t < s, both the prince and the duke will revolt.

Using the one-stage deviation principle, we now show that the king prefers payments of s_t = s and p_t = p in period t to any other alternatives in period t. At a cost of p + (n − 1)s, the king can prevent both the prince and the dukes from revolting by offering p_t = p and s_t = s. Because only offers with p_t ≥ p and s_t ≥ s will ensure that both the prince and the active duke will not revolt, any offer other than (p, s) that will prevent revolt by both the prince and the active duke must have a cost (n − 1)s_t + p_t > p + (n − 1)s and is dominated by (p, s). Furthermore, any offer with p_t < p or s_t < s triggers revolt and destroys the governance rent in the ongoing period and further exposes the king to the risk of being displaced, and is therefore dominated by (p, s).

Finally, we calculate the king’s rent as r = 1 − (n − 1)s − p, or

r = \frac{1}{1 + (n - 1) δ f_{s} + δ f_{p}} .

This completes the proof. ▪

Following lines similar to the ones in footnote 12, it can be shown that there does not exist any peaceful MPE in stationary strategies that will give the king a higher payoff than the equilibrium in Proposition 2. Such an MPE would have to be peaceful and would require either a lower s or a lower p. Together with a higher r, this is incompatible with both the prince and the duke prefer to abstain from revolting.

The equilibrium in Proposition 2 also has intuitive properties. Rents in the equilibrium are determined by the success probabilities of revolt. The probability by which the prince or a duke is successful in a coup if they initiate a coup determines the size of the king’s rent. The higher this probability, the more attractive a coup is for the respective type of player, and the higher the compensation that is required to prevent the respective type from initiating a coup, which, in turn, reduces the king’s rent. Further, r > s and r > p. The prince is better-off than a duke if f_p > f_s . The comparative static results are also intuitively plausible. The king’s period rent r is smaller if the prince or the duke imposes a higher threat, and if the future is less strongly discounted.

Comparing Regimes

We can now compare the king’s rents in the two regimes for the peaceful Markov Perfect Equilibria that are characterized in Propositions 1 and 2. We find:

Proposition 3: For the comparison of the leader’s rents, it holds that r > r^N if f_p + (n − 1)f_s < nf^N.

Proof: The result follows directly from comparing (2) to (9). ▪

Using a dynamic model, we show that under both the princeless regime and the regime with a prince, the total probabilities of success in a coup are proportional to the amounts of payments that are required to make a coup unattractive. This translates into the condition in Proposition 3: the king’s rent in the regime with a prince is higher than the rent in a regime without a prince, if the sum of revolt success probabilities of the potential challengers to the king is smaller in the regime with a prince. For n = 2, this condition can be rewritten as f_p − f^N < f^N − f_s. If the prince has a higher revolt success probability than a duke under a princeless regime, then the leader will appoint a prince only if appointing a prince leads to a significant enough offsetting decrease in the revolt success probability of the duke who is not appointed as the prince. Furthermore, because the benefit of the barrier effect is increasing in the number of dukes, an increase in n increases the king’s incentive to appoint a prince.¹³ When f_s < f^N, if n >> 2, then even if the prince is powerful and f_p is high, the regime with a prince becomes superior for the king.

Rent-maximizing versus Conflict-Minimizing Succession Arrangements

Our analysis highlights how the trade-off between the elevated threat effect and the barrier effect affects the leader’s decision regarding whether to appoint a successor. Many anecdotes can be found in early imperial Rome under the rule of Caligula, Claudius, Nero, Galba, Otho, and Vitellus. As documented by Christ (2009) and Winterling (2011), some incumbent emperors seemingly gave preference to leaving successorship open to the end of their rulership; sometimes a prince is appointed, and the prince was often a player with little own power (small f_p ).¹⁴

Our analysis so far focuses on when an incumbent who is only concerned about maximizing his lifetime rent will find it in his own interest to appoint a successor. The threat of conflicts that occurred after the incumbent’s departure can affect the potential challengers’ incentives to revolt against the incumbent, and our analysis shows how the incumbent can strategically manipulate the structures of such conflicts to his advantage when making succession decisions.¹⁵ The fact that conflicts among the contenders for successorship that occurred after his political or physical death impose significant costs on both the citizens and the elite is of no concern to such a purely self-interested incumbent. In short, the succession arrangement chosen by a strategic rent-maximizing incumbent ruler may often not minimize the conflict effort.

In contrast, the existing literature on succession emphasizes how succession arrangements may reduce the conflicts among contenders after the incumbent’s departure, often (implicitly) emphasizing how avoiding such conflicts is beneficial to the elite, the citizens, and an incumbent who has concerns about his legacy. For example, Tullock (1987, 162) argued that while hereditary succession does not always guarantee a peaceful transition from father to son, it “seems to be more peaceful than other methods.” Political scientists acknowledged that power transition may create a wasteful struggle for power (see Kokkonen and Sundell 2014, for a survey of this literature). The insecurity in the period of power transition is seen as a problem to be avoided (Herz 1952). Kurrild-Klitgaard (2000) suggests that an automatic and unambiguous succession rule may be a possible solution to this problem. Using a data set covering 961 monarchs ruling 42 European states between 1000 and 1800, Kokkonen and Sundell (2014) find that fewer monarchs were deposed in states practicing primogeniture than in states practicing alternative succession rules.

These observations suggest that the elite and the citizens face a political agency problem in the choice of succession arrangements. Suppose we relax the assumption that the incumbent only cares about his own lifetime rent, and allow for the possibilities that he may care about his legacy, has concerns about the survival of the regime beyond his own political survival, or has altruistic concerns for his designated successor (though still mindful about the threat posed by the successor). Even for an incumbent with concerns beyond his narrow self-interest, from the perspectives of the elite and the citizens, the incumbent will always be putting excessive weights on his own political survival and rent maximization while putting insufficient weights on how succession conflicts may impose costs on others.

Our current analysis is cast in the context of a self-interested incumbent deciding whether to appoint a successor when there is no socially-instituted rules governing succession, but it also sheds lights on when the incumbent may want to deviate from existing norms or institutional arrangements governing succession. The evolution of succession rules in many societies and organizations can be thought as driven to some extent by attempts of the elite—and to a lesser extent the citizens, through their threat of popular revolt—to impose constraints on the rulers and force them to internalize some of the costs borne by the citizens and the elite when succession conflicts occurred.

Our analysis suggests that if the formal appointment of a specific successor according to the existing succession rule increases the total probabilities of success in a coup compared to the alternatives of delaying the formal appointment or appointing an alternative successor, then the incumbent may prefer to deviate from the established rules of succession. This can occur if the appointment of the successor designated by the existing rules does not generate a significant barrier effect, or if this successor poses too much of a threat to the incumbent. In her study of successions in the Song dynasty in China, Ebrey (2006, 54) points out that Taizong—who succeeded his elder brother and founder of the dynasty Taizu when the latter died in somewhat suspicious circumstances—did not follow the established norm of appointing his eldest son as his successor. Furthermore, he was reluctant to appoint an heir at all, and when he finally appointed his third son as his heir, he was uneasy that the people were pleased with this appointment. Similarly, we alluded to the emperors in early imperial Rome and their strategic behavior with respect to appointing weak princes and deposing them once they become popular and powerful.

Future research can study more carefully this political agency problem in the choice of succession arrangements faced by the elite and the citizens under different environments to understand when socially-instituted rules are effective and when they fail in binding the behavior of the incumbent. For example, the literature on political transition emphasizes how splits among the elite (Przeworski 1991) and succession crises (Geddes 1999) open up windows of opportunity for democratic transition in autocracy. A careful study of the political agency problem in succession may generate useful insights and policy implications regarding how divergent interests and strategic interactions among the incumbent, the elite, and the citizens facilitate or prevent the emergence of such windows of opportunity for democratic transition.

Discussion

In this section, we consider some extensions, and discuss limitations of the current analysis and directions for future research.

Endogenous Probabilities of Successful Revolts

In this subsection, we sketch a two-period finite horizon variant of our model, which endogenizes the revolt success probabilities. This simple example illustrates how the king’s decision regarding whether to appoint a prince is determined by the trade-off between the elevated threat effect and the barrier effect in a non-stationary environment with endogenous revolt success probabilities.¹⁶ Main changes of the model are as follows.

We consider two periods t = 0, 1. There is an incumbent king at the beginning of period t = 0 and two further players. In one case, these are two dukes; in the other case, these are one duke and one prince. The king chooses non-negative payment offers to share the period-0 governance rent with the other two players. This governance rent is G ₀. These are (s^N , s^N ) in the regime without a prince and (p, s) in the prince regime. The governance rent that accrues in period 1 has a present value in period 0 which is denoted by δG₁ which may exceed or fall short of G ₀, but is normalized to δG₁ = 1, where δ is again the discount factor and is assumed to be the same for all players. In particular, the incumbent may have to take some investments to stabilize his regime in period 0, or period 1 may be longer.

In the no-prince regime, one of the dukes is randomly chosen to be active in period 0. This duke has the opportunity to revolt in this period. If he revolts, then the duke chooses revolt effort e ∈ [0, a/2] at a cost C(e) = e. We assume that this translates into a revolt success probability¹⁷:

f^{N} = a e - e^{2},

where we will restrict a to be from the open interval $(1, \sqrt{6} - 1)$ in order to obtain an interior equilibrium.

If the revolt is successful, the king is removed, and the revolting duke becomes the king in period t = 1. If the revolt fails, the revolting duke is removed and replaced by a new duke. The non-revolting duke remains in place in any case.

In the prince regime, the prince has an opportunity to revolt in period 0, and then the duke has an opportunity to revolt, also in period 0. Suppose the prince revolts. He chooses revolt effort e ∈ [0, a/2] at a cost of C(e) = e, and the revolt against the king is successful with probability:

π_{p} = λ a e - λ e^{2} .

Here, λ measures the revolt effectiveness of the prince. An elevated threat exists if λ > 1: for the same given effort, the prince has a higher likelihood to succeed against the king than a duke. If the prince fails, the prince is removed, and a new prince is appointed only in the next period. If the prince succeeds, the king is removed and the executive government consists of the former prince, now the king. In any case, if the prince revolts, this means there is no prince left as a result. Again, the parametric example requires a parameter restriction, which is that λ is from the open interval (4/(5 − a²),2/a).

Given that the prince revolted, the duke may or may not revolt. If the duke does not revolt, the game moves to the next period. If the duke revolts but fails, he is removed and replaced in the next period. If he revolts successfully, he becomes the king in the next period. Borrowing the technology from the no-prince regime, if the duke revolts and chooses effort e ∈ [0, a/2] his success probability is given by:

f^{N} = a e - e^{2} .

Success or failure of this revolt determines the survival of the players and their roles in t = 1.

Suppose the prince does not revolt. Then the duke has the opportunity to revolt against the prince and the king. The duke chooses revolt effort e ∈ [0, a/2] at a cost C(e) = e, and the revolt success probability of removing both the king and the prince is given by:

f^{s} = β a e - β e^{2} .

Again, for an interior equilibrium, we need to choose a parameter restriction and assume that β is from the interval (1/a, 1). The assumption that β > 1/a ensures that the duke’s chosen effort in the regime with a prince will be positive in equilibrium. The assumption of β < 1 says that it is more difficult to overthrow an executive government consisting of an incumbent and a prince than to overthrow a single incumbent king. This assumption will also ensure that 0 < f^s < 1 in equilibrium.

We made several assumptions about α, β, and λ. It is easy to confirm that these assumptions are internally consistent.¹⁸ This completes the description of period 0.

There is no period after period 1. Hence, it does not make sense to attack the incumbent in period 1. Whoever is the incumbent at the end of t = 0 receives the full governance rent in period 1. We assumed that the present value of this rent was δG₁ = 1 and the same for all players.

Proposition 4: The regime without a prince has a peaceful subgame perfect equilibrium. The king pays (a − 1)²/4 to each of the two dukes and his payoff is G₀ + 1 − 2((a − 1)/2)². The regime with a prince has a peaceful subgame perfect equilibrium in which the incumbent king pays (βa − 1)²/(4β) to the duke and (λaW − 1)²/(4λW) to the prince, where W = (5 − a²)/4. The king’s payoff is G₀ + 1 − ( βa − 1)²/(4β) − (λaW − 1)²/(4λW). The king’s equilibrium payoff will be higher under the regime with a prince, if 2((a − 1)/2)² > (βa − 1)²/(4β) + (λaW − 1)²/(4λW).

The proof of this proposition is given in the Online Appendix.

The comparison is very similar to the one in our main model: comparing the two regimes, in the regime with a prince the king gains from a barrier effect and loses from an elevated threat effect. If a prince is appointed, this reduces the rent that the king needs to give to the remaining duke from (a − 1)²/4 to (βa − 1)²/(4β), which is less, because β < 1. This is the barrier effect. But the amount (λaW − 1)²/(4λW) that needs to be given to the prince in order to appease him now exceeds (a − 1)²/4. This is the elevated threat effect. The king will prefer the regime with a prince when the gain from the barrier effect outweighs the elevated threat effect.

Bystander Dukes or Revolting Coalitions?

In order to focus on the barrier and elevated effects of appointing a prince, we removed a number of other aspects from the picture. We assume that a failed coup leads to a replacement of the revolting duke, but allows all other dukes to be bystanders and to continue in their position. Alternatively, dukes may be forced to take sides, and may be replaced in case of a failed coup or in case of a change in government, depending on the side they chose. Intuitively, what speaks in favor of this assumption is the uncertainty which a king may feel about who was involved in a failed coup and a general mistrust, which may make it in the interest of the king to remove all dukes.

Both the need to make the dukes take sides and the elements of uncertainty or loyalty considerations are absent in our formal model. Historical examples suggest that a large share of the dukes may survive a coup. The experience of the rule of the Chinese Communist Party under Mao and Deng was that the paramount leader survived all political conflicts with his designated successors, and not all the elite was removed from power when the designated successors were removed.¹⁹ Similar evidence exists for coups in imperial Rome (Christ 2009). If the share of the elite suspected to be connected to the revolt was large and the boundaries of the set were uncertain, it was not viable for the king who survived the coup to prosecute or eliminate the whole elite.

Successful coups may require the collaborative effort of a group of powerful players. The prince and the dukes themselves may, however, represent the heads of existing power groups that can be seen as coalitions. How such power groups themselves, or coalitions between them, are formed, is outside the scope of our analysis. The formal modeling of the endogenous formation of alliances or coalitions in a non-cooperative framework is challenging, because what is the natural equilibrium concept about coalition stability is not clearly defined already in a static context (for a discussion, see Hart and Kurz 1983, 1059-64; and Bloch 2009, for a more recent survey and contribution), and the dynamics of a multiple-period add further complexity. Within the context of the relationship between leaders and their power base, work by Bueno de Mesquita et al. (2003) has addressed this issue. While their work does not formally address these theoretical aspects of the stability of coalition structures in an otherwise non-cooperative framework, their work reaches interesting and important conclusions and highlights the importance of this issue.

Our analysis deliberately assumes that the players represent sufficiently powerful groups for a revolt. This also avoids the issue of endogenous formation of such groups, how coalitions coordinate their efforts, and how they solve the internal distributional conflict.²⁰ The dukes may represent the leaders of power groups within the society and may constitute a stable and powerful coalition with an internal governance structure which we take as given.

More Sophisticated Appointment Decisions

In our formal framework, the leader makes a zero-one choice regarding whether to elevate one of the dukes to be the prince. A more refined treatment can have the incumbent decide how much to elevate the duke. We are grateful to a reviewer who alerted us to the following example: the sultan in the Ottoman empire typically had many sons, who all were his potential heirs. The succession rule after his death was “first come first serve.” Having all of his sons at a equal distance from the capital would imply that they would all be “dukes”. Assigning the sons to governorships with different distances from the capital would be designating the one with the closest governorships as the “prince” (Goody [1966] 2010, 19). By manipulating the degree of asymmetry in the governorships for his sons, the sultan can decide how much to elevate the prince. Future work can investigate the implications of allowing for sophisticated appointment decisions that allow the king to manipulate endogenously the power of the prince.

Concluding Remarks

This analysis studies the strategic value of appointing a prince for the incumbent. We find that the appointment of a designated successor changes the elite members’ incentives for a coup. The coup may become more attractive for the prince and less attractive for other members of the elite. In turn, this changes the equilibrium distribution of rents. We studied how these two effects balance each other and identified conditions when the governance structure with a successor gives a higher rent to the incumbent.

Our framework has its limitations. In particular, we made it a simple and uncoordinated choice for members of the elite whether to revolt. In our current analysis, we made it easy for players to trigger a coup. We assume that the coup has an uncertain outcome, but can be triggered by a single duke. In a more sophisticated environment in which a coup may require a conspiracy and joint action, the outcome of the coup may depend on the choices of the majority of the dukes. In fact, a coup may require establishing a network or a group in a conspiracy. One of the consequences of this is that players care much more about which player launches a coup.

Evidently, there are a whole wealth of questions regarding how succession arrangements can affect the relationship between an incumbent and the elite who may support his regime or may stage a coup. Our formal analysis focuses on the single question regarding how appointing a successor affects the stability of a ruler’s regime and the allocation of governance rents, but the discussion in the fourth section illustrates how the framework can be extended to investigate other questions. We are hopeful that the framework can form the basis for further investigations concerning how the incumbent ruler’s strategic choices regarding succession arrangements alter the structure of rulership competition and shape the evolution of governance structures.

Footnotes

Acknowledgments

We thank Gary Magee, Kunal Sengupta, and Tridib Sharma for helpful conversations, and participants of the 2014 Australasian Public Choice Conference for helpful comments. We are also grateful to Paul Huth (the editor) and two anonymous reviewers. The usual caveat applies.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Supplementary Material

Supplementary material for this article is available online.

Notes

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