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Recall the Myersons model (the model of weak institutions) discussed in Section 6. We did not discuss so much the effect of the leaders exogenous payoff ?? for surviving the second period. Assume pr < c

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Assume a polity consisting of an incumbent leader and two captains, with the later indexed by ?? =

??, ??. Over two periods, ?? = 1,2, the incumbent leader faces a challenge to their rule that can only

be faced with help from the captains. That is, there are two challengers, and one challenger fights

against the leader in each period. At the start of period 1and again in period 2 if the leader is

still in powereach captain decides whether to exert observable effort. Observable means that

the other players can witness whether the captain exerts effort or not, while exert effort means,

for instance, participation in the conflict and contributions to the costs of conflict. The leader

survives (to fight another day in the first period) with probability 1 if both captains exert effort, but

with probability ?? if only one does (0 < ?? < 1), and with probability zero if neither does. The value
of ?? represents the leaders power, that is, the leaders ability to survive even when the leader has
only partial support from the captains. Each captain bears a cost ?? > 0 for each period in which

they exert effort. In period 1, if the leader survives, the leader may (partially) compensate for these

costs by choosing a wage ???? ? 0 to provide each captain ?? in return for the effort by that captain.

Critically, we do not assume that the leader can credibly commit to this wage ex ante. Rather, in

equilibrium, it must be in the leaders self-interest to provide this wage after the captains have

decided whether to exert effort. Note that this wage might not be monetary compensation; it can

include sharing power, no taxation on investments, as well as organizational privileges and resources

for the military. Beyond the possible wage, each captain receives an exogenous payoff ?? > 0 if the

leader survives through the second period and the captain exerts effort in the second period. The

leader receives an exogenous payoff ?? > 0 for surviving the second period. To focus on the

interesting case, we assume that ???? < ?? < ?? < 2??. If the captain does not exert effort, they cannot
get anything. If both captains exert effort, both can obtain ?? by paying the cost of effort ??. If the
captain exerts effort but not the other, the probability that the leader survives ??, so the captains
expected payoff is ???? ? ??. It is assumed that ???? < ?? < ??, which means that ?? ? ?? > 0 and ???? ? ??

< 0. Therefore, this game has two pure-strategy Nash equilibria: one where both agents exert effort,
and one where neither does. Consider the first version of the model with weak institutions. Due to
the presence of weak institutions, the captains can communicate with each other. Thus, if Captain
?? does not receive wages from the leader, Captain ?? could know this from communicating with
Captain ??, and vice versa. Hence, the captains can use the two equilibria of the second period as
rewards and punishment. Formally, the captains can condition their joint efforts in the second period
on whether the leader has paid each captain ?? a wage ???? ? ???? , whereby ???? is the minimum wage
that justifies effort across both periods. If both captains exert effort in both periods, they get ?? at
the end of the second period, but need to pay ?? in two periods; that is, the payoff is ?? ? 2??. Since
we assumed ?? < 2??, the captains each prefer not to exert effort in the first period unless the ruler
pays them a compensating wage. If the captain can get ???? , the total payoff becomes ?? ? 2?? + ???? ,
and it must be positive to induce them to exert effort, which means that ???? ? ???? ? ??. Thus, the
minimum wage that justifies effort across both periods is: ???? = 2?? ? ?? (6.2) It is optimal for the
leader to pay this wage to both captains if the expected payoff from doing so is greater than that
of deviation. With weak institutions, if the leader pays ???? ? ???? to both captains, they will coordinate
to exert effort in the second period. Therefore, the leader can survive for certain and obtain ??. In
this case, it is optimal to set ???? = ???? for the leader, so the leader gets ?? ? 2???? . Otherwise, both
captains coordinate to not exert effort, and the leader will be defeated in the second period and
become zero. If the leader expects their own defeat, it is optimal to set ???? = 0, so the payoff would
simply be zero. As such, the leader has an incentive to pay ???? to both captains when ?? ? 2???? ? 0.
Substituting (6.2) into it, this condition is the following: ?? + 2?? ? 4?? (6.3) This is the same as the
efficiency condition. The total benefits for the leader and the two captains (?? + 2??) are higher than
the total costs paid by the two captains (2 × 2?? = 4??). We assume that this condition is always
satisfied (otherwise, this conflict would be inefficient for this polity, so the incumbent should retire).
If this efficiency condition is satisfied, the leader can always survive with weak institutions.
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