Appendix for “The Feudal Revolution and Europe’s Rise: Political Divergence of the Christian and Muslim Worlds

To formalize the intuition presented in Section 4 using a simple framework, suppose that a perfectly myopic, risk-neutral sovereign imperfectly controls a polity that creates output of size one each period. Denote by γ the amount of land controlled by the military regardless of the actions of the sovereign (this can be interpreted as the percentage of the entire polity controlled by the military). Suppose that there are N perfectly myopic, risk-neutral members of the military (where N is suffi ciently large) and that γ is evenly distributed between the members of this class. We consider the parameter value γ exogenously given. A value of γ = 0 corresponds to a perfectly absolutist sovereign (who uses mamluks or mercenaries to staff his military) whereas higher values of γ denote more feudal arrangements. Note that our assumption of perfectly myopic agents allows us to abstract from the potentially important issue of how the sovereign compensates the military (i.e. monetary payments versus land grants).1 In addition, we abstract from other important issues in order to focus on the sovereign’s desire to prevent a successful revolt. We do so in order to highlight one mechanism that we believe contributes to the observed increase in ruler durations. The order of play in the game is as follows: after observing γ the sovereign moves first and decides whether to keep the entire amount of output he controls to himself or whether to divide it equally between himself and the military. Once the sovereign makes this decision, the military observes both γ and the decision of the sovereign. After observing this information, the military decides whether to revolt or not. If the military revolts the revolt will succeed with probability π ∈ (0, 1). If a revolt is undertaken and succeeds the sovereign gets nothing and the military get 1 N each. If it does not suceed the sovereign gets 1 and the military gets nothing. If revolt is not undertaken the payoffs (sovereign, military) are ( 1−γ N+1 , γ N + 1−γ N+1) if the sovereign shares and (1− γ, γ N ) if he does not. Let σS = {s} be the actions taken by the sovereign (share, not share). Similarly σM = {r(s)} are the actions of the military (revolt, not revolt). We solve for the subgame perfect Nash equilibrium using backwards induction (we assume that if indifferent the military does not revolt). First, note that if the sovereign shares the military will never revolt since π N < γ N + 1−γ N+1 when N is large. If the sovereign does not share, the military will revolt if and only if π > γ. We call this point the “mamluk (or absolutist) cutoff”since it implies that the probability of successful revolt is higher