As before, the two groups incur equal levels of campaign expenses but are characterised by different probabilities of success equal to their respective election campaign effectiveness. Comparison of (5) and (6) shows that we cannot say a priori whether fighting a violent conflict is more expensive than fighting an election, for this depends on the relative effectiveness parameters  and  and the destructive effect of the conflict , which tends to decrease the size of resources committed to winning the conflict. Substituting into (4.A) and (4.B), we obtain corresponding equilibrium net expected payoffs as VAG = [+k(1–2)]R (6.A) VBG = (6.B) A peaceful political arrangement will prevail if both rivals are better off under that arrangement, that is, when AGW VAG–VAW >0 and simultaneously BGW >0. Conflict arises if at least one of the two adversaries expects a higher payoff by engaging in a violent fight, that is, if (AGW<0, BGW>0), or (AGW>0, BGW<0), or (AGW<0, BGW<0). Upon substitution, we find AGW = [+k(1–2) – (1–)]R (7.A) BGW = {(1–) + k[2(2–)–1] – (1–)(1–)}R (7.B) With the coefficients ,  and  exogenously given, the size of the power-sharing parameter k emerges as a crucial variable in the decision to accept a peace settlement. Following Garfinkel and Skaperdas (2007), it is assumed that the rival groups agree to divide the gains that are realised when a settlement is agreed according to the formula, which postulates that each side obtains the same gain by avoiding war. Formally, this division rule implies that AGW= BGW. Substituting from the sets of equations (7), and solving with respect to k, we obtain the equilibrium size of k . (8) The latter is defined for values such that ½. Equation (8) shows that the power-sharing parameter depends on the fighting and electoral campaigning effectiveness of the rival groups and the destructive effect of war. More specifically, two components are
for ½, 0 when 0 ≤  < < (9.1) for ½, 0 when ≤  1 or ½ ≤  <  ≤ 1 (9.2) In order to gain a better understanding of what is involved, figure 1 shows the range of values of  (upper and lower limit) that yield  > 0 for parameter values of 0.1, 02, 0.3, 0.4, 0.6, 0.7, 0.8 and 0.9, and 0.1, 0.3, 0.5 and 0.9. The pattern that emerges is widely diverse. For 0.1 the interval of values of  (= war effectiveness of A) that lead to power-sharing shrinks as (= election effectiveness of A) rises for values of <½, but it increases as >½, while the opposite is true for =0.3 and =0.5; it is nevertheless widest (0, ½) as the destructive effect of war increases, =0.9, and <½. Finally, equation (8) can also be used to identify when the two rivals will agree to lay down their weapons and adopt majoritarian politics. Setting k* = 0 and solving we obtain 1– = . (9.3) That is, the two rivals will choose to establish a majoritarian democracy in the case where A’s election-war relative effectiveness equals the marginal income surviving the war. Since it is only by chance that this special condition is satisfied in practice, adoption of majoritarian democracy by the mutual agreement is unlikely. Figure 1: Values of which A and B choose power-sharing given ,  and R Figure 1 – continued 3.3 Conflict and power-sharing under risk aversion
campaign effectiveness of A and B (that is, they are independent of the size of rent from office). Both VAS and VBS are assumed to be positive. Comparing power-sharing to violent conflict we have ASC VAS – VAC (14.A) BSC VBS – VBC (14.B) For parameter values such that both ASC and BSC are positive simultaneously, both A and B choose a peaceful settlement. After the relevant manipulations and denoting and < 0, we derive the following critical values ASC > 0 when  < A (15.A) BSC > 0 when  > B where >0 . (15.B) The positive signs of A and follow from the positive signs of (13.A) and (13.B); I also assume that A<1and B>0. Thus, power-sharing is adopted for values of that satisfy (15.A) and (15.B) simultaneously. This requires that B < A, which in turn requires that the following inequality is satisfied: 3(lnR)2–2[E+–ln(1–)]lnR–[E++ln(1–)]2+4E > 0, which depends on the parameter values and cannot be solved analytically at this level of generality. Contrary to risk neutrality, in the present setting there is no power-sharing variable to solve for; we can only find the range of values B <  < A over which a power-sharing settlement may be adopted. A is the upper limit of A’s war effectiveness at which A compromises and shares power; analogously, B is the lower limit of A’s war effectiveness at which B settles for power-sharing – see figure 2. To put it another way, if  exceeds A, A expects a higher payoff from not compromising and pursuing the rewards of office through fighting, while, if  is below B, B’s fighting effectiveness is so high that he expects a higher payoff from fighting. If for some reason A rises or B falls, power-sharing becomes more likely. Table 2: Comparative static properties of the boundaries of power-sharing, A and B > (<) 0 as > (<) 0 > (<) 0 as < (>) 0