3. In the model of self-enforced property rights, it was shown that if the cost of conict 0 is a positive number less than 1, there were two subgame-perfect equilibria of the twostage game: one in which both players choose arms level 1/2 and always attack in the second stage, and one in which both players choose arms level (1 c) / (1+ 6) and do not attack in the second stage. A) Suppose that the players choose x A = x3 = 1/ 2 in the rst stage. Under what condition does the second-stage game have a PD normal form, and under what condition does it have a CO normal form? B) Suppose instead that the players choose x A = x3 = (1 c) / (1+ c) in the rst stage. Demonstrate that the second-stage game has a CO normal form. C) Verify that a necessary condition for the subgame-perfect equilibrium in which neither party attacks the other (so that de facto property rights are respected) is that xi. 2 [x_i -(l c)]/ (x. + c), i = A,B ]. The lowest arms levels that satisfy these two inequalities simultaneously are x A = x3 21 2c (assuming c S 1/2 ). Verify that these values are lower than x A = x3 = (lc)/ (1+c), then explain Why x A = x3 =l2c: deosn't satisfy the conditions for subgame perfect equilibrium. The expected payoffs of the two groups are then ' =(1-x;)-c+(x; / X)(1-x_) and 7"=(x/X)(1-x_;)-c. If both groups decide to attack, the probability that group i wins is again x / X, but the winning group receives all consumption goods, equal to 2-X . Thus, the expected payoff to each group is given by ' = [(x, / X)(2-X)]-cfor i=A or B. If neither group attacks, then the payoffs to each group are given by m' = (1-x ) for i=A or B. The normal form of the second-stage game is thus given by B doesn't attack B attacks A doesn't attack (1-XA), (1-XB) (x, / X)(1-XA) -C, (1-XB) - C+(XB / X)(1-XA) A attacks (1-x) -c+(x, / X)(1-x B), [(x, / X) (2- X)]-C, (XB / X) (1-XB) - C [(x B / X) (2- X) ]-c Finally, assume that groups act as Nash players against each other, but group members act as Kantian players within each group, doing whatever maximizes the expected payoff of the group as a whole. These assumptions imply that we must consider the subgame perfect equilibrium for the game played between the 2 groups, knowing that each group will act to maximize its expected payoff, given the strategy of the other group.3. Subgame perfect equilibrium Note rst that since [(xA fX)(2X)]c[(xA I'X)(1xA)c] = (x), / X)(1x3) 2 0, (Attack, Attack) is always a Nash equilibrium in the second stage for any choice arms levels. In this case, neither group respects the other group's property. As a result, the payoff function for group i = A,B is If = [(x, / X)(2 X )] c. To nd each group's best-response function, take the other group's arms levelx. as given and choose x, to maximize this payoff function. If the other group's arms level is x_, = 0, the maximizing group gets if: = [1 - (2 x,)]cfor any positive level of x, , but only if = [(1 / 2)(2)] c if x, = 0. Consequently, the sum of arms levels in this equilibrium can never be zero. Noting this, take the corresponding rst-order condition, which is given by air" I 6x, = [(x_,. f(x_,. + if) - (2 (x_, + 3] [,. / (x_,. + 33,- )] = O,i = A, B. Multiplying through by (x_,. + 526,.)2 then rearranging yields [x_, -(2 (x_, + 9] = [5%, -(x_, + 5%)], i = A,B. Since these two equations must hold simultaneously in Nash equilibrium, assume x, = xjv for both players and add the equations together to yield [X N -(2XN)] =[XN -XN )] :> X N =1. Substituting this result back into the modied rst-order condition then yields x: = x: = 1/2. Thus, the two groups will arm themselves and then attack, so there is no uniform protection of property in the unique equilibrium. Next, note that for (Don't attack, Don't Attack) to be a Nash equilibrium in the second stage, it must be that (1x,)[(1x,)c+(x, / X)(1x_,)] =c(x, / X)(1x_,) 2 0, t' = A,B. An immediate implication is that x; = x; = Ocannot be sustained as a subgame-perfect equilibrium unless c 21, since otherwise, if one player chooses not to arm, the other player could gain by choosing some positive level of arms and attacking. For lower values of 0, each group must allocate some positive amount to arms in order_to_ deter an attack from the other group, and thus sustain (Don't attack, Don't attack) as a subgame perfect equilibrium. W equilibrium can be attained only if x, 2 (1 c) / (1+ 0) for i = A,B. Since in this equilibrium each group prefers the lowest level of arms consistent with the other group having no incentive to arm and attack, the unique (Don't attack, Donit attack) equilibrium would be such that x: = x: = (1 c)/ (1+ 6). In this case, both groups arm but respect the other group's property, so \"property rights\" are sustained. Which equilibrium will obtain depends on the prior beliefs of the two parties. If both groups are sufficiently \"optimistic,\" then the \"property rights\" equilibrium is chosen. But if both groups are sufficiently \"pessimistic\