1. Suppose you have two nations, A and B, in a dispute over territory (or
something) of value Y in each time period. At the beginning of each time
period they must spend eA and eB to maintain their bargaining position
- - we'll assume that those influence their strength so that if they end up in
a conflict the probabilities of winning are qa and QB. In each time period
the nations must choose whether to come to a settlement (a division of
Y) or to fight it out (once and for all). Conflict is destructive and the
share 1 - 0 of Y is destroyed in the period conflict takes place (so in the
conflict period 6Y remains but in future periods the land is still worth
Y to the winner). Suppose in each bargaining round, A makes an offer
(S, for subsidy) and B can either accept or reject. If B rejects, conflict
ensues. As usually, future payoffs are discounted by s € [0, 1].
(a) Write the single period payoff to the players if they reach an agree-
ment in which A pays B a subsidy S in each period.
(b) Write the discounted sum of each players' payoff if they reach an
agreement in which A pays B a subsidy S in each period.
(c) Write the single period payoff to actor A if conflict occurs.
(d) Write the discounted sum of i's payoff if conflict occurs. (Note that
when i makes the decision to fight, it has already spent ei - SO it
doesn't affect its payoff in the first period - and if conflict occurs
the winner doesn't have to worry about maintaining its bargaining
position in the future).
(e) Using your answers to b) and d), derive the optimal subsidy S*.
(f) For S* to be an equilibrium offer, A must be willing to make that
offer. Show the conditions under which that is the case.
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