I'm struggling with part of a question from Kydd's IR and Game Theory book. Here's the question:
Consider two actors, 1 and 2, with utility functions defined over a single-dimensional issue space $X = [0,1]$, but where the actors can own sections or "provinces" of the interval. The boundaries between sections are defined by a vector $B=(b_1,b_2,...b_n,1)$. Thus, there are $n+1$ provinces. Define a "size" vector containing the lengths of each province, $S=(b_1,b_2-b_1,b_3-b_2,...,1-b_n)$. Define for each actor an "ownership" vector $O_i$ with $1$s in the place of the provinces that it owns and $0$s elsewhere.
Assume the utility of each actor is a sum of the squares of the provinces owned,
$$u_i = \sum\limits_{j=1}^{n+1}(o_js_j)^2$$
Here's the question: If the actors are free to consolidate multiple provinces into one province and trade provinces with each other, what set of boundaries is Pareto optimal with this utility function?
My instinct was that the only Pareto optimal outcome would be to trade and reduce until there were only two provinces, so the boundary would be $B=(b_1,1)$, but it also seems like if the set of boundaries is "lumpy," it may not be possible to trade down that far. For instance, if the initial allocation is
$$B = (.1,.4,1)$$ $$O_1 = (1,0,1)$$
There is no trade or consolidation in this case that will improve either player's utility without lowering the other player's.
It seems like the set of optimal boundaries is dependent on the initial division of provinces and I'm having trouble figuring out how to set up the different kinds of divisions that could occur.