Israel and Palestine, the Sacramento Delta, the Nile, and climate change. These conflicts will not end with a settlement as long as the parties to the conflict cannot find a common ground for agreement.
In economics, this is known as a game without a "core" -- in the sense that there's no overlap between parties' range of acceptable alternatives.
I've reviewed a book about conflict over water in the Middle East where it's clear that a solution is economically feasible, but perhaps not politically acceptable.
I've published a paper [pdf] on the Sacramento Delta in which I suggest that the conflict can be solved by forcing a core (a put-your-money-where-your-mouth-is vote among alternatives).
Blake et al. have a very interesting paper [pdf] in which they use experiments to show that a Coasian bargaining outcome* will not result when there are three or more participants in the game (as is true in the Delta).
Madani and Lund have a paper [pdf] modeling conflict in the Delta as a prisoner's dilemma game (in which both sides have an incentive to defect instead of cooperate) that's evolving into a game of chicken in which the State of California is likely to be the loser. They show that it's possible to reach cooperation by changing payoffs -- a solution that's different in detail but similar in structure to mine.**
Bottom Line: Some conflicts cannot be solved without outside intervention to change the incentives. That's the role for parents, teachers and referees managing kids and players. That's the role that politicians and leaders need to play, except that they often prefer to be part of the problem instead of the solution. Plan accordingly.
* It's possible to reach an efficient level of pollution if either the polluter or pollutee are given property rights to pollute or be free from pollution, respectively.
** Madani's Game theory and water resources [$] is a useful publication for engineers.