In Republic of Rome, if Rome falls, everyone loses. Some gaming groups I've been in grok this concept completely and players will make major sacrifices of their positions (donating lots of money from personal treasuries to the state to fight off the Punic Wars in the Early Republic, for example), while others regard Rome falling as a draw, and deliberately play to cause Rome to fall if they think they can't win.
If you have a mixture of both types of player, then the fall-of-Rome-is-a-draw players can exploit the fall-of-Rome-is-a-loss players by threatening to bring Rome down and demanding concessions for saving it. This tends to provoke resentment on both sides.
Does anyone have any suggestions for (a) how to prevent this conflict and (b) how to resolve it before punches start being thrown?