If we use correlated equilibria as our solution concept rather than Nash, convexity is always guaranteed. Also, this is usually the more realistic assumption for modeling purposes. Nash equilibria oddly assume certainty about which equilibrium a game will be in even as players are trying to reason about how to approach a game. So it’s really only applicable to cases where players know what equilibrium they are in, EG because there’s a long history and the situation has equilibriated.
But even in such situations, there is greater reason to expect things to equilibriate to a correlated equilibrium than there is to expect a nash equilibrium. This is partly because there are usually a lot of signals from the environment that can potentially be used as correlated randomness—for example, the weather. Also, convergence theorems for learning correlated equilibria are just better than those for Nash.
Still, your comment about mistake theorists believing in a convex boundary is interesting. It might also be that conflict theorists tend to believe that most feasible solutions are in fact close to Pareto-efficient (for example, they believe that any apparent “mistake” is actually benefiting someone). Mistake theorists won’t believe this, obviously, because they believe there is room for improvement (mistakes to be avoided). However, mistake theorists may additionally believe in large downsides to conflict (ie, some very very not-pareto-efficient solutions, which it is important to avoid). This would further motivate the importance of agreeing to stick to the Pareto frontier, rather than worrying about allocation.
If we use correlated equilibria as our solution concept rather than Nash, convexity is always guaranteed. Also, this is usually the more realistic assumption for modeling purposes. Nash equilibria oddly assume certainty about which equilibrium a game will be in even as players are trying to reason about how to approach a game. So it’s really only applicable to cases where players know what equilibrium they are in, EG because there’s a long history and the situation has equilibriated.
But even in such situations, there is greater reason to expect things to equilibriate to a correlated equilibrium than there is to expect a nash equilibrium. This is partly because there are usually a lot of signals from the environment that can potentially be used as correlated randomness—for example, the weather. Also, convergence theorems for learning correlated equilibria are just better than those for Nash.
Still, your comment about mistake theorists believing in a convex boundary is interesting. It might also be that conflict theorists tend to believe that most feasible solutions are in fact close to Pareto-efficient (for example, they believe that any apparent “mistake” is actually benefiting someone). Mistake theorists won’t believe this, obviously, because they believe there is room for improvement (mistakes to be avoided). However, mistake theorists may additionally believe in large downsides to conflict (ie, some very very not-pareto-efficient solutions, which it is important to avoid). This would further motivate the importance of agreeing to stick to the Pareto frontier, rather than worrying about allocation.