microconvexities

Microconvexities are a concept originating in game theory and economics, particularly in the study of cooperative games. A cooperative game is one where players can form coalitions to achieve a better outcome than they could individually. In such games, the "core" represents the set of feasible payoffs that cannot be improved upon by any subgroup of players. A game is said to have microconvexities if the core is non-empty and convex.

The existence of microconvexities implies that there are stable outcomes in the game. A non-empty core means that there is at least one allocation of the total value generated by the grand coalition (all players working together) that is Pareto efficient and cannot be undermined by any smaller coalition. Convexity of the core further refines this stability by ensuring that any weighted average of two core allocations is also a core allocation. This property is significant because it suggests a continuous range of stable outcomes, rather than isolated points.

The concept is important for understanding situations where cooperation can be sustained. If a game lacks microconvexities, it may be difficult to find stable agreements or enforce them, as there might be no payoff distribution that satisfies all potential coalition deviations. Research in this area often focuses on identifying conditions under which games exhibit microconvexities, or on developing mechanisms to ensure their existence in practical applications.