Two remarks on the inner core

An axiomatization of the inner core using appropriate reduced games

“…A key part of the proof of the main result relies on the following lemma that proposes a variant to Proposition 4 in de Clippel and Minelli (2005), dealing now with games instead of exchange economies. The proof is included for the sake of completeness.…”

“…Lemma 1 does not hold for convex-valued games that do not satisfy the two regularity conditions, as the following example shows. It is similar to Example A.1 in de Clippel and Minelli (2005), but it is included for the sake of completeness.…”

“…Yet, it may happen that S can improve the satisfaction of all its members by randomizing between the two types of objection. It turns out that the inner core is precisely the set of feasible allocations that are immune to these random objections if the utility functions are smooth and concave (see de Clippel and Minelli, 2005). Lemma 1 proposes a variant of this result for NTU-games, and will play a key role in the proof of the main result.…”

“…It is worth noting that competitive equilibria always belong to the inner core in exchange economies with concave utility functions. The inner core being a subset of the core, the converge and equivalence results established for the core also apply to the inner core (Qin, 1994;de Clippel and Minelli, 2005).…”

“…A first mention to the inner core can be found in Shapley and Shubik (1975). Though perhaps cryptic in its original constructive definition, the inner core admits a natural interpretation in terms of random blocking plans (Myerson, 1991, Section 9.8;Qin, 1993;de Clippel and Minelli, 2005). Consider for instance an allocation x in the core of an exchange economy.…”