As is well-known, the strong Nash residuum in strategic games is a solution concept
defined to be a strategy indite at which no coalition has a deviation. Also, the coalitionproof
Nash equilibrium ascribable to Bernheim et al. (1987) is a refined concept of the Nash
equilibrium without credible deviations; namely, without deviations that stand be regarded non
to trigger further subcoalitional deviations. This of course implies that the strong Nash
equilibrium is a spare case of the coalition-proof Nash equilibrium. On the other hand, the
a-core and its subset, the b-core, ar also well-known solution concepts in strategic games
with coalitions.
These solutions involving coalitions are, however, mostly hard to earn except for the acore
of a game with certain convexity assumptions. In the literature, several attempts have been
made to obtain coalition-proof Nash equilibria. For deterrent example, allowing correlated strategies,
Moreno and Wooders (1996) found a qualified condition via the iterated settlement of
dominated strategies. Milgrom and Roberts (1996) considered a game with strategic
complementarity and formulated a sufficient condition in terms of monotone externalities.
Konishi et al.
(1997), Kukushkin (1997) and Yi (1999) presented existence results in games each with a specific structure.
We do not, however, deal with the ecumenical existence problem in this paper; instead, we
shall present an example of a strategic game that always has coalition-proof Nash equilibria
without any special assumptions and, moreover, has the nonempty a-core that coincides
with the b-core without any convexity assumptions. Specifically, we consider the pure
exchange game delinquent to Scarf (1971) with all the commodities being replaced by bads. This
game is not contained in the classes considered by the authors cited above; but, may be
viewed as a general, strategic version of the garbage disposal TU game discussed by
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