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An allocation is in the core if there's no coalition that blocks it.

A strong equilibrium (Aumann, 1959) is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all members of the coalition.

How are these distinct? I usually don't think of the core as a solution concept, but I'm struggling to disentangle it from strong equilibrium.

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  • $\begingroup$ By singleton action spaces, do you mean there's one good and the action choice is a quantity? That is, the action choice is a continuum, whereas in Aumann, the action space is a finite set? $\endgroup$ – Shane May 6 '15 at 16:29
  • $\begingroup$ Sorry, I got it wrong. They're not singletons, which should be two actions, which can be interpreted as, participate and, refuse to parcipate. I think, in this case, two solution concepts should be the same. $\endgroup$ – Metta World Peace May 6 '15 at 16:35
  • $\begingroup$ In Aumann's setting, the players forming the coalition can deviate by more than one way, and at least one of the corresponding strategy profile should be profitable in terms of the sum of payoffs of these deviant players for them. $\endgroup$ – Metta World Peace May 6 '15 at 16:39
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    $\begingroup$ That would make sense, but I thought the core was defined in richer environments also. Consider a housing allocation problem. I'm not sure how you would define an action space in this setting. So maybe that's the difference -- the core is more related to the social planner's (i.e. centralized) problem whereas the strong equilibrium relates to the decentralized problem. $\endgroup$ – Shane May 6 '15 at 16:44
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    $\begingroup$ OK - thank you. If you want to synthesize your comments into an answer, I'll accept it. $\endgroup$ – Shane May 6 '15 at 17:28
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Strong Nash equilibrium is different from core mainly because of communication. In a Strong Nash, unlimited private communication is allowed. The core is a concept that is linked to Coalition-proof Nash equilibrium rather than Strong Nash. People can freely communicate but cannot make binding commitment before deciding.

In some games, both happen to be the same, but in generality, the core is a concept derived from Coalition-proof Nash equilibrium rather than from Strong Nash Equilibrium.

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