<p>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 <a href="http://en.wikipedia.org/wiki/Coalition-proof_Nash_equilibrium" rel="nofollow">Coalition-proof Nash equilibrium</a> rather than Strong Nash. People can freely communicate but <strong>cannot make binding commitment</strong> before deciding.</p>

<p>In <a href="http://link.springer.com/article/10.1007%2FBF01213445#page-1" rel="nofollow">some games</a>, 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.</p>

CC BY-SA 3.0
2015-05-18T14:38:55.340Z
2015-05-18T14:38:55.340Z
VicAche
<p>An allocation is in the core if there's no coalition that blocks it.</p>

<p>A strong equilibrium (<a href="http://www.ma.huji.ac.il/raumann/pdf/Acceptable%20Points%20in%20General.pdf" rel="nofollow">Aumann, 1959</a>) 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.</p>

<p>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.</p>

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2022-04-15T03:52:08.230Z
I have a similar question. First, I think I know why Core is a solution concept. As stated in Chwe (1994), if structure a is directly dominated by b via group S and S is capable of moving to b, structure a cannot be stable. But I also can not clearly distinguish the definitions of core and strong nash equilibrium. Maybe, the core just prevents the deviations via "effective groups"?
dre wang
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2015-05-06T16:29:30.447Z
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?
Shane
CC BY-SA 3.0
2015-05-06T16:35:45.080Z
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.
Metta World Peace
CC BY-SA 3.0
2015-05-06T16:39:28.777Z
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.
Metta World Peace
CC BY-SA 3.0
2015-05-06T16:44:42.640Z
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.
Shane
CC BY-SA 3.0
2015-05-06T16:46:30.070Z
Could you please specify what housing allocation problem is?
Metta World Peace
CC BY-SA 3.0
2015-05-06T16:49:15.873Z
It's the question of how to allocate finitely many students to finitely many housing places, such that there's a fixed quota of students for each housing place and each student may only be assigned to one housing place. See http://web.stanford.edu/~niederle/HouseAllocation.pdf for an introduction.  Hylland & Zeckhauser, JPE 1979, set it up well also.
Shane
CC BY-SA 3.0
2015-05-06T16:56:37.263Z
You're right. In this question, a player's action space is a function of other players' chosen actions. It's similar to the problem of general equilibrium.
Metta World Peace
CC BY-SA 3.0
2015-05-06T17:03:31.040Z
But in your second paragraph, strong equilbirum is a Nash equibrium with more stringent conditions, which means action spaces should be predetermined. So I think  strong equilibrium is not applicable to housing allocation problem.
Metta World Peace
CC BY-SA 3.0
2015-05-06T17:07:21.447Z
But you should aware,  the notion of Nash equilibrium  is not applicable to a problem in which a player's action space is determined by other players' actions.
Metta World Peace
CC BY-SA 3.0
2015-05-06T17:28:28.930Z
OK - thank you. If you want to synthesize your comments into an answer, I'll accept it.
Shane
<game-theory><nash-equilibrium><cooperative-game-theory>
How does the core relate to strong equilibrium?
Shane
# How does the core relate to strong equilibrium?

(Asked by: username_0 on 2015-05-06)

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

A strong equilibrium ([Aumann, 1959](http://www.ma.huji.ac.il/raumann/pdf/Acceptable%20Points%20in%20General.pdf)) 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.

**Tags:** game-theory, nash-equilibrium, cooperative-game-theory
> **Comments:**
> **username_0**: 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?  
> **username_2**: 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.  
> **username_2**: 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.  
> **username_0**: 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.  
> **username_2**: Could you please specify what housing allocation problem is?  
> **username_0**: It's the question of how to allocate finitely many students to finitely many housing places, such that there's a fixed quota of students for each housing place and each student may only be assigned to one housing place. See http://web.stanford.edu/~niederle/HouseAllocation.pdf for an introduction.  Hylland & Zeckhauser, JPE 1979, set it up well also.  
> **username_2**: You're right. In this question, a player's action space is a function of other players' chosen actions. It's similar to the problem of general equilibrium.  
> **username_2**: But in your second paragraph, strong equilbirum is a Nash equibrium with more stringent conditions, which means action spaces should be predetermined. So I think  strong equilibrium is not applicable to housing allocation problem.  
> **username_2**: But you should aware,  the notion of Nash equilibrium  is not applicable to a problem in which a player's action space is determined by other players' actions.  
> **username_0**: OK - thank you. If you want to synthesize your comments into an answer, I'll accept it.  
> **username_1**: I have a similar question. First, I think I know why Core is a solution concept. As stated in Chwe (1994), if structure a is directly dominated by b via group S and S is capable of moving to b, structure a cannot be stable. But I also can not clearly distinguish the definitions of core and strong nash equilibrium. Maybe, the core just prevents the deviations via "effective groups"?  

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## 1 answers

### Answer by username_3 on 2015-05-18. Score: 2 | Accepted answer

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](http://en.wikipedia.org/wiki/Coalition-proof_Nash_equilibrium) rather than Strong Nash. People can freely communicate but **cannot make binding commitment** before deciding.

In [some games](http://link.springer.com/article/10.1007%2FBF01213445#page-1), 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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