# Definition of a $k-$strong Nash Equilibrium

Consider a game $$G=(N, (A^i)_{i\in N}, (g^i)_{i\in N})$$, $$N=\{1,2,\dots,n\}$$, $$A=\Pi_{i\in N}A_i$$ is the set of actions and $$g^i:A\to \mathbb{R}$$ is the payoff function. The latter can be extended from $$\Delta(A)$$, which is the set of (correlated) strategies, to the real line. If $$S$$ is a calition, it is a non-empty member of $$2^N$$ and $$A^S=\Pi_{i\in S}A^i$$ is the actions set of the memebers of the coalition where a memeber of $$\Delta(A^S)$$ is called an $$S-$$strategy profile. Also, $$-S$$ denoted the complementary coalition. Suppose that $$U$$ is the set of uncorrelated strategy profiles and $$U^S$$ the set of the uncorrelated $$S$$-strategy profiles .Given $$q\in U$$, we write $$q = (q^S , q^{−S})$$ where: $$q^S\in U^S$$ , $$q^{−S} \in U^{−S}$$. I am trying to understand the intuition of the following definition

$$\mathbf{Definition:}$$ An uncorrelated strategy profile $$q\in U$$ is a $$k$$-strong Nash equilibrium if and only if for all coalitions $$S\subset N$$ satisfying $$|S|\leq k$$ and for every uncorrelated $$S$$-strategy profile $$p^S \in U^S$$, there exists a player $$i \in S$$ such that

$$g^i(q)=g^i(q^S,q^{-S})\geq g^i(p^S,q^{-S})$$

$$\textbf{Question:}$$ I can not understand the intuition of the $$k-$$strong N.E. It is an equilibrium where up to $$k$$ players do not have an incentive to deviate from the strategy profile $$q$$. What confuses me is the part where it says there exists a player $$i \in S$$" and I wonder it is enough for the coalition to choose the strategy profile $$q$$ if and only if one member of her, say $$i$$ gets weakly better with respect to any other $$(p^S,q^{-S})$$ strategy profile?

• Hi! What is your source for this definition? Sep 22 '21 at 14:08
• @Giskard Yuval Heller's paper Minority-proof cheap-talk protocol Sep 22 '21 at 14:16
• Can anyone tell me a textbook that it defines the coalition prtfrvtly? Sep 23 '21 at 16:41

In NTU games it is sensible that if a member of a coalition is worse off by implementing $$p^S$$ instead of $$q^S$$, then they will veto/object against this strategy.
Now weakly worse off is slightly stranger, but classic Nash-equilibrium uses the exact same notion. Players have no incentive to deviate if they do not profit from it. The situation is the same here - what has the coalition $$S$$ ever done for player $$i$$ that they should return the favour?