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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?

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  • $\begingroup$ Hi! What is your source for this definition? $\endgroup$
    – Giskard
    Sep 22 at 14:08
  • $\begingroup$ @Giskard Yuval Heller's paper Minority-proof cheap-talk protocol $\endgroup$ Sep 22 at 14:16
  • $\begingroup$ Can anyone tell me a textbook that it defines the coalition prtfrvtly? $\endgroup$ Sep 23 at 16:41
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A lot of cooperative game theory deals with transferable utility games, where players can easily transfer "utility" or "payoff" to each other. Another class is non-transferable utility (NTU) games where you get what you get after the strategy profile is implemented and that is that, there are no side transfers.

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?

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