<p>A lot of cooperative game theory deals with transferable utility games, where players can easily transfer &quot;utility&quot; or &quot;payoff&quot; 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.</p>
<p>In NTU games it is sensible that if a member of a coalition is worse off by implementing <span class="math-container">$p^S$</span> instead of <span class="math-container">$q^S$</span>, then they will veto/object against this strategy.</p>
<p>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 <span class="math-container">$S$</span> ever done for player <span class="math-container">$i$</span> that they should return the favour?</p>

CC BY-SA 4.0
2021-09-22T14:35:06.623Z
2021-09-22T15:23:00.887Z
2021-09-22T15:23:00.887Z
Giskard
<p>Consider a game <span class="math-container">$G=(N, (A^i)_{i\in N}, (g^i)_{i\in N})$</span>, <span class="math-container">$N=\{1,2,\dots,n\}$</span>, <span class="math-container">$A=\Pi_{i\in N}A_i$</span> is the set of actions and <span class="math-container">$g^i:A\to \mathbb{R}$</span> is the payoff function. The latter can be extended from <span class="math-container">$\Delta(A)$</span>, which is the set of (correlated) strategies, to the real line. If <span class="math-container">$S$</span> is a calition, it is a non-empty member of <span class="math-container">$2^N$</span> and <span class="math-container">$A^S=\Pi_{i\in S}A^i$</span> is the actions set of the memebers of the coalition where a memeber of <span class="math-container">$\Delta(A^S)$</span> is called an <span class="math-container">$S-$</span>strategy profile. Also, <span class="math-container">$-S$</span> denoted the complementary coalition. Suppose that <span class="math-container">$U$</span> is the set of uncorrelated strategy profiles and <span class="math-container">$U^S$</span> the set of the uncorrelated <span class="math-container">$S$</span>-strategy profiles .Given <span class="math-container">$q\in U$</span>, we write <span class="math-container">$q = (q^S , q^{−S})$</span> where: <span class="math-container">$q^S\in U^S$</span> , <span class="math-container">$q^{−S} \in U^{−S}$</span>. I am trying to understand the intuition of the following definition</p>
<p><span class="math-container">$\mathbf{Definition:}$</span> An uncorrelated strategy profile <span class="math-container">$q\in U$</span> is a <span class="math-container">$k$</span>-strong Nash equilibrium if and only if for all coalitions <span class="math-container">$S\subset N$</span> satisfying <span class="math-container">$|S|\leq k$</span> and for every uncorrelated <span class="math-container">$S$</span>-strategy profile <span class="math-container">$p^S \in U^S$</span>, there exists a player <span class="math-container">$i \in S$</span> such that</p>
<p><span class="math-container">$$g^i(q)=g^i(q^S,q^{-S})\geq g^i(p^S,q^{-S})$$</span></p>
<p><span class="math-container">$\textbf{Question:}$</span> I can not understand the intuition of the <span class="math-container">$k-$</span>strong N.E. It is an equilibrium where up to <span class="math-container">$k$</span> players do not have an incentive to deviate from the strategy profile <span class="math-container">$q$</span>. What confuses me is the part where it says ``there exists a player <span class="math-container">$i \in S$</span>&quot; and I wonder it is enough for the coalition to choose the strategy profile <span class="math-container">$q$</span> if and only if one member of her, say <span class="math-container">$i$</span> gets weakly better with respect to any other <span class="math-container">$(p^S,q^{-S})$</span> strategy profile?</p>

CC BY-SA 4.0
2021-09-22T14:08:48.057Z
Hi! What is your source for this definition?
Giskard
CC BY-SA 4.0
2021-09-22T14:16:09.237Z
@Giskard Yuval Heller's paper Minority-proof cheap-talk protocol
Hunger Learn
CC BY-SA 4.0
2021-09-23T16:41:32.437Z
Can anyone tell me a textbook that it defines the coalition prtfrvtly?
Hunger Learn
<mathematical-economics><game-theory><nash-equilibrium><definition>
Definition of a $k-$strong Nash Equilibrium
Hunger Learn
# Definition of a $k-$strong Nash Equilibrium

(Asked by: username_0 on 2021-09-22)

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?

**Tags:** mathematical-economics, game-theory, nash-equilibrium, definition
> **Comments:**
> **username_1**: Hi! What is your source for this definition?  
> **username_0**: @Giskard Yuval Heller's paper Minority-proof cheap-talk protocol  
> **username_0**: Can anyone tell me a textbook that it defines the coalition prtfrvtly?  

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## 1 answers

### Answer by username_1 on 2021-09-22. Score: 3 | Accepted answer

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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