N$, there exists a player $k \in C$ such that $x_k \geq v(C)$

An intuitive way to understand this definition is that players in the grand coalition are only satisfied with their current reward if there are no subcoalitions where a subset of players can choose to form privately and simultaneously earn a higher reward. As such, considering that the maximum payoff one can earn in any subcoalition is the value created by the subcoalition itself, there must be at least one player in every possible subcoalition $C$ who is contented with his current payoff under the grand coalition (i.e $x_k \geq v(C)$), preventing that subcoalition of players from deviating privately (since the refusal of a single player is sufficient to prevent a group of players from deviating together).

Definition of proportionality
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In Section 3, we introduced how the *Shapley value* derives a payoff vector $\phi(N) = <\phi_1, \phi_2, ... , \phi_n>$ $= <x_1,x_2,\dots,x_n>$ (assuming all $n$ players work together in coalition $N$) which dictates how the value generated from the grand coalition should be divided amongst its members to maintain some fair properties. In fact, the *shapley value* indicates the level of contribution of each member with respect 