# Bargaining Power in Cooperative Resource Allocations Games
## 2 Method
### Weighted Approaches  
The above constraints, respectively, evaluate the individual rationality, group rationality, and efficiency conditions. Equation (1) ensures that the benefit that player \(i\) can achieve under cooperation will be more than the benefit that the same player can gain under non-cooperation (status-quo). Equation (2) ensures that the sum of the cooperative allocationsof all players is greater than the total obtainable benefits of any coalition they can form together Equation (3) ensures that the grand coalition's overall gain equals the total benefits allocated to all the members of that grand coalition, ensuring that the incremental gains of cooperation are fully allocated to the cooperating parties. This set of equations establish the Core of the cooperative game [24]. A given cooperative game theory solution must belong to the Core to be potentially acceptable and stable. The Core allocation is a driving force for players to cooperate as it ensures that players can gain more through cooperation rather than acting individually. It also ensures that the grand coalition can provide the best gains to the players. The core can contain an infinite number of cooperative solutions but can be also empty when no cooperative solutions exist. The cooperative game theory methods, used in this study, select their solutions out of the Core, i.e., they can provide solutions that are potentially acceptable to the players only when the Core is non-empty. **Nash-Harsanyi (N-H)**  
The N-H solution [25; 26] is the expanded form of the Nash bargaining solution for an n-players bargaining game:  
\[ \begin{split} Max\ \prod\limits_{i=1}^{n}\left(U_{i}^{*}-U_{i} \right)\\ \text{Subject to Equations }(1)-(3)\end{split}  \tag{4}\]  
where \(\left(U_{i}^{*}-U_{i}\right)\) is player i's gain under cooperation. **Shapley value**  
The Shapley [27] value is a cooperative solution that considers the weighted average of a player's contribution to all possible coalitions as the fair and efficient gain of that player under cooperation:  
\[U_{i}^{*}=\sum\limits_{s\subset N,\ i\in S}\frac{(n-|S|)!(|S|-1)!}{n!}(V(s)-V (S-\{i\})) \tag{5}\]  
where \(V(S-\{i\})\) is the total value of coalition \(S\) without player \(i\). **Nucleolus value**  
The Nucleolus solution is another cooperative game solution concept first introduced by [PERSON] [23]. This method minimizes the worst-case dissatisfaction of the most displeased coalition. The benefit allocated to each beneficiary through the Nucleolus solution is determined by finding \(\varepsilon\) through the following optimization model:  
\[Max\ \ \varepsilon \tag{6}\]  
Subject to :  
\[\varepsilon\leq\sum\limits_{i\in s}U_{i}^{*}-V(s)\ \ \ \forall s\in S,\ S\subset N\]  
Equations (1)-(3)  
where \(\varepsilon\) is the maximum imposable tax on all coalitions to keep them in the Core. ### Weighted Approaches  
The conventional cooperative game theory methods, consider the endogenous weights of the players in negotiations. As specified above, these methods have different rationales for determining such weights including the players' status-quo gains and/or the value they can bring to different coalitions. In real-world situations, the players' negotiation power is not limited to only driven by endogenous factors, but also by some variables (e.g., economic, and political) that are exogenous to the game. Overlooking the exogenous weights of the players in negotiations can produce misleading and impractical results. Therefore, here, we propose new formulations for the three overviewed conventional cooperative game theory methods that take both exogenous and endogenous negotiation powers of the players into account to determine their cooperative shares.