# Cooperative Game-Based Energy Efficiency Management over Ultra-Dense Wireless Cellular Networks
## 3 Cooperative Game in Ultra-Dense Wireless Cellular Networks
### Definitions and Concepts of Cooperative Game  
Let K = {1, 2,..., K} be the set of players, which are base stations in this paper. Let \(S\) be the energy-efficient strategy of the players, with \(P_{n}\) being the power-allocation strategy space. Player \(k\) has the utility/payoff function \(U_{k}\), which can be derived from the allocated power. \(U_{k}^{\min}\) is the minimum desired payoff that player \(k\) expects, which is called the disagreement point. In cooperative games, players attempt to reach an agreement that provides a mutual advantage. If the minimum utility \(U_{k}^{\min}\) is not achieved, player \(k\) does not cooperate. In non-cooperative games, each player gets the minimum utility without collaboration [26]. Assuming that \(\{s_{1}^{*},\ldots,s_{n}^{*}\}\) is the strategy set composed of a strategy for each player, if \(U_{i}(s_{1}^{*},\ldots,s_{i-1}^{*},s_{i+1}^{*},\ldots,s_{n}^{*})\geq U_{i}(s_ {1}^{*},\ldots,s_{i-1}^{*},s_{j}^{*},s_{i+1}^{*},\ldots,s_{n}^{*})\), \(\forall s_{j}\in S_{i},\big{\{}s_{1}^{*},\ldots,s_{n}^{*}\big{\}}\) is the optimal stable solution, which is called the Nash equilibrium (NE). The NE is a fixed point; thus, no player can improve its utility by changing its strategy unilaterally [26]. Because non-cooperative games emphasize the individual rationality and optimal decisions for individuals, the NE is not efficient all the time [32]. Cooperative games focus more on the collaboration between players to pursue the most advantageous strategy, which is usually collectively optimal. Let \(\Gamma=\{U_{1},\ldots,U_{n}\}\subset\Re^{n}\) represent the feasible utility-allocation set if all the players cooperate, which is convex and closed; \(\big{\{}U_{k}\in\Gamma\big{|}U_{k}\geq U_{k}^{\min},\forall k\in K\big{\}}\) is a non-empty and bounded set, and \(d=\big{\{}U_{1}^{\min},\ldots,U_{k}^{\min}\big{\}}\) is the disagreement point. The pair \(\{\Gamma,d\}\) constructs a K-player bargaining problem. We define the Pareto-optimal point among multiple players. It is impossible to find another point that improves the utility for all the players simultaneously [26]. **Definition 1**: _(Pareto optimality): The resource-allocation point \(\Gamma\) is Pareto-optimal if and only if any allocation \(U_{k}^{\prime}\) satisfies \(U_{k}^{\prime}\geq U_{k},\forall k\in K\) when \(U_{k}^{\prime}=U_{k}\)._  
For multiple players, there may be an infinite number of Pareto-optimal points [32]. Thus, we must select the best Pareto point, which requires a criterion for the bargaining solution. Multiple bargaining selection criteria can be used for our energy-efficiency problem (e.g., they consider optimality and fairness) [32]. We investigate a well-known Nash bargaining solution (NBS), which can provide a fair and unique Pareto optimal point satisfying the following axioms. **Definition 2**: _Let F be a function \(F:(\Gamma,d)\rightarrow\Re^{n}\)). Let \(B\) be the set of Pareto-optimal points, i.e., all the individually rational utility points in the cooperative utility region. \(U^{*}=\Gamma\left(\Gamma,d\right)\) is said to be an NBS in \(\Gamma\) for \(d\) if the following axioms are satisfied [26]:_  
1. _Individual rationality:_ \(U_{k}^{*}\geq d_{k}\) _for_ \(U_{k}^{*}\in U^{*},\forall k\in\text{K}\)_._
2. _Feasibility:_ \(U^{*}\in\Gamma\)_._
3. _Pareto-optimality:_ \(U^{*}\) _is Pareto-optimal.__._
4. _Independence of irrelevant alternatives: if_ \(U^{*}\in\Gamma^{\prime}\subset\Gamma\) _and_ \(U^{*}=F\left(\Gamma,d\right)\)_, then_ \(U^{*}=F\left(\Gamma^{\prime},d\right)\)_._
5. _Independence of linear transformations: For any linear scale transformation_ \(\psi\)_,_ \(\psi\left(F\left(\Gamma,d\right)\right)=F\left(\psi\left(\Gamma\right),\psi\left( d\right)\right).\)__
6. _Symmetry: if_ \(\Gamma\) _is invariant under all exchanges of players (base stations),_ \(F_{i}\left(\Gamma,d\right)=F_{j}\left(\Gamma,d\right),\forall i,j\)_._  
Axioms (1), (2), and (3) define the bargaining set B. Hence, the NBS is located in the bargaining set. Axioms (4), (5), and (6) are called fairness axioms.