# An Efficient Dynamic Load Balancing Scheme Based on Nash Bargaining in SDN
## 3 Multi-Objective Decision Making Based on Nash Bargaining
### Nash Bargaining Model  
[PERSON] proposed to use the Nash bargaining model to solve the cooperative game problem [25]. Firstly, different targets are regarded as different players, and the initial strategy and the income functions are set. Each player party continuously negotiates and negotiates in the policy space and finally obtains a Nash equilibrium solution. This solution has Pareto validity, equivalent income invariance, and irrelevant selection independence [26]. [PERSON] uses the Nash product to represent global benefits and proves that the largest solution of the Nash product is the Nash equilibrium solution, which falls on the Pareto frontier and can achieve fairness and global optimization for multiple targets. The mathematical expression of game \(g\) is as follows: Suppose \(g\) has \(m\) game players, i.e., \(p_{1},p_{2},\ldots,p_{m}\), and all the alternative strategies of each game player in the strategy space are called the strategy set, i.e., \(s_{1},s_{2},\ldots,s_{m}\). The gain function obtained by the player according to the selected strategy is expressed as \(u_{1},u_{2},\ldots,u_{m}\). Then, the game can be expressed as \(g=\{p_{1},p_{2},\ldots,p_{m};s_{1},s_{2},\ldots,s_{m};u_{1},u_{2},\ldots,u_{m}\}\). Switch migration can be viewed as a remapping of the connection between the switch and the controller in the network. In an actual network, the switch does not form a mapping relationship with all controllers. If the connectivity between the node devices is not considered, the new mapping relationship obtained can only be an ideal state. Based on the above considerations, the remapping of the switch (that is, the policy set) needs to satisfy the connectivity constraint, as shown in Equation (17),  
\[x_{kj}= \begin{cases}0\text{ or }1&\sum\limits_{s=1}^{N}x^{\prime}_{ki}a_{ks}x^{ \prime}_{kj}>0\\ 0&\sum\limits_{s=1}^{N}x^{\prime}_{ki}a_{ks}x^{\prime}_{kj}=0\end{cases}  \tag{17}\]  
where \(x^{\prime}_{kj}\) is the original mapping relationship between switch \(s_{k}\) and controller \(c_{j}\), and \(x_{kj}\) is the mapping relationship after network remapping. If switch \(s_{k}\) in the management domain of controller \(c_{i}\) has connectivity with any switch \(s_{s}\) in the management domain of controller \(c_{j}\), then switch \(s_{k}\) can form a new connection relationship with controller \(c_{j}\) (that is, \(x_{kj}=1\)) or maintain the original mapping relationship (i.e., \(x_{kj}=0\)). If there is no connectivity, switch migration cannot be achieved (that is, \(x_{kj}\) can only be taken as 0). In this paper, the load balancing degree and migration cost are regarded as two players. \(LBD\) and \(Cost\) are the gain functions of these two players, respectively. The possible mapping relationship between the switch and controller is used as the strategy space of both sides of the game, and the [PERSON] bargaining game model is established, as shown in Formulas (18)-(23),  
\[\max\ (LBD-\mathrm{a}_{LBD})(Cost-\mathrm{a}_{cost}) \tag{18}\]  
\[L_{j}\leq\delta\omega_{j};\ 1\leq j\leq\mathrm{M} \tag{19}\]  
\[x_{kj}= \begin{cases}0\text{ or }1&\sum\limits_{s=1}^{N}x^{\prime}_{ki}a_{ks}x^{ \prime}_{kj}>0\\ 0&\sum\limits_{s=1}^{N}x^{\prime}_{ki}a_{ks}x^{\prime}_{kj}=0\end{cases}  \tag{20}\]\[\sum_{j=1}^{M}x_{kj}=1;\ \forall s_{k}\in S \tag{21}\]  
\[\mathrm{a}_{LBD}\leq LBD \tag{22}\]  
\[\mathrm{a}_{Cost}\leq Cost \tag{23}\]  
where \(\mathrm{a}_{LBD}\) and \(\mathrm{a}_{Cost}\) are the bargaining breakdown points for the load balance and the migration cost, respectively, i.e., the worst possible return for both sides of the game. The goal of the game is to achieve a game agreement at least at this breakpoint. In the game process, each game player will tend to change their own breakpoint to improve their own income. If there is no restriction on this selfish behavior, each player will continuously change the breakpoint value, causing the bargaining to fail.