## 5 Game-Theoretical Routing Scheme for Video-Streaming in MANETs  
A _game_ can be described by listing the players participating in the game, a set of strategies for those players, and a specification of payoffs for each combination of strategies. Let \(S\) be a finite set of \(N\) players \(1,...,N\). Each player \(i\) has a finite set of available actions \(A_{i}\). Let \(a_{i}\in A_{i}\) be each particular action chosen by player \(i\). The action space, \(A\), is the cartesian product of all \(A_{i}\), _i.e._, \(A=A_{1}\times A_{2}\times...A_{N}\). An _N_-tuple action, \(a\), is a point in the action space \(A\). A _pure strategy_ provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation it could face. A _mixed strategy_ of player \(i\), \(\alpha_{i}\), is an assignment of a probability, \(p_{i}\in P=[0,1]\), to each pure strategy. This allows a player to randomly select among the set of pure strategies. Let \(\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{N})\) be the mixed strategy profile, then the probability that a particular _N_-tuple action, \(a=(a_{1},a_{2},...,a_{N})\), will occur, \(p(a)\), is formed from the product of the probabilities assigned to \(a\) by \(\alpha\). Let \(u_{i}\) be the _utility function_ of player \(i\) in the strategic form game occurring in each stage. The utility function is a mathematical description of preferences that maps the action space to a set of real numbers. A utility function for a given player assigns a real number for every possible outcome of the game, so that a higher number implies that the outcome is more preferred. \[u_{i}:A\to R \tag{5}\]  
\(U_{i}(\alpha)\) is the expected utility for player \(i\) for the mixed strategy profile \(\alpha\), and has the following expression:  
\[U_{i}(\alpha)\equiv\sum_{a\in A}p_{a}\cdot u_{i}(a) \tag{6}\]  
A strategic game \(G\) can be expressed using three primary components: the set of players \(S\), the action space \(A\), and the set of individual utility functions for player \(i\), \(u_{i}\). \[G=(S,A,u_{i}) \tag{7}\]A mixed strategies extension to \(G\) is given by the next expression, where \(\Delta(A_{i})\) is the set of all probability distributions over \(A_{i}\) and \(U_{i}\) is the set of all expected utilities to \(i\). \[G^{\prime}=(S,\Delta(A_{i}),U_{i}) \tag{8}\]  
A _best response_ is a strategy which produces the most favorable outcome for a player, taking other players' strategies as given. A _Nash Equilibrium_ (NE) [32] is a solution in which each player plays a best response to the strategies of other players. Each player is assumed to know the strategies of the other players, and no player has incentive to unilaterally change their current strategy while the other players keep theirs unchanged. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if they remained with their current strategy. It is a mathematical fact that every mixed extension of a strategic game has at least one _mixed strategy Nash equilibrium_[32]. To define best responses more generally, we need a notation for the set of strategies used by all players other than player \(i\), named \(\alpha_{-i}\):  
\[\alpha_{-i}=(\alpha_{1},...,\alpha_{i-1},\alpha_{i+1},...,\alpha_{N}) \tag{9}\]  
Strategy \(\alpha_{i}^{*}\) is a best response for player \(i\) to the strategies of all players except \(i\), \(\alpha_{-i}^{*}\), if:  
\[U_{i}(\alpha_{i}^{*},\alpha_{-i}^{*})\geq U_{i}(\alpha_{i},\alpha_{-i}^{*}), \ \ \forall\ \alpha_{i}\ \in\ \Delta(A_{i}) \tag{10}\]  
This means that if \(\alpha_{i}^{*}\) is a best response for player \(i\) to the assumed set of strategies \(\alpha_{-i}^{*}\) played by the other _N-1_ players, then it must give player \(i\) a payoff at least as large as the player would get if they placed any other strategy \(\alpha_{i}\) from their set of allowed strategies.