## 2 Typical Categories of Game Theory
### Basics of Game Theory  
Game theory is increasingly attracting more attention as a mechanism to solve various problems in WSNs [71; 72; 73; 74; 75; 76]. Generally, a game consists of a set of players, a set of strategies for each player and a set of corresponding utility functions. A norm form game of a WSN of \(n\) sensor nodes is given by a 3-tuple G = \(<\)_N_, \(S\), _U\(>\)_. Here, \(G\) is a particular game, where \(N\) = {\(n_{1}\), \(n_{2}\),..., \(n_{n}\)} is a finite set of the sensor nodes. \(S\) = {\(S_{1}\), \(S_{2}\),..., \(S_{n}\)}, is the strategy space of the sensor node \(i\) can select from is represented by \(S_{i}\) (\(i\) = 1, 2,..., \(n\)). \(U\) = {\(u_{1}\), \(u_{2}\),..., \(u_{n}\)} is the corresponding payoff function of node \(i\) represented by \(u_{i}\) (\(i\) = 1, 2,..., \(n\)), \(u_{i}\) is a utility value of each node receives at the end of an action. A strategy for a player is a complete plan of actions in all possible situations in the game. The players try to act selfishly to maximize their consequences according to their preferences. We have to formulate the payoff functions in a way that will help node \(i\) to select a strategy \(S_{i}\) that represents the best response to the strategies selected by the other \(n\)-1 nodes. Here, \(s_{i}\) is the particular strategy chosen by node i and \(\textbf{s}_{\text{-}i}\) is the particular strategies chosen by all of the other nodes in the game. For strategies \(\textbf{s}\) = {\(s_{\text{i}}\), \(\textbf{s}_{\text{-}i}\)}, it is called a strategy profile or sometimes a strategy combination. Every different combination of individual choices of strategies can produce a different strategy profile. The strategy profile \(\textbf{s}\) = {\(s_{1}\), \(s_{2}\),..., \(s_{\text{n}}\)\(|\)\(s_{\text{i}}\)\(\in\)\(S_{\text{i}}\), \(i\) = 1, 2,..., \(n\)} needs to place the nodes responding to a Nash Equilibrium (NE). It is a solution concept that describes a steady state condition of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only its own strategy unilaterally. NE is identified wherein no nodes will rationally choose to deviate from his chosen strategy otherwise it will diminish its utility, _i.e._, \(u_{i}\) (\(s_{i}\), \(\textbf{s}_{\text{-}i}\)) \(\geq\)\(u_{i}\) (\(s_{i}^{\text{-}*}\), \(\textbf{s}_{\text{-}i}\)) for all \(s_{i}^{\text{-}*}\)\(\in\)\(S_{\text{i}}\). A utility function describing player preferences for a given player assigns a number for every possible outcome of the game with the property that a higher number implies that the outcome is more preferred. In literature [59], the utility is defined as:  
\[u_{i}(s_{i},\textbf{s}_{\text{-}i})\,{=}\,\frac{br}{F_{S_{i}}}\,f(\gamma_{j}) \tag{1}\] \begin{table}
\begin{tabular}{l l l} \hline  & **Terminology** & **References** \\ \hline (1) & Nash Equilibrium (NE) & [37; 38; 39; 40] \\ (2) & Pareto Optimal & [41] \\ (3) & Nash Bargaining Solution & [27; 35; 36; 42; 43; 44; 45; 46; 47] \\ (4) & Shapley Value & [30; 48] \\ (5) & Core & [49; 50; 51] \\ (6) & Mechanism Design (Computational) & [52; 53; 54] \\ (7) & Incentive compatible & [3] \\ (8) & Strategyproof Mechanism & [55] \\ (9) & Auction & [56] \\ (10) & Vickrey-Clarke-Groves (VCG) Mechanism & [57] \\ (11) & Utility Function & [7; 29; 55; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68] \\ (12) & Bayesian Nash Equilibrium (BNE) & [37; 58; 69; 70] \\ \hline \end{tabular}
\end{table} 
Table 2: Common Terminologies in \(\mathrm{GT}\). where \(u_{i}\) (\(s_{i}\), \(s_{i}\)) can be considered as the utility of node \(i\) transmitting information to a node \(j\), \(b\) is the number of information bits in a packet of size \(F\) bits, \(r\) is the transmission rate in bits/sec using strategy \(s_{i}\), and \(f\) (\(\gamma_{j}\)) is efficiency function which increase with expected SINR of the receiving node.