# Quantum Game Application to Recovery Problem in Mobile Database
## 3 Quantum Game-Based Recovery Model
### Recovery Modeling Using Quantum Game
#### 3.2.3 Build Payoff Matrix  
In the proposed model's game, see Table 2, a finite collection of players \(N=3\) was used. Three players are allocated \(P_{1}\), \(P_{2}\), and \(P_{3}\) in the three-player game. \(P_{1}\) selects pure strategies from \(S_{1}\), \(P_{2}\) selects pure strategies from \(S_{2}\), and \(P_{3}\) selects pure strategies from \(S_{3}\). If \(P_{1}\) chooses the pure strategy \(i\), \(P_{2}\) chooses \(j\), and \(P_{3}\) chooses \(k\), then the reward to \(P_{1}\) is \(a_{ijk}\), \(P_{2}\) chooses \(b_{ijk}\), and \(P_{3}\) chooses \(c_{ijk}\). Define the \(S_{1}\times S_{2}\times S_{3}\) payoff \"cubes\" \(A\), \(B\), and \(C\) as [32; 33]:  
\[A=\left[a_{ijk}\right]\in R^{S1\times S2\times S3},\mathcal{B}=\left[b_{ijk} \right]\in R^{S1\times S2\times S3},\,C=\left[c_{ijk}\right]\in R^{S1\times S2 \times S3} \tag{1}\]  
We create a matrix for each of player 3's actions (strategies); accordingly, player 1 selects a row, player 2 selects a column, and player 3 selects a table. Table 3 illustrates the bi-matrix for a three-player game with its payoff. As a result, each third player strategy \(k\) is represented separately in a matrix together with its associated reward in terms of player 1 and player 2's strategies. Here, \(a_{111}\) represents the payoff value for player 1's plan or strategy given the return function \(u_{1}(s_{1},\,\,t_{1},c_{1})\) if \((s_{1},\,\,t_{1},c_{1})\) is selected, \(b_{111}\) denotes the payoff value for player 2's strategy provided the reward function \(u_{2}(s_{1},\,\,t_{1},c_{1})\) if \((s_{1},\,\,t_{1},c_{1})\) is chosen, and \(c_{111}\) denotes the payoff value for player 3's strategy with payoff function \(u_{3}(s_{1},\,\,t_{1},c_{1})\). The proposed recovery model's game is a non-cooperative game in which all participants choose their own tactics in order to maximize their profit [15; 32; 33]. These kinds of games are amenable to formalization via the use of normal-form representations [34]. In a normal-form game, player \(i\)'s strategy \(S_{i}^{\prime\prime}\) strictly trumps player i's strategy \(S_{i}^{\prime}\) if and only if:  
\[U_{i}\left(S_{i}^{\prime\prime},\,\,S_{-i}\right)\geq U_{i}\left(S_{i}^{ \prime},\,\,S_{-i}\right) \tag{2}\]  
for every list of \(S_{-i}\) of the other players that represents all players' strategies except player \(i\). Calculating the payoff in a game is complicated since it is dependent on the actions of other players. As a result, the strategy chosen by one player has an effect on the gain value of the other player. Three utility functions are included in this proposal as performance and assessment benchmarks for the candidate protocols. The functions are as follows: the amount of time consumed by the protocol during operation \((TIME_{i})\) for each strategy, the amount of memory consumed during operation \((MEMO_{i})\) for each strategy, and the rate \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Symbol** & **Meaning** \\ \hline \(G\) & & Quantum game model \\ \(N\) & & Number of players \\ \(i\) & & Game player \\ \(S\) & & Player’s strategy \\ \(U\) & & Payoff function or utility \\ \(B_{i}\) & & Best response for a player \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The symbols used for description game theory model.