# Quantum Game Application to Recovery Problem in Mobile Database
## 3 Quantum Game-Based Recovery Model
### Recovery Modeling Using Quantum Game
#### 3.2.4 Quantum Nash Equilibrium for Selection  
Thus, the player's overall gain in this game is equal to the sum of the reward values associated with the variables (\(C_{1,i}\), \(C_{2,i}\), \(C_{3,j}\)). The ultimate solution may be obtained in one of two ways: (1) by achieving a single and exclusive dominant equilibrium method in the game, or (2) by using NE [15]. The strategies produced via the first method, dubbed iterated elimination of strictly dominated strategies, reflect the optimal actions that each player might rationally take, and therefore comprise the game's (rational) solution. Regrettably, this alluring approach yields no prediction at all for some kinds of situations in which no strategy survives the elimination phase. In this situation, it is unclear which course of action would be deemed reasonable and best. This is a unique game in which there are no absolutely dominant strategies. NE is a game theory concept that determines the optimal result in a non-cooperative game in which each player has little incentive to change his or her initial strategy. Under the NE, a player wins nothing by departing from their initial strategy, guaranteeing that the strategies of the other players stay constant as well. A game may include several NE states or none at all [35]. Unfortunately, the majority of games lack dominating strategies. Thus, if there are many solutions (more than one NE) to a given issue, the alternative is to find another handling mechanism. To address this issue, all values in the payoff matrix are subject to an addition or subtraction mechanism based on one of the critical factors, such as execution time, so that more points may be awarded to the quickest element and vice versa (normalization and reduction phase). Finally, the updated payoff matrix is utilized to identify a more optimal solution (Pure Nash) that matches the various environmental factors. Since all players are assumed to be rational, they make their preferred decisions which maximize their rewards. Consequently, one strategy for a player is dominant over another strategy for another player if it always provides a greater payoff to that player regardless of the strategy played by the opposing player. Therefore, it is very important to determine the method of calculating the return for each strategy for each player. A particular algorithm is selected when its strategy achieves the highest NE in the reward matrix. In contrast to the classical situation, where the theory is incapable of making any unique prediction, the application of quantum formalism will show a new property: the emergence in entangled strategies of a NE reflecting the unique solution to the game. In the quantum version of this three-player game, players execute their strategies by applying the identity operators they possess with probability \(p\), \(q\), and \(r\) to the starting quantum state, respectively. The three players apply the inversion operator \(\sigma\) with probability (1-\(p\)), (1-\(q\)), and (1-\(r\)). If \(\rho_{in}\) in is the density matrix corresponding to the initial quantum state, then the final state after players have implemented their strategies is [17; 18]. \[\rho_{fin}=\sum\limits_{U=l,\sigma}P(H_{A})P(H_{B})P(H_{C})H_{A}\otimes H_{B} \otimes H_{C}\;\rho_{in}\;H_{A}^{\intercal}\otimes\;H_{B}^{\intercal}\otimes H _{C}^{\intercal} \tag{6}\]  
where either \(l\) or \(\sigma\) may be used as the unitary and Hermitian operator \(H\). \(P(H_{A})\), \(P(H_{B})\), and \(P(H_{C})\) are the probability that players \(A\), \(B\), and \(C\), respectively, will apply the operator \(H\) to the initial state. \(\rho_{in}\) is a convex combination of all quantum processes. Assume the arbitrator creates the following pure initial quantum state with three qubits (two strategies for each player for simplicity):  
\[ \begin{array}{l}|\psi_{in}\rangle=\sum\limits_{i,j,k=1,2}C_{ijk}|ijk\rangle \\ \sum\limits_{i,j,k=1,2}\left|C_{ijk}\left|ijk\right|^{2}=1\right.\end{array}