# State-Of-The-Art and Prospects for Peer-To-Peer Transaction-Based Energy System
## 5 Frameworks for TE
### Classification Based on Optimization Techniques
#### 5.3.4 Classification Based on Variational Inequality Theory  
While optimization models are useful in the study of TE, game theory may be viewed as an integration of a set of optimization problems. Sometimes, both optimization models and game theory models may fail. Thus, a more robust and general technique for nonlinear analysis when classical game theory may fail is the variational inequality (VI) theory. VI-theory finds applications in typical convex optimization problems and delves strongly into game theory [102].  
\begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Ref.** & **Structure** & **Game Type** &
\begin{tabular}{c} **Optimization** \\ **Techniques** \\ \end{tabular} & **Algorithm Design** & **Desired Outcome** \\ \hline
[24] & Distributed & N/A & \begin{tabular}{c} Convex \\ optimization \\ \end{tabular} & \
\begin{tabular}{c} Sub-gradient-based \\ cost minimization \\ algorithm \\ \end{tabular} &
\begin{tabular}{c} Minimize global \\ operational cost. \\ \end{tabular} \\ \hline
[36] & \begin{tabular}{c} Centralized \\ control/P2P \\ energy trading \\ \end{tabular} & N/A & \
\begin{tabular}{c} Non-convex \\ MINLP/Pareto \\ optimality \\ \end{tabular} & N/A &
\begin{tabular}{c} Cost optimization \\ and optimal MG \\ energy and price for \\ P2P trading. \\ \end{tabular} \\ \hline
[78] & \begin{tabular}{c} Centralized \\ control through \\ ESP \\ \end{tabular} & N/A & \begin{tabular}{c} Bi-level \\ programming \\ \end{tabular} & \
\begin{tabular}{c} Distributed iterative \\ algorithm \\ \end{tabular} &
\begin{tabular}{c} Saving prosumer’s \\ cost and improving \\ energy sharing. \\ \end{tabular} \\ \hline
[96] & Distributed & \
\begin{tabular}{c} Coalition \\ game/Shapley \\ value \\ \end{tabular} & N/A & N/A &
\begin{tabular}{c} To investigate price \\ of electricity in direct \\ trading. \\ \end{tabular} \\ \hline
[21] & \begin{tabular}{c} Centralized \\ control \\ \end{tabular} & \begin{tabular}{c} Cooperative \\ and \\ non-cooperative \\ \end{tabular} & N/A & \
\begin{tabular}{c} Distributed and \\ iterative \\ \end{tabular} &
\begin{tabular}{c} To reduce monetary \\ expense. \\ \end{tabular} \\ \hline
[68] & Distributed & \
\begin{tabular}{c} Leader-follower \\ Stackelberg \\ game/Nash \\ equilibrium \\ \end{tabular} & N/A & N/A &
\begin{tabular}{c} To study the \\ economic benefits of \\ the leader-follower \\ energy trading \\ mechanism. \\ \end{tabular} \\ \hline
[83] & \begin{tabular}{c} Distributed \\ control \\ \end{tabular} & Stackelberg \\ game & N/A & \
\begin{tabular}{c} Distributed \\ management \\ algorithm \\ \end{tabular} &
\begin{tabular}{c} They proved that a \\ Nash equilibrium \\ exist in the two-stage \\ Stackelberg game. \\ \end{tabular} \\ \hline
[25] & \begin{tabular}{c} Electbay /central \\ market \\ \end{tabular} & \
\begin{tabular}{c} Non-cooperative \\ game/Nash \\ equilibrium \\ \end{tabular} & N/A & N/A &
\begin{tabular}{c} Proposed a \\ customer-to-customer \\ business model for \\ P2P energy trading. \\ \end{tabular} \\ \hline
[69] & Distributed/P2P & N/A & \begin{tabular}{c} Convex \\ optimization \\ \end{tabular} & \
\begin{tabular}{c} Distributed \\ algorithm/ADMM \\ \end{tabular} &
\begin{tabular}{c} Achieved economy \\ savings/cost \\ optimization. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Required framework and desired outcome for TE. ESP: energy sharing provider; ADMM: alternating direction method of multipliers. P2P: peer-to-peer; ESP: energy sharing provider; MINLP: mixed integer non-linear programming; MG: micro-grid.