# A Game Theoretic Approach for Energy Sharing in the Italian Renewable Energy Communities
## 3 Game Theoretic Approach for Value Redistribution
### Non-Cooperative Game  
In non-cooperative games the players are independent and choose their strategy, the maximization of their utility function, with no communication or exchange of information with the other players. The outcome of the decisions will lead to conflicting interests in reaching the desired objectives. A classical representation of static non-cooperative games is given by a set of players \(N\), a set of possible actions \((A_{i})_{i\in N}\) and the utility functions of the players \((u_{i})_{i\in N}\). The players have to select an appropriate action in the set \(a_{i}\in A_{i}\) so that its utility function will be maximised. Nevertheless, the utility function does not only depend on the individual action of the player \(a_{i}\), but also on the array of actions of the other \(N\backslash\{i\}\) players. One important solution in non-cooperative games is the Nash equilibrium, a stable state in which the players cannot improve their utility function by changing their action \(a_{i}^{*}\), if the other players \(N\backslash\{i\}\) keep the same array of decisions \(a_{-i}^{*}\).  
Non-cooperative game theory is often used to model those situations in microgrids or small power systems in which users act individually and aim to optimize their own benefit. In [21], a framework to regulate the percentage of domestic users owning a storage device is presented. The problem is modelled as a non-cooperative game in which players act to minimise their costs. A Nash equilibrium that maximises social welfare is found as the number of battery owners with respect to the considered population. The authors of [22] provide a methodology to enhance the stability and the efficiency of a microgrid, using an algorithm based on non-cooperative game theory. A Nash equilibrium is found as a solution of this dynamic game, that optimises the power regulation of the load and the sources within the system. Moreover, it is studied how loads and sources can react by adjusting their power to the actions of the other players. Some specific declinations of non-cooperative game theory are also considered in literature, in order to better analyse some particular situations. As an example, [23] studies an isolated microgrid with users benefiting of PV electricity generation and batteries. The system is analysed using two concepts from non-cooperative game theory: the Poltuck problem and an auction game. The interaction between consumers and producers is modelled as a Potluck problem, a dynamic non-cooperative problem in which players have no communication but act rationally. The problem is known in literature for not possessing a Nash equilibrium. As game theory predicts, due to the rational thinking of the players, the system oscillates between a state in which there is excess demand or excess production. As a consequence, a non-rational thinking is adopted for the users to find an equilibrium in which the system can work. An auction game is then implemented to simulate the internal market under different conditions of market clearing strategies. For every condition, a different Nash equilibrium is found. Another example of non-cooperative game is reported in [24], where a mechanism to encourage users in a smart grid to actively participate in energy trading with a central power station is proposed. The system is modelled as a single-leader multiple-follower Stackelberg game (a theory first developed in economics in which a leader has the advantage of a first move and the follower plays the best response to optimize its utility) in which the aggregator is the leader that sets the internal price of energy within the microgrid and the users of such microgrid have to decide the amount of energy to sell in response to the determined price. The objective of the aggregator is to set a price that incentivises active users to sell electricity during peak hours in order to meet the demand of the microgrid, while maximizing the overall benefit (generalized Nash equilibrium).