#### Summary

In this LP we learn everything there is about simultaneous games. These games, used when considering a game where players move or play their strategies simultaneously, are commonly used in many fields. From military strategies to collusion agreements, the analysis of these situations as simultaneous games can help us discover the best way to act.#### Nash equilibria and dominant strategies:

- Prisoner’s dilemma
**Nash equilibrium**- Dominant strategies

#### Mixed strategies:

#### Continuous strategies:

Nash equilibria are defined as the combination of strategies in a game in such a way, that there is no incentive for players to deviate from their choice. This is the best option a player can make, taking into account the other players’ decision and where a change in a player’s decision will only lead to a worse result if the other players stick to their strategy. One of the best known Nash equilibriums can be found in the *prisoner’s dilemma*. This concept belongs to *game theory*, specifically to non-cooperative games, and was named after *John Nash *who developed it.

There are a few consistency requirements that must be taken into account when dealing with Nash equilibria. One of them is known as *common knowledge*, which extends the necessity of *complete information*. Therefore, expectations about other player’s strategies must be *rational*.

A Nash equilibrium is therefore a combination of beliefs about *probabilities* over strategies and the choices of the other player. It is quite easy to understand this using an example, in this case the prisoner’s dilemma as depicted in the adjacent *game matrix*.

Prisoner 1 (P1) has to build a belief about what choice P2 is going to make, in order to choose the best strategy. If P2 confesses (P2_{C}), he will get either -8 or 0, and if he lies (P2_{L}) he will get either -10 or -1. It can be easily seen that P2 will choose to confess, since he will be better off. Therefore, P1 must choose the best strategy given that P2 will choose to confess: P1 can either confess (P1_{C}, which pays -8) or lie (P1_{L}, which pays -10). The rational thing to do for P1 is to confess. Proceeding inversely, we analyse the beliefs of P2 about P1’s strategies, which gets us to the same point: the rational thing to do for P2 is to confess. Therefore, [P1_{C}, P2_{C}] is the Nash equilibrium in this game (underlined in red).

Nash equilibria can be used to predict the outcome of *finite games*, whenever such equilibrium exists. Furthermore, the best equilibrium outcome can be found by using the method of *elimination of dominated strategies*, which will help us find the best Nash equilibrium by excluding ‘unreasonable’ Nash equilibria.

On the downside, we find the issue that arises when dealing with a Nash equilibrium that is neither social nor ethical, and where efficiency may be subjective, which is the case in the prisoner’s dilemma, where the Nash equilibrium does not meet the criteria for being *Pareto optimal* (underlined in green). Also, the possibility of multiple equilibria causes the outcome of the game to become less predictable.