#### 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.**Simultaneous games**

#### Nash equilibria and dominant strategies:

#### Mixed strategies:

#### Continuous strategies:

Simultaneous games are those where decisions are simultaneous: both we and the other ‘player’ choose at the same time. The simplest example of this is probably ‘rock, paper, scissors’. *Complete information* means that we know what we stand to win or lose: we know that rock beats scissors and that this will give us some form of *utility* (we might get to pick the plan for that evening or simply feel a warm glow of superiority). We also know that our opponent has this same information, this is, the rules of the game and each player’s payoffs are *common knowledge*. Simultaneous games are usually described using the *strategic (or normal) form*.

What this all means is that we only have one chance to get it right, but that we can play smart by knowing what our opponent will do and acting accordingly. An equilibrium is reached when both players will *rationally make a decision* that they have no reason to change: whatever else they do, they will only be worse off. Our situation can only be improved if our opponent chooses to do something else. These equilibria are known as *Nash equilibria* after *John Nash*, an economist from the early/mid 20^{th} century. The most famous example of this is the *prisoner’s dilemma*.

In this game, two prisoners are locked up in separate cells where they have no chance of communicating (*imperfect information*). They are both given the same (complete) information, which is summarised in the following *game matrix*:

In this example, the outcomes represented are years of freedom lost. If both prisoners confess, they will both receive an 8 year sentence. However, if neither confess, charges will have to be mitigated and both will receive low sentences. If one confesses and the other doesn’t, only one will receive punishment. Obviously, the best social outcome is for neither to confess, which would be the *Pareto optimal* situation. However, we can see that this is not a stable equilibrium. Both prisoners have an incentive to confess: if we are prisoner 2 (the vertical player) and we assume that prisoner 1 is going to confess, we have two possible outcomes: -8 if we confess and -10 if we lie. If we assume that prisoner1 is going to lie, we again have two outcomes: 0 if we confess or -1 if we lie. This means that, independently of the choice our opponent makes, we always have an incentive to tell the truth. Because we know this, we will choose to confess, and, because our opponent is in the same situation, chances are they will do the same. Once at this equilibrium, neither opponent has an incentive to change. This means we have one clear Nash equilibrium: where both players confess.

The process we have just followed in order to reach this conclusion is based on finding a *dominant strategy* (confess) which always overpowers the alternative under any choice the other player makes (whether our opponent confesses or lies). This allows us to rule out the overpowered strategy. In static games, we can have countless options, but this rational strategy will allow us to see our options more clearly and eliminate undesirable options in a systematic way.

What this process and what Nash equilibria do not allow us necessarily to do are to reach social optimums. The social optimum in this case would be for both players to lie and have charges mitigated: they would both be out within a year. But game theory tells us that social inefficiencies are common when dominant strategies are not for the common good.