#### 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:

Dominant strategies are considered as better than other strategies, no matter what other players might do. In *game theory*, there are two kinds of strategic dominance:

-a strictly dominant strategy is that strategy that always provides greater *utility* to a the player, no matter what the other player’s strategy is;

-a weakly dominant strategy is that strategy that provides at least the same utility for all the other player’s strategies, and strictly greater for some strategy.

A dominant strategy equilibrium is reached when each player chooses their own dominant strategy. In the *prisoner’s dilemma*, the dominant strategy for both players is to confess, which means that confess-confess is the dominant strategy equilibrium (underlined in red), even if this equilibrium is not a *Pareto optimal* equilibrium (underlined in green).

It must be noted that any dominant strategy equilibrium is always a *Nash equilibrium*. However, not all Nash equilibria are dominant strategy equilibria.

The elimination of dominated strategies is commonly used to simplify the analysis of any game. The way to proceed is to eliminate for each player every strategy that seems ‘unreasonable’, which will greatly reduce the number of equilibria. This method is quite easy to use when only strictly dominated strategies are in place, but the elimination of weakly dominated strategies can turn problematic, ending up with a game that does not resembles the original one from a strategic point of view.

A good example of elimination of dominated strategy is the analysis of the Battle of the Bismarck Sea. In this game, as depicted in the adjacent game matrix, Kenney has no dominant strategy (the sum of the payoffs of the first strategy equals the sum of the second strategy), but the Japanese do have a weakly dominating strategy, which is to go North (the payoffs are equal for one strategy but strictly better for the other). Since only one of them has a dominant strategy, there is no dominant strategy equilibrium. We must then proceed by eliminating dominated strategies. As we’ve already mentioned, for the Japanese strategy ‘go North’ weakly dominates strategy ‘go South’. Therefore, we eliminate the strategy ‘go South’ for the Japanese, who will go North. Now that we only consider the Japanese going North, Kenney’s strategy ‘go North’ is strictly dominant over strategy ‘go South’, which will be eliminated. Therefore, North-North is the weak-dominance equilibrium.

However, as it was mentioned before, it can be easily seen that the game has lost its strategic nature.