# Game Theory¶

**Game theory** is a branch of mathematics concerned with the study of strategic interaction among rational decision-makers. While inherently mathematical at its foundation, game theory has numerous applications in several social science disciplines, including economics. Game theory originated with the study of equilibria in zero-sum games by John von Neumann and has since expanded into many other paradigms, including applications such as a method of examining and strategizing for interactions between the US and the USSR during the Cold War and explaining the evolution and prevalence of 1:1 sex ratios in biology. In this chapter, we will study a few particular aspects of game theory and their application to economics.

There are many different ways to conceive of games and to divide them along different attributes.

**Cooperative vs. non-cooperative:**A game is*cooperative*if the players are able to form commitments that can be externally enforced. A game is*non-cooperative*if the players cannot form agreements or if those agreements need to be self-enforced.**Symmetric vs. asymmetric:**A game is*symmetric*if the payoffs depend only on the strategies used and not on the players using those strategies. It is*asymmetric*if changing the identities of the players changes the payoffs.**Simultaneous vs. sequential:**A game is*simultaneous*if players move at the same time without being aware of the other players’ actions. A game is*sequential*if moves occur one after the other and players have some knowledge of the earlier actions of their competitors.**Perfect vs. imperfect information:**Games of*perfect information*occur when all players know the moves previously made by all other players. If this is not the case, the game is an*imperfect-information*game.

In this chapter, we will concern ourselves with simultaneous games of imperfect information, examining both cooperative and non-cooperative games as well as symmetric and asymmetric ones. We will also discuss different theoretical aspects, like strategies and payoffs. We will look at various common game paradigms, including one of the most popular games: the prisoner’s dilemma. Lastly, we will study the economic application of game theory, including topics like equilibria and oligopolies.