Game Theory
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Abstract
Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send email to hari@csa.iisc.ernet.in. A more thorough version would be available soon in this space. In this chapter, we introduce the all important notion of pure strategy Nash equilibrium. We provide several examples to illustrate this notion. We show that pure strategy Nash equilibrium may not exist in many cases while in many other cases, there could exist multiple Nash equilibria. We also show that the payoffs that players get in a Nash equilibrium may not be socially optimal. Dominant strategy equilibria (strongly dominant, weakly dominant, very weakly dominant), if they exist, are very desirable but rarely do they exist because the conditions to be satisfied are too demanding. A dominant strategy equilibrium requires that each player's choice be a best response against all possible choices of all the other players. If we only insist that each player's choice is a best response against the best response strategies of the other players, we get the notion of Nash equilibrium. This solution concept is named after John Nash, one of the most celebrated game theorists of our times. In this section, we introduce and discuss the notion of pure strategy Nash equilibrium. In the following section, we discuss the notion of mixed strategy Nash equilibrium. That is, each player's Nash equilibrium strategy is a best response to the Nash equilibrium strategies of the other players.