Lecture notes 2 - Economics 112 Lecture Notes Lecture 2...

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Economics 112 Lecture Notes Lecture 2. Equilibrium This lecture sets out a number of solution concepts for games and illustrates their application in a set of classic games. Key terms: Dominant strategy; best response; cooperative solution; Nash equilibrium; focal point; zero sum game. So far we have concentrated on turning a description of a strategic situation into a formal representation, but have not said much about what happens in the game. Here we begin to solve games, to make a prediction about what happens. The solutions we find are referred to as equilibriums (or equilibria) of one kind or another. For the most part, we focus on non- cooperative solutions: that is, we assume that the players choose strategies looking to their own interest, and are unable to commit to making choices that do not work in their own interest. We will also concentrate for the time being on rational players, choosing strategies to achieve preferred outcomes. You will see that there are other interpretations of game theory, notably evolutionary models. Dividing the analysis of a game into a specification step followed by a solution step is useful. Getting the specification correct is often the hardest part, and criticizing the results of an application of game theory by trying to deny the logic of the solution method is usually less fruitful that reexamining the specification. But this separation is not absolute: in some cases there will be a number of potential solution concepts. In these cases, the choice of a solution concept is itself part of the game specification. 1. Best responses Rational players in non-cooperative games survey the available strategies, and choose the one that leads to the most preferred outcome. A player’s best response is the strategy that yields the highest payoff, given the strategies chosen by the other players. For example, consider the two player game illustrated below. First assume that player 2 is playing Left. Then player 1 can choose Up, for a payoff of 2, or Down for a payoff of 3. Since Down yields the highest payoff, we say that Down is the best response to Left. You should work out the other possible best responses. For example, you should convince yourself that Right is player 2’s best response to Down. Copyright David Scoones 2010 Page 1
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If you did what I just told you to do, you will have noticed that best responses need not be unique: two (or more) strategies may lead to the same payoff, given the strategies chosen by other players. For this reason, the assumption that players choose best responses does not always isolate a specific choice. Less obvious is that best responses may not exist. That is, it may be that no strategy is truly best, given the choices made by other players. This won’t happen in examples like the one above, but can in some not so well formed games. Technical assumptions that guarantee a game will have well defined best responses are
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Lecture notes 2 - Economics 112 Lecture Notes Lecture 2...

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