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Unformatted text preview: Lecture VI: Existence of Nash equilibrium Markus M. M¨obius February 26, 2004 • Gibbons, sections 1.3B • Osborne, chapter 4 1 Nash’s Existence Theorem When we introduced the notion of Nash equilibrium the idea was to come up with a solution concept which is stronger than IDSDS. Today we show that NE is not too strong in the sense that it guarantees the existence of at least one mixed Nash equilibrium in most games (for sure in all finite games). This is reassuring because it tells that there is at least one way to play most games. 1 Let’s start by stating the main theorem we will prove: Theorem 1 (Nash Existence) Every finite strategicform game has a mixed strategy Nash equilibrium. Many game theorists therefore regard the set of NE for this reason as the lower bound for the set of reasonably solution concept. A lot of research has gone into refining the notion of NE in order to retain the existence result but get more precise predictions in games with multiple equilibria (such as coordination games). However, we have already discussed games which are solvable by IDSDS and hence have a unique Nash equilibrium as well (for example, the two thirds of the average game), but subjects in an experiment will not follow those equilibrium prescription. Therefore, if we want to describe and predict 1 Note, that a pure Nash equilibrium is a (degenerate) mixed equilibrium, too. 1 the behavior of realworld people rather than come up with an explanation of how they should play a game, then the notion of NE and even even IDSDS can be too restricting. Behavioral game theory has tried to weaken the joint assumptions of rationality and common knowledge in order to come up with better theories of how real people play real games. Anyone interested should take David Laibson’s course next year. Despite these reservation about Nash equilibrium it is still a very useful benchmark and a starting point for any game analysis. In the following we will go through three proofs of the Existence Theorem using various levels of mathematical sophistication: • existence in 2 × 2 games using elementary techniques • existence in 2 × 2 games using a fixed point approach • general existence theorem in finite games You are only required to understand the simplest approach. The rest is for the intellectually curious. 2 Nash Existence in 2 × 2 Games Let us consider the simple 2 × 2 game which we discussed in the previous lecture on mixed Nash equilibria: D U L R 1,1 0,4 0,2 2,1 We next draw the bestresponse curves of both players. Recall that player 1’s strategy can be represented by a single number α such that σ 1 = αU + (1 α ) D while player 2’s strategy is σ 2 = βL + (1 β ) R ....
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This note was uploaded on 05/19/2010 for the course 412 002 taught by Professor Dingli during the Spring '10 term at École Normale Supérieure.
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