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lecture4-2 - Lecture IV Nash Equilibrium II Multiple...

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Lecture IV: Nash Equilibrium II - Multiple Equilibria Markus M. M¨obius March 3, 2003 Readings for this class: Osborne and Rubinstein, Chapter 2.1- 2.3 1 Multiple Equilibria I - Coordination Lots of games have multiple Nash equilibria. In this case the problem arises how to select between different equilibria. 1.1 New-York Game Look at this simple coordination game: 0,0 1,1 1,1 0,0 E C E C This game has two Nash equilibria - (E,E) and (C,C). In both cases no player can profitably deviate. (E,C) and (C,E) cannot be NE because both players would have an incentive to deviate. 1
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1.2 Voting Game Three players simultaneously cast ballots for one of three alternatives A,B or C. If a majority chooses any policy that policy is implemented. If the votes split 1-1-1 we assume that the status quo A is retained. Suppose the preferences are: u 1 ( A ) > u 1 ( B ) > u 1 ( C ) u 2 ( B ) > u 2 ( C ) > u 2 ( A ) u 3 ( C ) > u 3 ( A ) > u 3 ( B ) Claim 1 The game has several Nash equilibria including ( A, A, A ) , ( B, B, B ) , ( C, C, C ) , ( A, B, A ) , and ( A, C, C ) . Informal Proof: In the first three cases no single player can change the outcome. Therefore there is no profitable deviation. In the last two equilibria each of the two A and two C players, respectively, is pivotal but still would not deviate because it would lead to a less desirable result. 1.3 Focal Points In the New York game there is no sense in which one of the two equilibria is ’better’ than the other one. For certain games Schelling’s (1961) concept of a tipping point can be a useful way to select between different Nash equilibria. A focal point is a NE which stands out from the set of NE - in games which are played frequently social norms can develop. In one-shot games strategies which ’stand out’ are frequently played. In both cases, players can coordinate by using knowledge and information which is not part of the formal description of our game. An example of a social norm is the fact that Americans drive on the right hand side of the road. Consider the following game. Tom and Jerry drive in two cars on a two lane road and in opposite directions. They can drive on the right or on the left, but if they mis-coordinate they cause a traffic crash. The game can be represented as follows: 2
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0,0 1,1 1,1 0,0 R L R L We expect both drivers to choose (R,R) which is the social norm in this game.
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