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class12_Game_ - Econ 51 Winter 2011 Class#12 Nice...

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1 Econ 51, Winter 2011 Class #12 • Nice performance in the midterm! mean 70.8, standard deviation 19.2 your graded midterms are in the Academic Office in the department regrades: in writing only. Should write to both me and Director of Undergraduate Studies, following department course syllabus (read it carefully) •PS #5 is posted on Thursday, #4 is due on Friday. •I asked a lecture to be videotaped on Thursday (for teaching improvement purposes only). Please excuse their presence. • Today: Continue with game theory, primarily talking about the concept of Nash Equilibrium. Tuesday, February 15, 2011

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2 Game theory We finished last class by talking about dominated strategies. Today we’ll introduce the concept of Nash Equilibrium. We will spend most of today on illustrating how to apply it. Tuesday, February 15, 2011
3 Coordination games Game theory Many (most) strategic situations (games) don’t have dominant or dominated strategies. Coordination game is a common example. Once out of jail, Rob and Tom plan to relax. Rob likes hockey, Tom likes baseball. But most important, they want to hang out together (and plan the next robbery). Hockey Baseball Hockey 8,4 0,0 Baseball 0,0 4,8 This game has the famous name of “Battle of the Sexes.” It has the feature of both common and conflicting interests: on one hand we want to be with the other guy, on the other hand they’d rather be in their favorite event than in the other. Tuesday, February 15, 2011

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4 Coordination games (cont.) Game theory Hockey Baseball Hockey 8,4 0,0 Baseball 0,0 4,8 Now we do not have dominant (or dominated) strategies anymore: • If Tom goes to Hockey, Rob wants to go to Hockey. • If Tom goes to Baseball, Rob wants to go to Baseball. • The same is true for Tom. Can we say anything? We expect that the two end up going to the same event. It is harder to figure out which one of the activities they will end up attending, but it seems reasonable that they will hang out together. This is explained by the the concept of Nash Equilibrium (John Nash, 1950). Tuesday, February 15, 2011
5 Nash Equilibrium Game theory A Nash equilibrium is a particular kind of “steady state.” It is a set of strategies, from which none of the players wants to deviate. Formally, a Nash equilibrium is a set of strategies (s 1 , s 2 , , s N ) such that for every player i we have that s i is a best response against s -i , that is: π i (s i ,s -i ) ≥π i (s i ,s -i ) for every s i . Tuesday, February 15, 2011

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6 Nash Equilibrium (cont.) Game theory Remark: we do not try to ask how players reach a Nash equilibrium, but only ask the question: if we are at this point, does anyone want to do any else? If the answer is no, we are at a Nash equilibrium. Nash showed that under certain assumptions a Nash equilibrium always exists. So we can predict an outcome of any game!
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