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## Game Theory Lecture 3

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**Game Theory**Lecture 3 Game Theory Lecture 3**0**6 1 1/6 4 5 3 2 1 1 1 1 1 1 N.S. S D,W D,W D,W D,W D,W D,W L,W 2 2 2 2 2 W,D W,D W,D W,D W,D W,L 1 1 1 1 D,W D,W D,W D,W L,W 2 2 2 W,D W,D W,D W,L 1 1 D,W D,W L,W 2 W,D W,L**A Lottery**2 2 2 W,D W,D W,D S. W,L 1 1 N.S. S D,W D,W L,W N.S. 2 N.S. W,D W,D S W,L W,L**** A Lottery consider the lottery ? for S. N.S. assume that**2**2 2 W,D W,D W,D S. W,L 1 1 N.S. S D,W D,W L,W N.S. 2 W,D W,L**N.S.**2 2 2 S. W,D W,D W,D W,L W,L 1 1 N.S. S D,W D,W L,W L,W 2 W,D W,D W,L**** N.S. ? S.**von Neumann - Morgenstern**utility functions A consumer has preferences over a set of prizes and preferences over the set of all lotteries over the prizes**4.**if then:**we now look for a utlity function**representing the preferences over the lotteries**take a lottery:**Replace each prize with an equivalent lottery**define:**the expected utility of the lottery clearly U represents the preferences on the lotteries**** <**If**is a vN-M utility function then is a vN-M utility function iff A utility function on prizes is called a von Neumann - Morgenstern utility function if the expected utility function : represents the preferences over the lotteries. i.e. if U is a utility function for lotteries.**If**is a vN-M utility function then is a vN-M utility function iff • It is easy to show that if u( ) is a vN-M utility function • then so is au( )+b a>0 2. Let v() be a vN-M utility function. Choose a>0 ,b s.t.**since f( ) is a vN-M utility function, and since for all j**It follows that: But by the definition of f( ) hence:**Neumann Janos**John von Neumann 1903-1957 Oskar Morgenstern 1902-1976 Kurt Gödel**Information Sets**and Simultaneous Moves 1 2 2**Some (classical) examples of simultaneous games**Prisoners’ Dilemma +6**Free Rider**(Trittbrettfahrer)**Some (classical) examples of simultaneous games**Prisoners’ Dilemma The ‘D strategy dominates the C strategy**Strategy s1strictly dominates strategys2**if for all strategies tof the other player G1(s1,t)> G1(s2,t)**Successive deletion of**dominated strategies X X Nash Equilibrium (saddle point)