1 Game Theory Lecture 3 Game Theory Lecture 3 Game Theory Lecture 3

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S N.S / D,W W,D D,W W,D D,W W,D W,L L,W W,L L,W

S N.S. S W,D D,W W,D W,L L,W W,DW,D W,L N.S. S. A Lottery

N.S. S. A Lottery   ? consider the lottery for assume that

S N.S W,D D,W W,D W,L L,W N.S. S. 

S N.S W,D D,W W,D W,L L,W N.S. S. W, L L, W W, D

N.S. S.   ?

von Neumann - Morgenstern utility functions 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

for each prize w j there exists a unique  j s.t

if then: 4.

5.

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 r epresents the preferences on the lotteries

  <

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. Ifis a vN-M utility function then is a vN-M utility function iff

Ifis a vN-M utility function then is a vN-M utility function iff 2. Let v() be a vN-M utility function. Choose a>0,b s.t. 1.It is easy to show that if u( ) is a vN-M utility function then so is au( )+b a>0

since f( ) is a vN-M utility function, and since for all j It follows that: But by the definition of f( ) hence:

John von Neumann John von Neumann Oskar Morgenstern Oskar Morgenstern Neumann Janos Kurt Gödel

Information Sets and Simultaneous Moves 1 22

Some (classical) examples of simultaneous games C not confess D confess C not confess -3, -3 -6, 0 D confess 0, -6 -5, -5 Prisoners’ Dilemma +6 3, 3 0, 6 6, 0 1, 1

Free Rider (Trittbrettfahrer) C Cooperate D defect C cooperate 3, 3 0, 6 D defect 6, 0 1, 1

Some (classical) examples of simultaneous games C Cooperate D defect C cooperate 3, 3 0, 6 D defect 6, 0 1, 1 Prisoners’ Dilemma The ‘ D strategy dominates the C strategy

Strategy s 1 strictly dominates strategy s 2 if for all strategies t of the other player G 1 (s 1,t) > G 1 (s 2,t)

1, 5 2, 3 7, 4 3, 3 4, 7 5, 2 X X Nash Equilibrium (saddle point) Successive deletion of dominated strategies