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

2. 2

4. S N.S. 0 1 2 11 111 1 2 2 1 1/6 2 1 2 1 2 1 2 2 1 2 D,W W,D D,W W,D D,W W,D W,L L,W W,L L,W 1 4 23 5 6

5. S N.S. S 2 2 1 2 1 2 W,D D,W W,D W,L L,W W,DW,D W,L N.S. S. A Lottery

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

7. S N.S. 2 2 1 2 1 2 W,D D,W W,D W,L L,W N.S. S. 

8. S N.S. 2 2 1 2 1 2 W,D D,W W,D W,L L,W N.S. S. W, L L, W W, D

9. N.S. S.   ?

10. 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

12. for each prize w j there exists a unique  j s.t. 1. 2. 3.

13. if then: 4.

14. 5.

15. we now look for a utlity function representing the preferences over the lotteries

16. take a lottery: Replace each prize with an equivalent lottery

18. define: the expected utility of the lottery clearly U r epresents the preferences on the lotteries

19.   <

20. 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

21. 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

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

23. John von Neumann 1903-1957 John von Neumann 1903-1957 Oskar Morgenstern 1902-1976 Oskar Morgenstern 1902-1976 Neumann Janos Kurt Gödel

24. Information Sets and Simultaneous Moves 1 22

25. 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

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

27. 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

28. 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)

29. 1, 5 2, 3 7, 4 3, 3 4, 7 5, 2 X X Nash Equilibrium (saddle point) Successive deletion of dominated strategies