1. GAME THEORY Mathematical models of strategic interactions COMPETITIVE GAMESCOOPERATIVE GAMES 39

2. Forms –normal –extensive –characteristic I \ IIB1B1 B2B2 A1A1 0, 00, 1 A2A2 1, 0-1, -1 I II (0, 0) (0, 1) (1, 0)(-1, -1) SD SDSD 38

3. 1928John von Neumann MIN MAX THEOREM 1944John von Neumann & Oskar Morgenstern “The Theory of Games and Economic Behaviour” Princeton University Press ECONOMICS 1950John F. Nash Jr. EQUILIBRIA – BARGAINING THREAT 1968Guillermo Owen GUTEMBERG A PRIORI UN. MULTILINEAR 37

4. Nobel prizes in Economics 36 1994 2005 2007 2012 John F. Nash Jr. John Harsanyi Reinhard Selten Y. Robert J. Aumann Thomas C. Schelling Roger Myerson Leonid Hurwicz Eric Maskin Lloyd Shapley Alvin Roth PERFECT EQUILIBRIUM MECHANISM DESIGN COOPERATION & CONFLICT MARKET DESIGN & STABLE ALLOCATIONS

5. WARGulf,… ECONOMICSOligopolies,… MARKETINGCoca-Cola,… FINANCEFirms’ Control,… POLITICSElectoral Systems,… CLUB GAMESBridge, Poker, Chess,… SPORTSAttack-Defence Strategies,… SOCIOLOGYMigrations,… ENGINEERINGSafety in mechanical and civil en.,… MEDICINENeurons,… PSYCHOLOGYPrisoner’s dilemma,… BIOLOGYEvolution,… ENVIRONMENTPollution,… … LOGIC – PHILOSOPHY – RELIGION … 35

6. Marketing Game Market FIRM A4 units of capital FIRM B2 units of capital The winnings are referred to A 4, 01+0=1 3, 1 2, 2 1+1=2 1, 3-1+1=0 0, 4 STRATEGIES OF OF A STRATEGIES OF B 2, 0 1, 1 0, 2 34

7. Marketing Game- 2 - 2, 01, 10, 2 4, 0100 3, 1210 2, 2121 1, 3012 0, 4001 A B 33

8. 2, 01, 10, 2 3, 1210 2, 2121 1, 3012 MIN of A 0 1 0 A B MAX MIN of B Minmax Solution MIN of B -2 MAX MIN of A Marketing Game- 3 - 32

9. Courtesy of Silver/MCK 31

10. Courtesy of Silver/MCK 30

11. Courtesy of Silver/MCK 29

12. Courtesy of Silver/MCK 28

13. 754 263 801 MIN of A 4 2 0 A B MIN of B -8-6-4 Saddle Points MAX MIN of A MAX MIN of B 27

14. 26

15. 25

16. Constant sum games (2, 8)(5, 5) (-5,15)(10, 0) - 5 10-sum game (-3, 3) (0,0) (-10, 10) (5,- 5) zero-sum game 24

17. Terrorist’s Dilemma C (-5, -5)(-1, -10) NC (-10, -1)(0, 0) -10 -5 Min A CNC 23

18. Terrorist’s Dilemma -10-5 Min B (-5, -5)(-1, -10) (-10, -1)(0, 0) CNC C 22

19. Terrorist’s Dilemma (-5, -5)(-1, -10) (-10, -1)(0, 0) MaxMin A Max Min of B C C NC 21

20. Terrorist’s Dilemma COMPETITIVE SOLUTION NASH COOPERATIVE SOLUTION 20

21. USA vs URSS winning1200 – expense arm. 200 = earning1000 (0, 0)(-∞, 1000) D -∞-∞-200 Min URSS -∞-∞ -200 Min USA (1000, -∞)(-200, -200) A DA 19

22. Overtaking Game A\BNSS (-10, -10)(-10, 0) S (0, -10) Min A -10 -∞-∞ (-∞, -∞) Min B -10-∞-∞ (-∞, -∞) (-10, 0) (0, -10) Competitive solution 18

23. Overtaking Game - 2 - (-∞, -∞) (-10, 0) (0, -10) Cooperative solution 17

24. The battle of the Sexes soccerdancing soccer (2, 1)(-1, -1) dancing (-1, -1)(1, 2) Pure Maxmin: (-1, -1) Mixed Maxmin: (1/5, 1/5) (x 1 = 2/5, x 2 = 3/5, y 1 = 3/5, y 2 = 2/5) (1, 2) (2, 1) (-1, -1) (1/5, 1/5) Pure Maxmin Mixed Maxmin 16

25. Christian IV of Denmark XVI – XVII century The captain has to declare the value of the cargo. The king can decide: - to apply taxes - to buy the cargo at the declared price 15

26. Christian IV of Denmark XVI – XVII century V = value of the cargo (=100) D = value declared by the captain (80, 90, …) T = Tax [0, 1](=10%) CAPTAIN declares 8090100110120 B20100-10-20 NB89101112 KING 14

27. The revenue Inspector InspectorControlled IR + PE - C-R - PE NIR - E-R + E R = Real amount of the tax (=100) E = Evasion C = Cost of the examination (=20) P = Penality (=2) Evasion 0, …, 91011, …, 100 I 100+20-20-100-20 NI 100-10-100+10 13

28. Three players STRATEGIESOF CSTRATEGIESOF C STRATEGIES OF B STRATEGIES OF A 3, 12, -9 12

29. (1, 2)(0, 0) (7, 1) Nash Equilibria 11

30. A beautiful mind 10

31. Pollution Current situation: (-100, -100) Cost of the project: -150 CNC C (-75, -75)(-150, 0) NC (0, -150)(-100, -100) 9

32. Pollution - 2 - CNC C (-75, -75)(-150, 0) NC (0, -150)(-100, -100) ( -150,0 (0,-150) (-75, -75) 8

33. Games in Extensive Form 7

34. 6 5 8 4 2 17 3 3 ->4 3 -> 5 6 ->5 1 ->3 2 ->3 4->6 8->6 …… Winner: 7->6 Winner: 6

35. 6 5 8 4 2 17 3 3 ->4 3 -> 5 6 ->5 2 ->3 1 ->3 4->6 5->1 Winner: …… 5

36. winner 3  4 3  5 6  56  4 1  3 2  3 4  6 5  6 4  2 1  3 2  3 5  1 7676 7676 8  6 8  4 5  7 2  3 4  2 1  3 4  8 7  5 winner 4

37. ECONOMICSOligopolies,.. FINANCEFirms’ Control,… POLITICSElectoral Systems,… SOCIOLOGYMigrations,… MEDICINENeurons,… ENVIRONMENTKyoto,… Games in characteristic function form 3

38. He and she 2 sons Pentagon Pens Formulae Blonde The Speech I need… 2

39. 39 ed. Giappichelli - Torino

40. 40 ed. EDISES - Napoli

41. 41 ed. Campanotto - Pasian di Prato (UD) POESIE

42. MY WARMEST THANKS TO... gianfranco.gambarelli@unibg.it 1