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GAME THEORY Mathematical models of strategic interactions

GAME THEORY Mathematical models of strategic interactions. COMPETITIVE GAMES. COOPERATIVE GAMES. 39. I. S. D. II. II. S. D. S. D. (0 , 1). (0 , 0). (1 , 0). (-1 , -1). Forms. normal extensive characteristic. 38. 37. Nobel prizes in Economics. PERFECT EQUILIBRIUM.

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GAME THEORY Mathematical models of strategic interactions

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

  2. I S D II II S D S D (0 , 1) (0 , 0) (1 , 0) (-1 , -1) Forms • normal • extensive • characteristic 38

  3. 37

  4. Nobel prizes in Economics PERFECT EQUILIBRIUM COOPERATION & CONFLICT MECHANISM DESIGN MARKET DESIGN & STABLE ALLOCATIONS 36

  5. WAR Gulf,… ECONOMICS Oligopolies,… MARKETING Coca-Cola,… FINANCE Firms’ Control,… POLITICS Electoral Systems,… CLUB GAMES Bridge, Poker, Chess,… SPORTS Attack-Defence Strategies,… SOCIOLOGY Migrations,… ENGINEERING Safety in mechanical and civil en.,… MEDICINE Neurons,… PSYCHOLOGY Prisoner’s dilemma,… BIOLOGY Evolution,… ENVIRONMENT Pollution,… … LOGIC – PHILOSOPHY – RELIGION … 35

  6. Marketing Game BG BS STRATEGIES OF B Market Market 2 , 0 1 , 1 0 , 2 S T R A T E G I E S OF A FIRM A 4 units of capital FIRM B 2 units of capital The winnings are referred to A 34

  7. Marketing Game - 2 - B A 33

  8. MAX MIN of B Marketing Game - 3 - Minmax Solution B A MAX MIN of A 32

  9. 31 Courtesy of Silver/MCK

  10. 30 Courtesy of Silver/MCK

  11. 29 Courtesy of Silver/MCK

  12. Saddle Points 28 Courtesy of Silver/MCK

  13. MAX MIN of A MAX MIN of B Saddle Points B A 27

  14. 26

  15. 25

  16. - 5 Constant sum games 10-sum game zero-sum game 24

  17. Terrorist’s Dilemma Min A C NC -5 -10 23

  18. Terrorist’s Dilemma C NC C NC Min B -5 -10 22

  19. Terrorist’s Dilemma C NC MaxMin A C NC Max Min of B 21

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

  21. Min USA -200 -∞ Min URSS -200 -∞ USA vs URSS winning 1200 – expense arm. 200 = earning 1000 A D A (-200, -200) (1000, -∞) D (-∞, 1000) (0, 0) 19

  22. (-10, 0) (0, -10) (-∞, -∞) Overtaking Game (-∞, -∞) Competitive solution 18

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

  24. (1, 2) (2, 1) (-1, -1) Mixed Maxmin The battle of the Sexes Pure Maxmin: (-1, -1) Mixed Maxmin: (1/5, 1/5) (x1 = 2/5, x2 = 3/5, y1 = 3/5, y2 = 2/5) (1/5, 1/5) Pure 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%) K I N G 14

  27. The revenue Inspector R = Real amount of the tax (=100) E = Evasion C = Cost of the examination (=20) P = Penality (=2) 13

  28. Three players S T R A T E G I E S O F C , -9 3, 12 STRATEGIES OF B STRATEGIES OF A 12

  29. Nash Equilibria 11

  30. A beautiful mind 10

  31. Pollution Current situation: (-100, -100) Cost of the project: -150 9

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

  33. I S D II II S D S D I I I I S D S D S D S D II II II II II II II II S D S D S D S D S D S D S D S D (3,-1) (2,2) (3,4) (1,3) (0,-1) (-2,0) (5,-2) (3,8) (4,2) (1,2) (0,4) (1,-2) (0,1) (5,5) (2,-8) (7,-3) Games in Extensive Form 7

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

  35. 8 2 5 3 6 1 4 7 3 ->4 3 -> 5 6 ->5 4->6 1 ->3 2 ->3 …… …… 5->1 Winner: 5

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

  37. Games in characteristic function form ECONOMICS Oligopolies,.. FINANCE Firms’ Control,… POLITICS Electoral Systems,… SOCIOLOGY Migrations,… MEDICINE Neurons,… ENVIRONMENT Kyoto,… 3

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

  39. ed. Giappichelli - Torino 39

  40. ed. EDISES - Napoli 40

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

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

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