GAME THEORY Mathematical models of strategic interactions COMPETITIVE

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

Forms – normal I II B 1 B 2 A 1 0, 0 0, 1 A 2 1, 0 -1 , -1 S – extensive I D II S (0 , 0) II D (0 , 1) S (1 , 0) D (-1 , -1) – characteristic 38

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

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

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

Marketing Game STRATEGIES OF B Market FIRM A 4 units of capital FIRM B 2 units of capital The winnings are referred to A 2, 0 S T R A T E G I E S OF A 1, 1 0, 2 4 , 0 1+0=1 3, 1 2, 2 1+1=2 1 , 3 -1+1=0 0, 4 34

Marketing Game - 2 B 2, 0 1, 1 0, 2 4, 0 1 0 0 3, 1 2 1 0 2, 2 1 1, 3 0 1 2 0, 4 0 0 1 A 33

Marketing Game - 3 Minmax Solution 2, 0 1, 1 0, 2 MIN of A 3, 1 2 1 0 0 2, 2 1 1 1, 3 0 1 2 0 MIN of B -2 -2 -2 A B MAX MIN of A MAX MIN of B 32

Courtesy of Silver/MCK 31

Courtesy of Silver/MCK 30

Courtesy of Silver/MCK 29

Courtesy of Silver/MCK 28

Saddle Points A B MIN of A 7 4 4 2 6 3 2 8 MIN of B 5 0 1 0 -8 -6 -4 MAX MIN of A MAX MIN of B 27

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

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

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

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

Terrorist’s Dilemma NASH COOPERATIVE SOLUTION COMPETITIVE SOLUTION 20

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

Overtaking Game AB NS S Min A NS (-10, -10) (-10, 0) S (0, -10) (-∞, -∞) -10 -∞ Min B -10 -∞ (-10, 0) Competitive solution (0, -10) (-∞, -∞) 18

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

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

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

Christian –IV century of Denmark XVII V = value of the cargo (=100) D = value declared by the captain (80, 90, …) T = Tax [0, 1] (=10%) CAPTAIN declares 80 K B 20 I N G NB 8 90 100 110 120 10 0 -10 -20 9 10 11 12 14

The revenue Inspector R = Real amount of the tax (=100) E = Evasion C = Cost of the examination (=20) P = Penality (=2) Inspector Controlled I R + PE - C -R - PE NI R-E -R + E Evasion 0, …, 9 I NI 10 11, …, 100+20 -20 -100 -20 100 -10 -100+10 13

Three players S T R A T E G I E S 3, 12 , -9 O F STRATEGIES OF B ST R AT EG IE S O F A C 12

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

A beautiful mind 10

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

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

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

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

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

5 6 5 5 6 4 3 3 4 1 3 3 2 8 4 6 5 winner 2 4 2 5 1 winner 7 4 8 5 3 winner 2 winner 1 3 7 6 7 6 8 4 8 6 5 1 2 3 6 1 3 6 4 7 winner 4

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

ü He and she ü 2 sons üPentagon ü Pens ü Formulae ü Blonde üThe Speech ü I need… 2

ed. Giappichelli - Torino 39

ed. EDISES - Napoli 40

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

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