GAME THEORY Mathematical models of strategic interactions COMPETITIVE GAMESCOOPERATIVE GAMES 39

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

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

Nobel prizes in Economics 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

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

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

Marketing Game , 01, 10, 2 4, , , , , 4001 A B 33

2, 01, 10, 2 3, , , 3012 MIN of A A B MAX MIN of B Minmax Solution MIN of B -2 MAX MIN of A Marketing Game

Courtesy of Silver/MCK 31

Courtesy of Silver/MCK 30

Courtesy of Silver/MCK 29

Courtesy of Silver/MCK 28

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

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

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

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

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

Terrorist’s Dilemma COMPETITIVE SOLUTION NASH COOPERATIVE SOLUTION 20

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

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

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

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

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 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 B NB KING 14

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 NI

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

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

A beautiful mind 10

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

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

Games in Extensive Form 7

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

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

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

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

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

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