Gianfranco Gambarelli University of Bergamo, Italy

72

1) APPLICATIONS

2) THEORY

1) APPLICATIONS

- Finance- Politics- Collusions- Interfering Elements

71

1.1) Applications of G.T. to FINANCE

70

1

0

0

51

48

1

A

B

C

1/3

1/3

1/3

49

49

2

A

B

C

?50

30

20

A

B

C

1/3

1/3

1/3

40

30

30

A

B

C

POWERSSEATSPLAYERS

3/5

1/5

1/5

69

w1+ w2 = 100

68

w1+ w2 + w3 = 100

67

wA

wB

wC

061 3

1

32

1

66

Theorem : Let w be a function of the real variable s, so that

0s

wh

(h =1, …, n; h i)

0s

wi

Let be a strongly monotonic PI. The power i(s) of the i-th player is a monotonic nondecreasing step function of s, whose discontinuity points are given by:

n

hhhwbqs

1(h i j)

n

hhhwbqts

1(h i j)

65

Gambarelli G. (1983), International Journal of Game Theory

64

Open problem

- Other share tradings within the game

POWER

BIG

BIG

BIG

SMALL SHAREHOLDERS

63

2

1

3

Y

s

62

Theorem : Let be a strongly monotonic P.I. The power of the i-th player is a monotonic nondecreasing step function with discontinuity points:

ii

n

ihhhh

wqtb

wqtb

s

,

)(1

61

60

Gambarelli G. (1996), Modelling Techniques for Financial Markets and Bank Management

59

Open problems

- Oceanic Games

- Applications

Indirect control of corporations

58

10

45

45

i1

20

80

51 49

25 25

25 25

i2

57

Open problem

- An efficient algorithm

56

Gambarelli G. and G. Owen (1994), International Journal of Game Theory

STABILITY INDEX

32

1

211

ccrc

r aaaa

ad

availability of shares in the market

closeness to the absolute majority quota

ar= shares of the raider

ac= shares of the control group

55

Gambarelli G. (1993), Modelling Reality and Personal Modelling

Open problem

- Calibration

54

STATE THE QUOTA TO BE INVESTED IN CONTROL

EXEC THE MODEL OF CONTROL OPTIMIZATION The model will answer:-which firms to attack-which stocks to buy-what capital will remain (Kr)

MODIFY THE MATRIX VARIANCE-COVARIANCE

by leaving out the candidates to be controlled and those which are strongly correlated with them

EXEC THE MODEL OF PORTFOLIO MANAGEMENT

As you obtain the capital,

PERFORM THE ESCALATION

of the firms to conquer

53

Open problem

- Unified Model of Portfolio

52

Gambarelli G. (1982), Finance

Gambarelli G and S. Pesce (2004), Theory and Decision (G. Gambarelli Ed.)

1.2) Applications of G.T. to POLITICS

- Simulations

- Regulations

- Forecasting

51

SIMULATIONSSIMULATIONS

2

1

3

Y

s

w

A

w

B

w

C

061 3

1

32 1

immigrants, emigrants…

shifts of votes

50

Gambarelli G. and I. Stach (2009), Homo Oeconomicus

Open problem

- Applications

49

REGULATIONSREGULATIONS

48

PROPORTIONALPROPORTIONAL MAJORITY MAJORITY SYSTEMS SYSTEMS SYSTEMS SYSTEMS

ROUNDINGS THRESHOLDSROUNDINGS THRESHOLDSBONUSESBONUSES

REPRESENTATION REPRESENTATION GOVERNABILITYGOVERNABILITY

ROUNDINGSROUNDINGS

VotesVotes SeatsSeats

AA 5050 ??

BB 3030 ??

CC 2020 ??

TotalsTotals 100100 1010ssii = v = v

ii •• S / V S / V

BUT...BUT...

5

3

2

47

VotesVotes SeatsSeats

AA 5050

BB 3030

CC 2020

TotalsTotals 100100 55

2.51.5

1.0

???

46

CRITERIA OF ROUNDINGSCRITERIA OF ROUNDINGS

- Equal votes → Equal seats- Equal votes → Equal seats- Monotonicity - Monotonicity ((more votesmore votes → → not less seatsnot less seats))-Symmetry Symmetry - Hare (- Hare (roundingsroundings))- Super-additivity- Super-additivity- Majority (- Majority (power indicespower indices))

45

VOTESVOTES AA BB CC TotalsTotals

District .IDistrict .I 5050 6060 1010 120120

District IIDistrict II 1010 1010 6060 8080

National National TotalsTotals

6060 7070 7070 200200

BreaksBreaks

- District I: power index criterionDistrict I: power index criterion

- Totals:- Totals: symmetry andsymmetry and monotonicity criteriamonotonicity criteria

Hamilton Hamilton (1)(1)

AA BB CC TotalsTotals

II 33 33 00 66

IIII 11 00 44 55

TotalsTotals 44 33 44 1111

AA BB CC TotalsTotals

II 66

IIII 55

TotalsTotals 1111

44

HondtHondt AA BB CCTotalTotal

ss

II 33 33 00 66

IIII 00 00 55 55

TotalsTotals 33 33 55 1111

Breaks:Breaks:- District I: power index criterionDistrict I: power index criterion- District II: Hare maximum criterion- District II: Hare maximum criterion- Totals: Hare maximum and symmetry criteria Totals: Hare maximum and symmetry criteria

Bal.&Young Bal.&Young (1)(1)

AA BB CC TotalsTotals

II 33 33 00 66

IIII 00 11 44 55

TotalsTotals 33 44 44 1111

Breaks:Breaks:- District I: power index criterionDistrict I: power index criterion

43

OLD METHODS

Use the techinique Cry on breaks

NEW METHODRanking of criteria: 1) 2) …

Existence Theorem

42

MinimaxMinimax AA BB CC TotalsTotals

II 22 33 11 66

IIII 11 11 33 55

TotalsTotals 33 44 44 1111

The minimax The minimax apportionment respects apportionment respects both at a local and a national level: both at a local and a national level:

- Symmetry- Symmetry- Monotonicity- Monotonicity- Hare minimum- Hare minimum- Hare maximum- Hare maximum- Equal seats for equal votes- Equal seats for equal votes- Power Indices- Power Indices

41

THE ADVANTAGESTHE ADVANTAGES

To the party (or to the coalition) To the party (or to the coalition) having relative majorityhaving relative majority 1) the uninominal voting system1) the uninominal voting system 2) the majority prize.2) the majority prize.

To the remaining average and large partiesTo the remaining average and large parties 1)1) thresholds thresholds 2) greatest divisors.2) greatest divisors.

To the smaller parties with peculiar linguistic or ethnical characteristicsTo the smaller parties with peculiar linguistic or ethnical characteristics 1) the respect for such minorities.1) the respect for such minorities.

To the remaining small partiesTo the remaining small parties 1) quota and jump greatest divisors1) quota and jump greatest divisors 2) the proportional voting system.2) the proportional voting system.

40

No partyNo partygets any advantagegets any advantage

from the Minimax Methodfrom the Minimax Method

because this method:because this method:

1)1) respects all the principal equity criteria respects all the principal equity criteria

2)2) minimizes distortions as far as possibleminimizes distortions as far as possible. .

39

Probably Probably

it will never be adoptedit will never be adopted

38

GOVERNABILITYGOVERNABILITY

37

VOTES SEATS

+1

-1

majority prize

Italian Chamber 2013

36

DISTRICTSPARTITO DEMOCRATICO

SINISTRA ECOLOGIA LIBERTA'

CENTRO DEMOCRATICO SVP

IL POPOLO DELLA LIBERTA' LEGA NORD

FRATELLI D'ITALIA

MOVIMENTO 5 STELLE BEPPEGRILLO. IT

SCELTA CIVICA CON MONTI PER L'ITALIA

UNIONE DI CENTRO

VALLEE D'AOSTE

MOV. ASSOCIATIVO ITALIANI ALL'ESTERO

USEI (Unione sudamericana emigrati italiani) Totals D.P.R.

Piedmont 1 11 2 0 0 3 1 0 4 2 0 0 0 0 23 23Piedmont 2 10 1 0 0 3 1 1 4 2 0 0 0 0 22 22Lombardy 1 21 2 0 0 5 2 1 6 3 0 0 0 0 40 40Lombardy 2 20 2 0 0 7 6 0 6 4 0 0 0 0 45 45Lombardy 3 8 1 0 0 2 1 1 2 1 0 0 0 0 16 16TTA 3 1 0 5 1 0 0 1 1 0 0 0 0 12 11Veneto 1 13 1 0 0 5 3 0 6 2 1 0 0 0 31 31Veneto 2 9 1 0 0 2 2 0 4 2 0 0 0 0 20 20FVG 6 1 0 0 1 1 0 2 1 0 0 0 0 12 13Liguria 9 1 0 0 2 0 0 3 1 0 0 0 0 16 16Emilia R. 28 2 0 0 5 1 0 7 2 0 0 0 0 45 45Tuscany 23 2 1 0 4 0 1 5 2 0 0 0 0 38 38Umbria 5 0 0 0 1 0 0 2 1 0 0 0 0 9 9Marche 9 1 0 0 2 0 0 3 1 0 0 0 0 16 16Lazio 1 21 3 0 0 6 0 1 8 2 1 0 0 0 42 42Lazio 2 7 1 0 0 3 0 1 3 1 0 0 0 0 16 16Abruzzo 6 1 0 0 3 0 0 3 1 0 0 0 0 14 14Molise 2 0 0 0 0 0 0 0 0 0 0 0 0 2 3Campania1 14 2 1 0 7 0 1 5 1 1 0 0 0 32 32Campania2 12 2 0 0 6 0 1 4 2 1 0 0 0 28 28Puglia1 15 5 1 0 9 0 1 8 2 1 0 0 0 42 42Basilicata 3 1 0 0 1 0 0 1 0 0 0 0 0 6 6Calabria 9 1 1 0 4 0 0 4 0 1 0 0 0 20 20Sicily 1 10 1 0 0 6 0 0 6 1 1 0 0 0 25 25Sicily 2 10 1 1 0 6 0 0 7 1 1 0 0 0 27 27Sardinia 8 1 1 0 3 0 0 4 1 0 0 0 0 18 17Valle D'Aosta 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1Italian overseas districts 5 0 0 0 1 0 0 1 2 0 0 2 1 12 12

Totals 297 37 6 5 98 18 9 109 39 8 1 2 1 630 630

PARTIES

Italian Chamber

Trentino Alto AdigeSardinia

Friuli Venezia GiuliaMolise

Veneto 1Piedmont 2

Elections 2008

+1+1

Sicily 1Trentino Alto Adige

-1-1

Elections 2013

+1+1

-1-1

Seats

35

Gambarelli, G. (1999), Group Decision and Negotiation

Gambarelli, G. and A. Palestini (2007), Homo Oeconomicus (G. Gambarelli Ed.)

Open problems

- Optimize the algorithms for Minimax Apportionments

- Applications (VIOLANTE)

34

FORECASTINGFORECASTING

33

Multicameral Cohesion Games

Cohesion gamesMulticameral

games

FORECASTING

Applications

Algorithm

Multicameral Cohesion Games

32

Europe

Bulgaria Lithuania

Cyprus Latvia

Denmark Luxemburg

Estonia Malta

Finland Portugal

Greece Slovakia

Hungary Sweden

Austria

(Germany)

Ireland

Spain

Slovenia

U.K.

UNICAMERAL BICAMERALbut asymmetic

Belgium

Czech Republic

France

Italy

Netherlands

Poland

Romania

CONSIDEREDCOUNTRIES

31

Belgium

30

Open problem

- Applications of Multicameral Cohesion Games outside Europe

29

Gambarelli, G. and A. Uristani (2009), Central European Journal of Operations Research

1.3) Applications of G.T. to COLLUSIONS

28

BARAlexei Nemov

Athens 2004

27

- GymnasticsGymnastics- DivingDiving- Figure skating- Figure skating- Synchronized swimming- Synchronized swimming- ...- ...

SPORTS:

APPLICATIONSAPPLICATIONS

BANKING:

- LIBORLIBOR- EURIBOREURIBOR- EONIASWAP- EONIASWAP- EUREPO- EUREPO- ...- ...

EVALUATION OF PROJECTS

…………..26

2, 7, 7, 8, 9, 92, 7, 7, 8, 9, 9  

Common Sense:Common Sense:Arithmetic Mean:Arithmetic Mean:TTrimmed rimmed mean:mean:Median: Median: Coherent Majority Av.:Coherent Majority Av.:

88777.757.757.507.5088

25

66 judges → majority = 4judges → majority = 4

2, 7, 7, 8, 9, 92, 7, 7, 8, 9, 9 └─── └──────┘─┘ 8-2 = 8-2 = 66 └─ └──────┘───┘ 9-7 = 9-7 = 22 └─ └──────┘───┘ 9-7 = 9-7 = 22  

Minimum difference: Minimum difference: 22Corresponding scores: 7,7,8,9,9Corresponding scores: 7,7,8,9,9

Arithmetic mean of such scores (= CMA): 8 Arithmetic mean of such scores (= CMA): 8

24

Execution Artistry Difficulty

THE PROBLEM (gymnastics)

23

Environmental costs

Building costs

Disease costs

THE PROBLEM (project eval.)

22

21

Open problem

- A method of evaluation of the averages, via Cooperative Games

Gambarelli, G. (2008), Journal of Sport Sciences

Bertini, C., Gambarelli G. and A. Uristani (2010), Studies in Fuzziness and Soft Computing

Gambarelli G., Iaquinta G. and M. Piazza (2012), Journal of Sport Sciences

20

1.4) Applications of G.T. to INTERFERING ELEMENTS

Diseases: epilepsy (A), brain tumors(B)

Drugs: carbamazepine (A), CCNU (B)

19

• Zootechnique optimal breeding

• Agricolture optimal dosages of pesticides, etc…

• Social Choice fiscal policies

• Industrial Economics cannibalism of products

18

Open problems

- Decision Theory Game Theory

- Other applications

- n > 2

Carfì D., Gambarelli G. and A. Uristani (2013), Zeszyty Naukowe Szczecin University Press

Gambarelli G. and A. Lanterna (2014), "From Game Theory to Joint Best for Medical Interactions" (forthcoming)

17

2) OPEN PROBLEMS IN THE THEORY

- Values as baricenters- Comparing Power Indices- A link between two main forms of

games

16

2.1) VALUES AS BARICENTERS

15

v(1) = 0.1

v(2) = 0.3

v(3) = 1

14

x3

x2

x1

13

Open problem

- Extension to other values

Gambarelli G. (1990), Optimization

12

POWER INDICES(FROM VALUES) a Q b(s) fi(s)

SHAPLEY–SHUBIK 1 N ci(s)

ci(s): number of coalitions (of s members)

for which the i-th player is crucial

BANZHAF– COLEMAN 1 1 ci

LEMAIRE L B 1 xi(BS)B = set of bipartitions of Nxi(BS) = payoff of player i with respect to

bipartition BS

HARSANYI-NASH n 1 1 1

If player i wins alone i(v) = 1 and

j(v) = 0 j i

TIJS) 1 1 1If player i Jv (set of veto players) i(v) = 0

Qs

ii nisfsba

v ...,,, 211

!

!!1

n

sns

12 1 n

vj

Nj

jc

2.2) COMPARING POWER INDICES

11

Properties Shapley Banzhaf SC-”Banz

haf”Banzhaf-Penrose

SC-”Banzhaf-

Penrose”Tijs SC-”Tijs”

Efficiency

Dummy player

Additivity

Individual rationality

Local monotonicity

Global monotonicity

Symmetry

Properties of valuesYES

?

NO

10

Properties Shapley Banzhaf SC-”Banz

haf”Banzhaf-Penrose

SC-”Banzhaf-

Penrose”Tijs SC-”Tijs”

Efficiency

Dummy player

Additivity

Individual rationality

Local monotonicity

Global monotonicity

Symmetry

Properties of valuesYES

?

NO

9

Properties of values

Properties Shapley Banzhaf SC-”Banz

haf”Banzhaf-Penrose

SC-”Banzhaf-

Penrose”Tijs SC-”Tijs”

Efficiency

Dummy player

Additivity

Individual rationality

Local monotonicity

Global monotonicity

Symmetry

YES

?

NO

8

Bertini, C., Gambarelli G. and I. Stach (2008), Power, Freedom, and Voting

Bertini C., Gambarelli, G., Freixas, J. and I. Stach, (2013 a), International Game Theory Review

(Fragnelli, V. and G. Gambarelli Eds.)

Bertini C., Gambarelli G., Freixas, J. and I. Stach (2013 b),

International Game Theory Review

(Fragnelli, V. and G. Gambarelli Eds.)

7

Open problem

- Complete the table

– extensive form

I \ II B1 B2

A1 0 , 0 0 , 1

A2 1 , 0 -1 , -1

I

II II

(0 , 0) (0 , 1) (1 , 0) (-1 , -1)

S D

S D S D

zero

su

m

gam

es

– normal form

– characteristic form

2.3) A LINK BETWEEN TWO MAIN FORMS OF GAMES

6

The New Approach

How much should I demand, as my price, for taking part in each coalition?

If I ask too little, …

If I ask too much, …5

The New Approach

Characteristic

function form

v G(v)

Normal

form

4

Solution

(Pareto ottima

Nash equilibria)

–all subadditive games

–all inessential games

–all 2-person games

–all 3-person simple games

–every superadditive game havingthe interior of the core not-empty

THE TG-SOLUTION IS NOT-EMPTY FOR:

THEOREMS OF EXISTENCE

3

Gambarelli, G. (2007), International Game Theory Review

Open problems

- Existence

- Uniqueness

→ Value

2

MY WARMEST THANKS TO...

1gianfranco.gambarelli@unibg.it