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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