On the Ellipsoidal Core

for Cooperative Games

under Ellipsoidal Uncertainty

Gerhard-Wilhelm Weber

Institute of Applied Mathematics METU AnkaraTurkey

Rodica Branzei

Faculty of Computer Science Alexandru Ioan Cuza University Iaşi Romania

S Zeynep Alparslan-Goumlk

Department of Mathematics Suumlleyman Demirel University Isparta Turkey

5th International Summer School

Achievements and Applications of Contemporary Informatics

Mathematics and Physics

National University of Technology of the Ukraine

Kiev Ukraine August 3-15 2010

Games

Games

Optimal Energy Management

according to Joint Implementation

Aims

1 The instrument must work on the micro level with minimal costs

2 It should be protected against misuse on the macro level

Ad 1 Solved by the control theoretic approach

Ad 2 Can be solved by adding constraints to the feasible set

of our control parameters

Cost games are very important in the practice of OR

Ex

airport game

unanimity game

production economy with landowners and peasants

bankrupcy game etc

There is also a cost game in environmental protection (TEM model)

The aim is to reach a state which is mentioned in Kyoto Protocol

by choosing control parameters such that

the emissions of each player become minimized

For example the value is taken as a control parameter

Cost Games

Cost Games

The central problem in cooperative game theory is how to allocate the gain

among the individual players in a ldquofairrdquo way

There are various notions of fairness and corresponding allocation rules

(solution concepts)

Here we use the notation

Any with is an allocation

So a core allocation

guarantees each coalition to be satisfied

in the sense that it gets at least what it could get on its own

( )w w N i N

( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR

( ) ( )

i

i S

x S x S N coalition

Nx R ( ) x N w

( )x Core wS N

bull Method

Determine the core by this difference and

with the value which has to lie in it

bull Approach

Steer the system

bull Main idea

Take the value as a core element

and as a control parameter

( ) ( ) w Core w

Kyoto Game

Only 3 time points are assumed

The 2 players have

2 alternative strategies to invest

The origin of the coordinate system

is the starting point

Each player tries

to reach the blue square

This is the level of

reductions of emissions

in a given number of steps

mentioned in Kyoto Protocol

Kyoto Game

o The strategy (21) leads to a greater

reduction at the beginning and a

smaller investment at the end

of the period

o The costs are lower than in the

(12) case reflecting the fact that

early innovations are favourable

o This simple model can be transferred

to a simple matrix game

our Kyoto Game

Kyoto Game

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

Cost games are very important in the practice of OR

Ex

airport game

unanimity game

production economy with landowners and peasants

bankrupcy game etc

There is also a cost game in environmental protection (TEM model)

The aim is to reach a state which is mentioned in Kyoto Protocol

by choosing control parameters such that

the emissions of each player become minimized

For example the value is taken as a control parameter

Cost Games

Cost Games

The central problem in cooperative game theory is how to allocate the gain

among the individual players in a ldquofairrdquo way

There are various notions of fairness and corresponding allocation rules

(solution concepts)

Here we use the notation

Any with is an allocation

So a core allocation

guarantees each coalition to be satisfied

in the sense that it gets at least what it could get on its own

( )w w N i N

( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR

( ) ( )

i

i S

x S x S N coalition

Nx R ( ) x N w

( )x Core wS N

bull Method

Determine the core by this difference and

with the value which has to lie in it

bull Approach

Steer the system

bull Main idea

Take the value as a core element

and as a control parameter

( ) ( ) w Core w

Kyoto Game

Only 3 time points are assumed

The 2 players have

2 alternative strategies to invest

The origin of the coordinate system

is the starting point

Each player tries

to reach the blue square

This is the level of

reductions of emissions

in a given number of steps

mentioned in Kyoto Protocol

Kyoto Game

o The strategy (21) leads to a greater

reduction at the beginning and a

smaller investment at the end

of the period

o The costs are lower than in the

(12) case reflecting the fact that

early innovations are favourable

o This simple model can be transferred

to a simple matrix game

our Kyoto Game

Kyoto Game

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

Cost Games

The central problem in cooperative game theory is how to allocate the gain

among the individual players in a ldquofairrdquo way

There are various notions of fairness and corresponding allocation rules

(solution concepts)

Here we use the notation

Any with is an allocation

So a core allocation

guarantees each coalition to be satisfied

in the sense that it gets at least what it could get on its own

( )w w N i N

( ) | ( ) ( ) ( ) ( ) NCore w x x N w x S w S S NR

( ) ( )

i

i S

x S x S N coalition

Nx R ( ) x N w

( )x Core wS N

bull Method

Determine the core by this difference and

with the value which has to lie in it

bull Approach

Steer the system

bull Main idea

Take the value as a core element

and as a control parameter

( ) ( ) w Core w

Kyoto Game

Only 3 time points are assumed

The 2 players have

2 alternative strategies to invest

The origin of the coordinate system

is the starting point

Each player tries

to reach the blue square

This is the level of

reductions of emissions

in a given number of steps

mentioned in Kyoto Protocol

Kyoto Game

o The strategy (21) leads to a greater

reduction at the beginning and a

smaller investment at the end

of the period

o The costs are lower than in the

(12) case reflecting the fact that

early innovations are favourable

o This simple model can be transferred

to a simple matrix game

our Kyoto Game

Kyoto Game

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

bull Method

Determine the core by this difference and

with the value which has to lie in it

bull Approach

Steer the system

bull Main idea

Take the value as a core element

and as a control parameter

( ) ( ) w Core w

Kyoto Game

Only 3 time points are assumed

The 2 players have

2 alternative strategies to invest

The origin of the coordinate system

is the starting point

Each player tries

to reach the blue square

This is the level of

reductions of emissions

in a given number of steps

mentioned in Kyoto Protocol

Kyoto Game

o The strategy (21) leads to a greater

reduction at the beginning and a

smaller investment at the end

of the period

o The costs are lower than in the

(12) case reflecting the fact that

early innovations are favourable

o This simple model can be transferred

to a simple matrix game

our Kyoto Game

Kyoto Game

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

Only 3 time points are assumed

The 2 players have

2 alternative strategies to invest

The origin of the coordinate system

is the starting point

Each player tries

to reach the blue square

This is the level of

reductions of emissions

in a given number of steps

mentioned in Kyoto Protocol

Kyoto Game

o The strategy (21) leads to a greater

reduction at the beginning and a

smaller investment at the end

of the period

o The costs are lower than in the

(12) case reflecting the fact that

early innovations are favourable

o This simple model can be transferred

to a simple matrix game

our Kyoto Game

Kyoto Game

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

o The strategy (21) leads to a greater

reduction at the beginning and a

smaller investment at the end

of the period

o The costs are lower than in the

(12) case reflecting the fact that

early innovations are favourable

o This simple model can be transferred

to a simple matrix game

our Kyoto Game

Kyoto Game

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -01 30 60 160 1 -0525 -0475

2 -01 20 60 160 -0475 1 0525

3 -01 10 60 160 -01 -01 02

The actors have not reached the limiting Kyoto level 0

at the beginning of the time period

The em-matrix has positive and negative entries which means

that we deal with both cooperative and competitive behaviour

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

Every player is at Kyoto Level

This example is an extraordinary case

The memory parameter φ is chosen very big

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 0 03 1 001 1 -05 -05

2 0 05 1 001 -05 1 -05

3 0 02 1 001 -05 -05 1

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

Every actor reaches the Kyoto Level without any control parameter

But it takes a 100 year time period which is a too long time

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

player i Ei (0) Mi (0) Mi λi 160 x em matrix

1 -1 03 1 082 1 -07 -03

2 06 01 1 025 -08 1 -02

3 05 02 1 04 -09 -01 1

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

TEM Model

Here a chaotic behaviour occurs after a 20 year time period

the first bifurcation can be observed after 20 time steps

This behavior requests the determination of optimal control parameters

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

TEM Model

ui(t) control parameters

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

The figure shows the behavior

of the 3 players

after the control parameters added

It is seen that every actor can reach

the Kyoto Level in a short time period

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

( 1) ( ) ( )

( )

k k k

kM

E E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

TEM Model

time-continExpression data

time-discr

kkk EE M1

)(Μ jik em M

0)0( EE

Ex Euler Runge-Kutta

( ) ( )E M E E C E

environmental effects

TEM Model Gene-Environment Networks

( ) ΜkM E

E

TEM Model Gene-Environment Networks

( )NI R

( ) ΜkM E

E

TEM Model Gene-Environment Networks

( ) ΜkM E

E

TEM Model Gene-Environment Networks

( )NI R

( ) ΜkM E

E

TEM Model Gene-Environment Networks

( ) ΜkM E

E

TEM Model Gene-Environment Networks

Gene-Environment Networks Errors and Uncertainty

Errors uncorrelated Errors correlated Fuzzy values

Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics

θ1

θ2

Identify groups (clusters) of jointly acting

genetic and environmental variables

stable clustering

disjoint

overlapping

Gene-Environment Networks Errors and Uncertainty

2) Interaction of Genetic Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Environmental Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Genetic amp Environmental Clusters

Determine the degree of connectivity

Gene-Environment Networks Errors and Uncertainty

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters Pre-Coalitions

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Identify groups (clusters) of jointly acting

genetic and environmental variables

stable clustering

disjoint

overlapping

Gene-Environment Networks Errors and Uncertainty

2) Interaction of Genetic Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Environmental Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Genetic amp Environmental Clusters

Determine the degree of connectivity

Gene-Environment Networks Errors and Uncertainty

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters Pre-Coalitions

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

2) Interaction of Genetic Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Environmental Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Genetic amp Environmental Clusters

Determine the degree of connectivity

Gene-Environment Networks Errors and Uncertainty

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters Pre-Coalitions

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

3) Interaction of Environmental Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Genetic amp Environmental Clusters

Determine the degree of connectivity

Gene-Environment Networks Errors and Uncertainty

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters Pre-Coalitions

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

3) Interaction of Genetic amp Environmental Clusters

Determine the degree of connectivity

Gene-Environment Networks Errors and Uncertainty

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters Pre-Coalitions

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters Pre-Coalitions

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Clusters and Ellipsoids

Genetic clusters C1C2hellipCR

Environmental clusters D1D2hellipDS

Genetic ellipsoids X1X2hellipXR Xi = E (μiΣi)

Environmental ellipsoids E1E2hellipES Ej = E (ρjΠj)

Gene-Environment Networks Ellipsoidal Calculus

Clusters Pre-Coalitions

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

The Regression Problem

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Measures for the size of intersection

bull Volume rarr ellipsoid matrix determinant

bull Sum of squares of semiaxes rarr trace of configuration matrix

bull Length of largest semiaxes rarr eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr E

r

Gene-Environment Networks Ellipsoidal Calculus

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval GamesC ollaborativecooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval Games

cooperative

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval Games

cooperative

Robust Optimization

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval Gamescooperative

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

I Ni i

i N

w I I I w N I w i i NR

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval GamesC ollaborative

( ) | ( ) ( ) ( )

Ni i

i N

w I I I w N I w i i NRI

= 12

2 ( ) ( ) [0

0]

lt gt

N

NnN I ww

N IGw

R

cooperative

1 2 1 2

1 2

1 2

( )( ) ( ) ( )

( )( ) ( )

N

lt N gt

lt N gt

S S S

S S

w w

w w IG

w

w w

w

w w

w

Interval Glove Game

0 euro 0 euro

10 - 20 euro

(13) (23) (123) [10 20]

( ) [00] else

( ) = ([00][00][10 20])

= 1 23

= 1 2

L R

w w w

w S

Co

N

L

re w

L R

Interval core

( ) | ( ) ( )

ii S

Core w I w I w S S N SI

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval GamesC ollaborative

2

1

1 2

( ) = | ( ) ( ) 1

( ) = + | || || 1

E

E

Tpc x x c x c

c u c u

R

cooperative

( ) ( ) E E TA c +b = Ac+b A A

1 2

11

2 2

2

11 ( ( )

( ) (1 )

)

) (1

E EE E E+ = c +c

s s

s

s

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Glove Game

0 euro 0 euro

( ) euro

(13) (23) (123) ( )

( ) 0 else

= 123

= 12

L R

c

w w w c

w

R

S

N L

L

Ellipsoid ellipsoidcore value

E

E

Kyoto Game

(individual roles in TEM Model)

(individual role in TEM Model)

Ellipsoidal core

iv

Ellipsoid Glove Game

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Malacca Police Game

R

Ellipsoid Games Interval GamesC ollaborativecooperative

= 12 2

)

( 0

lt gt

N

N

wn wN

EGN w

set of all ellipsoids

Ellipsoid Malacca Police Game

R

Some results related with the ellipsoidal core

rer

rr

r

Some results related with the ellipsoidal core

rer

rr

r

Thank you very much for your attention

gwebermetuedutr

httpwww3iammetuedutriamimages773Willi-CVpdf

References

Some results related with the ellipsoidal core

rer

rr

r

Thank you very much for your attention

gwebermetuedutr

httpwww3iammetuedutriamimages773Willi-CVpdf

References

Thank you very much for your attention

gwebermetuedutr

httpwww3iammetuedutriamimages773Willi-CVpdf

References

rr

Appendix