A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
A Game Theoretic Approach to Insurance Sharing in
Environmental Pollution Risks
Vito FRAGNELLI
Dipartimento di Scienze e Tecnologie Avanzate
Università del Piemonte Orientale
[email protected]
Maria Erminia MARINA
Dipartimento di Economia e Metodi Quantitativi
Università di Genova
[email protected]
Second FIMA International Conference 2008
”Energy and Environment: new challenges to mathematical modelling and applications”
January 21-24, 2008 - Champoluc
1
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Summary
Setting
Problem
Co-Insurance Games
General Results
Constant Quotas
Proportional Allocation
Case Study
Concluding Remarks
2
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
3
Setting
A large risk, e.g. environmental pollution risk, has to be insured
It may depend on firms that in their production processes may have as a side effect the release of polluting wastes,
can alter the standard environment, influence the possibilities of using it damaging public or private goods, compromise directly or indirectly the human health, contaminate biological resources and ecosystems
Consequences may be described as damages to persons and/or materials or interruption of various activities
(industrial, agricultural or recreational)
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Possible environmental pollution risks:
air pollution
emissions
harmful gases
exhaust fumes
stenches
waste disposals
chemical productions
water pollution
effluents, when discharged in the rivers
soil pollution
rubbish
solid wastes
industrial (chemical manures or pesticides)
marine pollution wastes from coastal firms
sewage
oil tankers accidents
acoustic pollution high noise level
vibrations
third party liability for damages caused to persons and/or materials
→ insurance policy (Bazzano, 1994)
clean up costs for removal of pollutants
4
5
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Losses for environmental pollution can be very heavy
Costs refunded after main accidents in OCSE countries 1976-1988
Naval transportation accidents are not included
Year
1976
1978
1982
1984
1985
1986
1987
1988
1988
1988
Place
Seveso (Italy)
Los Alfaques (Spain)
Livingstone (USA)
Denver (USA)
Kenora (Canada)
Basilea (Switzerland)
Herborn (Germany)
Floreffe (USA)
S. Basile (Canada)
Piper Alpha (Northern Sea)
Cost ∗
103
15
41
20
7
16
8
67
39
111
Cause (pollutant)
Chemical plant (dioxin)
Tanker truck explosion (propylene)
Derailment (toxics)
Tank (gasoline)
Spill (PCB)
Fire with river Rhein pollution
Tanker truck
Tank explosion (oil)
Fire (toxic wastes)
Explosion (gas)
* in millions of euros
SCOR NOTES (1989)
Asbestos pollution damages refused by Lloyd’s in about twenty years: 15 billions of euros (some trials are still
lasting)
Pool of companies
6
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Division Plan for the Italian Pool for Environmental Risk Insurance
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
COMPANY
ALLIANZ SUBALPINA
LE ASSICURAZIONI DI ROMA
ASSICURAZIONI GENERALI
ASSIMOCO
ASSITALIA-LE ASSICURAZIONI D’ITALIA
AUGUSTA ASSICURAZIONI
AURORA ASSICURAZIONI
AXA ASSICURAZIONI
BAYERISCHE RUCK (*)
BERNESE ASS.NI-COMP. ITALO-SVIZZERA
BNC ASSICURAZIONI
COMPAGNIA ASSICURATRICE UNIPOL
COMPAGNIA DI ASSICURAZIONE DI MILANO
IL DUOMO
ERC - FRANKONA AG (*)
F.A.T.A.
LA FONDIARIA ASSICURAZIONI
GAN ITALIA
GENERAL & COLOGNE RE (*)
GIULIANA ASSICURAZIONI
ITALIANA ASSICURAZIONI
ITAS ASSICURAZIONI
ITAS SOC. DI MUTUA ASSICURAZIONE
LEVANTE NORDITALIA ASSICURAZIONI
LIGURIA
LLOYD ADRIATICO
LLOYD ITALICO ASSICURAZIONI
MAECI - SOC. MUTUA DI ASS.NI E RIASS.NI
MAECI ASSICURAZIONI E RIASSICURAZIONI
LA MANNHEIM
QUOTA %
1.286
0.286
5.263
0.429
5.263
0.717
1.071
2.460
2.857
0.429
0.286
2.231
5.263
0.574
5.714
1.429
5.263
0.791
2.714
0.286
0.857
0.529
0.529
1.029
0.429
1.340
0.429
0.286
0.429
0.429
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
TOTAL
COMPANY
MEDIOLANUM ASSICURAZIONI
METE ASSICURAZIONI
MUNCHENER RUCK ITALIA (*)
LA NATIONALE
NATIONALE SUISSE
NAVALE ASSICURAZIONI
NEW RE (*)
NUOVA MAA ASSICURAZIONI
NUOVA TIRRENA
PADANA ASSICURAZIONI
LA PIEMONTESE SOC. MUTUA DI ASS.NI
LA PIEMONTESE ASSICURAZIONI
RISPARMIO ASSICURAZIONI
RIUNIONE ADRIATICA DI SICURTA’
ROYAL & SUN ALLIANCE
SAI
SARA ASSICURAZIONI
SASA
SCOR ITALIA RIASSICURAZIONI (*)
S.E.A.R.
SIAT-SOCIETA’ ITALIANA ASS.NI E RIASS.NI
SOCIETA’ CATTOLICA DI ASSICURAZIONE
SOCIETA’ REALE MUTUA DI ASSICURAZIONI
SOREMA (*)
SWISSE RE - ITALIA
TICINO
TORO ASSICURAZIONI
UNIASS ASSICURAZIONI
UNIVERSO ASSICURAZIONI
VITTORIA ASSICURAZIONI
WINTERTHUR ASSICURAZIONI
QUOTA %
0.429
1.143
3.286
0.429
0.429
0.963
2.571
0.429
1.743
2.143
0.429
0.429
0.286
5.263
0.857
5.263
0.429
0.429
2.571
0.286
0.429
1.186
1.429
2.571
7.714
0.306
2.857
0.686
0.429
0.840
0.857
100.000
(*) Reinsurance company
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
7
Problem
The pool of companies that insures the risk look for being as competitive as possible → two intertwined questions:
• Which premium should they charge? (Economic question)
• How should they allocate the risk and the premium in order to obtain a fair division? (Game theoretic
question)
Applications of game theory to insurance:
Borch (1962a, 1962b)
Lemaire (1977, 1991)
Suijs, Borm, De Waegenaere, Tijs (1999)
Suijs (2000) - Survey
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
8
N = {1, ..., n} set of companies
L
set of random variables (insurable risks)
functional providing a measure of risk according to company i ∈ N
Hi : L → R
Hypothesis 1
a) a loading for a degenerate risk is not justified
Hi (w) = w, ∀ w ∈ R, ∀ i ∈ N
b) translation invariance
Hi (w + X) = w + Hi (X), ∀ w ∈ R, ∀ X ∈ L, ∀ i ∈ N
Many classical principles satisfy these hypotheses:
• the net premium principle H(X) = E(X), where E(X) is the expectation of X
• the variance principle H(X) = E(X) + aV (X), where V (X) is the variance of X and a > 0
• the standard deviation principle H(X) = E(X) + β V (X), where β > 0
• the zero utility principle H(X) = H̄, where H̄ satisfies E[u(z + H̄ − X)] = u(z) and u is the utility function
of the insurance company which has an initial surplus z
in particular for exponential utility u(x) = 1a (1 − e−ax), with a > 0 we have H̄ = a1 lnE(eaX )
• the ε-percentile principle H(X) = min {x|F (x) ≥ 1 − ε}, where F is the distribution function of X
9
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
The n companies have to insure a given risk R and receive a premium π
They look for ”fair” allocations of (π, R)
For any subset (coalition) of companies S ⊆ N let D(S) be the set of feasible divisions of the premium π:
di = π
D(S) = (di )i∈S ∈ R|S| s.t.
i∈S
and let A(S) be the non-empty set of the feasible decompositions of the risk R:
Xi = R
A(S) = (Xi )i∈S ∈ L|S| , s.t.
i∈S
According to the allocation (di , Xi)i∈S ∈ D(S) × A(S), the company i ∈ S receives the amount di and pays the
random variable Xi
Hypothesis 2
for each subset S ⊆ N it is possible to compute an optimal decomposition of the risk
∀ S ⊆ N there exists
min
Hi (Xi ) = P (S)
(Xi )i∈S ∈A(S)
i∈S
P (S) can be seen as the evaluation that the companies in S (as a whole) give of the risk R
10
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Example 1 For each i ∈ N the variance principle holds:
Hi(Y ) = E(Y ) + ai V ar(Y )
∀ Y ∈ L, 0 < a1 ≤ ... ≤ an
It is possible to prove (Deprez - Gerber, 1985) that P (N) = i∈N Hi (qi R); moreover as in Fragnelli-Marina
(2003) we have:
qi
Hi
R = E(R) + a(S)V ar(R)
P (S) =
q(S)
i∈S
a(N)
1
where qi = ai , q(S) = i∈S qi and a(S) = i∈S a1i
∀S⊂N
♦
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Consider the allocations of (π, R) with a premium high enough to cover the risk
For any subset of companies S ⊆ N let B(S) be the set of individually rational allocations:
B(S) = {(di , Xi)i∈S ∈ D(S) × A(S)| di − Hi (Xi ) ≥ 0, ∀ i ∈ S}
• B(S) = ∅ ⇐⇒ P (S) ≤ π
• B(S) = ∅ ⇒ B(T ) = ∅, ∀ T ⊃ S. In fact P (T ) ≤ P (S) because Hi (0) = 0, ∀i ∈ N
• We suppose that π > P (N), i.e. the companies all together may obtain a positive gain
11
12
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
For any subset of companies S ⊆ N s.t. B(S) = ∅ let O(S) be the set of allocations of (π, R) corresponding to
optimal risk decompositions:
O(S) =
(di , Xi )i∈S ∈ B(S) |
Hi (Xi ) = P (S)
i∈S
and let PO(S) be the set of Pareto optimal allocations of (π, R):
PO(S) = {(di , Xi)i∈S ∈ B(S) | ∃ (di , Xi)i∈S ∈ B(S), s.t. di − Hi (Xi ) > di − Hi (Xi ), ∀ i ∈ S}
Theorem 1 O(S) = PO(S) (see Proposition 3.5 in Suijs-Borm, 1999 and Proposition 1 in Fragnelli-Marina,
2003)
• If B(S) = ∅ then if we take (Xi )i∈A(S) s.t.
i∈S
Hi (Xi ) = P (S) and define di = Hi (Xi ) +
P (S)), ∀ i ∈ S then (di , Xi)i∈S ∈ PO(S)
1
|S| (π
−
Let Q(N) be the set of allocations s.t. there do not exist Pareto optimal allocations for S ⊂ N preferable to them
and let CO(N) be the set of allocations s.t. any S ⊆ N cannot do better acting separately by itself:
Q(N) = {(di , Xi )i∈N ∈ PO(N) | ∀ S ⊂ N s.t. B(S) = ∅,
∃ (di , Xi)i∈S ∈ PO(S) s.t. di − Hi (Xi ) > di − Hi (Xi ), ∀ i ∈ S}
CO(N) = (di , Xi )i∈N ∈ PO(N) | ∀ S ⊆ N
i∈S (di − Hi (Xi )) ≥ max {0, π − P (S)}
Theorem 2 CO(N) = Q(N) (see Theorem 2 in Lari-Marina, 2000)
13
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Co-Insurance Games
A cooperative game in characteristic function form with transferable utility (TU-game) is a pair (N, v) where:
N
set of players (companies)
v : 2N → R characteristic function, with v(∅) = 0
v(S), S ⊆ N is the worth of coalition S
Given a TU-game (N, v) the core is the set of allocations that give to each coalition S at least its worth:
|N|
xi ≥ v(S), ∀ S ⊆ N and
xi = v(N)}
Core(v) = {(xi )i∈N ∈ R s.t.
i∈S
i∈N
When a game has non-empty core it is said to be balanced
For our co-insurance problem we define a game whose characteristic function v is:
v(S) = max {0, π − P (S)}
∀S⊆N
The allocations of the co-insurance problem are related to the allocations of the co-insurance game:
Theorem 3 CO(N) = ∅ ⇐⇒ Core(v) = ∅
Let us suppose that the players are ordered as:
So
P (N) ≤ P (N \ {n}) ≤ ... ≤ P (N \ {1})
j∈N
P (N \ {j}) − (n − 1)P (N) ≥ P (N \ {i}), ∀ i ∈ N
14
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
General Results
Lemma 1 π ≤ P (N \ {1}) ⇒ the game is balanced
Lemma 2 π > i∈N P (N \ {i}) − (n − 1)P (N) ⇒ the game is not balanced
Theorem 4 There exists π
such that the game is balanced if and only if π ≤ π
(see Theorem 1 in Fragnelli et al.,
2000)
Hypothesis 3 The cost function P satisfies the (reduced concavity) hypothesis:
for each S ⊂ N s.t. P (S) < π
,
P (S) − P (S ∪ {i}) ≥ (P (N \ {i}) − P (N), ∀ i ∈ N \ S
Theorem 5 Suppose that P satisfies Hypothesis 3, then π
= j∈N P (N \ {j}) − (n − 1)P (N)
If P satisfies Hypothesis 3 and the premium is precisely π
=
j∈N
P (N \ {j}) − (n − 1)P (N) then:
• the core is the singleton whose only element is the marginal solution x = (P (N \ {1}) − P (N), ..., P (N \
{n}) − P (N))
• the marginal solution is also the nucleolus of the game (Schmeidler, 1969)
• the marginal solution is also the τ -value of the game (Tijs, 1981)
15
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Constant Quotas
Let us suppose that there exist a convex function H and n real numbers q1 ≥ ... ≥ qn > 0,
Y
,
Hi (Y ) = qiH
qi
(In Example 1 H(Y ) = E(Y ) + a(N)V ar(Y ))
i∈N
qi = 1
s.t.:
∀ i ∈ N, ∀ Y ∈ L
An optimal decomposition can be obtained sharing the risk according constant quotas, represented by the numbers
q1, ..., qn and the cost function P satisfies the reduced concavity hypothesis
Proposition 1 The function P verifies Hypothesis 3
16
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Proportional Allocation
The proportional (problem) allocation (qi π, qiR)i∈N has the following properties:
• belongs to B(N)
1 − qn
• belongs to CO(N) iff π ≤
qn
H
R
q(N \ {n})
− H(R) = π
1
(P (N \ {n}) − P (N)) the corresponding game solution is
If the premium is exactly π
, then v(N) =
q
n
qi
(P (N \ {n}) − P (N)
qn
i∈N
Player n gets exactly its marginal contribution, while each player i ∈ N \ {n} gets the marginal contribution of
player n times the ratio among qi and qn
If the premium is exactly π
, then v(N) =
j∈N (P (N \ {j}) − P (N)) the proportional solution is
, while the marginal solution is (P (N \ {i}) − P (N))i∈N
qi
j∈N P (N \ {j}) − P (N)
i∈N
The proportional solution divides each marginal contribution proportionally among all the players (plus the ”refund”
of the risk assumed)
The marginal solution assigns to each player exactly his marginal contribution (plus the ”refund” of the risk
assumed)
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
17
Case Study
We suppose that the 61 companies in Example 1 express their evaluation of a random variable X according
to the variance principle
Next we suppose that qi , i ∈ N are the quotas of the risk as in Table 2 and a(N) = 0.1
1
1
and V ar(R) = 2
µ
µ
and V ar(R) = 1.1025) (in millions of euros) and consequently:
Finally we suppose that the distribution function of the risk R is F (x) = 1−e−µx, so E(R) =
We assume that E(R) is 1.05 (µ =
P (N) = 1.160250
π
= 1.274612
π
= 1.270816
1
1.05
18
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Comp,(i)
55
15
3
5
13
17
44
46
33
9
57
19
37
49
54
8
12
40
39
16
53
26
1
52
32
7
24
36
21
45
%
7.71
5.71
5.26
5.26
5.26
5.26
5.26
5.26
3.29
2.86
2.86
2.71
2.57
2.57
2.57
2.46
2.23
2.14
1.74
1.43
1.43
1.34
1.29
1.19
1.14
1.07
1.03
0.96
0.86
0.86
qi
0.07714
0.05714
0.05263
0.05263
0.05263
0.05263
0.05263
0.05263
0.03286
0.02857
0.02857
0.02714
0.02571
0.02571
0.02571
0.02460
0.02231
0.02143
0.01743
0.01429
0.01429
0.01340
0.01286
0.01186
0.01143
0.01071
0.01029
0.00963
0.00857
0.00857
qi π
98,031
72,614
66,883
66,883
66,883
66,883
66,883
66,883
41,759
36,307
36,307
34,490
32,673
32,673
32,673
31,262
28,352
27,234
22,150
18,160
18,160
17,029
16,343
15,072
14,525
13,610
13,077
12,238
10,891
10,891
qi π
98,324
72,831
67,083
67,083
67,083
67,083
67,083
67,083
41,884
36,416
36,416
34,593
32,770
32,770
32,770
31,355
28,437
27,315
22,216
18,214
18,214
17,080
16,392
15,117
14,569
13,651
13,116
12,275
10,923
10,923
marg
98,717
72,978
67,189
67,189
67,189
67,189
67,189
67,189
41,872
36,391
36,391
34,565
32,739
32,739
32,739
31,323
28,401
27,279
22,179
18,178
18,178
17,045
16,357
15,084
14,536
13,620
13,085
12,245
10,896
10,896
Comp,(i)
61
60
18
6
58
14
22
23
4
10
25
27
29
30
31
34
35
38
41
42
47
48
51
59
56
2
11
20
28
43
50
%
0.86
0.84
0.79
0.72
0.69
0.57
0.53
0.53
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.31
0.29
0.29
0.29
0.29
0.29
0,29
qi
0.00857
0.00840
0.00791
0.00717
0.00686
0.00574
0.00529
0.00529
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00306
0.00286
0.00286
0.00286
0.00286
0.00286
0,00286
qi π
10,891
10,675
10,052
9,112
8,718
7,294
6,723
6,723
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
3,889
3,635
3,635
3,635
3,635
3,635
3.635
qi π
10,923
10,707
10,082
9,139
8,744
7,316
6,743
6,743
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
5,468
3,900
3,645
3,645
3,645
3,645
3,645
3.645
marg
10,896
10,680
10,057
9,115
8,721
7,296
6,724
6,724
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
5,452
3,889
3,635
3,635
3,635
3,635
3,635
3.635
Some allocations, in euros, for the problem (marg = qiP (N ) + P (N \ {i}) − P (N ))
19
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
Comp,(i)
55
15
3
5
13
17
44
46
33
9
57
19
37
49
54
8
12
40
39
16
53
26
1
52
32
7
24
36
21
45
%
7.71
5.71
5.26
5.26
5.26
5.26
5.26
5.26
3.29
2.86
2.86
2.71
2.57
2.57
2.57
2.46
2.23
2.14
1.74
1.43
1.43
1.34
1.29
1.19
1.14
1.07
1.03
0.96
0.86
0.86
qi
0.07714
0.05714
0.05263
0.05263
0.05263
0.05263
0.05263
0.05263
0.03286
0.02857
0.02857
0.02714
0.02571
0.02571
0.02571
0.02460
0.02231
0.02143
0.01743
0.01429
0.01429
0.01340
0.01286
0.01186
0.01143
0.01071
0.01029
0.00963
0.00857
0.00857
− P (N ))
qi (π
8,529
6,318
5,819
5,819
5,819
5,819
5,819
5,819
3,633
3,159
3,159
3,001
2,843
2,843
2,843
2,720
2,467
2,369
1,927
1,580
1,580
1,482
1,422
1,311
1,264
1,184
1,138
1,065
948
948
− P (N ))
qi (π
8,822
6,535
6,019
6,019
6,019
6,019
6,019
6,019
3,758
3,267
3,267
3,104
2,940
2,940
2,940
2,813
2,551
2,451
1,993
1,634
1,634
1,532
1,471
1,356
1,307
1,225
1,177
1,101
980
980
marg
9,216
6,681
6,125
6,125
6,125
6,125
6,125
6,125
3,746
3,242
3,242
3,076
2,909
2,909
2,909
2,781
2,516
2,414
1,956
1,598
1,598
1,497
1,436
1,323
1,275
1,194
1,146
1,072
953
953
Comp,(i)
61
60
18
6
58
14
22
23
4
10
25
27
29
30
31
34
35
38
41
42
47
48
51
59
56
2
11
20
28
43
50
%
0.86
0.84
0.79
0.72
0.69
0.57
0.53
0.53
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.43
0.31
0.29
0.29
0.29
0.29
0.29
0.29
qi
0.00857
0.00840
0.00791
0.00717
0.00686
0.00574
0.00529
0.00529
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00429
0.00306
0.00286
0.00286
0.00286
0.00286
0.00286
0.00286
− P (N ))
qi (π
948
929
875
793
758
635
585
585
474
474
474
474
474
474
474
474
474
474
474
474
474
474
474
474
338
316
316
316
316
316
316
− P (N ))
qi (π
980
961
905
820
785
656
605
605
491
491
491
491
491
491
491
491
491
491
491
491
491
491
491
491
350
327
327
327
327
327
327
Some allocations, in euros, for the game (marg = P (N \ {i}) − P (N ))
marg
953
934
879
796
762
636
586
586
475
475
475
475
475
475
475
475
475
475
475
475
475
475
475
475
338
316
316
316
316
316
316
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
20
Concluding Remarks
• if π = j∈N P (N \ {j}) − (n − 1)P (N ) the proportional allocation is not in the core of
the game because it assigns too much to last players, w.r.t. what is assigned to the first
players
• the unique core allocation, the marginal solution, is favorable to the first players, i.e. those
who assume larger quotas of risk
• the proportional solution is more ”leveled” so that in order to have a core allocation it is
necessary to have a lower premium that produces an amount assigned to the last player by
the proportional allocation at most equal to the amount of the marginal one
A Game Theoretic Approach to Insurance Sharing in Environmental Pollution Risks
21
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