Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
Chapitre 2. Theorie des jeux et cycles de marché
1.0
Prob. Leadership
0.8
0.6
0.4
1.4
1.6
Leader
Cap. Sol. Ratio
1.8
2.0
0.4
0.7
0.6
0.5
Market Share
0.4
Prob. Leadership
0.3
0.2
0.1
(a) NBLN
0.0
1.4
1.6
Outsider
Cap. Sol. Ratio
1.8
2.0
0.4
0.7
0.6
0.5
Market Share
1.0
Prob. Leadership
0.8
0.6
0.4
1.4
1.6
Leader
Cap. Sol. Ratio
1.8
2.0
0.4
0.7
0.6
0.5
Market Share
0.4
Prob. Leadership
0.3
0.2
0.1
(b) PGLN
0.0
1.4
1.6
Outsider
Cap. Sol. Ratio
Figure 2.4: Leadership probabilities of Leader (Insurer 1) and Outsiders (Insurers 2 and 3)
is essential for its survival. Unlike the classic risk theory where the collection of premiums
is fixed per unit of time, the main source of risk for an insurance company in our game is a
premium risk. We extends one-player model of Taylor (1986, 1987) and subsequent extensions
based optimal control theory. To our knowledge, the use of a repeated of noncooperative game
to model non-life insurance markets is new.
The game can be extended in various directions. A natural next step is to consider adverse
selection among policyholders, since insurers do not propose the same premium to all
customers. In practice, insurers do not propose the same premium to all customers. Considering
two risk classes of individuals would be an interesting extension of the game, but we would
also have to find a way to differentiate premium between individuals. A second extension is to
model investment results as well as loss reserves. We could also consider reinsurance treaties
for players in addition a catastrophe generator. However, we must avoid not to complexify
too much the game as the game is already complex and deriving theoretical properties is an
issue.
2.6 Appendix
2.6.1 On the one-period game
Implicit function theorem
Below the implicit function theorem, see, e.g., (Zorich, 2000, Chap. 8).
Theorem. Let F be a bivariate C1 function on some open disk with center in (a, b), such that
F (a, b) = 0. If ∂F
∂y (a, b) = 0, then there exists an h > 0, and a unique function ϕ defined for
]a − h, a + h[, such that
ϕ(a) = b and ∀|x − a| < h, F (x, ϕ(x)) = 0.
Moreover on |x − a| < h, the function ϕ is C1 and
ϕ ′
(x, y)
(x) = −
(x, y)
124
∂F
∂x
∂F
∂y
y=ϕ(x)
.
1.8
2.0
0.4
0.7
0.6
0.5
Market Share