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
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − ā|| 2 ||x ⋆ − ¯m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3 Nb
1 1 1 0.1132 0.0084 -19 -16 35 1
1.3041 1.025 1.0283 0.0497 0.0645 -2479 1264 1216 13
1 1.3183 1.0065 0.0964 0.0754 1001 -1899 898 4
1 1.0001 1.3427 0.1162 0.0896 722 701 -1423 3
1.0185 1.3694 1.3993 0.133 0.2215 2646 -1507 -1139 114
1.3856 1.0844 1.4501 0.1144 0.2696 -1729 2758 -1029 142
1.419 1.4541 1.1247 0.1121 0.3004 -1564 -1233 2797 111
1.0449 1.3931 3 3.4379 3.9075 3787 -1490 -2297 1
3 1.1738 1.5381 3.2767 4.0418 -4490 5412 -922 3
Table 2.7: Premium equilibria - PLN price ratio function
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − a|| 2 ||x ⋆ − m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3 Nb
1.3644 1.0574 1.0661 0.1239 0.0611 -2635 1397 1239 10
1 1.3942 1.0208 0.1201 0.1003 1315 -2258 943 1
1 1.001 1.4206 0.1398 0.1192 851 818 -1670 3
1.0044 1.4216 1.4569 0.0887 0.1781 3333 -1923 -1411 109
1.4875 1.1726 1.5792 0.1836 0.2781 -1622 2696 -1075 116
1.555 1.6092 1.2508 0.2323 0.3598 -1369 -1210 2579 97
1.561 1.2526 3 3.0865 3.5394 -1405 3695 -2291 4
1.7346 3 1.4348 3.2546 3.7733 -955 -3174 4129 5
3 1.4664 3 6.226 6.8384 -4485 6746 -2261 2
3 1.3699 1.7658 3.4794 3.7789 -4482 5299 -817 4
3 1.9041 1.5497 3.6941 4.0712 -4462 -743 5205 12
3 3 1.7542 6.407 7.0956 -4354 -2970 7324 4
Table 2.8: Premium equilibria - NBLN price ratio function
multipliers (not reported here). Those are not zero when a constraint g i j
i = 1, 2, 3 and j = 1, . . . , I).
is active, (where
Tables 2.12 and 2.13 in Appendix 2.6.1 report the computation when we use the price
difference function ˜ fj. The number of different premium equilibria is similar as in the previous
case.
This numerical application reveals that in our refined game, we have many generalized
premium equilibria. In our insurance context, a possible way to deal with multiple equilibria
is to choose as a premium equilibrium, the generalized Nash equilibrium x ⋆ that is closest to
the average market premium ¯m. This option is motivated by the high level of competition
present in most mature insurance markets (e.g. Europe and North America) where each
insurer sets the premium with a view towards the market premium.
However, this solution has drawbacks: while a single Nash equilibrium may be seen as
a self-enforcing solution, multiple generalized Nash equilibria cannot be self-enforcing. We
will not pursue this refined one-shot game further and focus on the simple insurance game of
Section 2.2.
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