running time and the success rate in relation of
solved instances by PATH solver. Table 3 summa-
rizes the numerical results for different sizes of the
chance-constrained games. Column 1 presents the
size of the game instances. Columns 2-4 show the
conﬁdence level α, success rate and average CPU
time for problems under Cauchy distribution, respec-
tively. Columns 5-7 provide the same information as
Columns 2-4 for problems under normal distribution.
As shown in Table 3, the average CPU time for
the instances up to (4 × 4 × 4 × 4) is within 1 sec-
ond, whilst (5 ×5 ×5 ×5) instances are solved within
49 seconds. Games with Cauchy distributions have
100% success rates for all the instances except (5 ×
5 × 5 × 5) instances where the success rates range
from 81% up to 94%. As for normal distribution
games, the success rate ranges from 52% for the large
instance to 99% for the smallest instances. In addi-
tion, we also solve game instances with size (6 × 6 ×
6×6 ×6 ×6) within 30 minutes. PATH failed to solve
game instances with more than (6 ×6 ×6 ×6× 6 ×6).
5 CONCLUSION
In this paper, we solved the Nash equilibrium prob-
lem with n-player chance-constrained games. We
proved the existence of Nash Equilibrium for stochas-
tic games with Cauchy and normal distributions.
We derive a deterministic equivalent NCP for these
games.
In order to show the performances of our ap-
proaches, we generated random instances and used
the PATH solver to solve the related NCPs.
For future work, we will consider different dis-
tributions for the addressed stochastic games and ap-
ply our approach to real-life applications, e.g., au-
tonomous vehicles.
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