By Alain Haurie, Shigeo Muto, Leon A. Petrosjan, T. E. S. Raghavan
This e-book, an outgrowth of the tenth overseas Symposium on Dynamic video games, provides present advancements of the speculation of dynamic video games and its functions. The textual content makes use of dynamic video game versions to procedure and resolve difficulties bearing on pursuit-evasion, advertising and marketing, finance, weather and environmental economics, source exploitation, in addition to auditing and tax evasions. It comprises chapters on cooperative video games, that are more and more drawing dynamic techniques to their classical solutions.
Read or Download Advances in Dynamic Games: Applications to Economics, Management Science, Engineering, and Environmental Management (Annals of the International Society of Dynamic Games) PDF
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Extra info for Advances in Dynamic Games: Applications to Economics, Management Science, Engineering, and Environmental Management (Annals of the International Society of Dynamic Games)
From the continuity of the value function, it follows that int Wc (t¯) = ∅ for c > c1 . Then also int Wc (t) = ∅ for c > c1 when t ∈ [t∗ , T ]. According to the fact proved above, the set Wc (t∗ ) completely sweeps the set Wc2 (t∗ ) for c ∈ (c1 , c2 ). It follows from this that the set Wc1 (t∗ ) completely sweeps the set Wc2 (t∗ ). 7 Is It Possible to Weaken the Dimension Assumption? 1 is formulated for the case when the payoﬀ function ϕ depends on two components of the phase vector at the terminal instant T .
Patsko Figure 3: Example of a convex function which does not possess the level sweeping property. 2 Description of the Main Result Let us consider a linear antagonistic diﬀerential game x˙ = A(t)x + B(t)u + C(t)v, ϕ xi (T ), xj (T ) → min max u t ∈ [t0 , T ], x ∈ Rn , u ∈ P, v ∈ Q, (1) v with ﬁxed terminal time T , convex compact constraints P , Q for controls of the ﬁrst and second players, and continuous quasi-convex payoﬀ function ϕ depending on two components xi , xj of the phase vector x at the terminal time.
The term “complete sweeping” is based on the concept of geometric diﬀerence (Minkowski diﬀerence) and is known in convex analysis and in diﬀerential game theory. It is proved that, in the game class considered, the level sweeping property is inherited by the value function. That is, if the payoﬀ function possesses the level sweeping property, then the same property is true for the constriction of the value function to any time instant from the game interval. It is shown (by a counterexample) that this holds only when the payoﬀ function depends on at most two components of the phase vector.
Advances in Dynamic Games: Applications to Economics, Management Science, Engineering, and Environmental Management (Annals of the International Society of Dynamic Games) by Alain Haurie, Shigeo Muto, Leon A. Petrosjan, T. E. S. Raghavan