By David González-Sánchez, Onésimo Hernández-Lerma
There are numerous concepts to review noncooperative dynamic video games, similar to dynamic programming and the utmost precept (also known as the Lagrange method). It seems, despite the fact that, that a technique to signify dynamic strength video games calls for to research inverse optimum regulate difficulties, and it's the following the place the Euler equation method is available in since it is very well–suited to resolve inverse difficulties. regardless of the significance of dynamic strength video games, there isn't any systematic examine approximately them. This monograph is the 1st try to offer a scientific, self–contained presentation of stochastic dynamic strength games.
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Extra resources for Discrete–Time Stochastic Control and Dynamic Potential Games: The Euler–Equation Approach
2. Our main results are stated in Sect. 3, and illustrated in Sect. 4 with a detailed example. In Sect. 5 we deal with the nonstationary case. 35 D. Gonz´alez-S´anchez and O. 1. We will use the line integrals introduced in the section on notation and acronyms. That is, if f : Rn → Rn is measurable with component functions f1 , f2 , . . , fn and φ : [0, 1] → Rn is a C 1 function with components φ1 , φ2 , . . , φn , then φ (1) φ (0) f (x)dx := 1 0 d φi n ∑ fi (φ (t)) dt (t) dt. i=1 The function f is said to be exact when this integral does not depend on the path φ .
We also have to require convexity of the set Φ (x0 , s0 ) and concavity of the function gt (·, ·, st ) for each st ∈ St (t = 0, 1, . ). 2 (A stochastic LQ problem). In this example we consider a stochastic version of the OCP studied in Sect. 5. We assume the dynamics xt+1 = α xt + γ ut + ξt , t = 0, 1, . . d. random variables with zero mean and variance σ 2 . Let q, r > 0, and 0 < β < 1. 59). Note that gt (x, y, s) = β t [qx2 + rγ −2 (y − α x − s)2 ]. 61) where Q := (α 2 r + γ 2 q)β . Let x¯t denote the expected value of xˆtπ for t = 0, 1, .
Where utj is chosen by player j in the control set U j ( j = 1, . . , n). In general, the set U j may depend on time t, the current state xt , the action uti of each player i = j, and the value st taken by ξt , for each t = 0, 1, . .. We suppose that player j wants to maximize a performance index (also known as reward or payoff function) of the form ∞ E ∑ rtj (xt , ut1 , . . 1) and the given initial pair (x0 , s0 ), which is supposed to be fixed throughout the following. At each time t = 0, 1, .