By David González-Sánchez, Onésimo Hernández-Lerma

​There are a number of recommendations to review noncooperative dynamic video games, reminiscent of dynamic programming and the utmost precept (also referred to as the Lagrange method). It seems, even if, that a method to symbolize dynamic strength video games calls for to research inverse optimum keep an eye on difficulties, and it's right here the place the Euler equation technique is available in since it is especially well–suited to unravel 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 and supply a scientific, self–contained presentation of stochastic dynamic power games.

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Extra resources for Discrete–Time Stochastic Control and Dynamic Potential Games: The Euler–Equation Approach

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We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain OCP. Our results extend to the stochastic case the work of Dechert [21]. In particular, we present a stochastic version of an important principle in welfare economics. The presentation of this chapter is based on Gonz´alez–S´anchez and Hern´andez–Lerma [36]. 1 Introduction As in Chap. 2, in an OCP we impose conditions for the existence of solutions and then we look for such solutions by some method.

2 The Euler Equation Approach In this section we give conditions to characterize Nash equilibria. 2 The Euler Equation Approach 51 utj = ht (xt , xt+1 , ut− j , ξt ), t = 0, 1, . . 5) where ut− j := (ut1 , . . , utj−1 , utj+1 , . . , utn ). 2), the performance index for player j becomes ∞ E ∑ gt (xt , xt+1 , ut , ξt ). 6) t=0 Markov–Nash Equilibria We first assume that players take their actions according to Markov strategies, say utj = φ j (t, xt , st ). 1), a Markov multi-strategy φ = (φ 1 , .

3. 15) with ϕ = ϕˆ . 4 holds for examples in Sects. 2 can be used. However, for the example in Sect. 4(c) is not satisfied. 1 we want to maximize ∞ ∑ β t log(At ktα − kt+1), t=0 k0 given. 3 Solving Optimal Control Problems 21 We assume that At (t = 0, 1, . ) belongs to some compact interval of positive numbers. The corresponding EE is the nonlinear difference equation −1 α At kˆtα −1 +β = 0, α At−1 kˆt−1 − kˆ t At kˆtα − kˆ t+1 t = 1, 2, . . 22) Let zt := kˆt+1 , At kˆtα t = 0, 1, 2, . . 22) becomes −zt−1 αβ + = 0, 1 − zt−1 1 − zt t = 1, 2, .

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