By Andrzej S. Nowak, Krzysztof Szajowski

"This ebook specializes in quite a few features of dynamic online game conception, providing cutting-edge learn and serving as a consultant to the energy and development of the sector and its purposes. A invaluable reference for practitioners and researchers in dynamic video game thought, the publication and its assorted purposes also will profit researchers and graduate scholars in utilized arithmetic, economics, engineering, structures and regulate, and environmental technological know-how.

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Additional info for Advances in dynamic games: Applications to economics, finance, optimization

Example text

The rule takes a simpler form in the proportional-rewards game where we let X1 = x − a + Y (a+b) , (4) for a random variable with distribution q(x, a, b). In either game play begins at a state X0 = x and proceeds in stages that we often call “days”. If on day n ≥ 1, the state of the game is Xn−1 = xn−1 and player I chooses action an ∈ Axn−1 while simultaneously player II chooses bn ∈ Bxn−1 , then I receives the daily reward rn = r(xn−1 , an , bn ) and II receives r˜n = r˜ (xn−1 , an , bn ). The game then moves to the new state Xn with probability distribution q(xn−1 , an , bn ).

This implies the statement. 1 1 β ≤ α n+1 V + d + β 1−α 1−α ✷ Let Tw be the operator given by Tw u(x, a, b) := (1 − ϑ)(ϑk(x, a, b) + w(x)) + ϑpu(x, a, b) for all u ∈ V, x ∈ X, a ∈ A, b ∈ B. We note that Tw has essentially the same structure as the cost operator T used in stochastic dynamic programming and stochastic game theory (see (3)). This implies that some of our proofs are very similar to known proofs. Therefore, we restrict ourselves to only a few remarks in these cases. ) Obviously, Tw u = (1 − ϑ)ϑT u 1−ϑ + (1 − ϑ)w.

Equality (7) will be established with the help of a lemma. 1. Let Q : S → [0, ∞) be non-decreasing and bounded above by u(1)/(1 − β). Then ˜ (i) (LQ)(x) = r(x, b(x), b(x)) + βEQ(X(x, b(x))), for x ∈ S, (ii) LQ is non-decreasing and bounded above by u(1)/(1 − β). 45 A Simple Two-Person Stochastic Game with Money Proof. To prove (i), define ˜ ψ(x, a) = r(x, a, b(x)) + βEQ(X(x, a)) ⎧ ˜ ⎪ u(1) + βEQ(x − a + Y (a) ) if a > b(x), ⎪ ⎨ ˜ ˜ ˜ = u(1/2) + βEQ(x − b(x) + Y (2b(x)) ) if a = b(x), ⎪ ⎪ ⎩ ˜ ˜ if a < b(x), u(0) + βEQ(x + Y (b(x)) ) (8) for x ∈ S, a ∈ Ax .