Advances in Dynamic Games: Applications to Economics, by Ioannis Karatzas, Martin Shubik, William D. Sudderth

By Ioannis Karatzas, Martin Shubik, William D. Sudderth (auth.), Andrzej S. Nowak, Krzysztof Szajowski (eds.)

This ebook specializes in a variety of features of dynamic online game thought, proposing state of the art examine and serving as a consultant to the power and development of the sector and its functions. the chosen chapters, written by way of specialists of their respective disciplines, are an outgrowth of displays initially given on the 9th overseas Symposium of Dynamic video games and purposes. Featured all through are valuable instruments for researchers and practitioners who use online game idea for modeling in lots of disciplines.

Major themes coated include:

* repeated and stochastic games

* differential dynamic games

* optimum preventing games

* purposes of dynamic video games to economics, finance, and queuing theory

* numerical tools and algorithms for fixing dynamic games

* Parrondo’s video games and comparable topics

A beneficial reference for practitioners and researchers in dynamic online game idea, the booklet and its assorted functions also will gain researchers and graduate scholars in utilized arithmetic, economics, engineering, structures and keep watch over, and environmental science.

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Extra resources for Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control

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Suppose, by way of contradiction, that a stationary equilibrium does exist, with wealth distribution μ and optimal stationary strategies {π α , α ∈ I } corresponding to consumption functions {cα (·), α ∈ I }. Let B = cα (s) μ(ds) be the total bid in each period, so that the price pn = B/Qn in period n is B/3 if Qn = 3 and is B if Qn = 1. , when Qn = 3), the optimal bid for an agent is c(s) = s. Thus, we must have cα (s) = s for all α and s. e. when Qn = 1), an agent who spends one unit of money receives in utility (1/B), whereas an agent who saves the money and spends it in the next period expects to receive β [(1/2B) + (3/2B)] = (2β/B).

29 Let w ∈ V arbitrary, π ∈ E, ρ ∈ F. Then the functional equation u = πρTw u (12) has a unique solution uw ∈ V and it holds for uw := Sπρ w, ∞ Sπρ w = lim (πρTw )n u = (1 − ϑ) n→∞ ϑ n (πρp)n (ϑπρk + w), (13) n=0 for every u ∈ V. Proof. We note that πρTw V ⊆ V. From (5) it follows that πρTw is contracting on V with modulus λ. The rest of the proof follows by Banach’s Fixed Point Theorem. ✷ We can consider Sπρ as an operator Sπρ : V → V. Let Sγ ,π,ρ be the operator defined by Sγ ,π,ρ w := −(1 − IC )γ + Sπρ w − IC μw (14) for π ∈ E, ρ ∈ F, w ∈ V.

This time the books obviously balance, since no one borrows and no one pays back. In fact, the bank has no role to play. 1, part (b)) and show that a stationary equilibrium need not exist. 4. For simplicity, we return to the no-lending model of Section 3 for this example. Assume that the utility function is u(b) = b, and let the distribution ζ of the variables {Qn } be the two-point distribution ζ ({1}) = ζ ({3}) = 1/2. Suppose that when Qn = 1, the variables {Znα , α ∈ I } are equal to 0 or 2 with probability 1/2 each, but that when Qn = 3, each of the Znα is equal to 1.

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