By Andrzej S. Nowak, Krzysztof Szajowski

"This booklet makes a speciality of quite a few points of dynamic online game thought, featuring state of the art learn and serving as a consultant to the power and development of the sphere and its purposes. A useful reference for practitioners and researchers in dynamic online game idea, the publication and its different purposes also will profit researchers and graduate scholars in utilized arithmetic, economics, engineering, platforms and keep watch over, and environmental technology.

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

**Example text**

In the winner-takes-all game, the daily reward function for player I is ⎧ ⎪ if a > b, ⎨u(1) (1) r(x, a, b) = u(1/2) if a = b > 0, ⎪ ⎩ 0 if a < b or a = b = 0, whereas the daily reward function for player II is defined symmetrically as r˜ (x, a, b) = r(M − x, b, a), for every (x, a, b) ∈ Z. Thus, if a > b, player I receives all the good which has price p = a; if a < b, player II receives all the good at price p = b; and, if a = b > 0, each player receives half the good and the price is p = a + b.

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. The following lemma gives some properties of this operator. 3. (a) Sγ ,π,ρ V ⊆ V. (b) Sγ ,π,ρ is isotonic. (c) Sγ ,π,ρ is contracting. Proof. (a) is obvious. (b) Using (13) we get Sγ ,π,ρ w = −(1 − IC )γ + (1 − ϑ) = −(1 − IC )γ + (1 − ϑ) ∞ n=0 ∞ n=0 ϑ n (πρp)n (ϑπρk + w) − IC μw ϑ n+1 (πρp)n πρk + (Qϑ,π,ρ − IC μ)w. 1 (a) it follows the statement.

The stationary wealth-distribution, namely, the distribution of Z − 2, assigns mass μ({−2}) = 1/2 at −2 and mass μ({0}) = 1/2 at 0. 1 is satisfied. Clause (ii) is also satisfied, because every agent borrows one unit of money and spends it; one-half of the agents receive no income and pay back back nothing, whereas the other half receive an income of 2 units of money, all of which they pay back to the bank since the interest rate is r1 = 2. As there are no lenders, the books balance. 1 now says that we have a stationary equilibrium, in which half of the agents are in debt for 2 units of money, and the other half hold no money at the beginning of each period.