# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

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## Journal of Dynamics & Games

January 2016 , Volume 3 , Issue 1

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2016, 3(1): 1-24 doi: 10.3934/jdg.2016001 +[Abstract](642) +[PDF](538.3KB)
Abstract:
In this paper, we study a robust-entropic optimal control problem in the presence of inside information. To be more precise, we consider an economic agent who is allowed to invest her wealth in a classical Black-Scholes type financial market. From the beginning of the trading interval, the agent exclusively possesses some inside information concerning the future realization of the stock price process. However, we assume that she is uncertain as to the validity of this information, thus introducing in this way robust aspects to our model. The aim of the economic agent is to solve an expected utility maximization problem under the worst-case scenario, taking into account her enlarged information set. By formulating this problem as a two-player, zero sum stochastic differential game, we are able to provide closed form solutions for the optimal robust strategies and the robust value function, in the case of the exponential and the power utility functions.
2016, 3(1): 25-50 doi: 10.3934/jdg.2016002 +[Abstract](1772) +[PDF](566.5KB)
Abstract:
We consider infinite-horizon zero-sum stochastic differential games with average payoff criteria, discount -sensitive criteria and, infinite-horizon undiscounted reward criteria which are sensitive to the growth rate of finite-horizon payoffs. These criteria include, average reward optimality, strong 0-discount optimality, strong -1-discount optimality, 0-discount optimality, bias optimality, F-strong average optimality and overtaking optimality. The main objective is to give conditions under which these criteria are interrelated.
2016, 3(1): 51-74 doi: 10.3934/jdg.2016003 +[Abstract](1358) +[PDF](446.2KB)
Abstract:
We consider a nonlinear degenerate parabolic equation containing a nonlocal term, where the spatial variable $x$ belongs to $\mathbb{R}^d$, $d \geq 2$. The equation serves as a replicator dynamics model where the set of strategies is $\mathbb{R}^d$ (hence a continuum). In our model the payoff operator (which is the continuous analog of the payoff matrix) is nonsymmetric and, also, evolves with time. We are interested in solutions $u(t, x)$ of our equation which are positive and their integral (with respect to $x$) over the whole space $\mathbb{R}^d$ is $1$, for any $t > 0$. These solutions, being probability densities, can serve as time-evolving mixed strategies of a player. We show that for our model there is an one-parameter family of self-similar such solutions $u(t, x)$, all approaching the Dirac delta function $\delta(x)$ as $t \to 0^+$. The present work extends our earlier work [11] which dealt with the case $d=1$.
2016, 3(1): 75-100 doi: 10.3934/jdg.2016004 +[Abstract](1630) +[PDF](450.3KB)
Abstract:
We develop a theoretical framework to study the location-price competition in a Hotelling-type network game, extending the Hotelling model, with linear transportation costs, from a line (city) to a network (town). We show the existence of a pure Nash equilibrium price if, and only if, some explicit conditions on the production costs and on the network structure hold. Furthermore, we prove that the local optimal localization of the firms are at the cross-roads of the town.
2016, 3(1): 101-120 doi: 10.3934/jdg.2016005 +[Abstract](1601) +[PDF](489.3KB)
Abstract:
We study games with finitely many participants, each having finitely many choices. We consider the following categories of participants:
(I) populations: sets of nonatomic agents,
(II) atomic splittable players,
(III) atomic non splittable players.
We recall and compare the basic properties, expressed through variational inequalities, concerning equilibria, potential games and dissipative games, as well as evolutionary dynamics.
Then we consider composite games where the three categories of participants are present, a typical example being congestion games, and extend the previous properties of equilibria and dynamics.
Finally we describe an instance of composite potential game.