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Volume 1, 2014

Journal of Dynamics & Games

2014 , Volume 1 , Issue 1

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Bio-inspired paradigms in network engineering games
Eitan Altman
2014, 1(1): 1-15 doi: 10.3934/jdg.2014.1.1 +[Abstract](547) +[PDF](410.5KB)
Abstract:
Network Engineering Games (NEGs) is an emerging branch of game theory developed in Electrical Engineering Departments. It concerns games that arise in all levels of telecommunication networks. There has been a growing interest among researchers in this community in bio-inspired methodologies in recent years due to two reasons. First, many problems in networking have much in common with problems in biology. Examples are (i) propagation of information in networks, that has similar dynamics as propagation of epidemics; (ii) energy management issues in wireless networks and competition over resources are often similar to issues by biologists; (iii) both equilibria concepts as well as replicator dynamics that arise in evolutionary games are quite relevant to NEGs. In this paper we present an overview of applications and tools used in network engineering games, we then describe in more depth bio-inspired tools used in or relevant to network engineering. We present finally an example of a stochastic epidemic game arising in wireless networks that involves competition over the relaying of information.
Collective attention and ranking methods
Gabrielle Demange
2014, 1(1): 17-43 doi: 10.3934/jdg.2014.1.17 +[Abstract](493) +[PDF](528.2KB)
Abstract:
In a world with a tremendous amount of choices, ranking systems are becoming increasingly important in helping individuals to find information relevant to them. As such, rankings play a crucial role of influencing the attention that is devoted to the various alternatives. This role generates a feedback when the ranking is based on citations, as is the case for PageRank used by Google. The attention bias due to published rankings affects new stated opinions (citations), which will, in turn, affect the next ranking. The purpose of this paper is to investigate this feedback by studying some simple but reasonable dynamics. We show that the long run behavior of the process much depends on the preferences, in particular on their diversity, and on the used ranking method. Two main families of methods are investigated, one based on the notion of handicaps, the other one on the notion of peers' rankings.
Tail probabilities for triangular arrays
Drew Fudenberg and David K. Levine
2014, 1(1): 45-56 doi: 10.3934/jdg.2014.1.45 +[Abstract](493) +[PDF](379.4KB)
Abstract:
Different discrete time triangular arrays representing a noisy signal of players' activities can lead to the same limiting diffusion process yet lead to different limit equilibria. Whether the limit equilibria are equilibria of the limiting continuous time game depends on the limit properties of test statistics for whether a player has deviated. We provide an estimate of the tail probabilities along these arrays that allows us to determine the asymptotic behavior of the best test and thus of the best equilibrium.
On the Euler equation approach to discrete--time nonstationary optimal control problems
David González-Sánchez and Onésimo Hernández-Lerma
2014, 1(1): 57-78 doi: 10.3934/jdg.2014.1.57 +[Abstract](759) +[PDF](473.6KB)
Abstract:
We are concerned with deterministic and stochastic nonstationary discrete--time optimal control problems in infinite horizon. We show, using Gâteaux differentials, that the so--called Euler equation and a transversality condition are necessary conditions for optimality. In particular, the transversality condition is obtained in a more general form and under milder hypotheses than in previous works. Sufficient conditions are also provided. We also find closed--form solutions to several (discounted) stationary and nonstationary control problems.
Optimal control indicators for the assessment of the influence of government policy to business cycle shocks
John Leventides and Iraklis Kollias
2014, 1(1): 79-104 doi: 10.3934/jdg.2014.1.79 +[Abstract](524) +[PDF](978.4KB)
Abstract:
We consider idealised dynamic models isolating the relationship between GDP and government expenditures. In this setting we assess the possibility of smoothing the effect of business cycle shocks via government expenditure alone and propose optimal control indicators measuring the control potential of this government action. This provides with new indicators and indices refining the dynamic relationship obtained by ARMA or similar type of macro - modeling.
Average optimal strategies for zero-sum Markov games with poorly known payoff function on one side
Fernando Luque-Vásquez and J. Adolfo Minjárez-Sosa
2014, 1(1): 105-119 doi: 10.3934/jdg.2014.1.105 +[Abstract](539) +[PDF](452.9KB)
Abstract:
We are concerned with two-person zero-sum Markov games with Borel spaces under a long-run average criterion. The payoff function is possibly unbounded and depends on a parameter which is unknown to one of the players. The parameter and the payoff function can be estimated by implementing statistical methods. Thus, our main objective is to combine such estimation procedure with a variant of the so-called vanishing discount approach to construct an average optimal pair of strategies for the game. Our results are applied to a class of zero-sum semi-Markov games.
Dynamics of human decisions
Renato Soeiro, Abdelrahim Mousa, Tânia R. Oliveira and Alberto A. Pinto
2014, 1(1): 121-151 doi: 10.3934/jdg.2014.1.121 +[Abstract](727) +[PDF](2417.1KB)
Abstract:
We study a dichotomous decision model, where individuals can make the decision yes or no and can influence the decisions of others. We characterize all decisions that form Nash equilibria. Taking into account the way individuals influence the decisions of others, we construct the decision tilings where the axes reflect the personal preferences of the individuals for making the decision yes or no. These tilings characterize geometrically all the pure and mixed Nash equilibria. We show, in these tilings, that Nash equilibria form degenerated hysteresis with respect to the dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hysteresis can help to explain the sudden appearance of social, political and economic crises. We observe the existence of limit cycles for the dynamics associated to situations where the individuals keep changing their decisions along time, but exhibiting a periodic repetition in their decisions. We introduce the notion of altruist and individualist leaders and study the way that the leader can affect the individuals to make the decision that the leader pretends.
Structure of approximate solutions of dynamic continuous time zero-sum games
Alexander J. Zaslavski
2014, 1(1): 153-179 doi: 10.3934/jdg.2014.1.153 +[Abstract](517) +[PDF](431.0KB)
Abstract:
In this paper we study a turnpike property of approximate solutions for a class of dynamic continuous-time two-player zero-sum games. These properties describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

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