# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

All Issues

## Journal of Dynamics & Games

April 2017 , Volume 4 , Issue 2

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2017, 4(2): 87-96 doi: 10.3934/jdg.2017006 +[Abstract](2060) +[HTML](54) +[PDF](310.9KB)
Abstract:

In this paper we study TU-games where the cooperation structure among the players is modeled by a forest. Using the classical component efficiency axiom and a generalized version of the component fairness axiom we obtain a family of solutions. We show that every solution in this family is based on a process of transfers among the players, and the average tree solution belongs to the family. Finally, we obtain a solution based on the degree of the nodes and we study a set of properties satisfied by this family.

2017, 4(2): 97-124 doi: 10.3934/jdg.2017007 +[Abstract](2231) +[HTML](48) +[PDF](923.0KB)
Abstract:

With the growing collection of sales and marketing data and depth of detailed knowledge of consumer habits and trends, firms are gaining the capability to discern customers of other firms from the potential market of uncommitted consumers. Firms with this capability will be able to implement a strategy where the advertising effort towards customers of competing firms may differ from that towards uncommitted consumers. In this work, dynamic models for advertising in an oligopoly setting with fixed total market size and sales decay are presented. Two models are described in detail: a nontargeted model in which the advertising effort is the same for both categories of prospective customers, and a targeted model that gives firms the capability to allocate effort across the two categories differently. In the differential game setting, open-loop and closed-loop Nash equilibrium strategies are derived for both models. Several strategic questions that a firm may face when practicing targeted advertising on a fixed budget are discussed and addressed.

2017, 4(2): 125-148 doi: 10.3934/jdg.2017008 +[Abstract](2344) +[HTML](51) +[PDF](538.8KB)
Abstract:

We consider a family of mirror descent strategies for online optimization in continuous-time and we show that they lead to no regret. From a more traditional, discrete-time viewpoint, this continuous-time approach allows us to derive the no-regret properties of a large class of discrete-time algorithms including as special cases the exponential weights algorithm, online mirror descent, smooth fictitious play and vanishingly smooth fictitious play. In so doing, we obtain a unified view of many classical regret bounds, and we show that they can be decomposed into a term stemming from continuous-time considerations and a term which measures the disparity between discrete and continuous time. This generalizes the continuous-time based analysis of the exponential weights algorithm from [29]. As a result, we obtain a general class of infinite horizon learning strategies that guarantee an \begin{document}$\mathcal{O}(n^{-1/2})$ \end{document} regret bound without having to resort to a doubling trick.

2017, 4(2): 149-173 doi: 10.3934/jdg.2017009 +[Abstract](2675) +[HTML](53) +[PDF](457.8KB)
Abstract:

We study a classic international trade model consisting of a strategic game in the tariffs of the governments. The model is a two-stage game where, at the first stage, governments of each country use their welfare functions to choose their tariffs either (ⅰ) competitively (Nash equilibrium) or (ⅱ) cooperatively (social optimum). In the second stage, firms choose competitively (Nash) their home and export quantities. We compare the competitive (Nash) tariffs with the cooperative (social) tariffs and we classify the game type according to the coincidence or not of these equilibria as a social equilibrium, a prisoner's dilemma or a lose-win dilemma.

2017, 4(2): 175-190 doi: 10.3934/jdg.2017010 +[Abstract](1580) +[HTML](46) +[PDF](391.3KB)
Abstract:

We present a simple model of international trade (IT) and growth. The model yields a unique equilibrium path in which the relationship between exogenous and endogenous variables does not resemble the equations estimated by the empirical literature: Ours are not linear, despite the fact that technology and demand are linear, they do not include variables used in this literature like shares of IT and investment and include variables that have never been used in this literature such as comparative and absolute advantage, specialization patterns, saving habits and technology of partner countries. Finally, the impact of the initial level of income and the number of years that the economy has been open is far more complicated than had been assumed by the literature.