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Journal of Dynamics & Games

October 2020 , Volume 7 , Issue 4

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Foundations of semialgebraic gene-environment networks
Erik Kropat, Gerhard-Wilhelm Weber and Erfan Babaee Tirkolaee
2020, 7(4): 253-268 doi: 10.3934/jdg.2020018 +[Abstract](455) +[HTML](241) +[PDF](735.08KB)
Abstract:

Gene-environment network studies rely on data originating from different disciplines such as chemistry, biology, psychology or social sciences. Sophisticated regulatory models are required for a deeper investigation of the unknown and hidden functional relationships between genetic and environmental factors. At the same time, various kinds of uncertainty can arise and interfere with the system's evolution. The aim of this study is to go beyond traditional stochastic approaches and to propose a novel framework of semialgebraic gene-environment networks. Foundation is laid for future research, methodology and application. This approach is a natural extension of interconnected systems based on stochastic, polyhedral, ellipsoidal or fuzzy (linguistic) uncertainty. It allows for a reconstruction of the underlying network from uncertain (semialgebraic) data sets and for a prediction of the uncertain futures states of the system. In addition, aspects of network pruning for large regulatory systems in genome-wide studies are discussed leading to mixed-integer programming (MIP) and continuous programming.

Approachability in population games
Dario Bauso and Thomas W. L. Norman
2020, 7(4): 269-289 doi: 10.3934/jdg.2020019 +[Abstract](437) +[HTML](198) +[PDF](393.63KB)
Abstract:

This paper reframes approachability theory within the context of population games. Thus, whilst a player still aims at driving her average payoff to a predefined set, her opponent is no longer malevolent but instead is extracted randomly at each instant of time from a population of individuals choosing actions in a similar manner. First, we define the notion of 1st-moment approachability, a weakening of Blackwell's approachability. Second, since the endogenous evolution of the population's play is then important, we develop a model of two coupled partial differential equations (PDEs) in the spirit of mean-field game theory: one describing the best-response of every player given the population distribution, the other capturing the macroscopic evolution of average payoffs if every player plays her best response. Third, we provide a detailed analysis of existence, nonuniqueness, and stability of equilibria (fixed points of the two PDEs). Fourth, we apply the model to regret-based dynamics, and use it to establish convergence to Bayesian equilibrium under incomplete information.

On class of non-transferable utility cooperative differential games with continuous updating
Zeyang Wang and Ovanes Petrosian
2020, 7(4): 291-302 doi: 10.3934/jdg.2020020 +[Abstract](424) +[HTML](170) +[PDF](589.04KB)
Abstract:

This paper considers and describes the class of cooperative differential games with the non-transferable utility and continuous updating. It is the first detailed paper about the application of continuous updating approach to the non-transferable utility differential games. The process of how to construct Pareto optimal strategy with continuous updating and Pareto trajectory is described. Another important contribution is that the property of subgame consistency is adopted for the class of games with continuous updating. The resource extraction game model is used as an example. The Pareto optimal strategies and corresponding trajectory are constructed, and the set of Pareto optimal strategies satisfying the subgame consistency property is presented. The results of numerical simulation are demonstrated in the Matlab environment, and the conclusion is drawn.

On the grey Baker-Thompson rule
Mehmet Onur Olgun, Osman Palanci and Sirma Zeynep Alparslan Gök
2020, 7(4): 303-315 doi: 10.3934/jdg.2020024 +[Abstract](250) +[HTML](118) +[PDF](329.69KB)
Abstract:

Cost sharing problems can arise from situations in which some service is provided to a variety of different customers who differ in the amount or type of service they need. One can think of and airports computers, telephones. This paper studies an airport problem which is concerned with the cost sharing of an airstrip between airplanes assuming that one airstrip is sufficient to serve all airplanes. Each airplane needs an airstrip whose length can be different across airplanes. Also, it is important how should the cost of each airstrip be shared among airplanes. The purpose of the present paper is to give an axiomatic characterization of the Baker-Thompson rule by using grey calculus. Further, it is shown that each of our main axioms (population fairness, smallest-cost consistency and balanced population impact) together with various combina tions of our minor axioms characterizes the best-known rule for the problem, namely the Baker-Thompson rule. Finally, it is demonstrated that the grey Shapley value of airport game and the grey Baker-Thompson rule coincides.

Real option value and poverty trap
Giorgio Calcagnini, Edgar J. Sanchez Carrera and Giuseppe Travaglini
2020, 7(4): 317-333 doi: 10.3934/jdg.2020025 +[Abstract](488) +[HTML](125) +[PDF](458.65KB)
Abstract:

In recent years concerns about poverty traps have risen to the forefront of policy. Accordingly, the decision on investing or waiting in specific sectors or locations of poor countries is in part assigned to the government of that country. We study the optimal timing of a foreign direct investment (FDI) where the returns are stochastic and the cost irreversible. A model of real option value compares the benefits and costs of a risky FDI with those of a riskless official development assistance (ODA). Once FDIs take place, the local government can shift ODAs towards different sectors or locations to hinder poverty. We show that with uncertainty and irreversibility, the policy decision has an opportunity value that must be included as a part of the full value of the FDI. This option value is highly sensitive to uncertainty over the future returns, so that changing actual economic conditions in poor countries can have a large impact on the poverty trap. Simulations show that this option value can be significant to explain the prevalence of hysteresis, that is the tendency of a poor country to persist in poverty.

Pricing equilibrium of transportation systems with behavioral commuters
Jaimie W. Lien, Vladimir V. Mazalov and Jie Zheng
2020, 7(4): 335-350 doi: 10.3934/jdg.2020026 +[Abstract](285) +[HTML](113) +[PDF](510.81KB)
Abstract:

We study Wardrop equilibrium in a transportation system with profit-maximizing firms and heterogeneous commuters. Standard commuters minimize the sum of monetary costs and equilibrium travel time in their route choice, while "oblivious" commuters choose the route with minimal idle time. Three possible scenarios can arise in equilibrium: A pooling scenario where all commuters make the same transport choice; A separating scenario where different types of commuters make different transport choices; A partial pooling scenario where some standard commuters make the same transport choice as the oblivious commuters. We characterize the equilibrium existence condition, derive equilibrium flows, prices and firms' profits in each scenario, and conduct comparative analyses on parameters representing route conditions and heterogeneity of commuters, respectively. The framework nests the standard model in which all commuters are standard as a special case, and also allows for the case in which all commuters are oblivious as the other extreme. Our study shows how the presence of behavioral commuters under different route conditions affects equilibrium behavior of commuters and firms, as well a the equilibrium outcome of the transportation system.

Behavior-based pricing in service differentiated industries
Dong-Sheng Ma and Hua-Ming Song
2020, 7(4): 351-364 doi: 10.3934/jdg.2020027 +[Abstract](251) +[HTML](110) +[PDF](433.79KB)
Abstract:

Firms often upgrade service level to enhance their profitability, which leads to competing firms at service disadvantages using behavior-based pricing (BBP) strategy to fight back. The interaction between service differentiation and BBP affects the profits of both competitors. In order to explore the impact of BBP on the competition of firms with service differentiation, we use game theory method to construct a two-period dynamic pricing model. We explore the optimal BBP strategy by comparing and analyzing firms sub-game equilibrium profits. The main conclusions are as follows: (ⅰ) the degree of service differentiation and the relative service cost interact to influence firms optimal pricing strategy. Specifically, when the degree of service differentiation is low (high) and the relative service cost is small (large), both firms do not adopt (adopt) BBP. When the degree of service differentiation is low (high) but the relative service cost is large (small), competing firms have mixed strategic Nash equilibrium, and both firms have a certain probability to adopt BBP. (ⅱ) BBP can help low-service firms to make up for the profits loss caused by the service disadvantage under certain conditions. However, it can lead to fierce price competition, which will damage the profits of both.

Replicator dynamics: Old and new
Sylvain Sorin
2020, 7(4): 365-386 doi: 10.3934/jdg.2020028 +[Abstract](183) +[HTML](104) +[PDF](488.56KB)
Abstract:

We introduce the unilateral version associated to the replicator dynamics and describe its connection to on-line learning procedures, in particular to the multiplicative weight algorithm. We show the interest of handling simultaneously discrete and continuous time analysis.

We then survey recent results on extensions of this dynamics as maximization of the cumulative outcome with alternative regularization functions and variable weights. This includes no regret algorithms, time average version and link to best reply dynamics in two person games, application to equilibria and variational inequalities, convergence properties in potential and dissipative games.

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