# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

All Issues

## Journal of Dynamics & Games

October 2018 , Volume 5 , Issue 4

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2018, 5(4): 265-282 doi: 10.3934/jdg.2018017 +[Abstract](1284) +[HTML](538) +[PDF](604.97KB)
Abstract:

This paper investigates the dynamic stability of an adaptive learning procedure in a traffic game. Using the Routh-Hurwitz criterion we study the stability of the rest points of the corresponding mean field dynamics. In the special case with two routes and two players we provide a full description of the number and nature of these rest points as well as the global asymptotic behavior of the dynamics. Depending on the parameters of the model, we find that there are either one, two or three equilibria and we show that in all cases the mean field trajectories converge towards a rest point for almost all initial conditions.

2018, 5(4): 283-309 doi: 10.3934/jdg.2018018 +[Abstract](1178) +[HTML](494) +[PDF](455.63KB)
Abstract:

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

2018, 5(4): 311-329 doi: 10.3934/jdg.2018019 +[Abstract](964) +[HTML](357) +[PDF](418.43KB)
Abstract:

We explore the strategic equivalence between the delegated menu contracting procedure and the centralized mechanism contracting procedure in general pure strategy multi-agency games under ex post equilibrium. We allow information externalities, contract externality, correlated types, and primitive constraints across the contracts for different agents. Our delegation principle identifies that even under this general setting ex post menu design is strategically equivalent to bilateral ex post mechanism design, which simplifies collective ex post mechanism design by ignoring relative information reference. Moreover, one can restrict attention to product menu design problems out of general menu design problems if the contract constraint sets have product structures. We provide conditions for when the principal can do strictly better by using the collective mechanism. Our results still hold if we include individual rationality or any degenerated form of our general model.

2018, 5(4): 331-341 doi: 10.3934/jdg.2018020 +[Abstract](1129) +[HTML](369) +[PDF](338.82KB)
Abstract:

We provide a polynomial algorithm to find the value and an optimal strategy for a generalization of the Pig game. Modeled as a competitive Markov decision process, the corresponding Bellman equations can be decoupled leading to systems of two non-linear equations with two unknowns. In this way we avoid the classical iterative approaches. A simple complexity analysis reveals that the algorithm requires \begin{document}$O(\mathbf{s}\log\mathbf{s})$\end{document} steps, where \begin{document}$\mathbf{s}$\end{document} is the number of states of the game. The classical Pig and the Piglet (a simple variant of the Pig played with a coin) are examined in detail.

2018, 5(4): 343-355 doi: 10.3934/jdg.2018021 +[Abstract](822) +[HTML](206) +[PDF](364.19KB)
Abstract:

In this paper we study technology transfer in a duopoly model with heterogeneous goods under quantity and price competition. We obtain three conclusions: Firstly, under product heterogeneity, technology transfer can take any form. Secondly, the properties found in the homogeneous case generalize to the heterogeneous case with various degrees of generality. Finally, if demand is not symmetric we may find full technology transfer between firms with different sizes and that firms with similar technology do not engage in technology transfer. We find evidence of the latter in the market for antidepressants.