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Journal of Dynamics & Games

January 2020 , Volume 7 , Issue 1

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Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments
Max-Olivier Hongler
2020, 7(1): 1-20 doi: 10.3934/jdg.2020001 +[Abstract](197) +[HTML](86) +[PDF](373.59KB)

The collective behaviour of stochastic multi-agents swarms driven by Gaussian and non-Gaussian environments is analytically discussed in a mean-field approach. We first exogenously implement long range mutual interactions rules with strengths that are weighted by the real-time distance separating each agent with the swarm barycentre. Depending on the form of this barycentric modulation, a transition between two drastically different collective behaviours can be unveiled. A behavioural bifurcation threshold due to the tradeoff between the desynchronisation effects of the stochastic environment and the synchronising interactions is analytically calculated. For strong enough interactions, the emergence of a soliton propagating wave is established. Alternatively, weaker interactions cannot overcome the environmental noise and evanescent diffusive waves result. In a second and complementary approach, we show that the emergent solitons can alternatively be interpreted as being the optimal equilibrium of mean-field games (MFG) models with ad-hoc running cost functions which are here exactly determined. These MFG's soliton equilibria are therefore endogenously generated. Hence for the classes of models here proposed, an explicit correspondence between exogenous and endogenous interaction rules leading to similar collective effects is explicitly constructed. For non-Gaussian environments our results offer a new class of exactly solvable mean-field games dynamics.

Sequencing grey games
Serap Ergün, Osman Palanci, Sirma Zeynep Alparslan Gök, Şule Nizamoğlu and Gerhard Wilhelm Weber
2020, 7(1): 21-35 doi: 10.3934/jdg.2020002 +[Abstract](215) +[HTML](94) +[PDF](569.71KB)

The job scheduling problem is a notoriously difficult problem in combinatorial optimization and Operational Research. In this study, we handle the job scheduling problem by using a cooperative game theoretical approach. In the sequel, sequencing situations arising grom grey uncertainty are considered. Cooperative grey game theory is applied to analyze these situations. Further, grey sequencing games are constructed and grey equal gain splitting (GEGS) rule is introduced. It is shown that cooperative grey games are convex. An application is given based on Priority Based Scheduling Algorithm. The paper ends with a conclusion.

Game theoretical modelling of a dynamically evolving network Ⅱ: Target sequences of score 1
Chris Cannings and Mark Broom
2020, 7(1): 37-64 doi: 10.3934/jdg.2020003 +[Abstract](218) +[HTML](98) +[PDF](630.88KB)

In previous work we considered a model of a population where individuals have an optimum level of social interaction, governed by a graph representing social connections between the individuals, who formed or broke those links to achieve their target number of contacts. In the original work an improvement in the number of links was carried out by breaking or joining to a randomly selected individual. In the most recent work, however, these actions were often not random, but chosen strategically, and this led to significant complications. One of these was that in any state, multiple individuals might wish to change their number of links. In this paper we consider a systematic analysis of the structure of the simplest class of non-trivial cases, where in general only a single individual has reason to make a change, and prove some general results. We then consider in detail an example game, and introduce a method of analysis for our chosen class based upon cycles on a graph. We see that whilst we can gain significant insight into the general structure of the state space, the analysis for specific games remains difficult.

Market games and walrasian equilibria
Carlos Hervés-Beloso and Emma Moreno-García
2020, 7(1): 65-77 doi: 10.3934/jdg.2020004 +[Abstract](222) +[HTML](77) +[PDF](374.59KB)

In this work, we recapitulate and compare the market game approaches provided by Shapley and Shubik [35] and Schmeidler [33]. We provide some extensions to economies with infinitely many commodities and point out some applications and lines for future research.

A regularization interpretation of the proximal point method for weakly convex functions
Tim Hoheisel, Maxime Laborde and Adam Oberman
2020, 7(1): 79-96 doi: 10.3934/jdg.2020005 +[Abstract](72) +[HTML](65) +[PDF](401.62KB)

Empirical evidence and theoretical results suggest that the proximal point method can be computed approximately and still converge faster than the corresponding gradient descent method, in both the stochastic and exact gradient case. In this article we provide a perspective on this result by interpreting the method as gradient descent on a regularized function. This perspective applies in the case of weakly convex functions where proofs of the faster rates are not available. Using this analysis we find the optimal value of the regularization parameter in terms of the weak convexity.



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